a! I. it‘ll-”I'Lllfil'l - ’ .. «mm a g -a \lflllllllllllzllgllflllllllllllllflljlllllllllllflll LIBRARY This is to certify that the thesis entitled Relevance and Application of Multiattribute Utility Theory and Risk Aversion Analysis to Participatory Group Decision Making presented by ALTAN cb'NER has been accepted towards fulfillment of the requirements for ___BHJ1L__degeein_Management Science flx/Lv/C'PQUX/{tfi/d. Major professor Date 8/28/1978 0-7 639 MiChigan State . ” University 4 a- A *4!“ 4 . e a .cngAlMfl fig‘-“ A‘U4u__~—-—v‘..—¢n__.n—.o-—A—9~— ® Copyright by . Altan Genet 1978 ii RELEVANCE AND APPLICATION or MULTIATTRIBUTE UTILITY THEORY AND RISK AVERSION ANALYSIS TO PARTICIPATORY GROUP DECISION MAKING by Altan CBner A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Management 1978 Group de able interest 501115 of soci £10115 by Flem these so Call tiVe assnmpti ProblEms, Ut utility theor cation to grc The theg groups 36 decj Supra decisic thEsis EOCUSG participatior metrical par-1 attribUte ut: group deels are deVElOpe 6:3Q9g?f'g/ ABSTRACT RELEVANCE AND APPLICATION OF MULTIATTRIBUTE UTILITY THEORY AND RISK AVERSION ANALYSIS TO PARTICIPATORY GROUP DECISION MAKING by Altan C5ner Group decision making behavior has been the subject of consider— able interest on the part of researchers in decision analysis. Various forms of social welfare functions were developed under different assump- tions by Fleming, Goodman, Nash, Harsanyi, Arrow and others. Most of these so called "social welfare" models were based upon many restric— tive assumptions and thus had limited applicability to real world problems. Utility theory and as a more recent development multiattribute utility theory aroused the interest of a few researchers in their appli- cation to group decision processes. The thesis, on theoretical side, develops a new classification for groups asdecision making bodies. Different kinds of groups, such as Supra decision maker and participatory group are investigated. The thesis focuses on participatory groups in no participation, partial participation or full participatiOn cases. Symmetrical and nonsym— metrical participatory groups are defined and investigated. Multi- attribute utility analysis is extended and applied to participatory group models that are developed in these sections. New methodologies are developed to evaluate the scaling constants in symmetric participatory plicative fon In the g related to ri ing and incre 0f participat functions in in the treatm tions. This literature. Finally with some of investigated 3m“? Chosen Lansing, Micl inteTViEWed ; to a COutrov. in East Lans Council. Th However the Wide diSpari ErOUp utilit than that O f the major CO apparent abi model to re; Altan CCner participatory groups whose utility functions are represented in multi— plicative forms. In the group risk aversion analysis chapter, various new theorems related to risk aggregation are developed. Cases of constant, decreas- ing and increasing risk aversion are investigated under different forms of participatory groups. Although Pratt has done some work on risk functions in unidimensional utility theory, a considerable vacuum exists in the treatment of risk in groups using multiattribute utility func- tions. This chapter on risk addresses itself to this vacuum in the literature. Finally a major real life application is undertaken to experiment with some of the multiattribute participatory group utility models investigated in previous chapters. The decision making participatory group chosen for this purpose is the city council of the city of East Lansing, Michigan. The five members of this council were extensively interviewed and their preferences were assessed. Two attributes related to a controversial issue of building or not building a new shopping mall in East Lansing were used to represent the overall objectives of the council. The council, with a four to one vote, voted against the mall. However the group utilities that were found did not indicate such a wide disparity between the two choices. Although it was found that the group utility figure for the alternative "don‘t build mall" was higher than that of "build mall", the two values were rather close. One of the major conclusions reached in this application chapter was the apparent ability of the multiattribute participatory group utility model to reasonably approximate the true preferences of the city council memb participator sion making Altan C6ner council members. On a rather limited scale it is demonstrated that participatory group utility models can be useful tools in group deci— sion making situations. TO My Mother and Father Nermin and Fettah C6ner I would for their su] fessor Richa: graduate stul corrections ; iIIStrumental been a privi Heushaw. Special HeSsers Larr- and MS~ Caro could not be The fin Organizatim of ManaEEmen I gratefully I also C3%ny the lining“? and Finally and Support ation for me ACKNOWLEDGMENTS I would like to thank the members of my dissertation committee for their support and valuable ideas. My greatest debt is to Pro- fessor Richard Henshaw, my major advisor during the course of my graduate studies. His patience, encouragement, valuable comments, corrections and above all his continuous friendly support were most instrumental in the preparation of this dissertation. It has indeed been a privilage to be guided in this work by Professor Richard Henshaw. Special thanks also go to the city council members of East Lansing, Messers Larry Owen, George Griffiths (mayor), Alan Fox, John Czarnecki and Ms. Carolyn Stell, without whose full cooperation the field research could not be made. The financial support for my graduate studies was provided by Organization for Economic Cooperation and Development and Department of Management, Graduate School of Business, Michigan State University. I gratefully acknowledge their contribution. I also thank Marie Dumeney and Jo McKenzie, who performed very capably the important task of fast and accurate typing of the pre- liminary and final drafts. Finally, I owe a lot to my parents, whose patience, encouragement and support from thousands of miles away, were great sources of inspir- ation for me. iv LIST OF TABLE LIST OF FIGUE Chapter 1- lN’l ._. N 'QNNNFJN TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . x Chapter I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . l 1.1 Descriptive Decision Making . . . . . . . . . . 2 1.2 Multiple Criteria Decision Making . . . . . 5 1.3 A Generalized Approach for Multiple Objective Decisions . . . . . . . . . . . . . 6 1.4 Multiple Attribute Decision Techniques. . . . 8 1.5 Group Decision Making and Probabilistic Problems. . . . . . . . . . . . . . . . . 9 1.6 Problems Related to Interpersonal Com- parison of Preferences. . . . . . . . . . . . 11 1.7 The Basic Approach of Thesis. . . . . . . . . . 13 II. GROUP WELFARE MODELS. . . . . . . . . . . . . . . . 14 2.1 Theories Related to Aggregation Preferences . . 14 2.2 Pareto Criterion and Principle of Pareto Optimality . . . . . . . . . . . . . . 15 2.3 Collective Choice Rules . . . . . . . 16 2.4 Arrow' 5 General Possibility (Impossibility) Theorem . . . . . . . . . . . . . 18 2.5 Fleming' 5 Social Welfare Theory . . . . . . . 22 2.6 Goodman-Markowitz Social Welfare Functions. . . 24 2.7 Nash's Approach . . . . . . . . . . . . . . . . 26 2.8 Harsanyi's Model. . . . . . . . . . . . . . . . 27 2.9 Rawls' 3 Theory of Fairness. . . . . . . . . , . 29 2.10 Sen' 3 Approach. . . . . . . . . . . . . . . . . 30 III. UTILITY AND DECISION THEORY . . . . . . . . . . . . . 32 3.1 VonNeumann Postulates and Their Extension . . . 32 3.2 Implications of the Axioms. . . . . . . . . . . 34 3.3 Decision Theory Framework . . . . . . . . . . 34 3.4 Previous Applications in Unidimensional Analysis. . . . . . . . . . . . . . . . 37 3.5 Monotonicity. . . . . . . . . . . . . . . . 38 3.6 Certainty Equivalent. . . . . . . . . . . . . 40 V r m p m I4. 4 A». “PL/4 74. AW [4. l4 4 Chapter 3.7 3.8 3.9 3.10 3.11 3.12 Strategic Equivalence . . . . . . . . . Risk Aversion . . . . . . . . . . . . . . Decision Analysis Incorporating Multiple Attributes . . . . . . The Characteristics of Decision Making Groups . . . . . . 3.10.1 The Supra Decision Maker. . . . 3.10.2 Participatory Groups. . . . . . Steps in Group Decision Analysis. . . . . Final Remarks on Chapter III. . . . . . . IV. MULTIATTRIBUTE UTILITY THEORY AND DECISION MAKING GROUPS. . . . . . . . . . . Part 4.1A 4.2A 4.3A Part 4.13 4.23 4.3B 4.5B 4.6B 4.7B 4.8B 4.93 4.10B 4.113 4.12B 4.13B 4.14B 4.15B A Introduction. . . . . . . . . . . . . Multiattribute Utility Analysis . . . . . The Main Assumptions of Multiattribute Utility Theory. . . . . . . . . . . . 4. 3. 1A Preferential Independence . . . . 4. 3. 2A Utility Independence. . . . . . . Additive (Value) Independence . . . . . . The Evaluation of Scaling Constants . . . Cardinal Group Utility Functions for Supra Decision Maker. . . . . . . . . . B Group Utility Theory. . . . . . . . . . . Participatory Groups. . . . . . Participatory Groups with Partial or No Interaction . . . . . . . . . Group Utility Functions for Nonsymmetric Participatory Groups with No or Partial Interaction . . . . . . . . . . . . . . A Definition of Symmetry in Groups. . . . Group Utility Functions for Symmetric Participatory Groups with No or Partial Interaction . . . . . . . . . . Utility Independence Condition. . . . . . Preferential Independence and Symmetry. . Meaning of Full Interaction . . . . . . . Assumption of Mutual Utility Inde- pendence for é and U7 . . . . . . . . . Utility Functions for Nonsymmetric Par— ticipatory Groups with Full Interaction Utility Functions for Symmetric Par- ticipatory Groups with Full Interaction . Evaluation of Scaling Constants in Symmetric Participatory Groups . Evaluation of Scaling Constants for a Member's Fully Interactive Utility Function . Final Remarks . . . . . . . . . . . . . . vi Page 41 42 43 50 50 50 52 52 53 55 56 58 60 62 63 65 68 69 70 75 76 78 80 83 88 9O Chapter M a ...... P—SSSSSS .m_ c o 5.3 Chapter V. VI. RISK AVERSION AND PARTICIPATORY GROUPS. . . . . Part A 5.1A Introduction . . . . . 5.2A Risk Aversion for Unidimensional Utilities . 5.3A Analysis of Risk Aversion. . . . . . . . . 5.4A Constant Risk Aversion . . . . . . . . t . 5.5A Decreasing Risk Aversion . 5.6A Increasing Risk Aversion . . . . . . . . . Part B 5.13 Introduction . . . . . . . 5.23 5.10B 5.11B Risk Aversion for Multiattribute Utility Functions. . Risk Aversion and Participatory Groups with No or Partial Interaction . . . . . 5.3.1B Constant Risk Aversion in Par— ticipatory Groups . . . Interpretation of Theorem 5. 3. 2B Results . A Generalized Application of Theorem 5. 3. 2B Constant Risk Aversion Under Multipli— cative Form. . . . Decreasing Risk Aversion in Participatory Groups . . . . . . . . . . . A Logarithmic Application of Theorem (5. 7.1B) . . . . . Increasing Risk Aversion in Participatory Groups . . . . Risk Aversion in Symmetric Participatory Groups . Risk Aversion in Nonsymmetric and Symmetric Groups with Full Interaction . . . . . . 5.12B Additional Remarks . . . . . . . . . . . . 5.13B Final Remarks. . . . . . . . . . . . . . . THE APPLICATION . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . 6.2 Background . . . . . . . . . . . . . . 6.3 Dayton-Hudson Mall Decision. 6.4 The City Council . . . . . . . . . . . 6.5 The Objectives and Attributes. . . . O\O\O\ o. C°\lO\ 6.5.1 X1, The Overall Economic and Environment Index . . . . 6.5.2 X2, The Political Support Index . . The Assessment of Utilities. . . . . . . . The Scaling Constants. . The Participatory Group Utility Function . Page 92 104 104 105 106 108 109 114 117 119 121 121 124 128 129 130 130 131 134 134 135 137 138 149 151 Chapter VII. CON RE 7 7 APPENDIX. . . BIBLIOGRAPHY . Chapter Page VII. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH . . . . . . . . . . . . . . . . . . . . . . 156 7.1 Conclusions. . . . . . . . . . . . . . . . . . 156 7.2 Suggestions for Future Research. . . . . . . . 161 APPENDIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . 166 viii e l . b a Ml. LIST OF TABLES Table Page 1 Number of Constants for Symmetry and Nonsymmetry . . . . . . . . . . . . . . . . . . . . 73 2 Commonly Used Decreasingly Risk Averse Utility Functions . . . . . . . . . . . . . . . . . 101 Figure l.lA 1.13 3.2.7.; 3.4.4A 3.5.1A 6.6.15 6.6.16 66.17 De De Pe Pe Fe Figure 1.1A 1.13 1.3 2.5 5.5.1A 6.6.15 6.6.16 6.6.17 6.6.18 6.6.19 6.6.20 6.6.21 6.6.22 6.6.23 6.6.24 LIST OF FIGURES Descriptive Decision Making . . . . . . . . . . . Decision Making Using Decision Theory . . . . . . An Approach for Multiple Criteria Decisions . . . Single or Multiple Peaked Preference Intensities. A Nonmonotonic Utility Function . . . . . . . . . A Model of Classification of Group Utility Theory Risk Averse, Neutral and Risk Prone Functions . . Constant Risk Aversion with Varying Parameter Values. . . . . . . . . . . . . . . . U(X) = log(X+b) for different values of 'b' . Overall Economic and Environmental Index for Stell. Percentage of East Lansing Voters Supporting Stell's Decision. . . . . . . . . . . . . . . Overal Economic and Environmental Index for Czarnecki . . . . . . . . . . . . . . . . . Percentage of East Lansing Voters Supporting Czarnecki's Decision. . . . . . . . . . . . . . Overall Economic and Environmental Index for Fox. Percentage of East Lansing Voters Supporting Fox's Decision. . . . . . . . . . . . . . . . . Overall Economic and Environmental Index for Griffiths . . . . . . . . . . . . . . . . . Percentage of East Lansing Voters Supporting Griffiths' Decision . . . . . . . . . . . . Overall Economic and Environmental Index for Owen Percentage of East Lansing Voters Supporting Owen's Decision . . . . . . . . . . . . . . 21 40 61 95 100 102 144 144 145 145 146 146 147 147 148 148 Simple humans. In People make making proc nate instin kinds of of Of "instinc describes a exPerience the latter ; VOIINeumann z of rational; decision mal for the ind: “ken follc he receiVes Mos: of purPOSes. ] but how Peo; of this the: of a descri; \ *NUIDber bibliograpm CHAPTER I INTRODUCTION Simple or complex decisions dominate the daily lives of all humans. In every facet of human endeavor, individuals, or groups of people make decisions which affect the lives of many. Human decision making process is enormously complex. Animals usually depend upon in— nate instinct to react to outward stimulus to survive. Under certain kinds of of environmental stimuli, humans can also react with a kind of "instinct" or "learned experience". Intuition is the term that describes a sort of decision process that is more based on instinct and experience than on "rational” considerations. In order to deal with the latter aspect of decision making, Bayes, Hurwicz, Savage [lll]*, VonNeumann and Morgenstern [130] and others have developed basic axioms of rationality and suggested that certain axioms should govern human decision making behavior. Many postulates or assumptions were given for the individual decision maker and they assumed that a decision maker, following these axioms will maximize the satisfaction or utility he receives from the outcome. Most of these axioms of rationality were developed for prescriptive Purposes. In other words the emphasis was not how people make decisions but how people E22219 make their decisions. Although the main emphasis Of this thesis is also in the normative realm, let's first get a glimpse 0f a descriptive view of decision making. *Numbers in brackets refer to numbered references 1n the bibliography. l.l Descri] In the things to Cl ment, basic; must balanci possible cor of the deci: culty to th. maker sees 1 stand. The In bus the environ finite. Th Gating What E . w j Undertaiu 1 Complex A Dynamic 1 1 Competitiv Finite COnfUS iOn Worry 1.1 Descriptive Decision Maki g In the descriptive view of decision making, one of the first things to consider is the environment of human decisions. The environ— ment, basically is characterized by uncertainty and the decision maker must balance judgments about uncertainties with his preferences for possible consequences of outcomes. This uncertainty is one attribute of the decision making process that adds enormous complexity and diffi- culty to the analysis. The environment is truly complex. The decision maker sees many factors interacting in ways that he often cannot under- stand. The environment is also dynamic and it evolves over time. In business problems, military problems, or national problems, the environment is also competitive. And finally, the resources are finite. The decision maker is usually faced with the problem of allo- cating what he has rather than expanding what he has. Environment Undertain Complex Dynamic Competitive Finite Confusion Worry l . . ' Creat1v1ty Choice j } Intuition I ' — Information} ' Logic Deci— Out- ' ——> H l . Preferences i Uncheckable Sion come I I ,' I Think Uneasiness Act Praise, Joy, Blame Sorrow Figure 1.1A Descriptive Decision Making The human I typically c like ingenu ceive and f phase of in grinding pr EHCES among sions. Int although thi analysis wi, 1119 main Ob. However in ; balance the tEChnical, . Now if flaking Prom Enviro w Uncertain lcollide: Dynamic lCOmPetitiVe finite 3 The human reaction to these characteristics of the environment is typically confusion or worry. However man is able to use some weapons like ingenuity, perception and philosophy. Ingenuity is used to con- ceive and formulate different courses of action. Perception is the phase of information gathering. And finally philosophy dictates the grinding principles of a decision makers life that give him prefer- ences among the various outcomes that he might obtain from his deci- sions. Intuition, as mentioned above, is often used in decision making although the employment of the tools and methodology of formal decision analysis will most likely result in more logical and consistent results. The main objection to intuition is that its logic is uncheckable [58]. However in formal decision analysis there is-an attempt to logically balance the factors that influence a decision. These factors might be technical, economic, environmental, competitive, etc. Now if we revise the previous chart for a generalized decision making process, we might have something like, Environment Uncertain M Ch01Ce—‘—‘—‘> Alternatives (U E o 0 Probability g ‘ Perception: Information—e-Assignments Logic —69 Structure g 'r-l Competitive g Preferences_;,value 8 Finite Philoso h ’ Assignment Q Time Preference Risk Preference Figure 1.1B Decision Making Using Decision Theory The f0 alternative tions like, What sort 0 tives? Wha analysis. The um different 6 Mt histor: there is th. assigns uti: courses of e the decisior Preferred tc PrfierencEs In tode are made by individuals State UHiver schoop this applicants. safety Stand this Will uu commercial a general PUbl new question to be anSwer 4 The formal decision theory indicates that there are various alternative courses of action whose outcomes might be uncertain. Ques- tions like, what choices can be made now? What choices can be deferred? What sort of information is needed for a better analysis of alterna- tives? What experiments are needed? are all part of the structural analysis. The uncertainty analysis is made by assigning probabilities to different events or outcomes. The subjective judgments coupled with past historical records play a great role at this phase. Finally there is the preference or utility analysis where the decision maker assigns utility values to consequences associated with alternative courses of action. The measurement at this stage not only reflects the decision maker's ordinal rankings for various Outcomes (e.g. x is preferred to y which is preferred to z) but also indicates the relative preferences for lotteries over these outcomes. [16] In today's complex world, many business or government decisions are made by individuals or groups and their decisions affect other individuals or groups. For example if Board of Trustees of Michigan State University decides to upgrade the admission standards of the school, this will affect the lives and decisions of many prospective applicants. To take another example, if FAA decides to change the safety standards for commercial flights by making them more strict, this will undoubtedly have a tremendous impact on the operations of commercial airlines and aircraft manufacturing companies besides the general public. The operations costs will undoubtedly increase and new questions like who will pay for the increases in costs will have to be answered. Taking Transportat Mexico City developed t from now to 1.2 Multip The abl businesses ‘ bute type. field of de‘ tions due ti theory being Sured outcor accepted th: 118th0ds deaj development: evalUfitiou . acCaptable 1 making peaci Criteria det Mllltipj set of Elite] Choosing aim butes as in: ample, a tVI ObjectinS c 5 Taking another example in the area of aviation [65], the Mexican Transportation Ministry has to decide how the airport facilities of Mexico City (now the largest in the world population wise) should be developed to assure adequate service for the region during the period from now to year 2000. 1.2 Multiple Criteria Decision Maki g The above situations and many more that public bodies or private businesses will encounter are all of multiple goal or multiple attri- bute type. Until recently however, the scientists investigating the field of decision making, rarely considered multiple criteria situa- tions due to the complexity of the analytical techniques and the main theory being at its infancy stage. In other words, they mostly mea- sured outcomes in terms of a single measure —- usually money -- and accepted this as a valid basis for making comparisons. Thus, the methods dealing with multiple criteria situations are rather recent developments of the last two decades. Determination of plant site, evaluation of alternative investment proposals, determination of an acceptable bid, selecting (or electing) a president, waging war or making peace, and so many others are all multiple objective, multi— criteria decisions. Multiple attribute decision problems deal with choosing among a set of alternatives which are described in terms of their attributes. Choosing among different brands of automobiles described by such attri— butes as initial cost, size, horsepower, and fuel economy is, for ex- ample, a typical multi—attribute decision problem. In some cases, the Objectives or the ensuing attributes might be in conflict. For example a company mi 'bpthmize th usually be a inconflict. ment on all achievement result in a Wthmlity,‘ tosueh a co Goals a Objectives 11 dent Kennedy e331111319 Of a 1'3 m C. D-M.'s Objective 3- Attributé PerCEiVE( DI D'M"S A( Processir 6' Actual 0L or Alterr 6 "minimize cost" and a company might have two main objectives like "optimize the quality of product." Since better quality product can usually be achieved for a price, these two objectives would seem to be in conflict. At certain thmes, a strategy might result in an achieve- ment on all objectives but at other times a limit is reached on the achievement in one objective where, increasing the gain in others will result in a reduction of achievement in the first one. Pareto Optimality, which we will discuss in detail later, is very much related to such a concept of variable attribute optimization. Goals are uSually more specific levels of achievement on different objectives in decision making. They are either achieved or not. Presid dent Kennedy's stated goal of reaching the moon by 1970 is a very good example of a goal. 1.3 A Generalized A roach for Multiple Objective Decisions Non Observable Observable Model C. D.M.'s Objectives '-*W“ I A. Multiple Attri- B. Attributes as bute Description Perceived by D.M.‘----' of Alternatives D. D.M.'s Actual - E. D.M.'s observable F. Analyst's Processing Model comparisons and Model of choices D.M.'s Choices G. Actual Outcomes H. Consequence I. Analyst's of Alternatives Measure Model of Outcome Figure 1.3 An Approach for Multiple Criteria Decisions This s: observable, sion making situation be identified. butes (A). Now the limited infc Pemeption, What he actu Observe the (C)- But he Sf-rltation of actual proce me by coll ratings, Etc IHalter from t and from the After t occur that w may not be m Surrogates f model (I), w 7 This simple chart clearly depicts the relationships among the non— observable, observable and modeling phases of a multi—objective deci- sion making process [78]. We start out at the stage where decision situation has been formulated and the available alternatives have been identified. These alternatives are characterized by multiple attri- butes (A). Now the decision maker observes these alternatives but due to limited information processing capacity, memory constraints, selective perception, etc. his internal representation may not be identical to what he actually observes. Unfortunately, the analyst cannot directly observe the objectives and internal perceptions of the decision maker (C). But he takes as inputs to his model the decision maker's repre- sentation of the attributes and his objectives (D). Although the actual processing cannot be observed, parts of it can be made access- ible by collecting data on final choices, evaluations, comparisons or ratings, etc. (E). The consultant can build a model for the decision maker from the observed multiple attribute description of alternatives and from the observed choices (F). After the implementation of the actual choice, various events will Occur that will result in different outcomes (G). While full outcomes may not be measurable, various outcome measures can be established as Surrogates for outcomes (H). This allows the analyst to form another model (I), which relates the basic multiattribute description (A) with the outcome measure (H). 1.4 Multi} There multiple or Methods, (2 Methods, at (l) u 8 1.4 Multiple Attribute Decision Techniques There are basically four kinds of techniques for dealing with multiple criteria decision making. These four groups are (l) Weighting Methods, (2) Sequential Elimination Methods, (3) Mathematical Programming Methods, and (4) Graphical Methods. An overview looks like: (1) (2) (3) (4) Multiple Attribute Decision Techniques Weighting Methods A. Inferred Preferences a. linear regression b. analysis of variance c. quasi-linear regression Directly Assessed Preferences a. trade—off technique b. additive utilities c. multiplicative utilities d. maximin e. maximax Sequential Elimination Procedures Alternative vs. Standard: Comparison across Attributes Alternative vs. Alternative: Dominance Alternative vs. Alternative: Comparison across Alternatives Mathematical Programming Methods A. B. C. Linear Programming Non—Linear Programming Goal Programming Graphical Methods It is to the firs directly as the proper decision th 1-5 GM; In the Progress in and uncerta analysis" i Principle 0 Present for only One ac In C00. to analee. mostly mn there are 8‘ group utili such functi‘ Use the app' fundamental To giVl and 3 Who a. 9 A. Iso Preference Graphs B. Multi—Dimensional Scaling C. Graphical Preferences It is easily observed that multiattribute utility theory belongs to the first main section weighting methods and to the subsection directly assessed preferences. This overview was given to determine the proper place of the multiattribute utilities among the various decision theory techniques that are available. 1.5 Group Decision Making and Probabilistic Problems In the past 25 or 50 years, there has been a relatively rapid progress in the theory of individual decision making under certainty and uncertainty. What is now called "decision analysis" or "Bayesian analysis" involves notions of subjective probability, utility, and the principle of expected utility maximization. This theory, in its present form, is an extension of the theory of games where there is only one active player. In contrast group decision making has proved to be very difficult to analyze. Attempts to formulate different theories about the groups mostly run into problems related to paradoxes of aggregation. Although there are some authors (e.g. Keeney and Raiffa [73]),do speak about group utility functions, there are others, like Arrow [2], who claim such functions, do not exist. Of course every theoretician tries to use the appropriate assumptions to support his theory but a few of the fundamental difficulties are subject to discussion even today. To give an example [16], suppose that there are two individuals A and B who are each in a position to choose a strategy between the two, '1 and '2' The con two events E his man esti has his own E E In the - Upper left 0: Siam131e, if 1 if went E té usthe expeci the upper rig (0f W0) for the average c have tOtal gr lO yl and y2. E E Expectation 10 7 5 y1 4 7 7.5 9 Y 2 8 6 6.4 The consequences of the actions yl and y2 can be influenced by two events E and E (E complement). Furthermore, each individual has his own estimate of the probabilities for these events and each also has his own assessment of the utilities of the outcomes. A's B's Probability Event Probabilities Probabilities Averages E .8 .2 .5 f .2 .8 .5 In the previous matrix, the utility assessments for A are in the upper left of the boxes and for B in the lower right of the boxes. For example, if he chooses strategy y2, B will have a utility of 6 units if event F takes place. The third and final column in the matrix gives us the expected utilities for each action yl and y2. In each cell at the upper right we have the average expected utilities for the group (of two) for each action. Also estimating the group probability as the average of individual estimates (namely .5, .5 for each event), we have total group expected utilities of 6.75 and 7.5 for each action, Y1 and y2, respectively. Now the comparison, 9 > 7.2 and 7.4 > 6.4 indicates individual the group and probab and how th. a discussi« 1.6 m Over a there is a ences. The bility of j SimPle exam Let us (Our group) the tEams a According t judge 2 tea this P011, ariSe' If We ‘ but bOth B ‘ judge 2 thou Alias a P001 totally Sat; Thus a Sillip] {W0 membel‘s 11 indicates to us that, action yl is preferred by both individual A and However if we take them as a group, then 6.75 < 7.5, and individual B. the group prefers y2 to yl. Here is an indication how uncertainty and probability estimates cause a great problem in utility aggregation and how the principle of Pareto optimality is violated. There will be a discussion of Pareto optimality later. 1.6 Problems Related to Interpersonal Comparisons of Preferences Over and above the problem of subjective probability aggregation, there is a more basic problem of interpersonal comparisons of prefer- ences. There is a great disagreement among researchers on the feasi— bility of interpersonal utility comparisons. Let us first consider a simple example. Let us say that there are three teams A, B, and C and two judges (our group) who are asked to rank their performance. If judge l ranks the teams as A, B, C and judge 2 as C, B, A than we have a problem. According to judge 1, team C is the least preferred and according to judge 2 team A is the least preferred. If according to the result of this poll, team B is declared the winner, then certain problems might arise. If we assume that judge I thought team A had a great performance but both B and C were almost equally very poor and at the same time judge 2 thought that team C and B were almost equally great teams but A was a poor performer, then, declaring team B as the winner would totally satisfy judge 2 but leave judge l extremely dissatisfied. Thus a simple and logical way of aggregating the preferences of the two members of the group result in a very undesirable result of a total dissatisfac In thi: In other we] totally ignr tially satis say from 1 t each alterna the least pr 3, and C as and A as S, 5 Points fro Satisfaction With the Out tion. Cruci have to be p We Will come in the next . aggregation. Flirthen EVery gI‘Oup T Cincial quEst dOHIinant in i If we take ti CorPOration v familiar Cage DresidthvS s prefhetence s 12 dissatisfaction of one of the members. In this particular example, only ordinal preferences were used. In other words the intensity of the preferences by the two judges were totally ignored. There are ways of overcoming this problem in par- tially satisfactory ways. One would be to assign preference scales, say from 1 to 10, to the degree of intensity of the preferences for each alternative. If we assume that 10 is the most preferred and l is the least preferred, and if we suppose that judge l evaluates team A, B, and C as 9, 7 and 5 respectively and judge 2 evaluates teams C, B and A as 5, 4 and 3, then a logical winner could be team C which gets 5 points from both judges. However the basic problem of group member satisfaction still remains because again one member's satisfaction with the outcome is counter balanced by another member's dissatisfac— tion. Crucial at this scaling process is the fact that the scales have to be perceived and interpreted identically by all group members. We will come back to the question of ordinal vs. cardinal preferences in the next chapter when we analyze Arrow's theories about preference aggregation. Furthermore there is the question of how much weight to attach to every group member's decision. This, in certain cases becomes a crucial question if one or more group members are or should be more dominant in the way they affect the final decision and its outcome. If we take the hypothetical example of the board of directors of a corporation which includes its president we might be able to see a familiar case. If the ultimate responsibility is to fall on the president's shoulders, he is going to attach more weight to his own preference structure than any of the other board member's when the final decis discussed a them. 1.7 M The th utility (se sion proble questions n Not only th. practical 0‘ lOgical aggj realistic g: 13 final decision is to be made. Later these important problems will be discussed and some recommendations will be offered on how to overcome them. 1.7 The Basic Approach of Thesis The thesis uses rather recently developed theory of multiattribute utility (see Keeney and Fishburn references) to attack the group deci- sion problems. The theory is still at its infancy stage and numerous questions need to be answered before it can become truly operational. Not only the theoretical difficulties suggested above, but also many practical or operational difficulties exist in the realistic and logical aggregation of multiattribute utilities on the one hand, and a realistic grouping of group member's preferences on the other. 2.1 m There i the problems economists c 0n the theor Viduals' pre then the ind individual I: Preference p SOCial welfa Mon which “ The the derivatiOu 0 but also gee This is bECa relevant for Befote Choice, let states that Utility fume U 'l"n.u b . II e CHAPTER II GROUP WELFARE MODELS 2.1 Theories Related to Aggregation of Preferences There is a very broad range of background work that deals with the problems social welfare or collective choices. Especially welfare economists created a vast, exciting but somewhat confusing literature on the theory of social welfare functions which are derived from indi- viduals' preferences. If different alternatives exist for the society, then the individuals comprising the social group express their own individual preferences which are aggregated to give us an overall preference picture for the society. What the welfare economists call social welfare function is very much analogous to group utility func- tion which will be defined in later chapters. The theory of collective choice not only concerns itself with the derivation of social preference from a set of individual preferenceS, but also goes into the formation of individual preferences themselves. This is because the formulation of individual preferences are mostly relevant for postulating rules for collective choice. Before going into the discussion of various theories of group choice, let us define s = {51, 52, ...sm} as the possible alternative states that become possible social choices. Let U represent the utility function of the societal group as a whole and also let ul, . h u2,....un be the utility functions of each of the members Of t e 14 decision ; then U(Si whole and 2.2 Pai A sin with the marized i (a) (b) The mEmber of and When rather th y. There aggrESate FrOm optimalit ChOOSe an as x and than x. there dOe able to a 15 decision group. Therefore, if the group indicates a social choice Si’ then U(Si) will be the utility of the alternative Si to the group as a whole and uj(Si) will be the utility of ith alternative to group member j. 2.2 Pareto Criterion and Principle of Pareto Optimality A simple criterion of comparison of social welfare is associated with the name of Pareto [93]. Pareto's basic contentions can be sum- marized in two rules. (a) If everyone in the society is indifferent between two alternative social situations x and y, then the society should also be indifferent. ‘ (b) If at least one individual strictly prefers x to y, and every individual regards x as to be at least as good as y, then the society should prefer x to y. The above criterion is very basic. When (a) is satisfied, no member of the group cares which alternative is adopted by the society and when (b) is satisfied, no one is really interested to be at y rather than x, and it is in someone's interest to be at x rather than y. Therefore it is very reasonable to assume that the society, as an aggregate of individuals, does prefer x to y. From the above criterion we arrive at the principle of Pareto optimality. We know an alternative x is Pareto—optimal if we cannot Choose an alternative that everyone will regard to be at least as good as x and which at least one person will regard to be strictly better than x. Stated in another way, a joint action is Pareto-optimal, if there does not exist an alternative action that is at least as accept- able to all and definitely preferred by some. In other words, a joint action is Pa better off w A great approach. I judged in te ever this ar prefers x to socially usi evaluate x v There a and Paretian assumptions SUbstitutabi as a decisio Optimality c for iIldividu in a group C (and VonNeUm again be dis 2.3 M The met: euce are 80111. majority dec social State; :6) if and 0n. The majOj 16 action is Pareto-optimal, if it is not possible to make one individual better off without making another individual worse off. A great deal of modern welfare economics has been based upon this approach. The "Optimality" of a system or of a policy has often been judged in terms of whether it achieves Pareto—optimality or not. How— ever this argument should not be stretched too far. If one individual prefers x to y, and another prefers y to x, then we cannot compare them socially using the Pareto criterion no matter how the rest of them evaluate x vis—a-vis y and no matter how many of them there are. There also seems to be an ongoing conflict between group Bayesians and Paretians. The group Bayesians would argue that the behavioral assumptions for individual rationality (for example transivity and substitutability) are equally compelling when applied to a group acting as a decision making unit. The Paretians would argue that Pareto- optimality cannot be violated, and therefore the behavioral assumptions for individual rationality need to be revised when they are interpreted in a group context. This possible area of conflict between Bayesian (and VonNeumann) rationality axioms and Pareto—optimality concept will again be discussed later in the thesis. 2.3 Collective Choice Rules The methods of going from individual orderings to social prefer— ence are sometimes called "collective choice rules." The method of majority decision is one such rule. Here, aSSuming that there are two social states, x and y, x is declared as socially at least as good as y, if and only if at least as many people prefer x to y as prefer y to X. The majority rule often yields intransitive social preference. For example, and s3 and I list like: for A: for B: for C: Condorc as above are lective cho: Anothe1 general con: there is une very satisf; Choice arise tions. One for a Chang. then Stick 1 SUI-Med UP a: and no One ( to y, and w] Preferred U Such a opposing a 1 6159. Wants. choice at t] to the Firs- vould have , 17 For example, if we suppose that there are three alternatives $1, $2 and s3 and three individuals A, B and C, then we can have a ranking list like: for A: s > 3 for B: 33 > 51 > 32 where > means is preferred to for C: 32 > 33 > 51 Condorcet [14], in an early work, determined that such rankings as above are intransitive and therefore cannot yield a meaningful col- lective choice in terms of simple majority rule. Another collective choice rule is the Unanimity Principle, whereby general consensus is considered as the basis of social action. When there is unanimity of views on some issue, clearly this provides a very satisfactory basis for social choice. Difficulties in social choice arise precisely because unanimity does not exist on many ques— tions. One answer to such a difficulty would be to insist on unanimity for a change, and if there is no such unanimity for any proposed change, then stick to the status quo. Such a rule for social choice can be summed up as: Given that some prefer alternative x to status quo y, and no one considers x to be worse than y, then x is socially preferred to y, and when this condition is not satisfied, the status quo y is preferred to the other alternative x. Such a rule is of extreme conservatism. Even a single person Opposing a change can block it altogether no matter what everybody else wants. For example, if this rule were to be used for social Choice at the time of French revolution, Marie Antionette's opposition to the First Republic would have saved the monarchy and the world would have seen very little change. Obviously such a decision rule is generall Finall with his fa 2.4 Arr—ow' Bergso of social w thought on question: Vidual pref lective cho In ord asSumptions 18 is generally unacceptable to the groups. Finally, considering various ranking schemes, Arrow [2] came up with his famous General Possibility Theorem. This is discussed next 2.4 Arrow's General Possibility (Impossibility) Theorem Bergson [4] and Samuelson [109], by developing a certain concept of social welfare function, cleared up several barriers of rational thought on social choice. In extending their ideas, Arrow asked the question: How would a social welfare function (W) depend upon indi- vidual preference orderings? Or in other words, what should the col- lective choice rule be? In order to arrive at his famous conclusion, Arrow used certain assumptions. These are: Assumption 1 (Unrestricted Domain) (a) At least three alternatives exist in the set {s}. (b) Group ordering is specified for all possible individual orderings. (c) The group consists of at least two individuals. Assumption 2 (Weak Pareto Principle or Positive Association of Social and Individual Orderings If the group ordering indicates that alternative 31 is preferred to alternative s2 for a certain set of individual rankings and if: (a) The individual's paired comparisons between alternatives other than 51 are not changed, and (b) Each individual‘s paired comparison between 51 and any other alternative remains unchanged or is modi— fied in 51's favor, then the group ordering will assert that 31 is still preferred to 82‘ ésggpp This a consid tives group to the éEEEEB For ea of ind Prefer NOW we that there satiSfy all Very iflnOco nonsreI tha the wOrld. Arrow' Parismls of to be relax PariSOHS‘ supPOSed to some Sort 0 There consideration and the rank orderings for the remaining alterna- f tives do not change for all the group members, then the new group ordering for the remaining alternatives should be identical to the original group ordering for these same alternatives. Assumption 4 (Citizen's Sovereignty) For each pair of alternatives s1 and 82, there is some set of individual orderings such that the group prefers S1 to 82' Assumption 5 (Nondictatorship) There should be no individual such that whenever he prefers l9 Assumption 3 (Independence of Irrelevant Alternatives) This assumption states that if an alternative is eliminated from S1 to s2, society must prefer S1 to $2, irrespective of the preference of everyone else. Now we come to the result. The stunning conclusion of Arrow is that there is no social welfare function that can simultaneously satisfy all these five conditions. Each of these assumptions looks Very innocous and reasonable, but put together they seem to create monster that gobbles up all the little social welfare functions in the world. Arrow's very general result does not admit any interpersonal com— parisons of preferences. The condition of irrelevant alternatives has to be relaxed if we want to include any interpersonal preference com— parisons. Many authors contend that, social choice rules, if they are Supposed to reflect individual strengths of preferences, must involve some sort of interpersonal preference comparisons. There is another avenue of relaxing the constraints in Arrow's theory. Arrow and Black [6] demonstrated that if the second part of the assumption 1 orderings that ! preferences," t7 be transitive. in the group ha The graphi stood by arrans and having peo' vertical axis. when single-pa and when this Figure 2 decision make that the util single peaked However, in t tion of singi 1101‘. use majo While t due. First ences are p( Peéikedness functions, arbitrarily axis will u that “fire utility Cu‘ 20 the assumption 1 (l(b)) is relaxed, to include only those preference orderings that satisfy a uni—modal pattern, called "single-peaked preferences," than simple majority rule can be used and it will always be transitive. One additional requirement is that the number of people in the group has to be odd. The graphic aspect of the expression "single peaked" can be under- stood by arranging the alternatives on a left-right horizontal line and having peoples welfare levels, or utilities, represented on the vertical axis. The graphs on the following page clearly illustrate when single-peakedness result in a majority rule that is transitive and when this condition is violated. Figure 2.5 shows the preference structure of a group of three decision makers. The first set of graphs, A, B and C indicate to us that the utility representation of the individual preferences are all single peaked and the majority decision in this case is transitive. However, in the second set of graphs, D, E and F we observe the viola- tion of single-peakedness condition for individual 1. This group can— not use majority rule and expect it to remain transitive. While this bit of pictography may be helpful, some warnings are due. First even if no utility representation of individual prefer- ences are possible, they can still be single—peaked, because single— peakedness is a property of a set of orderings and not of utility functions. Second, single—peakedness does not require that any arbitrarily chosen way of arranging the alternatives on the horizontal axis will make the utility curves of each individual uni-modal, but that there exists at least one method of sequencing them such that the utility curves will be uni—modal. .J Na — n:0..: - 33:3.- 9hu 0.. .- e X‘- um..:3~:~ 0,4: 3.- 3.5 3H.- moaunmeou:H ooconumuuh voxmom mamwuuaz no odu:fim m .~ 333m n. ~e:1«>ufi:_ ~a .a:=~>~e=~ : :fivgflus 3:353: «03:53 —< «953.53 ~< on mm mm _m an vm em mm mm ~m mm cm on mm mm mm mm on r A ‘ m a a * . xumm:ou=~ >umm=ousm xuumcou:_ 1 3:33?!— ouzohouohm ooeonou one 2 2 2333...: Na gangs: : 33.32:: mo>~ueskou_< mo>mua:hou_< mo>gas=kou~< a m n g N e o m m ~ N v o m m a N v m m . m a. m m . . m m m m m m m m m m m on _ _ x \ ‘ _ _ U a < 53:35 5 «2.35 33:35 3:23 one 3:9. om one 3:0,. 3 9:. Next the 2.5 neat Marcus] plausible po: tion to a na‘ sihle as the fnmus postu If s1 is p to 51' if ever prefer I 52 to Eggtul I Prefer Prefer is pre 22 Next the work of Fleming [45] will be examined. 2.5 Fleming's Social Welfare Theory Marcus Fleming has shown that if one accepts fairly weak and plausible postulates, one can restrict the type of social welfare func- tion to a narrow class of mathematical functions so as to be expres— sible as the weighted sum of the individual's utilities. Fleming's famous postulates are as follows: Postulate A: (Asymmetry of Social Preference) If from a collectivist social standpoint, social state s is preferred to social state 52, then 32 is not preferred 1 to 31' This is also the weak form of Pareto criterion, i.e., if everyone strictly prefers S1 to 52, then society must also prefer S1 to $2. Postulate B: (Transitivity of Social Preference) If from a social standpoint s1 is preferred to 32, and s to 33, then 31 is preferred to s3. 2 Postulate C: (Transitivity of Social Indifference) If from a social standpoint neither of S1 and 32 is preferred to the other, and again neither of 52 and s3 is preferred to the other then likewise neither of S1 and s3 is preferred to the other. Postulate D: (Positive Association of Social Preferences to Individual Preferences) If an individual i prefers alternative 51 to alternative 52, and none of the other individuals prefer 52 to 51’ then S1 is preferred to 52 from a collective aggregate standpoint. ThJ‘ ences c< social 1 relativ viduals least t the gro again. Fleming an aggregate f(sl) = where f f 1’ Such that a £1" (Si In otl‘ Sum can be The a: beCause the Furthermor. Fleming! S sure PIOSp to han be OUtcomeS e 23 Postulate E: (Independent Evaluation of Utility Distribution) This postulate states that on issues two individuals' prefer- - ences conflict, all other individuals' interests being unaffected, social preferences should depend exclusively on comparing the relative social importance of the preferences of the two indi- viduals concerned. For this postulate to hold true, we need at least those individuals in the group. Later on when we examine the group utility functions we will come back to this condition again. Fleming demonstrates that when these conditions hold, then we have an aggregate preference function of the form n f(sl) = jil fj(si) where fl, f2, ....fn are real valued functions on S = {$1, 32,....sm} such that an individual j prefers si to 51 if and only if fj (s1) > £34.91). In other words, there exists some ordinal utility functions whose sum can be adopted as social welfare function. The assessment of Fleming's utility functions are very difficult because they involve heavy use of interpersonal utility comparisons. Furthermore in accordance with prevalent usage in welfare economics, Fleming's postulates refer to social or individual preferences between sure prospects only. As it will be shown later, it is mose desirable to have both sorts of preferences defined for choices between uncertain outcomes as well. 2.6 none Goodman conparabilit not make ver umber of "1 Their I has a finite is the best person to p1 decision ma? before he 0 arriving at three impor M I 52, tr M the u: tives tion, 24 2.6 Goodman—Markowitz Social Welfare Functions Goodman and Markowitz [48] introduced the idea of interpersonal comparability by making the normative assumption that individuals can- not make very fine comparisons, so that each person has only a finite number of "levels of discretion." Their position is that there are n voters, from 1, 2,....n. Each has a finite number of levels of discretion, L1’ L2, ....Li. Level L1 is the best and Li worst. The number, Li’ of levels may differ from person to person. Here the levels correspond to alternatives and each decision maker can only compare a certain number of alternatives, L, before he or she starts being indifferent between some of them. Before arriving at this aggregate ability function, Goodman-Markowitz assume three important conditions. These are: Condition 1 (Pareto Optimality) If nobody prefers $2 to S1 and somebody prefers S1 to 32, then 31 is socially preferred to 32. Condition 2 (Symmetry) The group ordering of alternatives does not change if the utilities of any two individuals for all the alterna— tives are interchanged. Condition 3 (Stability of Social Ordering) Suppose that an individual j has Li levels of discre- tion. The social ordering between two alternatives 3k and s1 is unchanged if Uj(sk) and uj(sl) are replaced with Uj(sk)+C and Uj(Sl)+C where l f Uj (Si) + C 5 max Li 1 for all i. Here t] The di as Goodman observe all set of alte might alter dms the so independent The se the signifi tion level example, on "magnificie vidual has it will be the welfare to "magnif: from "aver; Goodm; They state a Simple c U(si) Wj'S are W Like Flemi Operationa 25 Here the constant C must be an integer. The difficulties with Goodman—Markowitz approach are many. First, as Goodman and Markowitz themselves point out, it is not possible to observe all the discrimination levels of an individual, given a fixed set of alternatives. If a new alternative, sm+l is introduced, this might alter the utility numbering system used for an individual and thus the social evaluation between alternatives 5k and 31 will not be independent of what other alternatives are available. The second difficulty lies in the assumption of symmetry whereby the significance for social welfare of a change from one discrimina- tion level to the next is the same for all group members. If, for example, one individual has extreme perceptions for alternatives, like "magnificient" or "horrible" and another, more middle of the road indi— vidual has finer perceptions like "good," "average" and "poor," then it will be unfair to make the assumption 2 (symmetry). This is because the welfare significance of moving the first individual from "horrible" to "magnificient" is not really the same as moving the second individual from "average" to "good." Goodman and Markowitz admit the difficulty with the 2nd condition. They state that, even if conditions 1 and 3 are required, there exists a simple class of social welfare functions where n U(s.) = z w.U.(s.) i j=l 3 3 1 Wj's are weighting constants that are positive but also arbitrary. Like Fleming‘s theory, Goodman—Markowitz results are not very useful operationally but they provide very good insights into group preferences. 2.7 as: Nash [i situation. mner certa behavior un individual nizing the operative o viduals. H and Raiffa In the function Ui where 80 re if there is exists an c Max [l = MEX SUbjeCt to In de Axiom \ “POn of U coope 26 2.7 Nash's Approach Nash [87] was especially interested in a two person bargaining situation. He tried to maximize the utilities of two person bargainers under certain conditions. Nash specified assumptions about individual behavior under uncertainty that permit a cardinal representation of individual preferences. He proposed a solution that is given by maxi— mizing the product of the differences between the utility from a co— . o . . operative outcome 3* and the status quo Outcome S for the two indi— viduals. His solution was readily generalized to n persons by Luce a and Raiffa [76]. In the n person generalization, if we assume a cardinal utility function Ui’ i=1, 2, ..., n over the alternatives So, 51, 82’ ...,Sm where S0 represents the status quo or "do nothing" social state, and if there is at least one Sj such that Ui(sj) > Ui(S°), then there exists an optimum alternative 3* that will maximize the function Max [Ul(S*) — Ul(SO)] [U2(S*) — U2*S°)] Un(S*) - Un(S°) = Max u'm o H O 1 mime) - Ui(s )1 subject to the constraint that that Ui(S*) > Ui(S°) for all i. In deriving the above results Nash uses a number of axioms. These are: Axiom l (Invariance of the Solution with Respect to Positive Linear Transformations) The alternative preferred by the group shall not depend upon the utility scales (origins and units of measurements) of Ui's. The solution proposed by Nash of the two person cooperative game is expressed not in terms of numerical payoffs Axiom 2 If no othe of play affect Axiom A A new al status The fi 0f utilitie ness" or "j then Nash's labor marke to accept 5 absence of POSsihly u, 2.8 % Harsa utility fu The group Can be giv 27 payoffs, but in terms of coordinated strategy mixture. Axiom 2 (Pareto Optimality) If the group prefers alternative Sf, then there exists no other alternative Sk such that Ui(sk) 3 Ui(Sf) for i=1, 2, ...., n. Axiom 3 (Symmetry) The solution should be invariant with respect to relabeling of players. In other words the identity of players does not affect the evaluation of alternatives and the final choice. Axiom 4 (Independence of Irrelevant Alternatives) As long as the status quo, So is kept fixed, then adding new alternatives to the system will not change the non—preferred status of an old alternative. The first axiom explicitly rejects the interpersonal comparison of utilities, but the symmetry axiom does not. When values like "fair- ness" or "just outcomes" are incorporated into the evaluation process, then Nash's solution becomes rather unattractive. For example in a labor market with lots of unemployment, the workers may be agreeable to accept subhuman wages and poor terms of employment, since in the absence of a contract they may starve (So). However such a solution, possibly under Nash's theory, does not make it a desirable outcome. 2.8 Harsanyi's Model Harsanyi[53] developed some assumptions under which individual's utility functions can be aggregated to produce group utility functions. The group utility function for the situation 3, under his assumptions, can be given by U(s) = where Wi's Harsan to develop possible if M S all th discus m T tions Postul stand; from ; Diamor cases of g‘ V idualS, A Lotte Lotte If 11 While 3 vi lr/Z that f! 28 n U(s) = Z wiUi(S) i=1 where Wi's are positive weights. Harsanyi uses the V “ fivrgeustern or Marschak postulates to develop his theory. The group utility function, given above, is possible if, Postulate A Social preferences or the group utility function satisfy all the axioms of VonNeumann-Morgenstern (these postulates are discussed in the next chapter). Postulate B The individual preferences or individual utility func- tions satisfy these postulates of VonNeumann—Morgenstern. Postulate C If two alternatives s1 and 52 are indifferent from the standpoint of every individual, they are also indifferent from a social standpoint (for the group). Diamond [21] considers the third assumption not to be valid for cases of groups. To prove his point he takes the case of two indi— viduals, A and B and two alternative "lotteries," I and II. l/2 Probability l/2 Probability Lottery I UA=l, UB=0 UA=0, UB=l Lottery II UA=l, UB=0 UA=l, UB= If II is chosen, then individual A will obtain a unit of utility while B will have none. With alternative I there is a probability of 1/2 that A will have one unit of utility and B none, while there is also a prob. none. In ter equally goo society (in ferent betw unfair to 1 alternative seriOus prc if all the 2.9 B3Yl§_ Rawls status, but In his own Princ \ by it compa m expe C Provi or fr The u but it tur fUnCtiOn l 29 also a probability of l/2 that B will have one unit of utility and A none. In terms of aggregate expected utility maximization, I and II are equally good, having an expected aggregate value of 1. Therefore the society (in this case the group of two, namely A and B) will be indif- ferent between the two alternatives. However alternative II is very unfair to individual B because of its "0" payoff in each case although alternative I is equitable between A and B. Obviously, there is a serious problem of possible unfairness or inequity to the participants if all the postulates of Harsangiare accepted as they are given. 2.9 Rawls Theory of Fairness Rawls [105] was not so much interested in the ordering of social status, but with finding just institution as opposed to unjust ones. In his own words, Rawls' principles of fairness are: Principle A "Each person participating in a practice, or affected by it, has an equal right to the most extensive liberty compatible with a like liberty for alf: Principle B "Inequalities are arbitrary unless it is reasonable to expect that they will work out for everyone's advantage, and provided that the positions and offices to which they attack, or from which they may be gained, are open to all." The meaning of the above principles is not altogether obvious, but it turns out that Rawls is essentially proposing a social welfare function based on the maximin principle, and always measuring the welfare of El The Rawlsian more weight assigns to t Rawls' of the relew mm princip] by using cor lead to nor: Finallj fore we tur and its ext 2.10 SVL'S Sen [1 should give riCh People by Strict r Wants the 1 but also 0. is that of This cauSe Sen w that Sen's against pe This 30 welfare of the society by the utility level of the worst off individual. The Rawlsian social welfare function would always assign infinitely more weight to the interests of poorest members of society than it assigns to the richer members. Most Rawls' theory came under heavy criticism from many sources. of the relevant criticism boiled down to the exclusive use of the maxi- mum principle for the sake of fairness. Harsanyi[54] also demonstrates, by using convincing examples, that Rawls' social welfare function might lead to morally wrong decisions. Finally, Sen's arguments on social choice will be considered be- fore we turn our attention to Y “ Hutguubtuiu utility theory and its extensions. 2.10 Sen's Approach Sen [14] tends to agree with Rawls that the social welfare funcion should give more weight to the peer people's utility functions than to rich people's. He also claims that the inequities that might be caused by strict utilitarianism might be reduced by his approach. Sen simply wants the social welfare or group utility depend, not only on the mean, but also on some measure of inequality. This measure, in his theory, is that of dispersion among the different individuals utility levels. This causes the group utility function to be nonlinear. Sen was heavily criticized, especially by Harsanyi [54]. He states that Sen's utility theory would give rise to unfair discrimination against people enjoying relatively high utility levels. This concludes the discussion of the background work in group preferences. Next VonNeumann-Morgenstern utility theory and its extensions w of the thesi 31 extensions will be considered at a somewhat greater length since much of the thesis revolves around this theory. This Morgenster Certain be also disco 3.1 EEEEE The i cal found; analysis. and P2 be Cussion, : “J “is ind: §§§pg If we decision r them, In §§§EE This that X is Preferred CHAPTER III UTILITY AND DECISION THEORY This chapter begins with the formal treatment of VonNeumann— Morgenstern axioms, unidimensional and multiattribute utility theory. Certain behavioralistic characteristics and their implications are also discussed. 3.1 VonNeumann Postulates and Their Extension The following eight assumptions pretty much provide the theoreti— cal foundations of the unidimensional and multiattribute utility analysis. Let there be three alternatives X, Y, and Z and let Px’ Py and P2 be the respective probabilities between 0 and 1. In this dis— cussion, > indicates "is preferred to", < "is not preferred to" and W "is indifferent to." Assumption 1: Closure and Orderability If we compare the alternatives in pairs,-then either the decision maker will prefer X to Y, Y to X or be indifferent between them. In other words either X > Y, Y > X or X N Y. Assumption 2: Transitivity This assumption states that if the decision maker determines that X is preferred to Y and Y is preferred to Z, then X must be preferred to Z. In other words X > Y, Y > Z => X > Z. 32 Assum‘ This (a) (b) Here the l Asfl If th then accor the letter between th (X, PX, Z) Assum If th denoted X the two lc eiiCh other Assun This L2 = (X, E Assun This Y is Prefe is Prefer] (PK; (l-Pj 915$ This Posed and 33 Assumption 3: Sure—Thing This assumption is composed of two subsections. (a) if X > Y, then X > [PXX, (l—PX) Y] (b) if X < Y, then X < [PXX, (l—Px) Y] Here the lottery [PxX, (l-PX)Y] is equivalent to L = (X, Px’ Y). ‘ Assumption 4: Continuity If the preference ordering exhibits the relationship X > Y > Z, then according to this assumption there exists a probability Px for the lottery L = (X, Px Z) such that the decision maker is indifferent between the outcome Y and the above lottery. This can be expressed as (X, Px’ Z) N Y. Assumption 5: Substitutabilipy If there exists an indifference relationship between X and Y, denoted X N Y, then X is substitutable for Y in the lotteries and thus the two lotteries L1 = (X, p, Z) and L2 = (Y, p, Z) are indifferent to each other. In other words X N Y = (pX; (l-p)Z) W (PY; (l-P)Z)- Assumption 6: Monotonicity This assumption requires that L1 = (X, Pl’ Z) is preferred to L2 = (X, P2, Z) if and only if Pl is greater than P2. Assumption 7: Utility Differences This assumption requires that, if X is preferred to Y and if Y is preferred to Z, then there exists a P such that L = (X, P, Z) is preferred to Y. In other words if X > Y > Z, then 3 P such that (PX; (l-P)Z) > Y. Assumption 8: Decomposability This assumption requires that component lotteries can be decom- Posed and be reduced to a simple lottery. Let a compound lottery be L1: (X: PX P , Y with x decomposabi the simple lottery L1' 3.2 RIPE If one relations 1' tions, the: U), = W), (l) 1 the expect Rm (1 Final axioms ab 0. tIllncticm t (3) In 0t linear tra 3-3 Denis \ The F lated as f 34 L1 = (X, Px’ Y, Py’ Z) where outcome X is obtained with a probability Px’ Y with probability Py’ and Z with probability l-Px—Py. Then decomposability states that there existsa probability P; such that the simple lottery L2 = (X, Px’ Z) is indifferent to the compound lottery L1' 3.2 Implications of the Axioms If one accepts the axioms described above and if the preference relations involving the outcomes X, Y, and Z satisfies these assump— tions, then there exists real valued functions U(‘) - (where Ux = U(X), Uy = U(Y), and U2 = U(Z)) - such that (l) X > Y > Z if only if UK > Uy > U2 (2) U(pX. (l-p) Y) = W (X) + (l-p) U (Y) Also if we decide to use a compound lottery and try to evaluate the expected utility of the lottery, we have E(U (L ) = PXU(X) + PyU(Y) + (l-PX-Py) U(Z) 1) Finally, if V is some other utility function that satisfies the axioms above, then it must be a linear function of any other utility function that satisfies the axioms. That is (3) V(X) = aU(X) + b. In other words utility functions can be written as positive linear transformations of each other. 3.3 Decision Theory Framework The problem of decision making under uncertainty can be formu— lated as follows: (b) ‘ Act (a) (b) (e) 35 There are a number of alternatives in the set A = (a1, a2, ..., am) There are a number of states of nature Si = (SI, 82’ ..., Sn) that the decision maker is aware of. What he doesn't know is which state will occur in reality. And finally there are the outcomes 0 = (0 0 O ) ll’ 12""’ mn which will take place if a certain act is followed and if a certain event takes place. A simple matrix will indicate these relationships more clearly (d) State Sl 82......Sj......Sn al 011 012 ..... Olj.....0ln a2 021 022. "02j" ..02n . ........... 0. a1 011 012 013' m . ........... 0 am 0mi omZ OmJ mn When there is an uncertainty about the occurrence of the outcomes Oij’ the decision maker associates a probability function P(Oiiaj) with the outcomes. This probability function encodes the feelings of the decision maker about the likelihood of different outcomes. The probability function is conditioned upon a particular action that is being carried out. For any given action aj, the sum of the probabilities for the outcomes would be 1. This is 1 The e: should cho< risky envii teries. T] has the hi; and Lj’ U(f Property i: the utilitj 0f its con There alternativ max J For the ab In the abo II r4 :3 "O H- ,_. This making it than“. is 36 called subjective probability function. (e) Finally, we are able to identify a utility function, U(Oij), with each outcome. This utility function expresses the relative preferences of the decision maker with respect to various outcomes. It is called the individuals cardinal utility function. The expected utility hypothesis states that the decision maker should choose the alternative with the highest expected utility. In a risky environment, all alternatives can be reduced to a set of lot— teries. The optimal alternative corresponds to that lottery which has the highest expected utility since, for any two lottereies Li and Lj’ U(Li) > U(Lj) if and only if Li is preferred to Lj. This property is called the expected utility hypothesis. In other words, the utility of a lottery is equal to the sum of the expected utility of its components. Therefore we can evaluate the maximum expected utility of an alternative aj by the formula _ n m:x EU (aj) — mix 1:1 U(Oi) Pij (Oilaj) (3.3.1) For the above expression, m alternatives and n outcomes are assumed. In the above analysis, the probability estimates are so chosen that n z Pij (Oilaj) = 1. This theory is very appealing and logically consistent thus making it a feasible normative tool. The main difficulties of the theory is not in its analytical development but in the operationalization phase. A d aspects of have more t more diffic assurance a diverse out The ur found some lying asses extensive. 3.4 m One 01 theory was Setting Wht By varying different 1 tion was d. °Ver Moste Curve for : laboratOry Problems w Later analysis t wild Catte investmeut not try to 37 phase. A decision maker may not be very clear with respect to diverse aspects of different outcomes. Furthermore many of the outcomes may have more than one attribute. These complications makes it more and more difficult for the decision maker to attempt to specify with assurance a utility function given his relative preferences for the diverse outcomes. The unidimensional decision theory incorporating utility functions found some applications during the past three decades. Due to under— lying assessment difficulties those applications have not been very extensive. Some of these important applications are given below. 3.4 Previous Applications in Unidimensional Analysis One of the earliest experiments in the unidimensional utility theory was done by Mosteller and Nogee [85]. They used a laboratory setting where an individual was to choose between several lotteries. By varying the probabilities associated with different payoffs seven different points were specified from which the subject's utility func- tion was derived. Davidson, Suppes and Siegel [l8] tried to improve over Mosteller's model by attempting to measure a subject's utility curve for money. These two models were both conducted in artificial laboratory settings and thus their relevance to real world decision problems were not very obvious. Later Grayson [50] tried to apply the unidimensional utility analysis to the quantification of the preferences for money of oil wild catters. Using responses to risky hypothetical oil drilling investments, Grayson developed twelve utility functions, but he did not try to analyze characteristics like risk aversion. P. E. managers at different 1 Anothe where he i1 rations, w: functions. toward risl given comp: Also ‘ preference. the intent agers were fied. Fin W) = a + POints. Mosko research a of a Simul managers a behavior 5 Refer We anal) iStlQS Of 3.5 m Mono 38 P. E. Green [51] tried to develop utility functions for sixteen managers at middle management level of one company. He also tested different probability distributions with his subjects. Another large scale project was undertaken by Ralph 0. Swalm [121] where he interviewed a hundred different executives from various corpo- rations, with the goal of trying to assess their corporate utility functions. This was a descriptive work. Swalm concludes that attitudes toward risk decisions vary even more widely among decision makers in a given company than we are inclined to think. Also there is the work of Spetzler [119] who quantified the preferences of a number of business executives from one company with the intention of developing a corporate risk policy. Thirty-six man— agers were interviewed and their attitudes towards risk were identi- fied. Finally a logarithmic utility function of the basic form U(X) = a + b log (X+C), b > 0 was developed and fitted to the utility points. Moskowitz [84] performed several experiments on a sample of research and development managers and MBA students within the context of a simulated business environment. Both research and development managers and students exhibited a significant degree of irrational behavior as evidenced by such factors as intransitivities. Before going into the multi—attribute utility and group prefer— ence analysis, let us briefly review the most important character— istics of utility functions. 3.5 Monotonicity Monotonicity is one of the most basic property of utility func- tions. Let U indicate the utilit function, a desirable attribute Y and Xd a PI the fact t? is greater other word. property 0 (Xdz If on dealing wi (Xud2 where Xu d An un SimPly red Attri mmber of attributes the road, Final column but If a Stand able level desirable, a "normal“ Outcgme_ always les 39 and Xd a particular level of XD. Here desirable attribute indicates the fact that more of XD is preferred to less of XD. Therefore if Xd2 18 greater than Xdl’ then the dec1sion maker prefers Xd2 to Xdl' In other words the utility of Xd2 is greater than that of Xdl‘ This property of utility functions can mathematically be expressed as (Xd2 > Xdl) <=> U(Xdz) > U(Xdl). If on the other hand we have an undesirable attribute that we are dealing with, then less of it is preferred. This is expressed as (x <=> u(x udl) > U(X ud2 > Xudl) ud2) where Xud denotes level of undesirable attribute. An undesirable attribute can be transformed into desirable one by simply redefining the attribute. Attributes like profits, number of people recovering from cancer, number of people winning Olympic gold medals, etc. are all desirable attributes. Undesirable attributes could be number of accidents on the road, amount of pollution in the air, number of people unemployed, etc. Finally there is one class of utility functions which are not very common but which do occur. These are nonmonotonic utility functions. If a standard or an average amount of an attribute is the most desir— able level and any deviation from this average becomes less and less desirable, than we have nonmonotonic utility functions. For example a "normal" level of blood pressure in a patient is the most preferred outcome. Lower and higher levels of blood pressure than normal are always less preferred to normal. This can be expressed by the graph 3.6 Certa Let u "s X . n With each Utility of exponent 40 Utility Normal Blood Pressure Level Figure 3.5 A Nonmonotonic Utility Function 3.6 Certainty Equivalent Let us assume that we have a lottery with outcomes like X1, X2, ...., Xn' Also let the probabilities P1, P2, ...., Pn be associated with each and every one of these outcomes. Let U(Xi) define the utility of outcome 1 and let the expectation of outcomes be _ _ n X = E(X) = Z P.X. . l 1 i=1 In this case, the expected utility of this lottery is _ n E[U(X)] = “Z PiU(Xi) i=1 Then we can define the certainty equivalent of the above lottery L as X, where U(x) = E[U(X)] or i = U’1 EU(X). To give an example, if the utility function is expressed by an exponential such as —e—'CX and if we have a 50—50 lottery for the outcomes X3 outcomes 1E In th: -c -e if we let lottery (X lent, X is 3.7 m Strat utility fu another wi Vidual. ] git Equivz identical means tha1 utility f. This ti°ns are functions linear tr ifa 41 outcomes X1 or X2, expressed as (X1, X2), then the expected value of outcomes is X + X - _l__2 x— .5x1+ .5x2- 2 In this case the uncertainty equivalent must be the solution to A 'CX -e = _ %-e -cxl l -cx --E e 2 = — e l + e 2 If we let c=l, X1 = 10 and X2 = 20, then the expected value of the lottery (X1, X2) is .50(lO) + .5(20) = 15 and the uncertainty equiva- lent, X is equal to 10.69 3.7 Strategic Equivalence Strategic equivalence is one of the important characteristics of utility functions. It helps us to transform one utility function to another without losing the relative preference orderings of the indi- vidual. In other words any two utility functions U1 and U2 are strate- gic equivalents of each other if and only if they both result in the identical preference rankings for any two lotteries. This simply means that the certainty equivalents of two strategically equivalent utility functions are identical. Or, mathematically Ul m U2 => Xl.= X2 for all X This relationship is the same thing as saying that utility func— tions are unique only up to their linear transformations. Two utility functions are strategically equivalent if one of them is a positive linear transformation of the other. Symbolically if aand b > O, thenVX [U103 As it will aggregatic tions . 3.8 w The c briefly, i are talkir of any lot expectatic A X<] if the 101 In C: and thus 4 A ri the utili than the tomes ' T U [P1 The dealt wit findings 42 [Ul(X) = a + b U2(X)] => Ul(X) m UX(X) As it will be observed later, this property will be useful in the aggregation of individual preferences to form group preference func— tions. 3.8 Risk Aversion The concept of certainty equivalence, which has been explained briefly, is very much related to the concept of risk aversion. If we are talking about a desirable attribute, then the certainty equivalent of any lottery over this attribute would be less than the mathematical expectation of the lottery. This is mathematicaly expressed as x < E (X) or x < Ple + (l-Pl) x 2 if the lottery has two outcomes, X1 and X2. In case of undesirable attributes we just reverse the inequality and thus obtain a > E(X) or x > Ple + (l-Pl) X2 A risk aversion utility function is concave and this implies that the utility of the mathematical expectation of the outcomes is greater than the mathematical expectation of the utility of the various out- comes. Therefore U[Plxl + <1-%BX2] > P1U(Xl) + (l-P£U(X2), O < P < l The concept of risk aversion for individuals and groups will be dealt with in detail later in a separate chapter where some new findings with respect to group risk behavior will be demonstrated. 3.9 Les; Since attribute u attribute d Multip extension c tially def: involve di: sion problr or needs t at hand co address it minimizati mization c the soluti sion maker this anal: COtlflict . The the situa of these Where the BY means able to j are avai; and the , arrived ‘ 43 3.9 Decision Analysis Incorporating Multiple Attributes Since most of the thesis is concerned with the extension of multi— attribute utility theory to group preference analysis, the multiple attribute decision theory will be presented at some length here. Multiple attribute decision analysis is a normal and expected extension of unidimensional utility theory whose premises were essen- tially defined by VonNeumann and Morgenstern. Many decision problems involve different outcomes. Different types of solutions to any deci- sion problem will satisfy with varying degrees the different criteria or needs that originally generate the problem. For example the problem at hand could be the design of a transportation system that will address itself to different goals. These goals or needs could be minimization of operational costs, minimization of travel time, mini- mization of discomfort, and minimization of all sorts of delays. Then the solution might be in the maximization of the utility(ies) of deci- sion maker(s) over these sometimes conflicting objectives. Naturally this analysis involves some preference tradeoffs because of the real conflict among some objectives. The decision maker defines the attributes that are relevant to the situation and then assess his utility function for varying amounts of these attributes. It is in the identification of these attributes where the greatest benefits of formal decision analysis are realized. By means of this formal theory, furthermore, the decision maker is able to identify the most desirable course of action among those that are available to him. Continuous revisions are made in the attribute and the action sets until the most desirable combination for both is arrived at. It is this feedback opportunity that is also one of the assets of There analysis. outcomes t results it ability 01 In on sion make: whose uti multiattr one attri utility f strate ho the past general u decision to a cert SPetzler general ; functiOm dECisiog PTEferen. supra de 3'10 Th Dif ature to 44 assets of multiattribute decision analysis. There are various difficulties associated with this type of analysis. One of them is the problem of uncertainty concerning the outcomes that are conditioned upon a certain course of action. This results in the inclusion of probabilistic models to improve the reli- ability of the results. In many decision making situations there is more than one deci- sion maker. Therefore the problems will arise when it is not clear whose utility function should be used in the analysis. Although multiattribute utility analysis is useful in incorporating more than one attribute into the analysis, it sill uses only one individual's utility function. The later developments in the thesis will demon- strate how this theory can meaningfully be extended to groups. In the past research in group decision making situations, only a very general utility function used to be developed and each individual decision maker's utility function was expected to conform, hopefully to a certain degree, to this general utility function. For example, Spetzler [119] develops a corporate risk function which is very general and which tries to approximate the 36 decision makers utility functions. In this thesis, general models are developed where the decision maker not only incorporates various attributes but also the preferences of others into his decision making process. This case of Supra decision maker's utilit functions is examined in greater detail. Y 3.10 The Characteristics of Decision Making Groups Different classifications of groups have been used in the liter- ature to analyze the group decision making behavior. Marschak [80] considers 1 his defini cause each tion the i sonal obje Finally it a group of goals. Sr formulati< process 0: us consid. 3.l0.l E Ther actual de decision maker" wl Words the Preferem Would be being (0. deciSion N indivi him the CrGated the supI Who triE 45 considers groups in terms of teams, foundations and coalitions. In his definition a team demonstrates cohesiveness and solidarity be— cause each member has the same group oriented interest. In a founda— tion the identity of purpose is less strong and individuals own per- sonal objectives enter greatly into the decision making process. Finally in coalitions the individuals do not necessarily subscribe to a group objective but are drawn together in pursuit of their own goals. Such classifications do not lend themselves to mathematical formulations very easily. Therefore, to analyze the decision making process of a group in terms of multiattribute utility functions, let us consider the following kinds of group decision making situations. 3.10.1 The Supra-Decision Maker There are many decision making situations where although the actual decision is really generated by the group, there is one apparent decision maker. This is the case where there is one "supra decision maker" whose decision is totally influenced by others. In other words the supra decision maker incorporates into his decisions the preferences of others in a certain group. The extreme case of this would be when the supra decision maker tries to maximize the well being (or utility) of N individuals in the group. In this case his decision function for the group is really a function of each of the N individuals' preference relationships. Therefore by trying to maxi- mize the group's well being, a certain group preference function is created in the person of the supra decision maker. If we assume that the supra decision maker is not a member of the group but an outsider who tries to maximize the utility to the total group by his decision, then this 1: "well being might have this case 1 function. groups) ut U g(_) where X re vidual grc SIOUp util function. the supra Supra dec the impli U X 8(‘ where UN+ decision Obvz‘ be Only I ences de. f0Inulat that of be dEmOn The example 46 then this model will be identical with a participating group whose "well being" is to be maximized. However, the "supra decision maker" might have his own preferences over the attributes to consider and in this case there is one more argument in the overall group utility function. The most general form of the supra decision maker's (or groups) utility function, in implicit form is Ug(X) = UD((_x), U2(X), ...., Un(_X_)) (3.10.11) where X represents the attributes set and each of the Ui's are indi- vidual group members multiattribute utility functions. Here Ug is group utility function and UD is the supra decision maker's utility function. In this formulation, the preferences over attributes for the supra decision maker is not taken into consideration. If the supra decision maker has a direct interest in the consequences, then the implicit group utility function can be expressed as Ug<§) = Unleé)’ U299, UN(§>.UN+1(29) (3.10.12) where U (X) is the multiattribute utility function for the supra N+l decision maker. Obviously in the above formulation the interaction is assumed to be only one way. In other words the supra decision maker's prefer- ences depend totally upon the preferences of others, eSpecially in formulation 3.10.11 but those others' preferences do not depend upon that of the supra decision maker. Later, a general implicit model will be demonstrated by dropping this particular assumption. The supra decision maker is found in all walks of life. For example the chief operating officer of a company might act like one fi he is t: arriving a1 if we assm sity has tr tion conmti considered final choi recommenda 3.10.2 35 Parti at througl decision I this case perhaps I: function U K E(- Where U_( 1 X1, X2... Conn Very mucl example, reSnonsi Process. Of a tow of One c 47 if he is to consider the preferences of the members of the board in arriving at a decision. In another type of decision making situation, if we assume that the chairman of a certain department at a univer— sity has to explicitly consider the preferences of his faculty selec- tion committee in hiring new people to the department, then he can be considered a type of supra decision maker if the responsibility for a final choice lies with him only and if he is obliged to act upon the recommendation of his faculty. 3.10.2 Participatory Groups Participatory groups are those where the decisions are arrived at through democratic procedures and the responsibility for the final decision lies with the total group and not with one individual. In this case the individuals collectively share the responsibility and perhaps the risks associated with the outcome. The group utility function for the participatory group will, in implicit form, be Ugg) = U (UICX), U2(X),...., Un(X)) (3.10.2) where Ui(X) is a group member's utility function over the m attributes X1, X2"""Xm' Committee type of deciSOn making, as classified by Marschak, very much corresponds to participatory group identification. For example, in a boxing match, if there are three referees equally responsible for the final decision, then there is a participatory group process. Another example would be a city council managing the affairs of a town. As in the above examples, usually there is no dominance of one or more individuals in the group by virtue of their positions. However e‘ functions pref erenc 3.ll & In a members 0 butes. F preferenc l. It 1‘ ences, tl bee-n pres the aggré 3.12 E Whet there is IEm, as greategt function tions to 48 However even if such a situation exists, formulation of group utility functions is still possible by attaching different weights to the preferences of various individuals. 3.11 Steps in Group Decision Analysis In a group we have to consider the preferences of the individual members of the group. These preferences develop over certain attri- butes. Finally there is the problem of aggregation of individuals' preferences. These steps can be summarized as: 1. Determination of the objectives and their attributes to the group. In other words X1, X2,....,Xm are specified. 2. Determination of the utility functions for each indi- vidual in the group. In other words U1, U2,....,Un are specified. 3. Determination of the actual explicit functional form for the group. Here the explicit Ug is specified. It is this last step, i.e. the aggregation of individual prefer— ences, that has been very controversial. These controversies have been presented in chapter two and the following chapters will analyze the aggregation process in terms of multiattribute utility theory. 3.12 Final Remarks on Chapter III When we talk about utility functions, we implicitly assume that there is a problem of uncertainty to be dealt with. It is this prob— lem, as it was analyzed more in depth in Chapter I, that causes the greatest difficulty in the development of group decision making functions. The group members may have different probability distribu— tions to describe the alternatives. Procedures that aid in the aggregatic to this d2 individual sion proce the group tainty asi chapters, preferenc unanimous butes. The for singl extended 49 aggregation of these distributions are very important. Unfortunately, to this day, no satisfactory generally applicable method of combining individual subjective probability estimates to arrive at a group deci- sion procedure, has been found. Although the uncertainty aspect of the group decision procedures, has been found. Although the uncer— tainty aspect of the group decision problem will be touched on in later chapters, to be able to apply multiattribute utility theory to group preferences, it will be assumed that the individual members of the group unanimously agree on the probability distributions over levels of attri— butes. The next chapter will discuss the multiattribute utility theory for single individual's decision function and how this theory can be extended for group decisions will be demonstrated. 4.1A 1312 The applicati sis. Thi “ (asses then it v theory 1; bute uti] 0f Keene} tions of theory a1 Ml E The ClaSsica down by Chapter multiple COmPlex and thus CHAPTER IV MULTIATTRIBUTE UTILITY THEORY AND DECISION MAKING GROUPS Part A 4.1A Introduction The main emphasis in this thesis is the analytical and empirical application of multiattribute utility theory to group decision analy— sis. This chapter first introduces the multiattribute utility theory -— (assessment of utility functions over more than one attribute) and then it will be shown that there are interesting ways to extend the theory into group preferences analysis. The treatment of multiattri— bute utilities in this chapter is mainly based upon various articles of Keeney, Raiffa and Fishburn. Although many authors laid the founda— tions of this theory, the above authors contributed the most to the theory and application in this field. 4.2A Multiattribute Utility Analysis The multiattribute utility analysis is a natural extension of the classical unidimensional utility theory whose assumptions were laid down by VonNeumann and Morgenstern. As extensively discussed in Chapter I, this development arose out of the necessity of introducing multiple criteria into the decision making situations. In today's complex world almost no major decision involves one single dimension and thus multiattribute utility models addressed themselves to such 50 ; situations In ti attributes X XX3X1 2 level of a X-. (X. cor J J X ) from ‘ u bute set ' condition the other The simplif ie Us) The Utility j 110: Where ea form. 3 the attr mIlltiatt U(F Here X. 1 Concepts 51 situations that are judged by multiple criteria. In the following analysis let x = (X1, X2"""Xn) be the n attributes that we are dealing with. The outcome space is X3 = (X1 x X2 x X3 x....x Xn)' Also let (x1, x2,....,xn) indicate a particular level of an outcome in terms of the n attributes. Furthermore let X3 (Xj complement) denote an outcome space (X1, X2”'°'Xj—l’ Xj+l"°" X“) from which Xj is missing and let Xij (Xij complement) be the attri- bute set which lacks Xi and Xj both. Finally let Ui(Xi) denote the conditional utility function over a particular attribute Xi when all the other attributes are at any arbitrary levels. The main idea in multiattribute utility theory is to find some simplified forms to the implicit utility function U(g) = U(Xl, x ...,xn) (4.2.1A) 2" The first step in this process is the decomposition of this utility function into the form U(Xl, X2, ....,Xn) = f[fl(xl), f2(x2),----fn(xn)] (4.2.2A) where each fi is a function over the attribute Xi and f has a simple form. Since fi's represent one attribute conditional utilities over the attribute Xi’ we can have fi(Xi) = Ui(Xi) and the final implicit multiattribute form looks like U(Xl, X2,....,Xn) = U[U1(Xl), U2(X2)....,Un(Xn)] (4.2.3A) Here Xi could be a scalar or a vector. Now the three independence concepts and the resultant u(§) forms will be discussed. 4.3A _T_h_€ The of prefe1 assumpth dealt Wit and to a systemat simplifi Since th these as them bri 4.3.].A Xi preferer and not Other w( level x. Which d OUtcome tainty. Cally b E} prefers Xij an 52 4.3A The Main Assumptions of Multiattribute Utility Theory The three most important concepts of independence, namely those of preferential, utility and additive independence, are the basic assumptions behind multiattribute utilities. Although many authors dealt with these concepts in a non—integrative manner, it was Keeney and to a certain degree Raiffa and Fishburn who analyzed them in a systematic and comprehensive manner with the goal of obtaining certain simplified forms for the implicit multiattribute utility functions. Since the developments in the later chapters are mainly based upon these assumptions, and the resultant functional forms, let us review them briefly. 4.3.lA Preferential Independence Xi is preferentially independent of other attributes (Xi)’ if preferences for the outcomes depend only on the level of attribute Xi and not on the levels ofcomplementary (Xi) set of attributes. In other words the complementary attributes can be set at any arbitrary level Xi and this will not affect the preference for the consequences which differ only in terms of Xi' Here we use only preferences for Outcomes and not preferences for lotteries as in the case of uncer- tainty. These are ordinal preferences. This assumption can symboli— cally be expressed as _ > | _ = I. > I I Extending the concept to (Xi, Xj) pair, we can say that Xij is preferentially independent of X33 if the preferences for outcomes in Xij are not at all affected by the abritrary levels of X13. Theo pendent 0 function V(X1 where V a scsling s and the l 4.3.2A U Util is utilit when )(-i a X? Ther U (Xi where g > utility f result of m independe U (X1 or multiI 1+ 53 Theorem 1. Given n 2 3 and if (Xi’ Xj) is preferentially inde- pendent of all other attributes Xij’ then there exists ordinal utility function (an additive value function) V such that n V(Xl, X2"""Xn) .2 kiVi(Xi) (4.3.lA) i=1 n where V and Vi are scaled from 0 to l and 2 k1 = l, ki > O. This i=1 scsling says that the most preferred outcome X: has a value Vi(X§) = l and the least preferred outcome X? has a value Vi(Xg) = 0. 4.3.2A Utility Independence Utility independence is concerned with cardinal preferences. Xi is utility independent of Xi if preferences over lotteries for Xi’ when Xi are held fixed, do not depend upon those fixed levels set for Xi' Therefore Xi is utility independent if and only if U(Xi’ Xi) = g(Xi) + h (Xi) Ui(Xi) (4.3.2A) where g > O and h > 0 are scalar functions and Ui(Xi) is a conditional utility function where Xi are fixed at a certain level Xi. The main result of this assumption is Theorem 2. For n 2 2, if X1, X2""”Xn are mutually utility independent, then the utility function is either additive n U(Xl, x2,....,x ) = U(X) = '2: kiUiOCi) (4.3.3A) 1:1 or multiplicative n l + kU(X) = H [l + kk.U.(X.)] (4.3.4A) —' i=1 1 :L l where U a: satisfyin: is a solui 1+? It 1 implies t (A. 3 . 4A) For U (g U0 where go each am for ever: others 11. level f0 and k, 1 Theorem (Xi, ij ADI U(z 54 where U and Ui are scaled from O to l, and ki are scaling constants satisfying 0 < ki < l and k > -1 is a non zero scaling constant which is a solution to, n l + k = H (l + k ki) (4.3.5A) i=1 I n It is easy to observe that when 2 k1 = 1, then R = 0. This i=1 implies that U(g) is of the additive form in (4.3.3A). The form (4.3.4A) is called the multiplicative form. For this theorem, the scaling is required in such a way that II 0 u (f) and (4.3.4A) II H U (ye) (4.3.5A) where £9 = (X10, X20,....,Xno) is the least desired outcome in terms of each attribute and g# = (Xl*, X2*,....,Xn*) is the most desired outcome for every attribute. Since certain attributes might be desirable and others not, this does not mean that every attribute is at its highest level for g}. Furthermore the scaling conventions also require that O — Ui (Xi ) — o (4.3.6A) * —- Ui (Xi) — 1 (4.3.7A) and ki = U(x,*, x3 ) (4.3.8A) l 1 Theorem 2 also holds true when X1 is utility independent of Xi and if (Xi’ Xj) is preferentially independent of (X33). Applying Theorem 2 to two attributes we have U(Xl, X2) = klul(Xl) + k2u2(X2) + k klk2U1(Xl)U2(X2) (4.3.9A) ‘ + k 1f k1 theorem 4 U(X] For exann 4.4A ggy Fis butes g over lot ability to depen attribut Theorem Ihg (Value) 1' fUthiOE 55 if k1 + k2 = l and k = 0, we have the pure additive form. Applying theorem 2 to three attribute case, we have U(Xl, X2, X3) = klul(Xl) + k2u2(X2) + k3u3(X3) + kklk2u1(Xl)u2(X2) + kklk3u1(Xl)u3(X3) + kk2k3u2(X2)u3(X3) 2 + k klk2k3ul(Xl)u2(X2)u3(X3) (4.3.10A) For example expression (4.3.10A) holds when X1 is utility independent of (X2, X3) and if X2 and X3 are preferentially independent of (X1, X3) and (X1, X2) respectively. If k # 0, then by multiplying each side of (4.3.10A) by k, adding 1, and factoring, we obtain the multiplicative form 3 kU(Xl, X2, X3) + l = 1:1 [kkiUi(Xi) + l] (4.3.llA) 4.4A Additive (Value) Independence Fishburn dealt with n attribute utility function. A set of attri- butes X = X ..X are additive (value) independent if preferences - n 1’ X2,.. over lotteries on X1, X2"""Xn depend only on their marginal prob- ability distributions. In other words such preferences are assumed not to depend on the joint probability distributions of the members of the attribute set. The fundamental result of additive independence is Theorem 3. Theorem 3. If attributes Xi’ i = l, 2, ....,n are additive (value)independent of each other, then there exists an additive utility function U(X] where U(Z = U x, ki ( 1‘ Let butes. < l/2 must be 1 U[§ The 00.1) are cont his set Utility utility meUtiOn k.'s l at 56 — _. n - U(Xl, X2,....,Xn) — U(é) - 1:1 kiUi(Xi) (4.4.lA) 0 = * = 0 = * = where U(§_) 0, U(§_) l, Ui(Xi ) O, Ui(Xi ) l and o . ki = U(Xi*, Xi) for 1 = l, 2,....,n. Let us see what value independence implies in case of two attri- butes. If we have X1 and X2 and they are value independent then (X , X) (X , Xi) l/2 1 1/ l and 1/2 1/2 (Xi’ Xi) (Xi, X2) must be equally preferable. In other words 1 i x' x')] m Lula: x') +l(x' x )1 (4.4.2A) UP§1 n + :1 Ki]. Ui (a) U3. (g) Ul (a) j>l l>j + ..... ..+ K123..”n U1(X)U2(X_)....Un(§) (4.6.3A) Finally, the mutual utility independence condition (which is equivalent to preferential independence gpd simple utility independence) as men— tioned above, is used to obtain the more specialized case of multiplica- tive utility function for the supra decision maker. This function, in its general form, is expressed as U(X) = U ..,Un) D (1’ U2,.. n n = . +.... ’2 Ki Uig) + K '2‘. Ki Kj Ui (X) U:1 (X) i=1 i=1 j>i -l + KN Kl K2....Kn elm) U2 ()9 Hmong) (4.6.4A) Here it is important to note that, the special case occurs when K = O and the supra decision maker's function becomes a simple additive utility function where n U (g) = r K. U. (g) (4.659.) i=1 1 1 Otherwise add 1 to 1G] i This utility 1 utility : lations, using mu mainly i particip 4.1 B S The tions a] theory 1 one ind atace 6O Otherwise if K = O, we can multiply both sides of (4.6.4A) by K and add 1 to obtain the multiplicative form, n H KU (X) + l = i=1 (KKi Ui(X) + l) (4.6.6A) This concludes the introduction to Chapter IV where multiattribute utility theory and its applications to supra decision maker's group utility function was discussed. Now the thesis will develop new formu— lations, theorems and demonstrations in the area of group utility theory using multiattribute utility concepts. These formulations will be mainly in the area of symmetrical groups, participatory groups, and participatory groups with full interaction. Part B 4.1 B Group Utility Theogy The group utility theory is still at its infancy and new contribu- tions are continuously being made. Part of the developments in group theory has been in the area of supra decision maker situation, where one individual tries to incorporate the preferences of many to arrive at a certain decision. Figure 4.1 B demonstrates the group utility theory approach that is developed by this thesis. The past development has been in section A, subsection D and sub- sub section J. In other words the main interest of Keeney, Raiffa, and Kirkwood has been supra decision maker with a one way interaction (in other words the group members preferences are not affected by the supra decision maker's preferences) and the whole setting being an asymmetrical group where each individual's preference is associated / >.~GO.P_. keg—~3— 45.0....» \ 61 5.85. :33: 9.86 do 33:83:23 so ~26: a a . .V 0.5th Aswaasesmfi $525.35 3335th 33335.35 “Nessa xnuofiim xhuossxm umuiugim nonhuoaaim z xkuoaaxm =02 xsuo§a>m :oz germane—Era :oz _ . z \ a 53253.: 33.3.. .3 oz a / 95.5 roams—3399. a >322. .333: $5.5 :Ow UUQHQHAF— — ‘3"— e a. n \ a :omuoahoufi >53 25 antenna: :3. / . hoe—3.. so: 32. man—.5 \. \. with his preferen action b are part comes ar instance particip preferen preferen Symmetry indicate alter it complete the ider minatior Of a grc Particu] not impc the aff; Weight ( In will be group d4 62 with his identity. However there are obviously many groups where preference holder's identity might not be important or where full inter— action by the members of the group is very desirable. Furthermore there are participatory groups where the responsibilities for the final out- comes are collectively shared and where full interaction might in some instances be more desirable than no interaction. In the context of participatory groups by full interaction we mean that each member's preferences are not only affected by the attributes Xi but also by the preferences U; (Ui complement) of all other n—l members of the group. Symmetry, on the other hand, in the context of participatory groups indicates that the vote or preference of an individual member does not alter its significance by the identity of the preference holder. In completely democratic groups where one—man one vote principle holds, the identity of the vote caster is not important as far as the deter- mination of the final decision is concerned. In other words, if, out of a group of three people, two vote "aye" and one votes ”no" for a particular resolution, the resolution passes. For this outcome it is not important which two of the three individuals in the group votes in the affirmative. The identities of the vote casters do not affect the weight of their votes and thus there is complete symmetry. In the next sections of the thesis some new ideas and formulations will be generated in the field of thus—far neglected participatory group decision analysis. 4.2 B Participatory Groups As mentioned before, most of the attention thus far (which itself is very scant and noncomprehensive) has been paid to the case of supra decision decision tions win the grou] types of areas in where th process ticipato within t councils governor fact, tr group p] dictatm in this 4.3 B Th 0f the care on each de inform This is like tk data b; of Wha' 63 decision maker incorporating the preferences of others into his (her) decision making process. However, there are a large number of situa- tions where the decisions are made collectively, with every member of the group participating and contributing to the final outcome. These types of situations are observed in many phases of the decision making areas in the public or private domain. Especially in those instances, where there is a purely democratic process that governs the decision process and where the decision makers are elected officials, the par— ticipatory group analysis is very useful. Various committees formed within the business organizations, various elected bodies like city councils, House of Representatives, the Senate, boards of trustees or governors, etc. are all good examples of participatory groups. In fact, today, more decisions are probably made by the participatory group process rather than by the supra decision maker or "benevolent dictator." Therefore the investigation and the extension of the theory in this respect will also have an abundant application possibilities. 4.3 B Participatory Gropps with Partial Interaction or No Interaction These are the kinds of groups considered at Part B subsection F of the Figure 4.1.B, where the group members are supposed to communi— cate only attributal information with each other. In other words, each decision maker or member of a group is assumed to give or receive information related to the various attributes under consideration. This is actually the most common type of interaction. In some cases, like those in which the committee is not working from an established data base, much of the initial interaction centers around the question of what information is to be used within the group. Individual members may pres or same set of c great de organizz sion ma} Thi there i: above bl members viduals but not ii the other i pref ere: Utility Where t interac Th actiOn unc Oflmm etc. 64 may present different versions of data relating to the same subject or same quantitative parameters. Resolution of the question of which set of data is valid for the decision problem at hand can take up a great deal of time. This is one of the most compelling reasons why any organization should establish a common data base for use in such deci- sion making. The condition of partial interaction here indicates the fact that there is an interaction related to attributal information as described above but that there is pg explicit consideration of the other group members' preferences by each individual. In other words the indi- viduals utility function is only over the attributes, Xi, i=1, 2,....,n, but not over Ui’ i=1, 2,..., i-l, i+l,...n, where there are n individuals in the group,_Xi attributes, and where U: is the utility set of all the other individuals whose preferences may influence individual i's own preference formulation. Thus every individual member's implicit utility function is expressed as Ui (Xi) (4.3.13) and not as Ui (gi, U3) (4.3.2B) where this second formulation (4.3.23) refers to the case of 22;; interaction which will be examined later. Thus, we clearly observe that, there is some (or partial) inter- action in this case among the individuals. These cases are not very uncommon and when there is not much conflict, personal considerations, etc. and when each member basically bases his judgment or preference with res be relev group me consider again on judging a boxing because views, c some grc views is COHSidEl Mali interac tYpe of 4.4 B ' of 5m Thus tt the Wej Fahd: a sunr and de 65 with respect to the attributes of the problem at hand, such a model will be relevant. If, furthermore, there is pp communication among the group members, where each member expresses his final preference only considering the merits of the case, then the same model applies because again only the attributes are evaluated. For example a jury or a panel judging a beauty contest or a group of referees judging the outcome of a boxing match belong to these types of groups with pp_interaction because the members of these groups do not exchange any information, views, opinions, etc. before they arrive at their own decisions. In some groups, in fact, such discussion and exchange of information or views is expressly forbidden so that only attributes Xi can be con- considered by the judges or group members. In conclusion when there is pply attributal information exchange type of interaction (partial interaction) or when there is pg information and/or opinion exchange type of interaction (no interaction), then the above model applies. 4.4 B Group Utility Functions for Nonsymmetric Participatory Egggps with No or Partial Interaction These are the above type of groups where there is no aSSumption of symmetry of the kind briefly discussed in the previous sections. Thus these are the more common type of groups where the identity or the weight of the individual members might be of some importance in the expression of the final outcome. V The development in this subsection relates to branch B, subbranch F and sub-sub section N in Figure 4.1.3. The previous theory, discussed at Part A of this chapter, assumes a supra decision maker, who incorporates other's preferences into his and decides for them. The group utility function of the supra decisio tion (4 where U incorpo Now let maker w group. group n that th only co and no (her) d In this leges a rates t same to tion of Member, Sion Wh Erences attribu Calling Therefo 66 decision maker over attributes (no or partial interaction) in formula- tion (4.6.1A) was egg) = UD (111 (x), U2 (x),....,Un (y) (4.4.113) where Ug is the supra decision maker's multiattribute utility function incorporating the preferences of others, which are denoted by Ui(§). Now let us assume that this "benevolent dictator" or supra decision maker who acts and decides for others is himself a member of the group. Also because we assume "no" or "partial" interaction among the group members as defined in the previous section, let us further assume that this supra decision maker, now as a full member of the group, only considers his (or her) own personal preferences over attributes and no longer tries to incorporate the preferences of others with his (her) decision function. (Otherwise he would be fully interactive). In this case he becomes the n+1 member of the group with equal privi— leges and responsibilities. In other words the group fully incorpo- rates the supra decision maker into itself and accords him with the same role as the other members. Consequently the "supra" identifica- tion of the decision maker disappears and he becomes a full-fledged member. Thus in this case the group utility function will be an exPres- sion which will try to indicate an aggregation of the individual pref— erences, U1(X), U2(§),....Un(X), Un+l(X) where Un+l(X) is the multi— attribute utility function of the new member (previous supra decision maker). Calling this group aggregation UG’ we have UGQ) = U(U1(X), U2(§_),....,Un(§), Un+1@) (4.4.213) Therefore expression (4 4.2B) is the new participatory group utility function New UGI 11: for the partial if addi functio dent oi (4.6.3 (01' El with iii), 62 function with pg or partial interaction. Now let n+1 = N, we therefore have a U like G tics) = U(Ulm), u2<§>,....,UN<_r)) (4.4.33) This formulation, therefore, does not differ from the formulation I for the supra decision maker, under the same circumstances of no or partial interaction, except for the substitution N=n+l. Thus, if additive independence (see Chapter III, 4.4A) holds, the utility function for the participatory group will look like 11 Ugg) = 1:1 Ki Ui (g) (4.4.43) On the other hand, given that any attribute Ui is utility indepen- dent of the other attributes U1, then we have, analogous to formulation (4.6.3A) UG(§) = U (U1, U2,....,Un) p n n = z 2 i=1 KiUiQ) "' i=1 KijUi (5) Uj (39 j>1 N + E K13.l Ui (xi) Uj (xj) Ul (x1) 1—l j>1 l>j +.. ..... + K123”.HHN U1 (3;) U3(X)....UNQ{_) (4.4.513) And finally, if the attributes Ui are mutually utility independent (or equivalently simple utility independence as in 4.4.5B above coupled S with preferential independence of (Ui’ Uj) of their complement U53 ifj), then for the participatory groups we have where, or othe functic T1 metry T are ve: same 0 case 0 4.53 mentio vidual SYmmet CQSQ 0 Symmet 68 UGQg) = U (U1, U2,....,Un) p N N = 2 K. U. (g) +K 2 k. k. U (x) U. (39+... i=1 1 1 i=1 1 3 i 3 j>1 +KN_1K K U (x) U (x) U (x) (4 4 613) D... 1 2....% l——- 2 _'... N O I where, in a special case when K = 0 we obtain the additive form U = z Ki U. (4.4.7B) or other wise we can transform (4.4.6B) to a multiplicative utility function by multiplying both sides by K and adding 1, and thus we have N l+KUG = .21 (Kki Ui + l) . (4.4.83) Thus under the conditions of no or partial interaction and asym- metry we have obtained the participatory group utility functions which are very much analogous to supra decision maker's utility under the same conditions. Since more of the simplification will come in the case of symmetry, it will be investigated next. 4.5B A Definition of Symmetry in Groupg In the previous sections, the property of symmetry was briefly mentioned as indicating a situation where the identity of the indi- vidual decision makers did not affect the final outcome. We can have symmetry with no, partial, or full interaction in the groups. In the case of supra decision maker, which will not be discussed here, the symmetry condition implies that the supra decision maker does not disting incorpo changea approac symmetr symmetr the uti changed his own by Luce utility include partici We have Symmetr Will nc 4.6B gmups1 In 0th( where 1 Here U Symmet UG mus 69 distinguish between the'ith or jth members' preference when he tries to incorporate them into his own. This relationship is also called inter- changeability. If we also recall from Chapter II, the Goodman—Markowitz approach to group preference aggregation ' also had the condition of symmetry as one of its requirements. Therefore, the definition of the symmetry was that the group ordering of alternatives is unchanged if the utilities of any two individuals for all the alternatives are inter- changed. Furthermore, recalling again from Chapter II, Nash also had his own definition of symmetry, which was used in the generalized model by Luce and Raiffa. Here in this section, for the purposes of group utility function derivation, this definition is broadened so as to include all the members of the participatory group. Such symmetry in participatory groups is not very rare. In all those occasions, where we have one-man one vote type of interaction, then we have complete symmetry simply because interchanging the identities of the vote casters will not affect the final form of the group utility function. 4.6B Group Utility Functions for Symmetric Participatory Groups with No or Partial Interaction The group utility function, as it was defined for participatory groups, had the individual utility functions, Ui's, as its arguments. In other words ) ' (4.6.lB) UG(U) = U (U1, U2,....,UN P where U denotes the implicit form of the functional relationship. Here U1, U2’°"'UN are attributes of UG. If we now assume that UG is symmetric with respect to Ui’ i=1, 2,....N, then the region over which UG must be assessed, can be considerably reduced. The reason for this is that will nm is poss: to asse: this in E Then Ué over the Q butes we way that A, B up then we so fortt W because simply h PM. Thi the Hum]: Now gm“? ut 4'73 Ut If Chapter Pendent 70 is that the functional arguments of UG can be interchanged and this will not affect the final form of UG' Thus since such an interchange is possible we can impose an ordering relationship on the Ui's and try to assess the UG over this specific region. Kirkwood [74] demonstrates this in the form of a theorem. Theorem 4.6.23. Assume UG (U) is symmetric with respect of Ui' Then Ué over U1 5 U2 5 U3....5 UN will be identical to any other UG over the same attributes in any arbitrary order. 2322f. Because UG(U) is symmetric with respect to all its attri- butes we can interchange all of its functional attributes Ui in such a way that we can obtain the sequence UA 5 UB~5...5 UN for individuals A, B up to N. Since the identity of the individuals is not important, then we can call the smallest U as U1, the second smallest as U2 and so forth. The fundamental argument in the proof is that since any permutation of Ui's in the functional space will result in the same UG because of symmetry, a specific ordering likevUl 5 U2 5....5 UN’ will simply be just one specific form of all possible permutations. End of This way of approaching symmetry would result in the reduction in the number of points assessed to obtain the group utility function. Now let's see how symmetry condition affects the general form of group utility function. 4.7B Utility Independence Condition If utility independence condition explained in part A of this chapter holds, then the attributes Ui’ i=1, 2,..., N are utility inde— pendent of Ui' In other words, this condition simply states that, if all the outcome these 0 N, whoe in Sect wh Thus, u (see fc Now unc Cate t‘: equati. and 71 all the N—l members of the group are indifferent among all possible outcomes, then the preferences of the group for any lotteries over these Outcomes are uniquely determined by the preferences of individual N, whoever he might be. Therefore, if this condition holds as explained in Section 4.3.2A, we have _ = _ - * Up(Ui’ Ui) 31(Ui) + h (Ui) U1 (U1) where Ui*(Ui(X)) = Ui due to strategic equivalence (4.7.13) Thus, under this assumption, an asymmetric group had a utility function (see formulation 4.4.5B) N N UG(U) = .2 KiUi + z KijUin +.....+ K12_NU1U2....UN (4.7.23) 1=l i=1 j>l Now under conditions of symmetry, it becomes Result 4.7.3B N N UG(U) = K1 2 Ui + K2 .2 Uin +.n...+ KNUlUZ...UN (4.7.3B) i=1 i=1 j>i where Ki, K2,....Kfi are scaling constants. Proof: Let Vi, Ui = O-be in the feasible set. And let (U1, U3) indi- "‘_“ i cate that all attributes except Ui are equal to 0. Therefore, from equation (4.7.2B), o = Ug(Ul’ Ui) KlUI (4.7.43) 0 .- Ug(U2, U5) — K U (4.7.53) Now becz other, 2 Th Applyin K1 Now, at Ug(Ui. Therefo and But fr< tion We equati} frOm w‘ IEpeat analyS 72 Now because U1 and U2 are symmetric, we can substitute them with each other, and afterwards equating (4.7.43) and (4.7.5B) O — = = = Ug(Ul’ UT) ‘ K1U1 K2U2 K2U1 K1U2 Therefore Kl = K2 (4.7.6B) Applying the above result repeatedly for U3, U4""‘UN’ we obtain K1 = K2 = K3 = ..... = K.N = Ki (in 4.7.33) (4.7.73) Now, at the second step, for eliminating Kij’ let us define o . UggUi’ Uj’ U13 ) as all attributes except Ui and Uj being equal to 0. Therefore from equation (4.7.2B) we have 0 —- Ug (U1, U2, Uli) - KlUl + KZUZ + KlZUlUZ O = Ug (U2, U3, U53) KZUZ + K3U3 + K23 U2U3 But from (4.7.73) we know that Kl = K2 = K3 and from the symmetry condi- tion we know that we can interchange U1 and U2 or U2 and U3. Therefore, equating the above equations we have, KlUl + K2U2 + KlZUlUZ = KlUl + K2U2 + K23U1U2 (4.7.8B) from where we determine that K12 = K23. Applying the above analysis repeatedly we can determine that, K = K34 = '00- = KN-‘l, N = K2 (in 4'7‘33) (4.7.9B) 12 = K23 In the third and ensuing steps, by similar application of the above analysis, we can determine the K5 and other constants. End of roof. Cc diminis partici (4.7.3E equatic stants indivié metry aSSess ME NOW, 1‘ 73 Consequently, the number of scaling constants to be assessed diminished considerably because of the condition of symmetry in the participatory groups with no or partial interaction. In equation (4.7.3B) there are N constants to be assessed. However in the original equation (4.7.23) for the case without symmetry, there are ZN—l con- Comparing the two situations, where N = number of stants to determine. individuals in the group, we have, Table 1 Number of Constants for Symmetry and Non-Symmetry No Symmetry Symmetry N 2N-1 N l l l 2 3 2 3 7 3 4 15 4 20 1,044,484 50 From the above table, we can readily observe the fact that sym— metry condition, if prevails, is very useful in reducing the load of assessing the necessary scale constants. 4.83 Preferential Independence and Symmetry In the above analysis only simple utility independence was assumed. Now, in line with the developments of multiattribute utility theory, _'¥— we add t group an pendent ential j N—Z menfl then the quences N-l and that th: analysi as it w; groups and pre additiv rapeat, 74 we add the preferential independence condition unto our participatory group and assume that the attributes (Ui’ Uj) are preferentially inde- pendent of their complement Uij for all i#j. In other words, prefer- ential independence condition simply indicates that if all the other N—Z members of the group are indifferent among all possible outcomes, then the preferences of the group for any lotteries over these conse- quences will be solely determined by the preferences of individuals N—l and N, whoever are designated as such. It is interesting to observe that this assumption that is brought in from multiattribute utility analysis is almost identical to one of the major assumptions of Fleming as it was mentioned in Chapter II. Now as mentioned for asymmetric groups in Section 4.6.A, in this case (joining the utility independence and preferential independence assumptions together), we have either an additive utility function or a multiplicative utility function. To repeat, for asymmetric groups, these formulations are N UG(U) = -E KiUi (4.8.13) 1—l or (4.8.23) N KUG(u) = 121 (KKiUi + l) - 1 Where K, K1, K ,....,KN are all scaling constants. Result 4.8.33 and 4.8.43. If the participatory group with no or Partial interaction is symmetrical with respect to its attributes, then we have, either N U¢(u) = K 1:1 Ui (4.8.33) or N KUG(u) = 1E1 (KkUi + l) -1 (4.8.43) _Pr_o£f_: 1 proof for (interchz as it was Fro: stants f< plicativ. Thu: metrical action w; interact. 4. 93 Q As the grou In other the attr 0f the o X and U‘. ‘1 set of a is What ProcessE In thOSe Opinions to attrj are in ( much m0] 75 3392:: The proof of this result is almost identical with the previous proof for the case with simple utility independence. By the symmetry (interchangeability) condition we show that k1 = k2 = k3 = ....kN = k as it was done in the previous.proof. From the nonsymmetric to the symmetric case the number of con— stants for the additive form diminishes from N to l and for the multi- plicative form from N+l to 2. Thus far the general functional forms of symmetrical and nonsym- metrical participatory groups under conditions of partial and no inter- action was discussed. Now let's turn our attention to the major full interaction case. 4.93 Meaning of Full Interaction As explained briefly in section 4.13, the individual members of the group might be assumed to have full interaction with each other. In other words, the individual members are no longer considering only the attributes of the problem, but they also consider the preferences of the other N—l members in the group. Thus each U1 is a function of §_and Q; where g indicates set of all attributes and 3; indicates the set of all preferences, other than the individuals own. Thus Ui(§, EE) is what we observe instead of a simple Ui(§). In certain interactive processes there is the element of persuasion, discussion and conflict. In those cases, where individual members are very much affected by the opinions of others, there is no longer partial interaction only related to attributes. In certain committees, Subgroups may be formed that are in conflict with each other, making the ultimate decision process much more difficult to achieve. On the other hand, a dominant member or subg1 gence o and doc In prefere there a in the i is as A1 or shm the so is nee plicm Howey about tions 4.10E 1’3 grouy and Sion 76 or subgroup may emerge that can recruit others to a team. The emer— gence of a leader of this sort in such proceedings has been observed and documented in some experimental research. In full interaction, each decision maker tries to incorporate the preferences of others into his decision analysis. Let us say that there are M attributes to a decision problem and there are N individuals in the participatory group. The attribute set for the decision maker 1 is as follows: Attribute Set S = (X1, X2"""XM’ U1, U2""'Ui-l’ Ui+l""'UN) or simply S = (E, U:) (4.9.13) Therefore, for each and every individual, a utility function of the sort Ung, U?) (4.9.23) is needed. The assessment of each of the Ui's become extremely comr plicated in this case because now we have an M+N dimensional function. However certain simplifying and reasonable assumptions can be made about the interaction of g and U3. The most relevant of these assump- tions is mutual utility independence of §_and Ui' 4.103 Assumption of Mutual Utility Independence for §_and U1 For any decision maker, who is interested in attributes §_and Ui’ it is reasonable to separate them into those categories. One group (E) is simply the attributes of the problem under consideration and the other group (U1) is the preferences of the other fellow deci— sion makers. If we assume utility independence between these two separat tion is Un decisio will no of USd 1 sion ma is held differs Result I Proof: multit attrit rasult form« COndi 77 separate categories of attributes, then the utility function formula- tion is more simplified. Under this assumption, if we hold U; at a fixed level UE, then the decision maker's relative preferences for different lotteries over x will not change for different levels of U2. In other words the level of US does not affect the evaluation of the attributes g by the deci- 1 sion maker. Now, again, by the mutuality of the same assumption, if g is held constant at E9, the decision maker's relative preferences for different lotteries over U: will not change with varying levels of §?. In other words how the decision maker evaluates others' prefer— ences is not affected by the level at which g is held constant. Result 4.10.13 If g and U; are mutually utility independent, then 00111 u(U—) (4. 10 . 113) , (Ui) + xi3U ix— Ui(_}_{_, U-i) = A (X) + A ilU ix- iZU iu Proof: The result directly follows from the application of more general multiplicative form of Theorem 2, at section (4.3.2A). Substituting attribute X1 = §_and X2 result 4.3.9A). = Ui’ we obtain the multiplicative form (see If Ail + X12 = l and ki3 = 0, then we have the purely additive form of Ui(§, Ui) = Xi lUix(§) + 112 Uiu(Ui) (4.10.23) In the above formulations (4.10.13 and 4.10.23), UK and Uu are conditional utility functions and 1's are the scaling constants. What is interesting to note here is the fact that each decision maker becau g and tion w The fo we mak Il U- i) thE 78 maker, in the case of full interation, acts like a supra decision maker because of trying to incorporate others preferences into their own. The difference is that, unlike the supra decision maker, they not only incorporate others preferences but also try to evaluate the attributes g on their own. 4.113 Utility Functign_s for Nowetric Participatgy Groups with Full Interaction In the following formulations, mutual utility independence between .§ and U} as discussed above, is assumed. When we have a nonsymmetric group, the implicit general formula- tion will be, UG(u) = UP(U1(§, UI)’ U2(§, U§)"""UN(§’ Ufi)) (4.11.13) The form of this general relationship will depend upon the assumptions we make about the individual Ui's. Result 4.11.23 If an attribute Ui is utility independent of the other attributes U3, then the group utility function looks like, N N UG(u) = 1:1 Kiui(§, U3) + 1:1 Kijui(§, U1) Uj(§. U3) N . + .Z KijUiQE’ U3.) Uj (Z? U3) Ul(§, U1) i=l j>i l>j + ...... .+ K123'H’NU1QE’ U3") U2(§, U2)....UN(§, Ufi) (4.11.23) and as of eac tion, Result above Eithe metrj 79 and assuming mutual independence between X and U; inside the argument of each Ui function and by substituting (4.10.13) into the above func- tion, we have Result 4.11.33 N Us“) = 1:1 K1(A11U1x(x) + A12U1uw1) H13171159111611?) N + .21 Kij (AilU ix(X) + Ai 21U u(U-) + Ai 3ixU (X)Ui u(Ui)) l: (A (X) + A (U?) U. le 1x 1'2 Ju J + Aj3] U. x(X)Uj u(Uj)) + . . . . . . + K123....N()‘11le(§) + A lZUluwI) + A13le(§-)Ulu(U1-)"' (X) UNU (UN) (4) + ANZUNUWN) + AN3U NX .. ..... (ANlUNX (4.11.33) Proof: Result (4.11.33) directly follows from group utility result (4.4.53 or 4.6.3A) under the appropriate assumptions. We have previ— ously shown that the result (4.4.53) follows directly from Keeney's multiattribute utility function for a single individual decision maker. This is the generalization and application of that multiattribute result to participatory groups. And now, we add the preferential independence assumption to the above utility independence condition among individual Ui’s, to obtain either the multiplicative or the additive utility function for nonasym— metric participatory groups having full interaction. plemeni mutuall Tl Beth of and adc FJ' with fl 51-1213 actiOH 01‘ Conc 80 Result 4.11.43 and 4.11.53 If, in addition to the above utility independence, attributes (U1, U3) in the argument of UG are preferentially independent of their com- plement Uij’ and if again, X and Ui in the argument of Ui's are mutually utility independent, then, we have either, the additive form, N l=l N = 2 Kim (K) + A . 1 ilUix iZUiuwi) + A13U1x®U1uwin (“11'”) l or the multiplicative form N = — + — 1 N = A 11 (“101111111 (9 + . 1 Uiuwi) + A13U1x®U1u ' 1 l: 12 (4.11.53) This result follows directly from results (4.6.4A and 4.4.63). Both of these previous results were derived from Keeney's multiplicative and additive utility functions for a single decision maker. Finally, we come to the case of symmetric participatory groups with full interaction. 4.123 Utility Functions for Symmetric Participatory Groups With Full Interaction The case of symmetric groups with either partial or no inter— action had been formulated at section (4.63—4.83). Now the property or condition of full interaction is introduced to these groupS. Resul! of otl argume group and vi 3329:: direct tion 0 member [17‘ 1 1n 81 Result 4.12.13 and 4.12.23 (a) If attribute Ui in the argument of UG is utility independent of other attributes, U1, (b) if the attribute groups X and U: in the argument of each Ui are mutually utility independent and (c) if the group is symmetric, then we have, N N UG(u) = K1 :1 Ui(X,U-i-) + K2 =2 Uig, Ui) Ujg, U3.) j>i (x, U— l) U. (x, U-.)....UN (X, U-) (4.12.13) +.... K12_ NU i- and with the appropriate substitution. N UG(U) = Kl .21 (Ail Uix(§) + AiZUiu(Ui) + xi3Uix(§)Uiu(Ui)) 1: N + K2 .21 ((A'11U11(X) + AiZU iu(Ui) + A13U ix(X)Ui u(Ui)) 1:1 371 (4 jl ij(X) + AjZU ju(uj) + A J3ij(X)J U uJ(u. )) + """" + K12... .N OilUixQQ + AiZUiuwi) + A1 3U1x®U1 11011)) (lNlU NX (X) + AN, UNu(U§) + Am UNXQQUNuWN” (4.12.213) Proof: As in the case of nonsymmetric groups, this result follows directly from the previous result (4.7.33), where there was no assump- tion of full interaction. Therefore full interaction of each group member and the mutual utility independence assumption batween 22 and ' s in the above formulation. U: in the argument of Ui’ result 311.12 F0: to the attribu of thei As brie if all then th _ individ under s or the the f (part 82 Result 4.12.33 and 4.12.43 For the symmetric groups with full interaction, if, in addition to the above assumptions of utility independence of Ui and U1, the attributes (Ui’ Uj) in the argument of UG are preferentially independent of their complement U13, we have the additive or multiplicative forms. As briefly mentioned before, this final assumption simply states that if all N—2 individuals are indifferent among all possible outcomes, then the group preference is solely determined by the remaining two individuals' preferences, namely (Ui, Uj). Thus, the additive form under symmetry and full interaction is, N UGg)Uiu(U—i)) (4.12.313) or the multiplicative form N - - - l KUG(u) — .n (KkUi(_)_(_, Ui) + 1) l: N 1 U (X)U (U-)) + l) - l = .111 (Kkaiiuixq) + A12U1uw'1) + 13 ix — in 1 1: (4.12.43) Proof: This result follows directly from result (4-8-33) and 4-8-4B)~ The only difference between these previous results and the Present one is the introduction of the concept of full interaction into the group. Thus if we look back at Figure 4.13 on page 61, we can see that the formulations of the group utility functions for the case B (participatory groups) are now complete. The cases of full interaction and pa nonsym and ne partic of gro N formul is on such g the sc maker, inter; 4.13B catiw that, addit fOr n. for f or pa 83 and partial or no interaction under the conditions of symmetry and nonsymmetry have now been covered. These formulations are important and necessary extensions of the overall group utility theory because participatory groups constitute a major branch of the different kinds of group formulations. Now let us see how some of the scaling constants in these new formulations can be evaluated. Since the main interest in this thesis is on the symmetric groups, the evaluation of scaling constants in such groups will be discussed. The previous work on the evaluation of the scaling constants has been mostly for the case of supra decision maker, in nonsymmetric analysis and under conditions of no or partial interaction. 4.133 Evaluation of Scaling Constants in Symmetric Participatomy Groups This section will concentrate on the constraints in the multipli- cative and additive forms under conditions of symmetry. We have seen that, for symmetric groups, under different kinds of interaction, the additive forms look like N UG(u) = K 1: Uiqp (4.13.13) for no or partial interaction, and it looks like N _ (4.13.23) 1: for full interaction. The corresponding multiplicative forms for no or partial interaction are, or it where need diffs multi CODSt butes Sents Thus and Forth 84 N KUG(u) = 1:1 (KkUi(§) + l) - 1 (4.13.33) or for full interaction N KUG(u) = 1:1 (KkUi(§, U3) + l) - l (4.13.43) where in both (4.12.23) and (4.12.43) Uig, U-i) = A (m) + AizUiu (UK) + 1 U. (X)Uiu(U-j:) (4.13.53) ilUix 13 1x — When the additive forms (4.13.13) and (4.13.23) hold, there is no need to evaluate K since this constant is always arbitrary, unless two different group utility functions are to be compared. However, in multiplicative formulations (4.13.33) and (4.13.43) there are two constants K and k which must be dealt with. Results 4.13.183 and 4.13.213 Assume Xi, X§,..... o o o butesXi where Ui's are =1, and also assume that x1, x2,....,xM repre— ,X§ represent the optimum values of the attri- sents the least desirable levels of the attributesgi where Ui's are = 0. Thus we have H (Xi’ x424,” ,x§) = Ui . L3. <32 , Ui for certain, or . * * o o > L40 ((E ’ 05): P2, (E 3 UI) letting Ui(L = Ui(L4) we have 3) * ’ HE) = A12 — * * — 1 + 1 + 1 — P ‘ P2Ui(§ ’ HI)" P2‘ 1 2 3) ‘ 2 (4.14.33) 112 = P2 Therefore, solving (4.14.13), (4.14.23) and (4.14.33) simul— taneously, we have A11 = P1*12 P1P2 A12 = P2 113 = l - Ail - A12 = 1—PlP2-P2 (4.14.43) Thus the scaling constants for the fully interactive Ui(§, U1) have been determined. This concludes the discussion of the scaling constants for the symmetric groups. 4.153 Final Remarks Thus we conclude the analysis of the utility functions for the participatory groups under varying conditions. The right branch of Figu util full Next tici 91 Figure (4.13) on page : has been fully analyzed. We saw how the group utility functions behave under asymmetry and symmetry and also under full, partial or no interaction of the individuals forming the group. Next the concept and implications of risk aversion analysis in par— ticipatory groups will be analyzed. tici will pati the difl film the act cla pas wou CHAPTER V RISK AVERSION AND PARTICIPATORY GROUPS Part A 5.1A Introduction This chapter introduces the concept of risk aversion into par- ticipatory groups. Again, like in the previous chapter, these groups will be divided into classes as no participation, partially partici- pating or fully participating. Again the main point of interest in the risk analysis will be nonsymmetric and symmetric groups. In case of fully participating groups where the individual mem- ber's utility function in its implicit form is Ui(§, Ui)’ the decision maker is interested in the preferences of other individuals explicitly. If the number of individuals in the group is large, it becomes extremely difficult, in terms of time or resources, to assess all their utility functions. This problem will especially get more complicated if also the number of attributes is large. In order to reduce the dimensions of this problem, certain approximations are made to the individual's actual utility functions. This is usually done by applying a certain class of utility functions with known properties. Research in the past has identified properties that utility functions of real-world would have and certain functional forms U(X) = U(XlCl, CZ"""Cm) having these properties were devised. Usually the value(s) of the 92 parame vidual tiona] indivi then 1 parmm appro initi asses ac tua risk utilj 5.2A PrOp. cert. cert ther Vah Whe and 93 parameter(s) affect the nature of the utility function and each indi- vidual might assign different values to the parameters, once the func— For example if, for a single attribute the individual's utility function is assumed to be of the form Ul(X) = e—CX, tional form is determined. then the assessment problem is reduced to the determination of the parameter C. If an appropriate functional form is selected, then the approximation would be good. Unless a certain functional form is initially determined, the number of points whose utilities are to be assessed is very large when we want to obtain the exact shape of the actual function. Thus certain classes of functions, with desirable properties like risk aversion, are frequently used to identify the decision makers utility functions. 5.2A Risk Aversion for Unidimensional Utilities In Chapter III, risk aversion was mentioned as one of the basic properties of utility functions. It is usually defined in terms of the certainty equivalent for a lottery where the decision maker prefers the certainty equivalent (for sure) to the lottery itself. If, for example, there is a 50-50 lottery with outcomes (:X', X'i> , the expected value of this lottery Ll will be _ l_ , l_ ,, = X' + X" = - E(Ll) — 2 X + 2 X _—_7f-__ X (5.2.1A) If the decision maker avoids the lottery and prefers the expected value for sure, then he is a risk avoiding or averse type of person. When the attribute is an undesirable (monotonicly decreasing) one, and when the certainty equivalent is greater than the mathematical expect indivi where is the tion . attri risk make lOttI cert Whe1 94 expectation of the lottery, then the decision maker is a risk averse individual. This above fact can symbolically be expressed as CE ). PX' + (l-P)X" = E(L) (5.2.2A) where X' and X" are the two different outcomes in the lottery, and P is the probability of X'. L is the lottery (X', P, X"). Furthermore, if a decision maker is risk averse, his utility func- tion is concave. Mathematically RU(X') + (l-P) U (X") < U[PX' + (l-P)X"] for O = E(L) — 3‘1 33(1) (5.2.53) where U-1 is the inverse of U and gives the certainty equivalent. When U(X) is a monotonically increasing utility function and when i is the expected value of the lottery L over X with a small variance 0%, then Pratt [98] has shown that, the risk premium H is approximated by l “"6“ 2 (5.2.6A) H=X—CE='§ U.(X) OX litt tior level make} 95 Before developing the concept of risk premium and risk aversion a little further, let us see the three different shapes of utility func— tions for desirable and undesirable attributes. U(X) U(X) U (X) Risk Premium Risk Premium A x B x C x Risk Averse - Risk Prone — Risk Neutral - Desirable Attribute Desirable Attribute Desirable Attribute X Risk + remium U(X) U(X) U(X) Risk \ D E Premium F Risk Averse — Risk Prone — Risk Neutral - Undesirable Attribute Undesirable Attribute Undesirable Attribute Figure 5.2.7A Risk Averse, Neutral and Risk Prone Functions Thus the risk premium H might depend upon the already achieved level of attribute X. For the case A in Figure (5.2.7A) the decision maker's risk premium is positive for all lotteries and he is risk averse. For case D, his risk premium is again positive and for an unde the Ana: C2154 abl1 for but for and th‘ th] at de; 96 undesirable attribute the decision maker is risk averse. For case B, the decision maker's risk premium H is negative and he is risk prone. Analogously, for case D, for an undesirable attribute, the decision maker's risk premium is negative and he has a risk prone function. In case of risk neutrality (cases C and F) for both desirable and undesir- able attributes, the risk premium is equal to O, the utility function U(X) being a straight line. It is possible, however, to transform an undesirable attribute form into a desirable attribute form. If X is the undesirable attri- bute, nggl and UX(O) = 1 and UX(1) = 0, then we can have the trans— formation Y = -X + 1 (5.2.8A) and Uu(Y) = UX(1-Y) (5.2.9A) which is equivalent to rotating the original utility curve about the utility axis and then shifting it to the right so that Uy passes through the origin. The rest of the analysis in this chapter is in terms of desirable attributes and a simple transformation can convert every attribute into desirable form. 5.3A Analysis of Risk Aversion In his work concerning risk aversion, Pratt [98] has shown that, a local risk aversion function, r(X), for a unidimensional (single attribute) utility function for a desirable attribute can be defined by — ”"00 (5.3.1.4) r(X) = 11' (X) anal tion U" (X impc infc risl att] for ind: pos: thu: set 810: fix WE whi 97 or r(X) = - 5% (log U'(X)) (5.3.23) This is called the risk aversion function in unidimensional utility analysis. It indicates how large the decision maker would like to pay in order to avoid the uncertainty of choosing the lottery. The func- tion also assumes that U(X) is twice continuously differentiable since U"(X) and U'(X) are the second and first derivatives of the function. The risk aversion function has many desirable properties, the most important of which is the fact that it preserves all the essential information about U(X), while eliminating everything arbitrary. A very important question in risk analysis is what happens to the risk premium, H, as X increases. Again with regard to a desirable attribute, if the decision maker's risk premium gets larger and larger for greater amounts of the attribute, then we can consider that the individual is increasingly risk averse. Decreasing risk aversion on the other hand, arises in those situations where the decision maker possesses a minimum substance level for the desirable attribute X. It thus implies that the individual becomes less and less cautious (con- servative) as the level of X increases. Finally, constant risk aver— sion implies that the risk premium of different lotteries would remain fixed as the level of X increases over some region. If we integrate (5.3.1A), exponentiate it and integrate again, we obtain U(X) = klfe-fr(x)dxdx + k2 (5‘3'3A) which is the general form from which one can generate the utility funct CODSI 5.4A sion eith for ma beco and and 98 functions that have the same risk aversion properties by varying kl and k2. Of all the classes of utility functions the ones that are in the constant and decreasing risk aversion categories are the most important. 5.4A Constant Risk Aversion Constant risk aversion occurs when the risk premium or risk aver- sion function r(X) is a constant function of X. This is the result of either a linear or an exponential type of utility function. When there is a constant risk aversion the decision maker's certainty equivalent for an uncertain but specified lottery involving attribue X is inde- pendent of his assets Xo at the time he makes a dec1sion. This property becomes very important in a sequential decision process where the amounts of each attribute change from decision point to decision point. In order to obtain the exponential form, we integrate both sides of (5.3.1A) to obtain f—r(X)dx = jlogU' (X)dx or -rX—kl = log U'(X) or ‘rx = log U'(X) + k1 and exponentiating, we have kze‘rx = U' (x) and finally integrating again, we have tion: fon Act WE. 99 _ —rx Cl + Cze r+0 U(X) = (5.4.1A) C1 + CZX r=0 If we scale this general form according to the boundary condi- tions U(X*) = l and U(Xo) = 0 where X* is the most desirable level and X0 is the least desirable, then we have the scaled utility function 1 - e U(Xs) - -————-—- _r : 0511351 (5.4. 23) for a desirable attribute. The parameter r is called the risk parameter for the exponential form. If a decision maker decides that his risk aversion is constant, then only one lottery is needed to assess his utility function. Actually three points are needed but since U(X*) = l and U(Xo) = 0, we need only one more point to determine the value of r. Howard [56] states that exponential utility functions are very ade— quate approximations to many utility functions. He notes that "the utility functions assessed by actual decision makers....are usually smooth functions that are concave downward and representable by an exponential at least over a limited range of outcomes." The support of empirical evidence and its versatility have justifiably promoted the popular use of exponential forms to represent the decision makers utility functions. To observe the flexibility of the exponential form let us see the plot for the utility function for a desirable attri- bute X, (5.4.3A) m 100 Figure (5.4.4A) Constant Risk Aversion With Varying Parameter Values The above plots signify the fact that representations of risk aversion, risk neutrality and risk proneness are all possible with the exponential form by simply varying the value of parameter r. Thus, in exponential utility functions, r(X) = r>04$ constant risk aversion (5.4.5A) r(X) = r = 0 =) risk neutrality (5.4.6A) r(X) = r<0 =9 constant risk proneness (5.4.7A) Concluding the arguments for the case of constant risk aversion, now let us consider the case of decreasing risk aversion. Unl ing dec tOW Spe lOl 5.5A Decreasing Risk Aversion Decreasing risk aversion is also one of the frequently observed characteristics of some of the empirically determined utility functions. Unlike the constant risk aversion case, there are families of decreas- ingly risk averse functions with many members each. Therefore, if a decision maker tends to be decreasingly risk averse in his attitudes towards the attribute X, then an appropriate member of a particular family could be chosen to represent the individuals utility function. Logarithmic forms or certain kinds of exponential forms have the desired characteristics of decreasing risk aversion. Some of the most commonly used functions for this case are given in the following table. Tab le 2 Commonly Used Decreasingly Risk Averse Utility Functions U(X) Restrictions r (x) Buggizggrnsk 1 log (x + b) -- m x 2 -b c _ (0‘1) - (X + b) O b -C C + l > _ (X + b) CI>0 x + b X b -ax - aze-ax + beze-cx -e - be CK a,b,c>0 —————————-——~ All X -ax -cx ae + bce For the above functions, U(X), the risk aversion function, r(X), is monotonically decreasing for the appropriate range of X that is specified at the table. tion the 102 To illustrate this point, let us generate the risk aversion func— tion, r(X), for different values of b in r(X) = Eifi’ which corresponds the utility function U(X) = log (x+b). (See Figure 5.5.1A). r(X) Figure 5.5.1A U(X) = log (x+b) for Different Values of "b" From the figure we can observe that r(X) decreases as X increases and therefore the logarithmic functional form for U(X) is considered to be decreasingly risk averse. the the COD uni ext par 103 5.6A IncreasingiRisk Aversion In single attribute utility theory, for the case of desirable attributes, one of the tempting forms to use is the quadratic function. If we consider the function 2 U(X) = a + bX — CX (5.6.1A) then _ _ U"(X) _ 2C r(X) - U'(X) — b-ZCX (5.6.2A) Since r(X)>O for all X, U(X) is risk averse. However as X increases r(X) also increases. Therefore the quadratic form gives us increas— ingly risk averse utility functions. Although this form is also used at times to represent the individual decision maker‘s utility function, the fact that it implies increasing risk aversion has to be taken into consideration. This concludes the discussion of the concept of risk aversion in unidimensional utility functions. The rest of the chapter will develop, extend and apply these concepts to multiattribute utility theory and to participatory group utility functions. 104 Part 3 5-lB W Because of the very recent nature of the developments in group utility theory using multiattribute functions, quantitative analysis of risk aversion in this particular field is virtually nonexistent. How— ever, it would be important to know what the implications of various forms of multiattribute participatory group utility functions are on the aggregation of risk. Although there are certain sociological and psychological theories about the individual's attitude towards risk in groups, never before such theories have been linked to the multiattri— bute group utility theory as discussed at great length in the previous chapter. Before the development and analysis of the concept of risk aver- sion in participatory groups, we need a new definition for risk aver- sion function for multiattribute utility functions. Next this new definition is introduced. 5.23 Risk Aversion for Multiattribute Utility Functions Definition 5.2.13 Local risk aversion for multiattribute functions of the form U(Xl’ X2,....,XM) will be defined, for an attribute Xi, by U"(X)i 22U(§)i Bug):L l = _ T311 [log moon (5.2.13) where U"(X)i is the second partial derivative of the utility function with respect to the attribute Xi and U'(X)i is the first partial deri attr Thus COIN fur C01 105 derivative of the multiattribute utility function with respect to the attribute Xi' In multiattribute utility~theory U(X) = U(X1,X2,...,XM). Thus the function Ri(Xi) applies to any attribute Xi' Furthermore for (5.2.13) to hold, we assume that U(X) is twice continuously differentiable with respect to every attribute Xi' In the above definition of the risk function, the other attributes Xi are assumed to be held constant at a fixed level and only the attri- bute Xi affects the outcome Ri(Xi). Thus, after defining the new risk function for the multiattribute case, it is possible to proceed with the group utility theory. 5.33 Risk Aversion and Participatory Groups With No or Partial Interaction In Chapter IV, various forms of participatory group utilities were discussed. It was shown that the participatory group utility function under conditions of no or partial interaction and asymmetry could implicitly be expressed by, UG = Up (Ul(_X_), U2 ()_<),....,UN (3)) (5.3.13) It was also shown that the above form could be reduced to either the additive form . N UG(u) = 1:1 kiUi(X) (5.3.13) or the multiplicative form N KUG(u) + 1 = 11:1 [1 + KkiUi(§)] (5.3.13) under the assumptions of preferential independence and utility f1 ltd 106 independence. Now let us see how the group will behave in terms of risk aversion with respect to a given attribute Xi' 5.3.13 Constant Risk Aversion in Participatory Groups For the case of constant risk aversion in single attribute utility functions, it was shown that (5.3.113) U(X) N -e-CX 5) r(X) = C > 0 constant risk aversion (5.3.123) U(X) m -X =) r(X) = 0 constant risk neutrality (5.3.133) U(X) m e-CX => r(X) = C < 0 constant risk proneness For the case of risk aversion, the parameter C of the exponential function is always positive and is =r. Now we can develop two theorems for participatory groups made up of constantly risk averse individuals. Theorem 5.3.23 For participatory groups, if U1(X), U2(X),....,UN(X) represent constantly risk averse multiattribute utility functions of the individ— ual group members over the desirable attributes X = (X1, X2,....,XM) and on the interval (Xi, X?) for each attribute Xi’ and if the group utility function is of the additive £239, igl kiUi(X), where kl’kZ’ ....,KN are positive constants, then the grbup utility function UG itself is either (a) constantly risk averse or (b) decreasingly risk averse over an attribute Xi' Proof: The general case, RG(Xi) for U N G = 1:1 kiUi(X), follows from the repeated application of UG = klUl + kZUZ. Therefore but 2 U U U" n n R (X ) _ '3 G1 / '5 G1 _ 61 _ k1U11 + k2U21 (5 3 2 13) ..- " — _' - —'_— - _ _——T——-_T . u c G 1 3x2. 2X1 U G. k1Ul1 + k21121 1 l U", U”. but we know that R1. = — —],'-1* and R21 = - E1 1 U . U 11 21 ' ' _ k1U11 R11 + k2U21 321 (5 3 2 23) — _-—-————' l C —T———'_ O I I u k1U11 + k2U21 k1U11 + k2U21 differentiating the above expression we have, n v I _ " U" , k U' . . k1U11(k1U11+k2U21) (k1U11 + 32 21) 1 11 R . RG(Xi) = 2 ll ' ' (k1U11 + k2 U21) I H ' ‘ _ ll " V R'. k1U11 k2U210‘1U11 + k2U21) (k1U11 + k2U21)k2U21 + 11 k U' +k U' + 2 ' o 0 ' I l 11 2 21 (klUli + kZUZi) k U' y . + R21 ———sz?1 + k U' (5.3.2.33) 1 11 2 21 ' ' ' ‘ _ k1U11R11 + k2U21R21 _ _______T________T____ k1U11 + k2U21 n I _ n v n v _ u I + R1[k1k2(U11U21 U21U11)] + R2[k1k2(U21U11 ”111121)1 2 (k U'. + k U'.) 1 11 2 11 (5.3.2.43) ' ' . . U11U21 and multiplying (5.3.2.43) by and rearranging U' U' 11 21 2 l v 1 t __ I I _ k1U11R11 + k2U21R21 k1k2(R11 R21) U11U21 — k. u! + k U1. — 2 (5.3.2.53) 1 11 2 21 (k U'.+k U' ) 111 22i R21 av [5‘ LIE mt ut 108 Now since E’s are desirable attributes, Ui's are increasing, and hence are constants, U' > O, U' > 0, Ri = 0, Ré = 0 because R1 and R 1i 21 i i 1 2i kl > 0, k2 > 0 as defined before. Therefore RGi = 0 if R11 = R2i and thus UG 15 also constantly risk 1 . . . . averse but RGis 0 if R1i # R21 and thUSw UG is decrea51ngly risk averse. Eng of proof. 5.4B Interpretation of Theorem 5.3.23 Results The above theorem 5.3.23 does produce some significant results in group risk theory. For a participatory group, this theorem states that, if the group utility function is expressed in an additive form under the appropriate assumptions, then whenever the individuals' multiattribute utility functions are constantly risk averse, the group utility function is either constantly or decreasingly risk averse. In fact the theorem clearly proves that only if each of the group members have identical multiattribute utility functions, with constant risk aversion, then the group utility function preserves the constantly risk averse characteristics. Otherwise, even though each group member has a similar but not identical constantly risk averse multiattribute utility function, the group utility function mg§£_bg decreasingly risk averse. Simply summarized, constant risk aversion for group members may result in decreasing risk aversion for the group as a whole. The fact that this result is achieved under the simple additive case makes it somewhat unexpected but all the more interesting. This result here is also in line with the diffusion of respon— sibility theory and the risky shift phenomenon [101, 116]. Such phenomenon has been observed under many experimental conditions and it concerns 109 the fact that decisions made by the groups are generally more risky then those that would be allocated by the individual members of the group. Research workers studying this effect conclude that it is mainly activities within a group faced with a decision involving risk that tend to support a more risky solution than would be chosen by individuals acting alone. On the other hand, the "diffusion of responsibility" theory attributes the risky shift to reduce concern about failure resulting from sharing the blame with others. For ex— ample, a board of directors of a company may be considering various alternative investment proposals with different attributes, the level of monetary payoffs being one of the most important. Then even if every individuals' utility function is constantly risk averse, then it is possible that the group may be decreasingly risk averse simply because the responsibilities for the final outcomes are shared by all the board members. 5.5B A Generalized Application of Theorem 5.3.23 We have seen, in section 5.4A of this chapter, how exponential utility functions characterize the case of constant risk aversion. Since, as in the case of the theorem, we can generalize the two person case to N person case, let us start with two individuals in a group, with utility functions Ul(§), U2(§) where §.= (X1, X2’°""XM)' Therefore, assuming an additive multiattribute utility function for U1 and the simple exponential form for each attribute, we have Ung) = U1 (X1, x2,....,xM) = blUl (x1) + b2U2(X2) + + bMUM(XM) .... — b e (5.5.13) a1 a1 a] b: a1 110 and similarly, U2 (x) = U2(Xl, X2,....XM) = h2U2(X2) +....+ hMUM(XM) = -h e - h e - ...... - hMe (5.5.23) and K k 2 O, bl’ b2,....,bM 2 O, and hl’ h2,....,hM Z 0. 1’ 2 Now let us demonstrate the two distinct cases (a) constant risk aversion for the group and, (b) decreasing risk aversion for the group, by using attribute X1. (3) In this case R = R 11 21 cal constantly risk averse functions for attribute X1. We can demon- or the individuals in the group have identi— strate this by letting -c X >211 (X) BU (x) -c 25 e l 1 l‘— l-— _ l l _ Rll = - —‘2— — — ' _—.E-§{_ ' c1 (5'5'33) \xl Bxl C b e l l l 1 and -d X BZUZQ) buz Q) _dl2hle 1 1 R21 = — W/ }X = - W = d1 (5.5.4B) 1 1 dlhle Therefore R11 = R21 =) cl = d1 (5.5.53) Thus, substituting cl for d , we have 1 111 H H RG _ b20539 3UGQS) _ k11111 + k21121 _ _ —_ " - __I——-__v’_ 1 3x12 5X1 k1U11 + k2U21 2 "C X 2 “chl - -klcl ble 1 l — k2c1 hle —c X -c X klclble 1 l + k c2hlel 1 l c X 2 l 1 cl (e ) (klbl + kZhl) - X l 1 c e ) (klbl + kzh 1‘ 1) Hence the group has constant risk aversion cl. (5.5.63) Q3) For case (b), we have Rll # R21 i.e. the individuals in the group have constantly risk averse functions but they are not identical. In this case, again, we have, as in (5.5.33) and in(5.5.4B), but, R11 7‘ R21 => cl ’4 d1 Therefore, ' 2 H H RG = _ 3 UGQS) BUGQQ _ k11111 + k21121 m " - ——I—_——l_ 1 3le 5X1 k1‘11 + k2U21 2 “C1X1 2 ’d1X1 -kl.cl ble - kzdl hle k c b e‘C1X1 + k d hle‘dlxl 1 1 1 2 1 (5.5.7B) Nc 112 k 2 _Clxl c b e = 111 + ' _ x kzdlhl ex1(c1'd1) klclble C11 1+———kcb 111 —x 2 11 kzdl hle X(d-a) -dX klclblelll kdhe 11 211 kzdlhl _ C1 + d1 = R0 kdh ex1 kcbl 1 1+2 1+-—1—1—————— k c b x (c —d ) 111 kdhelll 211 (5.5.8B) Now there are three cases, (1) cl = dl where RG = cl = d1 18 already solved. (ii) if cl — dl > 0 or C1 > dl and if in expression (5.5.8B) X1 in— creases without bound (i.e. lim Xl + 0°), then R —- —L= Gl — O + 1 + 0 dl (5.5.93) if, on the other hand, in expression (5.5.83) Xl decreases without bound (i.e. lim X + —M), then we have 1 G = ——-—- + O = C (5.5.10B) Therefore, since at the lower limit the function R achieves a higher G1 value of c1 and at the upper limit a lower value of d1, the group risk lrrt 113 function R is a monotonically decreasing function for C1 ) d1, and G1 finally, (iii) if cl — dl < O or cl < <11 and if in the expression(5.5.8B) Xl increases without bound (lim Xl +m), then C RC = l + o = c (5.5.113) and, if in the same expression Xl decreases without bound (lim Xl +-M), then we have, Re = o + . 1 = d (5.5.123) Again, at the lower limit the function RG1 achieves a higher value and a lower value at the higher limit. Therefore, for the case of c - d < 0, the group risk function RG over attribute X is also a 1 l l l monotonically decreasing function. End of proof. Thus we have demonstrated that for a general class of constantly risk averse exponential functions, if for each group member the con- stant risk parameter is unique, then the participatory group utility function is no longer constantly risk averse but it is decreasingly risk averse, provided we have an additive group utility function. The above case was generalized application of a special case of two individuals of those N group members for which theorem (5.3.2B) was developed. Its extension to the case of N individuals is trivial. 114 5.63 Constant Risk Aversion Under Multiplicative Form In Chapter IV, it was shown that when certain assumptions hold, the participatory group utility functions assumes the multiplicative form. This form was KUG(u) + l = .g [l + KkiUi(X)] (5.6.1B) l=l It would have been very encouraging if we could prove a theorem for the multiplicative form, similar to the one that was proved for the additive form (Theorem 5.3.28). However, this unfortunately, is not possible due to the interactive elements in the group utility func- tion. In other words, when the individual multiattribute utility func— tions are all constantly risk averse, we still cannot conclude anything abOut the rate of risk aversion for the group as a whole. Depending upon the nature of the utility functions and the values of the con- stants k k2,....,kN and K, the multiplicative group utility function 1’ may be decreasingly, constantly or increasingly risk averse, even though each U1 is constantly risk averse. Bearing the above remarks in mind, let us apply the multiplicative form to the simplest case of two member group whose multiattribute utility functions are approximated by the same constantly EEEE averse exponential functions. In other words, we have U1(X) and U2(X) for two members, and X = X1, X2"""XM' The group utility function by multiplicative form is, UG = klUl(§) + kzuzqg + Kklszl(X)U2(§) (5.6.23) N Z . where i=1 ki < 1 and kl, k2, K > O as a spatial case a1 115 and U109 = U2 (fi) Parenthetically, repeating one of Chapter IV's results, we can state that function (5.6.2B) is the general multiplicative form for two mem- ber nonsymmetric participatory groups under the condition of pp 95 partial interaction. Result 5.6.33 Under the conditions specified above, the group utility function UG will be increasingly risk averse over an attribute Xi’ if U199 > 0 or U2(X) > 0 (5.6.4B) Proof: In this proof, as before, U1i and U31, are the first and second partial derivatives of U1 With respect to Xi' Let UG = klU1 + k2U2 + KklkZUlUZ and letting Ul = U2, we have _ 2 UG - (k1 + k2) Ul + Kklszl and know that U" l V ' RG _ G1 _ [(k1+k2) U11 + (2Kk1k2U1U11)] - " " —I"" ‘ I I l UGi (kl+k2) Uli + ZKklkZUIUli H '2 ll — (kl+k2) Uli + ZKklszli + ZKklkZUlUli ' " I I (kl+k2) Uli + ZKklszlUli 2 H ' Uli (kl+k2 + ZKklszl) + ZKklkZUli " ' I Uli (kl+k2+2Kklk2Ul) U" 2Kk k U'2 1i 1 2 11 .-l—__——_—_ Uli kl+k2+2Kklk2Ul =R1: therefore. I RG but and therefore, we know t1 I . Uli . TherefOre U1 = U2, I Thus functiOn j Mm The a the risk 2 for grOUps analYZe t} 116 2 l = R _ 2Kklk2Uli = RG (5 6 53) 11 kl+k2+2Kklk2Ul 1 therefore, differentiating both sides we have I ll — ' 2 ' R; = ' _ ZIKklszliUli (klk2+2Kklk2Ul) (ZKklkzuli) U11 i 1 2 (kl+k2+2Kklk2Ul) but ', = 0 l _ Un li and R, . == _ _'___ l Uli therefore, .2 , . 1 kl+k2+2Kklk2Ul kl+k2+2Kk1k2U1 11 we know that U1i > O, kl, k2, K > O, and Rli > 0 (because of risk aversion) Therefore expression (5.6.6B) is always positive if Ul > 0. Since U=U 1 U also has to be > O. 2’ 2 Thus if Ul > 0, then R5 > 0 and therefore the group utility i function is increasingly risk averse. End of proof. The above result was a special case which demonstrated to us how the risk aversion can shift from constant for individuals to increasing for groups under certain specific assumptions. Now let us briefly analyze the case of decreasing risk aversion. 5.7B 93.5 What prising t the come: will be d1 Theorem 5 For : action, 1: multiattr: the desir; (X3, X:) : 0f the ad. constants Pm. The 1 Theorem (3 therefore In T] \I . LOW kl’ k. the indiv; A 'V RG( 117 5.73 Decreasing Risk Aversion in Participatory Groups What happens to the group risk aversion when the individuals com- prising the group are decreasingly risk averse? This is also one of the common and important classes of risk aversion and some results will be derived for this case. Theorem 5.7.lB For nonsymmetric participatory groups with no or partial inter- action, if U1(X), U2(X),....,UN(X) represent decreasingly EEEE averse multiattribute utility functions of the individual group members over the desirable attributes X = (X1, X2,....,XM) and on the interval (X° * i’ Xi) for each attribute Xi’ and if the group utility function is of the additive form, 121 kiUi(X) where kl, k2,....,kN are positive constants, then the ggppp utility function UG itself is decreasingly Eisk averse over an attribute Xi' The proof of this theorem is almost identical to the proof of Theorem (5.3.2B) presented for the case of constant risk aversion and therefore the intermediate steps will be omitted. In Theorem (5.3.2B), for the group risk aversion function over attribute Xi, we had obtained the result, (see result (5.3.2B» I I I I _ 2 I I . k1U11R11 + k2U21R21 k1k2(R11 R21) U11U21 RG(Xi) = k u'_ + k u'_ — ' ' 2 (5.7.2B) l 11 2 21 (klUli + kZUZi) I V , I Now kl, k2 > O, Uli > O, U2i > 0. Also, Rli < 0 and R21'( 0 because the individual's utilit functions are decreasin 1 risk averse. T y _________E_X.___________ hus Ré (X1) < 0 (5.7.33) Thus, the End of prc The a vidials' L rithmic ut utility ft with the c utility ft group uti] Now ] decreasing aversion, the indivi then the g under the and the Va risk avers aVersiOn e W If tw increas‘ \lng attributes aVer3e ove U1, U2 > 0 118 Thus, the group utility function, UG’ is also decreasingly risk averse. End of proof. The above result is significant in the sense that when the indi- vidials' utility functions are decreasingly risk averse (eg. loga— rithmic utility functions fit this category), then the additive group utility function preserves this property. This is in Sharp contrast with the conclusion of Theorem 5.3.2B where constantly risk averse utility functions for individuals resulted in decreasingly risk averse group utility functions. Now let us turn back to the case of multiplicative form and decreasing risk aversion. Again, as in the case of constant risk aversion, no broad generalizations are possible. In other words, if the individuals comprising the group are decreasingly risk averse each, then the group as a whole may or may not be decreasingly risk averse under the multiplicative form, depending upon the functional forms and the values of constants. However, as we did for the constant risk aversion case, let us see what happens in an increasing risk aversion case. Result 5.7.4B A Case of Increasing Risk Aversion If two members of the participatory group each have identical and increasingly risk averse multiattribute utility functions over the attributes X, then the group utility function is increasingly risk averse over an attribute Xi if it has a multiplicative form and if U1, U2 > 0. Proof. The proof is identical to that of result (5.6.3B) and is therefore omitted. Thus RC(Xi From positive 1 These because th. might cans. a conventi1 > 0. Thus. (5.6.3B) a: the conclu: 0f the thec 5.8B A Log One of is that the m fOrr: A103 (x+b) Let th 119 omitted. The final result, where Rli > O, is '2 l 2 R' = W + w . U11 + Rii (5.7.43) 1 kl+k2+2Kklk2Ul kl+k2+2Kklk2U Thus Ré(Xi) > O and UG is increasingly risk averse From the final result, it is again obvious that Réi will be always positive if Ul > 0, since K,kr kZ’Rli’ Uli > O. Rli is also positive. These two results, namely (5.6.33) and (5.7.43) are interesting because they indicate to us what certain conventions in utility theory might cause if we apply the same to group utilities. In other words, a convention is usually adopted to express the Ui's for individuals as > 0. Thus, under very specific conditions as expressed in results (5.6.33) and (5.7.43), these conventions will automatically validate the conclusions reached above. Now let us see a generalized application of the theorem (5.7.13). 5.83 A Logarithmic Application of Theorem (5.7.13) I One of the differences between theorem (5.7.13) and result (5.7.43) 1 is that the former assumes an additive form but the latter a multipli- cative form. Now assuming a logarithmic function of the form A log (X+b) for the individual decision makers, let us demonstrate these results. Let the multiattribute utility functions be, Ul(§) = cllullocl) + €12U12(X2) + ..... +clMUlM(XM) 21 21 U2(§) = c U (X1) + c22322(x2) + ..... +C2MU2MCXM) UN(§) = chUN1(Xl) + CNZUN2(X2) + ..... CNMUNM(XM) but we k: h1=hj = C 111(1) = C 112(1) . . = ( UNq) And also, %=kfl1 Thus, usi collecti: II “I TherEfore grOUp uti ber has a 120 but we know that for person h and attribute i Uhi = Ahi log (Xi+bi) which 18 decreas1ngly risk averse, and therefore, U1(X) = CllAll log (Xl+bl) + C12A12 log (X2+b2) + ..... + ClMAlM log (XM+bM) U2(§) = CZlAZl log (Xl+bl) + 022A22 log (X2+b2) + ..... +C2MA2M log (XM+bM) UNCE) = CNlANl log (Xl+bl) + CNZANZ log (X2+b2) +.....+CNMANM log (XM+bM) And also, UG = klUl(X) + k2U2(X) +.....+kNUN(X) Thus, using the partial derivatives for risk function, U" " l' n R (X ) _ 1 _ Hung) + k2U2i(§) +.....+kNUNi(§) - " _I- " I I I G 1 UGi klulig) + k2U21(§) +.....+kNUNi(X) leliAli _ k2021A21 _ _ KNCNiANi] ——2 ——————2 ..... ———2 (Xi+bi) (Xi+bi) (Xi-I-bi) k1011911 + k2C21‘3‘21 + + k3‘3111‘3‘111 (Xi+bi) (Xi+bi) (x 1+b1) collecting the terms and cancelling, = G. (5.8.23) Therefore, as Xi increases, R always decreases. Consequently the G. i group utility function is decreasingly risk averse. As suggested by theorem (5.7.13) this result is achieved when in fact each group mem- ber has a decreasingly risk averse logarithmic utility function. 5.9B .lpg A to of group creasing] Unfortuna possible of a the: (theorem result ( aversion R3 (X1) the grou if Ré = tions a: case of is one < creasing mUltiplj Patory cases a 5.103 In grOUps fOllowj tions. 121 5.93 Increasing Risk Aversion in Participatory Groppg A couple pages before, in Result 5.7.43, we have seen the effect of group aggregation on the risk aversion when individuals had in- creasingly risk averse utility functions. This was a very special case. Unfortunately, as in the case of decreasing risk aversion, it is not possible to generalize the case of increasing risk aversion in form of a theorem similar to that which applies to constant risk aversion (theorem 5.3.23). Although we would still arrive at the same final result (result 5.3.2.53) because Rli’ R2i > 0 due to increasing risk aversion, it would not be possible to obtain a positive or negative Ré (Xi) at all times. Thus for increasingly risk averse individuals, the group might behave as increasingly, decreasingly (or constantly if Ré = 0) risk averse dependent upon the nature of the utility func- tions and their constants. This same conclusion also holds for the case of multiplicative group utility functions. This result (5.7.43) is one of the rather specific conclusions that can be drawn for in— creasingly risk averse utility functions and group utility having the multiplicative form. Similar results for additive forms in partici— patory groups can be found but because the conditions leading to such cases are not very universal and common, they will be omitted here. 5.103 Risk Aversion in Symmetric Participatory Groppg In Chapter IV, it was demonstrated that symmetric participatory groups under the condition of no or partial interaction possessed the following additive or multiplicative forms depending upon the assump— tions. The symmetric group utility function in additive form is N U = K E U. (5.10.13) G i=1 i and in n Thus, th and symm unique k cal (or the case demonstr change f be simpl be demon in the p Let late a n. e M For and haV1] memb ers' are cons: .____ X2, . ° ' "I: the Symm or (b) d. ‘ U2(X),.. ‘ the“ the 122 and in multiplicative form it is, KU = g (KkUi + l) - l (5.10.23) i=1 Thus, the only difference between nonsymmetric (or asymmetric) forms and symmetric forms are in the constants ki' Whereas there is a unique ki for every Ui in the nonsymmetric forms, all ki's are identi- cal (or ki = k2 = .... = kN = k) in the symmetric forms. This being the case, none of the major or minor theorems or results that were demonstrated in this chapter under the conditions of nonsymmetry will change for the case of symmetry. The results and theorems will only be simplified in this case. Consequently, the case of symmetry will be demonstrated only in terms of the two most important theorems proven in the previous sections. Let us now combine the theorems (5.3.23) and (5.7.13) and formu- late a new theorem for symmetric participatory groups. Theorem 5.10.33 For symmetric participatory groups with no or partial interaction and having the additive form, K .gl Ui’ where K > 0, if the individual members' multiattribute utility—functions, Ul(§), U2(X),....,UN(§) are constantly risk averse over the desirable attributes X = (X1, X2,....,XM) and on the interval (Xi, x:) for each attribute Xi’ then the symmetric group utility function itself is either (a) constantly, or (b) decreasingly risk averse. If those utility functions U1(X), U2(X),....,UN(X) are instead decreasingly risk averse ceteris paribus, then the group utility function itself is (c) decreasingly risk averse. Eggpf. The with the omitting (5.3.23) appropri Analyzin Case (a) Ind ' — R21 0 are incr then Ré( averse. 9% All I fore, RG aVerSe. M We 1 risk aVeI ing funct quCtiong Symmetric 123 Proof. The proof of this theorem is almost identical to that of (5.3.23) with the exception of changes in group constants for symmetry. Thus omitting the intermediate steps which are demonstrated for theorem (5.3.23) and assuming kl= k2 = kN= K, we have, after the appropriate cancellations, the final result for a symmetric group, I I I I _ 2 I I R. _ U11R11 + U211521 (R11 R21) U11‘121 C(Xi) ~ W— - (5.10.33) 1 21 (Uii + 1121)2 Analyzing this result in three cases, namely: Case (a) Individual Ui's are constantly risk averse. Therefore Rli = O, R'. = ’ . . '. ' .' 21 0 Since R11 and R21 are constants U11 > O, UZi > 0 because U1 3 are increasing functions for desirable attributes. Now if Rli = RZi’ then Ré(Xi) = O and thus the symmetric group UG is constantly risk averse. Case (b) All the above conditions for case (a) hold but Rli # R21. There- fore, Ré(Xi) < 0 and thus the symmetric group is decreasingly risk averse. Case (c) We now assume that individual multiattribute Ui's are decreasingly r‘ . '. ' ‘ . - isk averse Therefore R1l < 0, R2].- < 0 Since R11 and R21 are decreas ing functions. As before, Uli > 0, U2i > 0 because Ui's are increasing functions of desirable attributes. Thus Ré(Xi) < 0. Consequently the symmetric participatory group, UG’ is decreasingly risk averse. The patory g repeated were def group me preferen a funo in the g vidual i In 1 above Ui group ut: (4.11 B) Here CatiVe fc tiofl is c Utility f nest, We attribute basiCall3 We have n 124 The other results that were demonstrated for nonsymmetric partici- patory groups also hold for symmetric groups and thus they will not be repeated here. 5.113 Risk Aversion in Nonsygmetric and Sympetric Groups with Full Interaction In Chapter IV, the concepts of no, partial and full interaction were defined and analyzed in detail. We now know that fully interactive group members fully incorporate the preferences of others into their own preference structure. Under such a condition, each Ui is only a function of_§ but also of other U3. Therefore each Ui = Ui(§, U3) in the group. In Chapter IV it was demonstrated that for every indi- vidual in the group, each Ui looks like Ui = Uiq. U;) = AiXUng) + xiuuiuw-Qg» + AiXuUiX(§-)Uiu(Ui(X)) (5.11.13) In the additive and multiplicative form we simply substitute the above Ui's into the equations to find the nonsymmetric or symmetric group utility function. This representation was given in section (4.11.3) in Chapter IV. Here, it is obvious that each Ui is in a two attribute multipli— cative form itself. The complexity of the ensuing group utility func— tion is quite apparent because there are three types of multiattribute utility functions nested within each other. First, in the innermost nest, we have each one of the Uix(X)'s and Uiu(Ui)'S which are multi- attribute. The second stage nest comprises the Ui's which have basically two groups of attributes, st and Ui's. In the final stage we have multiattribute UG(U1’ U2,.....,UN), whose attributes are the individu of the n tive for the stre: undergo 1 In other be risk 1 aversion aversion a simple tion in ( Example 5 In a vidual wj aSSUmptic attribute function U10: Where X' 3 Now attribute IV, Ulu(U 1X 125 individual Ui's. In this type of three stage complexity, and because of the necessary introduction of the two attribute (X, U1) multiplica— tive form for each Ui’ the nature (risk averse, neutral, prone) and the strength (decreasing, constant, increasing) of Eigk aversion will undergo unpredictable changes as we go from the individual to the group. 1 In other words the individuals could be risk averse and the group could be risk prone, etc. There might also be a shift from decreasing risk . . . . I . . . . . . averSion for the indiViduals utility functions to an increaSing risk aversion for the participatory group. Here, we would just like to give a simple example of what might happen to an individuaTs utility func- I tion in case of a simplified version full interaction. Example 5.11.23 In a fully interactive participatory group, let us assume an indi- vidual with a fully interactive utility function U1. Other simplifying assumptions are N=2, two person group, and that there is only one attribute, X. Under these assum tions the first individual's utilit P Y function looks like: Ul(X, U2) = AlUlX(X) + AZUlu(U2(X)) + 13U1X(X)Ulu(Ux(X)) (5.11.33) where X's are scaling constants and Al + 12 + 13 = 1. Now let us assume that both individuals are risk neutral on the attribute X. Also, because of strategic equivalence defined in Chapter IV, Ulu(U2(X)) = U2(X). Let this risk neutral function be: — l — U1X — B—:—; (X + a) for a < X < b and Ulu Therefor Ul( Therefor R(X We know Therefor if Thu: X and U . and OVEI‘ attribub neutral 3 0n 1 3Verse (1; (Al or and U =U‘=—l-—(X+a)for-a 0 but )3 can be positive or negative. Therefore, if 13 > 0 =) R < o 13 < O =) R > 0 if the denominator of (5.10.43) is > 0. Thus if A > 0, the individual is risk prone over the attributes 3 X and U taken together although he was risk neutral over attribute X and over attribute U2 separately. In this case risk neutrality over attribute U simply indicates that the other member of the group is risk 2 neutral in his preferences over the attribute X. 0n the other hand, if 13 < 0, then, in order to obtain a risk averse case, we have to have 1 1 (11 + 12) b + a + 213 (b + a)2 (x + a) > o (5.11.63) or 1 (11 + 12) > - 213 - b + a - (X + a) (XH The SubstitL or Thi risk net functior interact full int Preferer n6utralj Sibility Vant her Thi COnstant in the r ObSerVed action 6 Al] 127 (b + a) (11+12) -2A3 (X+a) < (b + a) (kl+12) + 213a X < -213 (5.11.73) The largest negative 13 can have is —l, and if 13 = -1, Al + 12 = 2. Substituting these into the inequality (5.11.73), we have x (W = 2713: b (5.11.83) or X < b, which is the upper limit for X. This very interesting example demonstrates to us the fact that two risk neutral utility functions can result in a risk averse or risk prone function depending upon the values of the scaling constants in the interactive component. This type of result was obtained because of full interaction. In other words, incorporating the other person’s preferences into one's utility function resulted in a shift from risk neutrality to risk proneness or risk aversion. The diffusion of respon- sibility theory mentioned in section 5.43 of this chapter is also rele- vant here and is one of the possible causes of such risk shifts. This example not only illustrates the importance of the scaling constants but also gives us an idea about the possibilities of changes in the nature of risk aversion. Such radical shifts in risk were not observed in the additive forms for the cases of no or partial inter— action as they were explained in the previous sections. All the remarks in this section (5.11.3) hold whether the group is symmetri 5.123 g The tion cas must be rately f form and ticular risk by tude of tion lin degree. Chapter represer ideal fc V Shifte towards under ce the gror goal of risk frc accepted utility rEmarks in expec utility 128 symmetric or nonsymmetric. 5.123 Additional Remarks These last few sections on multiplicative forms and full interac- tion cases demonstrate the fact that in some cases certain sacrifices must be made in the consistency in risk attributes in order to accu- rately formulate a group utility function. That is to say, the ideal form and nature of the participatory group utility function for a par- ticular problem might result in rather unexpected attitudes towards risk by the group. Thus if the nature and direction of the'risk atti— tude of the group is known or prescribed in advance, then this condi— tion limits the formulation of the group utility function to a certain degree. This is a very interesting relationship between the results of Chapter IV and Chapter V. In Chapter IV the only concern was to try to represent the group utility function in its most realistic and close to ideal form under certain assumptions. However our interest in Chapter V shifted to the demonstrations of what happens to group attitudes towards risk when certain forms of group utility functions are used under certain risk attitude assumptions for each individual comprising the group. It is interesting to note that, in certain instances, the goal of obtaining a certain type of observed group attitude towards risk from the group utility function is not compatible with the other accepted assumptions that determine the actual shape of the group utility function. Consequently, for these certain cases where these remarks apply, appropriate tradeoffs can be made between consistency in expected group risk attitudes and the most realistic form of group utility function. However the group utility analysis, being so complex as it i and coni 5.133 1 Che group u1 tions it was to 1 develope functior individL group ut chapter treatmer Th1 the app] Problem 129 as it is, makes the satisfactory resolutions of such inconsistencies and conflicts rather difficult if not impossible. 5.13B Final Remarks Chapters IV and V dealt with some hitherto neglected aspects of group utility theory. After generating some new concepts and formula- tions in group utility theory in Chapter IV, the attempt in Chapter V was to relate the risk theory to the group utility formulations developed in Chapter IV. Although Pratt has done some work on risk functions in unidimensional (single attribute) utility theory for individuals, a considerable vacuum exists in the treatment of risk in group utility theory using multiattribute utility functions. This chapter addressed itself to this vacuum in the literature on the treatment of risk in group utility theory. Thus we come to the final chapter of the thesis which deals with the application of some of the developments here to a controversial problem concerning the citizens of a small town. developu patory g demonstl decisio: 6.2 §§£ The members Michigar tYPe of its COIIm One of y This Sma Prevailj the City final de mot10ns_ teristic defined CHAPTER VI THE APPLICATION 6.1. Introduction This chapter mainly deals with the applications of some of the developments in previous chapters to a real life problem in partici- patory group decision making. The participatory group decision models, demonstrated in Chapter IV, were tested on a group responsible for the decision making in a small town. 6.2 Background The group of decision makers described in this section are the members of the city council of East Lansing, a small college town where Michigan State University is located. East Lansing has a city council type of self-government and all the decisions concerning the city and its community are made by this body. The council has five members, one of which is the city mayor and another member is the mayor pro-tem. This small group of five people form a cohesive unit and decide on the prevailing issues by the one-man one—vote principle as indicated in the city charter. Thus the process is completely democratic and the final decisions purely depend upon the votes cast for or against the motions. Thus in this form the city council demonstrates the charac- teristics of a symmetrical participatory group as these concepts are defined in sections (4.2.3) and (4.5.3). Basically this particular 130 group i decisim The gr01 mmortm Thus the equal vc The viewed 3 particul 6.3 9&2 Day develOp Partiall Proposed 275,000 Square f 1980. T ment Sto of 75,00 750,000 would at AftI Lansing, designatt to CGimme] fact: be LaHSing. 131 group is participatory because the decisions are not made by a supra decision maker, but are made collectively through a democratic process. The group is also symmetrical because the identity of the voters it not important as long as the composition of the votes remains the same. Thus the members of the East Lansing City Council have objectively an equal voice in the outcomes for their decisions. These five members of the city council were extensively inter— viewed with the idea of assessing their preference functions over a particular decision described below. 6.3 Dayton—Hudson Mall Decision Dayton—Hudson Properties of Minneapolis, Minnesota is proposing to develop a shopping center (a shopping mall) on an 86 acre site located partially in both Lansing Township and the City of East Lansing. The proposed development is planned in two phases. In the first phase 275,000 square feet of department store space (two stores) and 250,000 square feet of mall and tenant space would be developed for opening in 1980. The second phase would involve the addition of a third depart- ment store of 150,000 square feet and additional mall and tenant space of 75,000 square feet, bringing the total size of the development to 750,000 square feet. The entire development would be two stories and would at full development require 3,750 parking spaces. After Dayton-Hudson Corporation proposed such a mall for East Lansing, the city council was faced with a rezoning decision. The area designated for the mall development had to be rezoned from industrial to commercial. With the rezoning decision the city council would, in fact, be approving or disapproving the building of the mall in East Lansing. Th the ver against into th the mal Communi in the and int by buil Environ consult commiss the cou with ya raprese argumen Propose 0n the the 111431 hearing rezonin the rez Cial. their p Ho beginni Dayt0n_ 132 The mall issue began to interest the East Lansing residents from the very beginning. Many diverging opinions were expressed for or against the building of the mall. Hundreds of letters started pouring into the offices of council members. Furthermore, those who oppose the mall decided to organize themselves, and "Citizens For Livable Community" was formed. Naturally, Dayton—Hudson was very interested in the project and did their best to convince the city council members and interested citizens of the benefits that will accrue to the city by building the mall. Dayton-Hudson had a feasibility study [29] and Environmental Impact Study [106] prepared by a group of independent consultants and submitted to the city. Furthermore, city planning commission also prepared an impact study which was also submitted to the council. All these studies favored the construction of the mall with varying degrees of intensity. Citizens for a Livable Community representatives, on the other hand, also put forth very plausible arguments about the negative economic and environmental impacts of the proposed project. The city council held a number of public hearings on the project before the vote and every possible view for or against the mall was expressed by individuals or organized groups during these hearings. Finally, on August 3, 1977, the council took a vote on the rezoning issue. The outcome of the vote was three for and two against the rezoning of the Dayton-Hudson property from industrial to commer- cial. Thus the green light was lit for the builders to go ahead with their project. However, this vote did not resolve anything and it was the beginning of a long fight between the opponents of the project and Dayton—Hudson Corporation. Citizens for Livable Community group started Such ref Lansing, council the deci had won were hel council, election Dur organize Corporat until th Supreme at least tion cou WOuld be decision Poration At the t tiating November simply 1 Public c be a VET this int be sett; 133 started collecting signatures for a referendum on the mall issue. Such referendums are possible under the charter of the City of East Lansing, and the outcome of such referendums can overrule the city council decisions. Dayton-Hudson, sensing a possible overturn of the decision, challenged the referendum proposal in the courts and had won the initial rounds. Towards the end of 1977 new elections were held for two seats on the city council. Two members of the council, who originally voted in favor of the mall, did not seek re- election and two new council members were elected in late 1977. During all this time the court battles between the opponents organized under Citizens for a Livable Community and Dayton-Hudson Corporation were continuing. The courts gave injunction decisions until the matter could be finally settled, possibly at the Michigan Supreme Court. It began to appear that the court process would linger at least two, possibly more years, during which time the mall construc— tion could not be started because of injunctions. Sensing that it would be more detrimental to the company to wait for the final court decision than to risk the outcome of a referendum, Dayton-Hudson Cor- poration withdrew their objection to this process late July, 1978. At the time these pages were written, Dayton-Hudson Company was nego— tiating the final wording of the proposal to be put on the ballot in November, 1978. The question to be asked of East Lansing voters will simply be whether they want this proposed mall in their city or not. Public opinion polls during mid—summer of 1978 suggest that it will be a very close decision and the final vote could go either way. Thus, this interesting but rather controversial community issue will finally be settled by the voters themselves, after heated debates and arguments 6.4 1353 Afte Lansing 1‘ (mayor) , Fox, and council July 197 between] purpose in Novem involved and for until it the resi For concepts city Coy functior a detail see What 6.5 13! Tu Such as main ob traffic 134 arguments that have lasted more than eighteen months. 6.4 The City Council After the November 1977, elections, the city council of East Lansing is composed of five members. They are Mr. George Griffiths (mayor), Mr. Larry Owen (Mayor pro-tem), Mr. John Czarnecki, Mr. Alan Fox, and Ms. Carolyn Stell. The last two members have joined the council after the November elections. This city council voted in late July 1978 to appoint the city attorney to mediate the negotiations between Dayton—Hudson and Citizens for a Livable Community with the purpose of drawing up a clear-cut ballot proposal for the public vote J in November. Thus we observe that the city council was again directly involved with the mall issue and trying to finalize the decision once and for all. All the members named above actively pursue this issue until it will finally be settled in November 1978 by a public vote of the residents of the city of East Lansing. For a practical application of some of the participatory group concepts that were developed in the previous chapters, each of these city council members were interviewed extensively and their preference functions over certain attributes were assessed. Before going into a detailed discussion of the preference assessment process, let us see what the attributes are and why they were used. 6.5 The Objectives and Attributes The Dayton~Hudson mall decision basically involves many aspects, such as political, economical and environmental. Each one of these main objectives can also be divided into subobjectives, such as traffic decisions, air, water pollution, tax impact, employment impact, etc. 1 that c2 pants 1 the giv more me tant ob attribu Index," Support Wh of East envirom objecti life f0: are alS( their Vt they mig Citizen: hers. I rEflect Thu butes, X each One 6.5.1 X 135 etc. Most of these attributes do not carry objective measurement units that can be interpreted in the same manner by each one of the partici— pants in the assessment process. Thus, to evaluate the problem within the given time limitations of the researcher and to make the analysis more meaningful, two major attributes representing the two most impor- tant objectives were employed. The first one is a subjectively scaled attribute which is called "X1 = Net Overall Economic and Environmental Index," and the second one is "X2 = Percentage of East Lansing Voters Supporting the Council Member's Decision." Whether the Dayton-Hudson Mall is built or not, the city council of East Lansing as a group is first concerned with the economic and environmental future of the city. Thus the council has the major objective of bettering the economic and environmental aspects of the life for the city residents. However, being political figures, they are also concerned with the support they have for their decisions since their voting records are of great importance in the future elections they might be involved. Thus maintaining political support of the citizenry is also one of the important objectives of the council mem— bers. Parallel to this objective, there would be also a desire to truly reflect the preferences of the city residents in the voting process. Thus, to reflect these two important objectives, the two attri- and X mentioned above, were chosen in concurrence with butes, X1 2 each one of the five participants. 6.5.1 X1, The Overall Economic and Environment Index For the sake of facilitating the analysis and increasing the com- prehension, the extreme values were chosen as -100 and +100. When X1 = -l( of econc diminisf inflatic and big experien very sev in great pollutio is achie there W01 severe p( The where at maximum 1 its theor tion and services Prosperit envirOHme With VEry eral and For to be Som. this have which Can at this 16 X1 = -100, it is assumed that East Lansing would experience the worst of economic and environmental situations. The tax revenues would diminish, the services provided by the city would decrease, and possibly inflation would eat into revenues. The business revenue of the mall and big stores in constant dollars would decrease and the city would be experiencing insurmountable environmental problems. There would be very severe traffic problems with continuous traffic jams resulting in great loss of time and energy and environmental (air and water) pollution would be at its worst. When this level (~100) of the index is achieved, with economic breakdowns and unemployment at its maximum, there would be extreme citizen dissatisfaction resulting in very severe political problems. The second extremium of the index was chosen to be as X1 = +100, where at this level exactly the opposite events would happen. At this maximum level of the attribute the economic situation was assumed at its theoretical best with very abundant tax revenues, minimum infla— tion and unemployment levels. At this level of the index all the services are provided at close to optimum levels and there is great prosperity maximizing the businesses' and citizens' incomes. On the environmental side, the pollution problem is completely under control with very little problem. This level would naturally result in gen— eral and extreme citizenry satisfaction at all levels. For the purposes of analysis, the mid value of X1 = O was assumed to be somehwere close to status quo in the city of East Lansing. At this level of the index, there is a normal increase in tax revenues which can barely keep up with the inflationary cost increases. Also, at this level of X1 = O, the services are provided in a limited manner and poi are occ still c ment in Th points bers' p strated 6.5.2 X Thi Port can council Oates th mamber, indicate hYPOthet; by the V: Scale, t} Perc 137 and pollution is under control but still an undesirable levels. There are occasional traffic jams and the unemployment at moderate levels still continues to be a nagging problem. (At the time, the unemploy- ment in the state of Michigan was around 7.3 percent.) Thus this important attribute was anchored at these three pivotal points of -100, 0, and +100. Each and every one of the council mem— bers' preferences were assessed over this attribute, which is demon- strated on a horizontal scale below. —100 0 +100 Overall Economic and Environmental Level 6.5.2 X The Political Support Index 2, This attribute is much simpler to analyze because political sup— port can be measured in terms of percentages. Thus a minimum value of X2 = 0 percent is assigned to the level of political support for the council members' decision on the Dayton-Hudson Mall. This simply indi— cates that all voters unanimously reject the decision of the council member. At the other extreme, X2 is naturally +100 percent, which indicates a unanimous support for the member's decision. These are hypothetical extremes and naturally the actual percentage of support iddle. On a horizontal by the voters would lie somewhere in the m scale, this attribute is demonstrated as: X2 //—0 oz 50% 1004 Percentage of Political Support of Council Member's Decision 6.6 33 ll challet when, n analysi problen nifican plain, assessm his cor In ing pha 1) 2) 3) 4) 5) 6) 7) ment Wer 138 6.6 The Assessment of Utilities The assessment of an individual's preferences represents the most challenging and difficult task in the utility analysis. Especially when_ more than one attribute is involved (i.e., multiattribute utility analysis) as in the case of the Dayton-Hudson Mall decision, then the problem gets even more complicated. The challenges are mutually sig— nificant both for the analyst and the subject. The analyst has to ex- plain, in very simple terms, the purpose and process of preference assessment and the subject has to formulate, to the best of his ability, his correct preferences. In this research, with each one of the five subjects, the follow— ing phases were implemented in the given sequence: 1) Explanation of the purpose and nature of Decision Analysis; 2) Explanation of the nature of utility assessments; 3) Determination of the individual and group objectives for the Dayton-Hudson decision; 4) Determination and explanation of the two attributes and their relevant ranges; 5) Testing for utility independence between attributes; 6) Assessment of the actual utilities for each attribute; 7) Checking for consistency; 8) Selection of the appropriate utility function; 9) Determination of the scaling constants; 10) Final determination of the council members' multiattribute utility functions over the two specified attributes. Each one of these ten steps Were followed thoroughly with each member of the group. The first four steps of preparation for assess— ment were very useful and essential for the later stages. The W researc member attribu intervi the cou re-expl A summa Appendi Fc given i utility utility Ui T1 utilit} Pr0perl versat: thus ma T 0f uti tial fl to rep tions 139 researcher discussed the objectives of the decision with each council member and explained thoroughly the meaning and range of each of the attributes discussed in the previous sections. During these oral interviews, there was a lot of interaction between the researcher and the council members. Therefore, most of the questions were explained, re—explained and modified, especially during the assessment phases. A summary of these questions for utility assessment is presented in the Appendix section of this dissertation. For each council member, the more general multiplicative model given in Chapter IV (equation 4.4.5B) was used. In a multiattribute utility function, if there are two attributes, then the multiplicative utility function over these attributes is given by: U(Xl, X2) = kl U1(X1) + k2 U2(X2) + kl2 Ul(Xl)U2(X2) (6.6.1) This function has been derived under the conditions of mutual utility independence assumptions. The scaling constants have the property k1 + k2 + k12 = 1.0. Consequently, the function is very versatile and becomes an additive utility function if kl + k2 = 1.0, thus making k 0. 12= The second choice involved the determination of the general family of utility functions to represent the utility functions. The exponen- tial function, described in Chapter V, section (5.4.2A), was chosen to represent the individual's utility functions. This family of func— tions can be represented by: U(X) = 1—‘-e—— (6.6.2) l“ Where c As indi and whe there a functic heurist subject utility grossly given c in fitt utiliti Ea lOtteri Points- cedure, COnStan Employe. as to w mall ii say the Senting mthal 1 a value would i] envirom be the I 140 Where c would be the changing parameter from one individual to next. As indicated in Chapter V, when c > 0, the individual is risk averse and when c < O the decision maker is risk taking. In utility theory there are no clear cut procedures for choosing an appropriate utility function that fits the data and the choices are usually based upon heuristic search process. The utility assessments themselves are subjective and therefore, are usually valid within a range. Then, a utility function satisfying most of the constraints and not being grossly incompatible with the others would be appropriate under the given circumstances. The exponential family is extremely versatile in fitting the data, especially around attribute midpoints whose utilities have been assessed. Each council member was asked several questions in the form of lotteries or gambles (see Appendix) over each attribute. Thus several points were determined. Following the utility point assessment pro- cedure, several questions were used to determine the values of scaling constants kl for the general multiplicative function model that was employed. The final questions related to the council members' idea as to where the value of the attribute will be if the Dayton-Hudson mall 3§_built and if it is pg£_built. For example, a respondent would say that he believes East Lansing will achieve an index of "0" repre— senting status quo on the first attribute (i.e., Economic and Environ— mental Index ranging from ~100 to +100) if the mall is not built but a value of ”—20” on the index if the mall is built. This, of course, would indicate that he would be expecting unfavorable economic and environmental impact from the mall. The second similar question would be the percentage of support a council member expects from the city “l voters mall 1 cent 0 This n; belieV( issue: ll or don‘ was ob expecte probabi dent tc 0f the will vc be in f which i functic t0 the Substit Th decided and max form of and for 141 voters on his/her decision-—whether he or she votes for or against the mall issue. Thus a typical answer would be saying that about 55 per- cent of the voters would support my decision of "no" for the mall. This naturally implies that a council member giving such an answer believes that 55 percent of the voters will vote against the mall issue in the referendum. Thus for each of the two decisions or acts (i.e., build the mall or don't build the mall), an expected central value for each attribute was ob tained from each council member. These valuesrepresent the expected values or values of central tendency for each attribute whose probability is assumed to be 1.0. Thus, instead of forcing the respon— dent to say something like "there is one—third chance that 50 percent of the voters will vote no, one—third chance 55 percent of the voters will vote no, and finally one-third chance that 60 percent would not be in favor of the project," he was directly asked the mean value, which is 55 percent in this case. After forming the group utility function, to find the expected utility of each of the two decisions to the group, the appropriate expected values of the attributes were Substituted in the council members' utility functions. Thus, once the usage of exponential family of functions was decided, the general form had to be modified according to the minimum and maximum values of the two attributes. Therefore, the general form of the utility function for attributes X was determined to be: 1 e-C(Xl + 100) U 1- X1 -200c (6'6'3) 1-e and for X2 1—e-CX2 U = —-———-f:-—- (6.6.4) X2 1—e lOOc It w the data to arrive plex and prone to justified U X1 3 value c value of Thus both The utility i let us se are for e F°r attri For attri Whe re US‘ For attri FOI attrj 142 It was then determined that the above exponential functions fit the data points closely. Since the goal in the dissertation was not to arrive at a theoretically optimum function which could be very com- plex and since the utility points themselves are subjective and are prone to error and change, the usage of the exponential family seemed justified and appropriate in this particular case. UXl of (6.6.3) achieves a maximum value 1.0 when X1 is +100 and a value of 0.0 when X1 is —100. Similarly, UX2 achieves its maximum value of 1.0 when X2 = 100% and its minimum value of 0.0 when X2 = 0.0%. Thus both of the utility functions are normalized. The following ten graphs show the assessed utility points and the utility functions of each of the five council members. First, however, let us see what the component utility functions for the two attributes are for each one of the council members. Council Member Stell l_e—O.OO64(X1 + 100) For attribute X1: US(X1) = O 722 (6.6.5) l-e_0'0128X2 For attribute X2: US(X2) = W (6.6.6) Where US(X1, x2) = klUS(Xl) + szs(X2) + klZUs(xl)Us(X2) Council Member Czarnecki l_e0.oo9(xl + 100) For attribute X1: UC(X1) = —-—-::;RE;—-———— (6.6.7) . 1 F . = ____ or attribute X2. Uc(X2) 100 X2 (6.6.8) Wherel For at1 For att Where I For art For att Where L‘ For att F0r att Where U 143 Where UC(Xl, X2) = klUC(Xl) + szc(X2) + klZUC(Xl)UC(X2) Council Member Fox 1-e‘0' 00522(X1 + 100) For attribute X1: Uf(Xl) = 0.648 (6.6.9) l_e—0.0244X2 For attribute X2: Uf(X2) = -—-—6f§i§——— (6.6.10) Where UF(Xl, X2) = k1Uf(Xl) + szf(X2) + klZUf(Xl)Uf(XE)' Council Member Griffiths _e—0.00522(Xl + 100) . , = 1 For attribute X1. Ug(Xl) 0.648 (6.6.11) 1—e --0.0148X2 For attribute X2: Ug(X2) = 0.772 (6.6.12) = k Where UG(X1’ X2) lUg(X1) + kZUg(X2) + klZUg(Xl)Ug(X2) Council Member Owen l_e-0.00765(Xl + 100) For attribute X1: Uo(X1) = 0.783 (6.6.13) l_e-O.Ol8X2 For attribute X2: UO(X2) = —-—6T§§§—- (6.6.14) Where UO (X1, X2) = kon(Xl) + k2U0(X2) + k12U0(Xl)Uo(X2). AAMHm z k2 or kl < k2 for each decision maker. This was done by asking (see Appendix) whether the council member prefers U' (+100, 0%) or U" (—100, 100%). In this case it seemed logical that all the council members would prefer U' to U”, since they would be more concerned with the economic and environmental situation of East Lansing than the political support they would get. This was, indeed, the case. In this analysis, I a _ U (+100, 06) — klUl(+100) + k2U2(0) + klel(100)U2(0) (6.7.1) and, H a _ _ a _ c U (-100, 1004) — klUl( 100) + k2(1OOA) + k12U1( 100)U2(1004) (6.7.2) however, since U2(0) = O and Ul(-lOO) = O, we have, U' = R1 (6.7.3) and n__ U — k2 (6.7.4) After finding that all kl's are greater than all kz's for every council member, they were asked to express a value for X1' and X1" in the fol] U(X and U(X In attribut conseque equation klU and equa klU Aft each mem three eq for thes Constant utility by the c tIVe uti For US( 150 the following equations: U(Xl" 0.0%) = U(—100, + 100%) (6.7.5) and U(X ", 100%) = U(+100), 0.0%) (6.7.6) In the above equations, X1' and X1" indicate those levels for the attribute Xl at which the decision makers were indifferent between the consequences on the left and right sides of the equations. Thus equation (6.7.5) results in ' .. klUl(Xl ) — k2 (6.7.7) and equation (6.7.6) results in n n __ k1U1(X1 ) + k2 + 1:12 Ul(Xl ) - kl (6.7.8) After determining the actual shape of the utility functions for each member, both Ul(Xl') and Ul(Xl") were calculated. This gave us three equations involving the unknowns kl, k2, and klZ' The solutions for these simultaneous and independent equations gave us the scaling constants for every council member's multiplicative multiattribute utility function over the attributes X1 and X2. The constants found by the certainty equivalence method are substituted in the multiplica— tive utility functions below: For council member Stell: US(Xl, X2) = 0.91 UX1 + 0.75 UX2 - 0.66 UXl. UX2 (6.7.9) For l U (X. For U (X For UG() For U 6.8 The and whe K18 ch 151 For council member Czarnecki: UC(X1’ X2) = 0.69 le + 0.28 UX2 + 0.03 le. UX2 (6.7.10) For council member Fox: UF(X1’ X2) = 0.99 UX + 0.91 UX - 0.90 UX - UX (6.7.11) 1 2 1 2 For council member Griffiths: UG(X1’ X2) = 0.88 UX + 0.68 UX - 0.56 UX ' UX (6.7.12) 1 2 l 2 For council member Owen: Uo(Xl, X2) = 0.91 UXl + 0.62 UX2 - 0.53 le' UX2 (6.7.13) 6.8 The Participatory Group Utility Function The group utility functions for symmetric participatory groups were developed in section (4.6B) of Chapter IV. It was assumed that city council as a participatory group obeyed the preferential and utility independence assumptions that were defined and explained in sections (4.7B) and(4.8B) of Chapter IV. The formulation that is used for the city council is that of (4.8.3B), where the symmetric partici- patory group utility funttion looks like 2 UG(U) = K . 1 U. (6.8.1) 1 1 HM and where K is an arbitrary constant. In this particular application K is chosen to be one-fifth, so that the group utility function Ug is normal becaus the fa not ac fore, the ma each c at ind attrit loose] pretty senta1 issue: or ag sider judgm be a city Where bers (6.7. tiCi] 152 normalized between 0 and 1. Basically, the model (6.8.1) was chosen because of its simplicity and because all the decision makers expressed the fact that they formed their judgments based upon the attributes and not according to the other council members' possible votes. There— fore, the merits and demerits of the two alternatives--namely, build the mall or don't build the mall--were overwhelmingly important for each council member. Thus, their final judgments were basically arrived at independently of each other and considering mostly the values of attributes in this issue. In any case, the city council is a group loosely held together and its members in most of the decisions act pretty much independently of each other mainly because it is a repre- sentative body and only meet at periodic intervals to discuss the issues. Thus, there are minimal pressures, if any, to vote together or against each other and the general public also expects them to con- sider only the pros and cons of the issues and form their independent judgments. Consequently, the model indicated at (6.8.1) was found to be a good approximation for aggregating the decisions of such a group. Thus, the participatory group utility function, UG(u), for the city council, looks like, UG(u) = 1/5 (US + UC + UF + UG + U0) (6.8.2) Where the utility functions in the parenthesis belong to council mem— bers Stell, Czarnecki, Fox, Griffin, and Owen. Substituting formulas (6.6.5) through (6.6.14) into the formulas (6.7.9) through (6.7.13), we obtain the utility function of this par- ticipatory group. membe 153 .The utility functions in the parenthesis belong to council members Stell, Czarnecki, Fox, Griffin, and Owen- Substituting formulas (6.6.5) through (6.6.14) into the formula (6.8.2) we now obtain the final utility function for this participatory group. This function, UG, looks like: l_e—0.0064(Xl+100) l_e-0.0128X2 Us“) = ”5 (0°91 _5—.722—— + 0'75 W _ 0 66 l_e-0.0064(Xl+100) . l_e-0.0128X2 ) ' 0.722 0.722 0.009(X +100) 1-e 1 l + (0.69 _j-05_-_ + 0.28 ' m X2 + o 03 l_e0.009(xl+100) . 1 x ) ° -5.05 100 2 -0. 22 +1 - . + (0 99 l-e 005 (X1 00) + o 91 1—e 0 0244K2 ' 0.648 '- 0.913 90 _EO.00522(X1+100) . l_e-O.244X2 ) ‘ ' 0.648' 0.913 88 l_e-0.00522(Xl+100) + o 681_e-0.0148X2 + (0' 0.648 ‘ 0.722 _ ' + _ — 0 56 1-e O 00522 UG (Mall) since 0.7229 > 0.6624 for the group. This conclusion is pretty much in line with the votes of the group on the issue. Except council member fore, l was ag; cation 155 member Czarnecki, all four members voted against the issue and, there— fore, the group's overall feeling by way of a simple yes or no vote was against the mall. The implications and conclusions on this appli- cation are analyzed in the next chapter. 7.1 SEE The in Chap1 symmetr' in Chap clusion thus no interes utility It member Prefer. counci buildi same w Who v0 SidEre expeci major: WOuld CHAPTER VII CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 7.1 Conclusions The theoretical framework for the application chapter was developed in Chapter IV. In that chapter, hitherto neglected utility theory for symmetric and nonsymmetric participatory groups were developed. Then in Chapter 'V, the same theory in terms of risk was investigated. Con- clusions for both these chapters were given in those chapters and will thus not be repeated. Chapter VI, however, provides us with an interesting real life application of the symmetric participatory group utility model. In this application, the model's predictions for each city council member were completely in parallel with the actual expressions of preferences by a simple yes or no vote. In other words, for all those council members who voted against the mall, their utilities for not building theumfll were higher than those of establishing the mall. The same was also true in the opposite direction for one council member who voted for the mall. It was also interesting to observe that each council member con- sidered attribute X1 as more important than X2, which was really expected. In some of the cases, although they believed that the majority of the public would not support their particular vote, they would still go ahead with their decision. Their tradeoffs between 156 the first and sacrj each att] they stil the deci pressed 1 the city not much Wha for the ranked b an inapp factory not acco ever, by the inte It for the the mall Other as mall. '1 Mr. Lar: first vC ('no' Vc building region a in SUpp( 157 the first and second attribute were almost always in favor of the first and sacrificing from the second. Although the scaling constants for each attribute do not exactly indicate the importance of that attribute, they still give us an idea about how the attributes were compared in the decision maker's minds. One of the council members actually ex- pressed the belief that in decisions like the Dayton—Hudson Mall case, the city council should lead and educate the public even if there is not much support for the particular decisions. What this application shows us is the intensity of the preferences for the group. In section 1.6 of Chapter I, the example of three teams ranked by two judges was discussed. In that problem, it was shown that an inappropriate aggregation procedure may result in very unsatis— factory decision. This was because the intensity of preferences was not accounted for in the aggregation model. In our application, how- ever, by virtue of the measurement of the utilities of every individual, the intensities of their preferences were fully accounted for. It might seem somewhat unrealistic that the expected utilities for the "build the mall" decision (which is .6624) and "don't build the mall" decision (which is 0.7229) are not too far apart from each other as suggested by the four to one vote of the council against the mall. The answer to this apparent disparity lies in the remarks of Mr. Larry Owen, when he made a public statement on the day of the first vote in August of 1977. He said, "I have not made my decision ('no' vote) because of a belief that all of the consequences of building the mall will be negative, either for East Lansing or the region as a whole. In fact, I believe that the competing arguments in support and in opposition to the proposed mall are, when weighed agains reason views the be words, fortun to rev Thus a the ma reflec "elusi that i more I in the alter: clear about three sion v not be tive A this t becaus sectic lem it 158 against one another, quite close. This decision is one about which reasonable women and men can—-and obviously do--differ. The starting . . ."[92] This quotation and the responses given during the inter— views clearly suggested the fact that every council member considered the benefits and disadvantages of the mall rather close. In other words, in most cases the decisions were quite marginal. However, un- fortunately, a simple yes or no type of voting procedure was not able to reveal the true intensities of the decision makers' preferences. Thus although the city council as a whole voted four to one against the mall, the actual preference of the group as a whole was not really reflected by such lopsided voting. The researcher believes that the "elusive" true preference is much better approximated by the model that is used where the overall group utilities for both strategies are more representative of the group's tendencies. If there are two decision alternatives, A and B and five people in the decision group and if two of the group members think that alternative B is extremely inferior to A (i.e., for them A is a very clear choice) and if the other three think that both alternatives are about equally desirable and just for the sake of making a decision all three happen to choose B, then alternative B will be the group's deci- sion with a vote of three to two. In this type of situation, it would not be unreasonable to suggest that the group really favors alterna— tive A in reality, but a simple voting procedure is unable to reflect this true preference. Thus, it seems that group utility analysis, because it is based upon preferential independence assumption (see section 4.83 of Chapter IV), is able to cope with this important prob— lem in a much more satisfactory way than other simple procedures. result Theore cipat( utilfl the g is de the s indic prese there tion final among pato posa prop not part Part gov: 159 In addition to the formulations of Chapter IV, some of the basic results of Chapter V is also directly related to this application. Theorem 5.10.33 of Chapter V relating to the risk aversion in parti— cipatory groups directly applies to this case. Since the multiattribute utility functions are different but all constantly risk averse and since the group utility function has the additive form, the group as a whole is decreasingly risk averse over each attribute. Due to-its complexity, group utility analysis will not replace all the simple democratic voting procedures. However, this application indicates to us the possibility of doing such an analysis as it was presented here, in business firms or in governmental agencies where there are no strict rules or regulations governing preference aggrega— tion procedures for groups. Thus it seems that, for example, for a finance committee of five in a large company responsible for choosing among alternative capital budgeting proposals, a comprehensive partici- patory group utility analysis covering various attributes of these pro- posals would be much more desirable to a simple no or yes vote for each proposal. The utility analysis presented here is really prescriptive and not descriptive. Decision analysis is an aid to decision-making and participatory group multiattribute utility theory as being an integral part of the decision theory, would be very helpful to managers and government administrators in coming closer to the most desirable or "optimum" choice among many possible courses of action. Multiattribute group utility theory, like many other tools of decision analysis, uses many assumptions and limitations. These assumptions, like preferential or utility independence, strategic equivai There verif iI of att tion 0 of att makers utilit Howeve time a would of int it ha; 0n the jecti\ an am; into 5 would and c_' would Darth might mEmb e‘ nativ was t Shoul 160 equivalence and others are essential to build up a tractable theory. There are many real life situations where those assumptions can be verified or be approximated. Limitations related to the construction of attributes, to the number of decision makers, and to the aggrega- tion of probability distributions are to be recognized. As the number of attributes and their complexity grows and as the number of decision makers in the group increases, the participatory group multiattribute utility analysis becomes more and more complex and perhaps less useful. However, if the group size is within manageable limits in terms of time and cost constraints of the analysts, then the group members would possibly benefit a lot from being exposed to the grueling details of introspective utility analysis. During the course of analysis, as it happened in the city council case, the decision makers would improve on the specification of their objectives and how they relate those ob- jectives to certain attributes and their consequences. By way of such an analysis, different parts of the issue at hand would be decomposed into simpler and identifiable components. Since such an analysis would normally occur before the actual decision is made, a better and clearer understanding of the problem by individual decision makers would also result in better and more effective communication among the participatory group members. Furthermore, such group utility analysis might result in a better understanding of differences among the group members and thus would facilitate their resolution. Even new alter- native solutions to the problem can be created this way. One of the areas where the researcher encountered more difficulty was the assessment of multiattribute preferences. The respondent should be given the opportunity to change his/her answers if he/she feels shoul utili in de of gr assum uncer of we diffe very is a] group issue does inter been actue the c tigat applj 161 feels uncomfortable with the initial responses. Sensitivity analysis should always be conducted to check the consistency of responses. In conclusion, the researcher believes that multiattribute utility analysis of participatory groups is one of the valuable tools in decision analysis and it can be put to good use in different types of group decision-making situations. 7.2 Suggestions for Future Research On the theoretical side, more research is needed on the different assumptions that group utility theory is based upon. .Furthermore, the uncertainty aspects of this theory are still a virgin area where a lot of work can be done. Especially ressearch into the aggregation of different probability distributions in group utility theory would be very relevant. On the other hand, more research in the area of risk is also needed. Especially the analysis of risk for nonsymmetric groups under different cases of risk aversion can be one of the first issues to tackle in this area. On the practical side the author, except for the present study does not know of any real life group utility studies where an explicit interpersonal comparison of preferences of different group members has been made. Thus, this effort should be complemented by many more actual group utility analysis under varying circumstances. Especially the cases of full interaction and nonsymmetric groups should be inves— tigated. The theory can benefit a lot from many more practical applications. anaL sent: form decis answe of at of th is th if ta gambl4 econm gambl. membe] index, Prefer menta] Perity 3 APPENDIX QUESTIONNAIRE AND INSTRUCTIONS Utility assessments involve lengthy discussions between the analyst and the respondent. Thus no written questionnaire was pre- sented to the council members, but many questions were asked in one form or another to every decision maker. A detailed explanation of decision analysis, utility assessment procedures, nature of the answers, objectives, attributes and the meaning of different levels of attributes preceded all these questions. The more important ones of these questions are given below in summary form. 1) What is your vote on the mall issue, yes or no? 2) Let us say we have a coin. We flip the coin. If heads, it is the worst value of the index (attribute X1) which is -100, otherwise, if tails, it is the most desirable value, +100. In other words, a gamble can be played for a chance of 50-50 for the best and worst economic situations of East Lansing. You have a choice between this gamble and a definite value of this index for certain. Now as a council member, would you rather pick the gamble or the value of "0" for the index, representing the status quo? In other words, which do you prefer? A gamble with equal chances for ruinous economic and environ- mental conditions and for unlimited economic and environmental pros- perity in East Lansing, or, the status quo? 3) At approximately what level of this index would you be 162 H indif attr: held Supp affe tior a 5( As: indt ext for 50 163 indifferent between the above gamble and the value you have specified as a certain outcome? 4) Now, while you answer these questions, assume that the other attribute (percentage of public support for your yes or no vote) is held constant at, say 50 percent level. Suppose I change your public support from 50 percent to 70 percent. Does this change in any way affect your answer to question 3? 5) Now we have a similar gamble to the one offered in question 2. This gamble involves the value you indicated as a response to ques- tion 3. Suppose you have 50 percent chance to achieve this value and a 50 percent chance to get +100 on the index. This is the gamble. As an alternative to this gamble, you can pick a sure value of the index. Now what is that certain outcome on the index that will make you indifferent between the two alternatives? 6) Now we have another similar gamble. You have equal chance for a value of -lOO and for the value that you indicated as a response to question 3. What is the value of the index between these two extremes that will make you indifferent between that particular value for certain or the gamble given above? 7) If we again change fixed value of the second attribute from 50 to 70 percent, does this change in any way affect your answers to questions 5 and 6? 8) Now we deal with the second attribute, X2. You now have two choices. First is a gamble, with equal chances for no (0.0%) or full (100%) public support. The second choice is 50 percent of public support for certain. Which would you prefer? 9) What is the percentage of support from the people at which leve the Ther ques perc and bute your for1 Come: with attr: 164 level you would be indifferent between that percentage for certain and the gamble described in the previous question? 10) Now while you answer these questions, assume that the level of the first attribute (X1) was held constant at, say, "0." Does shifting this level to, for example, +25 in any way affect your answers to question 9? 11) Now we have again a similar gamble as in question 8. You have two choices. The first is a gamble where you have half a chance of obtaining the percentage of support figure you gave as a response to question 9 and half a chance for a full public support of 100 percent. The second choice is a definite value between these two percentages that you can have for certain for public support. Now what percentage of support should this figure be so that you will be indifferent between the two choices? 12) Again we have a similar gamble, but with different limits. There is now half a chance for the amount of support you indicated in question 9 and half a chance for nosupport (0.0%) at all. What is the percentage of support that will make you indifferent between this gamble and that support you indicate for certain? 13) Now, again suppose that we vary the level of the first attri- bute while you answer questions 11 and 12. Does this in any way affect your answers to these questins? 14) Now there are no longer gambles but levels of satisfaction for the two attributes taken together. Which of the following out- comes do you prefer? An index of +100 for the first attribute coupled with no public support (0.0%), or an index of —100 for the first attribute coupled with full public support (100%)? In other words, whict a (—J In tf of ti betwe to CC perce you 1 supp< coupj on t1 choic word: that to a( or d aSkEH unde: of t] of e: 165 which is preferable, U(+100, 0.0%) or U(-100, 100%)? 15) Assume that you have two choices. In the first one you have a (—lOO) value on the index coupled with full public support (100%). In the second choice you have no public support and a certain value of the index (Xl'). What is this value that will make you indifferent between these two choices? This is the level of X1‘ where U(X ', 0.0%) = UC-lOO, 100%).~ In other words, how much has X1 to increase from -lOO to compensate for the decrease in attribute X2 from 100 percent to 0.0 percent public support? 16) This is the opposite of the above situation. On the one hand you have a choice of an index of (+100) coupled with no (0.0%) public support and on the other you have a certain value of the index (Xl"), coupled with full (100%) public support. 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