MSU LIBRARIES .— ‘ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. l ‘ "it 2 :33 in EFFICIENCY COMPARISONS OF VOTING SYSTEMS WITH STRATEGIC VOTING By Laura M. Hayes A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1987 The 1 effect of different Sl’stems CC Horde SYSt votiT18. in is a Choic Effie theoretica SS’StemS’ a assumptiOn SfflCient, standard V Howev pOpUlatiOn it is Show equilibriu /i; c f/ \ ‘\ "\ v -[l ABSTRACT EFFICIENCY COMPARISONS OE VOTING SYSTEMS WITH STRATEGIC VOTING By Laura M. Hayes The purpose of this dissertation is to investigate the effect of strategic voting on efficiency measures of different multi-candidate voting systems. The voting systems compared include the standard plurality system; the Borda system, a weighted ranking voting system; and approval voting, in which the number of alternatives receiving a vote is a choice variable for the voter. Efficiency measures have already been developed theoretically and estimated via simulation for these voting systems, assuming voters use sincere strategies. Given this assumption, the Borda system is found to be the most efficient, followed by approval voting, followed by the standard voting system. However, a set of sincere strategies for the voting population does not always constitute a Nash equilibrium. It is shown that sincere strategies do converge to a Nash equilibrium as the voting population becomes large. Similarly, as the degree of information the voting population is assumed to have decreases, i.e. the standard error of their estimates of alternatives' total votes received increases, sincere strategies converge to a Nash equilibr voting ; assumpti A s confirms under th simulati 0f the v measures most eff Voting b equilibrium. Thus, for small, sufficiently knowledgeable voting populations, efficiency measures may change with the assumption of strategic voting as opposed to sincere voting. A simulation of the voting systems under consideration confirms that efficiency measures do change significantly under these conditions. In addition, the results of the simulation show that strategic voting can alter the ranking of the voting systems. For one of the two efficiency measures used, the standard voting system is found to be most efficient, followed by the Borda system, with approval voting being the least efficient of the three. This work i Miller, "it This work is dedicated to Michael Hayes, and to Jim and Anne Miller, with thanks for their encouragement and support. iv I would DaVidso: appreci. critici; obtaini: help of invalua' ACKNOWLEDGEMENTS I would like to acknowledge the help of Jack Meyer, Carl Davidson, John Goddeeris, and Larry Martin, with appreciation for their insightful comments and constructive criticism. Thanks are due to Tom Hammond, for his help in obtaining resource materials. I am also grateful for the help of Kathy Esselman, whose editorial comments were invaluable. Introdu Chapter 1.1 Ea 1.2 V0 13 V0 NNN ACON C410) Aha) TABLE OF CONTENTS Introduction Chapter 1 Historical Background 1.1 Early Work 1.2 Voting Systems as Ways of Aggregating Individual Preferences 1.2.1 Impossibility Theorems 1.2.2 Incentive Compatibility 1.3 Voting Systems, Equilibrium, and Individual Motivation 1.3.1 Voting Equilibria 1.3.1.1 Unidimensional Spatial Model (Median Voter Model) 1.3.1.2 Multidimensional Spatial Model 1.3.2 Individual Motivation, or Why Vote? 1.3.3 Randomness in Voting Models 1.3.4 Strategic Voting Chapter 2 Literature Review: Expected Outcomes 2.1 Comparison Measures 2.1.1 Condorcet Efficiency 2.1.1.1 Existence of a Condorcet Winner 2.1.1.2 Condorcet Efficiency and Pairwise Majority Voting 2.1.2 "Effectiveness" or Social Utility Efficiency 2.2 Relationship of Comparison Measures 2.3 Voting Systems and Expected Outcomes 2.4 Varying Other Parameters of the System 2.4.1 Varying Weight Sets 2.4.2 Voter Concordance and Correlation 2.4.3 Other Parameters Chapter 3 A Model for Simulation of Voting Systems With Strategic Voting 3.1 Assumptions of the Model 3.2 Voting Strategies 3.2.1 Sincere Strategies 3.2.2 Optimal Strategies 3.3 An Analytical Example 3.4 Information Conditions vi Ef: 5 -5 Inc Chapter V01 Im; Li: Op1 (”mm ong 5.5 3.5 Sincere Strategies and Nash Equilibria 3.5.1 An Infinite Voting Population 3.5.2 Information Conditions Again 3.6 The Simulation Program and Solving Algorithm 3.7 Restriction to Pure Strategy Equilibria 3.8 The Nature of Equilibria Found Chapter 4 Results 4.1 Theoretical Values 4.1.1 Sincere Voting 4.1.2 Strategic Voting 4.2 Social Utility Efficiency Rankings with Sincere and Strategic Voting 4.3 Condorcet Efficiency Rankings 4.4 Condorcet Efficiency and Social Utility Efficiency 4.5 Forecasted Values 4.6 Incomplete Information Chapter 5 Discussion and Suggestions for Further Research 5.1 Efficiency Measure Changes with Strategic Voting 5.2 Implications of the Results 5.3 Limitations on Nash Equilibria Found 5.4 Optimality Properties of Comparison Measures 5.4.1 Choice of the Level of a Pure Public Good 5.4.1.1 Social Utility Efficiency and Optimality in the Provision of Pure Public Goods 5.4.1.2 Condorcet Efficiency and Optimality in the Provision of Pure Public Goods 5.4.2 Choice Along a Pareto-Frontier 5.4.2.1 Social Utility Efficiency and Choice Along a Pareto-Frontier 5.4.2.2 Condorcet Efficiency and Choice Along a Pareto-Frontier 5.4.3 Implicit Equity Considerations 5.5 Suggestions for Further Research 5.5.1 Costs of Voting 5.5.2 Other Equilibria: The Competitive Solution 5.5.3 Social Welfare Functions 5.6 Conclusion Appendix A: Sincere Voting as a Nash Equilibrium with an Infinite Voting P0pulation A.1 Borda System A.2 Approval System Appendix B: Voting System Simulation Programs vii 101 106 112 115 119 121 122 125 126 129 129 130 134 134 135 135 136 136 136 140 143 143 144 144 145 146 Appe List Appendix C: Numerical Efficiency Estimates 160 Appendix D: Regression Results 166 Appendix E: Notes to Text 170 List of References 172 viii Tab. Tab. Tab} Tab} Tabl Tabl Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table N 03030031010 boom #44 LIST OF TABLES Voter Turnout as Percentage of Minority Registration Possible Preference Orders, Sincere Votes, and "Sophisticated" Votes on a Saving Amendment and Final Passage (Amendment Expected to Pass) Probabilities of the Existence of a Condorcet Winner for Various Numbers of Alternatives Condorcet Efficiencies for Various Voting Systems (%) Chamberlin and Cohen: Proportion of Condorcet Winners Selected - Impartial Culture Expected Correlations Among Voter Dimensions Chamberlin and Cohen: Proportion of Condorcet Winners Selected - Spatial Model Merrill: Proportion of Condorcet Winners Selected (X) Merrill: Social Utility Efficiency Outcomes for the Standard and Borda Voting Systems with Sincere Voting Possible Strategies and Outcomes Given Others' Strategies Outcomes for the Standard and Borda Voting Systems with Strategic Voting Possible Strategies and Expected Utility for Voters 1 and 2, Given Voter 3’s Strategy Equilibrium Outcomes of the Standard Voting System When One Specific Voter's Strategy is Known. Equilibrium Outcomes of the Standard Voting System When Two Voters' Strategies are Known. Probabilities of Voters' lst, 2nd, and 3rd Choices Being Chosen by the Standard Voting System Given the Information Structure of the Game. Strategies and Possible Outcomes of a ix 31 35 38 45 50 51 52 52 55 67 68 70 71 72 72 73 Table Table 1 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table 0' GOOD .10. .11. .12. HH OJN Three-Alternative,Three Voter Election. Vote Vectors and Corresponding Probability Vectors for a 3 Alternative, 3 Voter Election An Example of Preferences for Which a Pure Strategy Equilibrium is Not Found when Voters are Taken in a Specified Order Sequence of Strategy Changes Produced by the Solving Algorithm Expected Utility, Preference Orderings, and Sincere Strategies of Voters Using the Standard Voting System. Regression Results for Strategic Condorcet Efficiency Incomplete Information Efficiency Measures for 3 Alternatives and 3 Voters Summary Statistics for the 3 Voter 3 Alternative System When All Nash Equilibria are Found Numerical Efficiency Estimates Sincere Social Utility Efficiency Regression Results Sincere Condorcet Efficiency Regression Results Strategic Social Utility Efficiency Regression Results 79 80 85 85 87 112 120 127 156 166 167 168 FigL Figu Figu Figu. 183) F180: Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure OJNH LIST OF FIGURES Sequential Voting Equilibrium with Reintroduction of Issues. The Game Tree for Pairwise Majority Voting with Amendment. Possible Agendas and Outcomes for Pairwise Majority Voting Possible Strategies in the Standard Voting System with 3 Alternatives: Convex Subset of RB. Social Utility Efficiency for the Standard Voting System with 3 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Standard Voting System with 4 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Standard Voting System with 5 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Standard Voting System with 6 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Approval Voting System with 3 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Approval Voting System with 4 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Approval Voting System with 5 Alternatives: Strategic, Sincere, Limit Values Social Utility Efficiency for the Approval Voting System with 6 Alternatives: Strategic, Sincere, Limit Values xi a and and and and and and and and 26 34 38 83 94 95 95 96 97 97 98 98 Fi Fi Fl. Fig, Fig: ‘ Fig! Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure fihhhhfihhhbhkbhofikfihhhk .9. .10. .11. .12. .13. .14. .15. .16. .17. .18. .19. .20. .21. .22. .23. .24. .25. .26. .27. .28. .29. .30. .31. .32. .33. Social Utility Efficiency for the Borda Voting System with 3 Alternatives: Strategic, Sincere, and Limit Values Social Utility Efficiency for the Borda Voting System with 4 Alternatives: Strategic, Sincere, and Limit Values Social Utility Efficiency for the Borda Voting System with 5 Alternatives: Strategic and Limit Values Social Utility Efficiency for the Borda Voting System with 6 Alternatives: Strategic and Limit Values Sincere Social Utility Efficiency: 3 Alternatives Strategic Social Utility Efficiency: 3 Alternatives Sincere Social Utility Efficiency: 4 Alternatives Strategic Social Utility Efficiency: 4 Alternatives Sincere Social Utility Efficiency: 5 Alternatives Strategic Social Utility Efficiency: 5 Alternatives Sincere Social Utility Efficiency: 6 Alternatives Strategic Social Utility Efficiency: 6 Alternatives Sincere Condorcet Efficiency: 3 Alternatives Strategic Condorcet Efficiency: 3 Alternatives Sincere Condorcet Efficiency: 4 Alternatives Strategic Condorcet Efficiency: 4 Alternatives Sincere Condorcet Efficiency: 5 Alternatives Strategic Condorcet Efficiency: 5 Alternatives Sincere Condorcet Efficiency: 6 Alternatives Strategic Condorcet Efficiency: 6 Alternatives Actual and Estimated Condorcet Efficiency: Standard Voting System Actual and Estimated Condorcet Efficiency: Approval Voting System Actual and Estimated Condorcet Efficiency: Borda Voting System Sincere Social Utility Efficiency: 7 Alternatives (Estimated) Strategic Social Utility Efficiency: 7 Alternatives (Estimated) xii 99 100 100 101 102 103 103 104 104 105 105 106 108 108 109 109 110 110 111 111 114 114 115 117 117 Figure 4.34. Sincere Condorcet Efficiency: 7 Alternatives (Estimated) 118 Figure 4.35. Strategic Condorcet Efficiency: 7 Alternatives (Estimated) 118 Figure 5.1. Mean Expected Utility and Corresponding Levels of the Public Good (G) Produced. 132 xiii C00 the INTRODUCTION It is often necessary to make decisions which will affect a group of individuals. Arrow [1] in his General Possibility Theorem, proved the impossibility of constructing a social welfare function (without using cardinal utilities) which fulfilled the following conditions: (1) unrestricted domain; (2) consistency with the Pareto principle; (3) independence of irrelevant alternatives; and (4) nondictatorship. Certainly if such a social welfare function could be constructed it could be used to determine which of the possible alternatives to choose. Despite the fact that no such social welfare function exists, the decisions remain to be made. In lieu of using a social welfare function with these characteristics, voting systems are often used. There are many different voting systems to choose from, and different voting systems may produce different outcomes. The voting systems considered here are the standard plurality system, the Borda system, and the approval voting system. The standard voting system is the one commonly used in the United States, where each voter casts one vote for the alternative of his choice. The Borda system is a weighted ranking system in which alternatives are ranked and the in. ind For alp? The Frel Pro: 180( 800: Cur] metl 0the eQui 2 then assigned points according to their rank. For example, in an election with five alternatives (A, B, C, D, and E) an individual would rank the alternatives from first to last. For simplicity, let the alternatives be ranked in alphabetical order. Then points are assigned as follows: Alternative Rank Points Assigned A 1 4 B 2 3 C 3 2 D 4 1 E 5 0 The Borda system was presented for the first time to the French Academy in 1784 by Jean—Charles de Borda, and was promptly adopted by the Academy. It remained in use until 1800, when it was challenged by a new member and modified soon afterward. The new member was Napoleon Bonaparte.1 Currently, a modified Borda system is used as the selection method for the Heisman trophy winner, as well as for several other athletic awards. In the approval voting system, voters are allowed to vote for as many of the alternatives as they find acceptable or approve of. In the example above, a voter could cast from zero to five votes, although zero and five are equivalent strategies in the sense that neither affects the outcome of the election. Approval voting was first discussed by S. Brams in 1976 [17], and there have been efforts to have this system adopted for use in the Massachusetts primary. al it as re 2'8 The 3 The Institute of Management Sciences tested the approval voting system against the standard voting system in its 1985 annual elections. 85% of the 1,851 voters, or 1,579 voters returned the test ballot. Members were also asked to rank the candidates, and 82% provided at least some rankings. Three elections were used for comparison. The results of the first election are presented here. Candidate Official Approval Vote Vote A 166 417 B 827 1038 C ___208_ 1828 2363 (1,562 voters) The outcome of the election is C under the standard voting system, while B wins under approval voting. This difference is caused by the pattern of second choices. There is no scope for information about second choices in the standard voting system, but some of this information is used in the approval voting system. lst choice 2nd vote A B 36% A C 23% B C 27% C B 45% As shown above, among A's followers, more approve of B than C (36% to 23%), and more of C's followers approve of B (45%) than B's followers do of C (27%). Using the ranking data submitted, Little and Fishburn [87] extrapolated to obtain the result of a hypothetical pairwise race between B and C. Interestingly, the expected outcome of such a race is a tie, In I‘G‘ 4 with both B and C obtaining 914 votes. Clearly, the choice of voting system used impacts directly on the outcomes achieved. The question now becomes one of determining which voting system is "best," and the criterion which should be used in making this determination. A brief outline of the dissertation is presented here. In chapters 1 and 2, the literature on voting systems is reviewed. The literature focuses on three major areas: 1) voting systems as ways of aggregating individual preferences, and their characteristics, e.g. Arrow's General Possibility Theorem, work on incentive compatibility; 2) how voting systems work in terms of individual motivation and equilibrium: voting equilibria, and why individuals vote; and 3) comparisons of voting systems in terms of expected outcomes. Chapter 1 outlines the historical background of voting system research, while Chapter 2 defines the comparison measures for voting systems and reviews voting system comparisons in terms of expected outcomes. Chapter 3 presents the formal model used for simulation as well as investigating some of the implications of the model, such as equilibria found. Also presented is a discussion of when sincere strategies constitute a Nash equilibrium. Chapter 4 presents the results of simulations run under the complete information assumption and an intermediate level of 5 information. A discussion of the results and their policy implications is presented in Chapter 5, along with possible extensions and areas for further research. VO‘ th. 3P} 8x; st: COL; 1.1 (3033 CHAPTER 1 HISTORICAL BACKGROUND Approaches to the study of voting systems vary widely. The earliest work, beginning with Jean-Charles de Borda in 1781 [11] and continuing through the 19th century, appears for the most part to be a continuing ideological debate on the subject. Later work can be categorized into three major areas. The first of these focuses on voting systems as a means of aggregating individual preferences and the characteristics of the aggregation process. The second looks rather at individual motivation and equilibria in a voting system, usually one specific voting system. The third area, which can be characterized as a strictly modern approach, compares voting systems in terms of outcomes or expected outcomes. A great deal of the literature falls strictly into one class or another, although there is of course some work which crosses these lines. 1.1 W Jean—Charles de Borda's work [11], the earliest commonly cited on voting and voting systems, begins with an example to show that the "single vote“ (the standard voting system), may select the "wrong" candidate. In this example, he makes implicit use of the Condorcet criterion, showing that the standard voting system may select a candidate who can be beaten by another candidate in a pairwise race. Borda then shows that this "defect“ can be remedied either by his method of ranking or by pairwise voting. He defines hi ac eq goc def to V0! eXp: eXan 7 his method of ranking as giving points to each candidate in accordance with their rank on a preference scale, which is equivalent to assuming a linear utility function for voters. During the same period, Condorcet [26],[118] discussed the "paradox of voting" and internal consistency of social choices. Condorcet motivated his work as follows: "...it is in the interest of those who dispose of the public power to employ that power only to sustain decisions that conform to the truth, and to give, to the representatives they have charged to decide on their behalf, rules which guarantee the goodness of their decisions."2 He focuses on how to determine the best rules by applying the laws of probability to the voting process. Condorcet's own description of his work explains much more fully his reasoning: "...we shall first suppose assemblies composed of voters possessing equal soundness of mind and equal enlightenment. We shall suppose that none of the voters influences the votes of others and that all express their opinion in good faith. Supposing then that one knows the probability that the opinion of each voter will be in conformity with the truth, the form of the decision, the hypothetical majority and the number of voters, one seeks to discover (1) the probability of not having an decision contrary to the truth; (2) the probability of having a true decision; (3) the probability of having any decision (true or false); (4) the probability that a decision that one knows to have been taken will be true rather than false; and, finally, the probability of this decision when the majority by which it has been taken is known. Such is the subject of the first part of this book.“3 In the second part of his work, he deals more explicitly with the standard voting system. He uses an example in the same manner as Borda to show that the Obj: 8 standard voting system "can result in a decision really contrary to the opinion of the majority.“4 "...to have a majority decision that merits confidence, it is absolutely necessary to reduce all opinions in such a way that they represent in a distinct manner the different combinations that can arise from a system of simple propositions and their opposites; ...every complex proposition is reducible to a system of simple propositions, and that all the opinions that can be formed in deliberating upon this proposition are equal in number to the combinations that one can make of these propositions and those contradicting them.”5 Pairwise comparisons of candidates were to be used to determine a social ranking, and the Condorcet criterion, although used implicitly by de Borda, was made explicit for the first time. The candidate (or alternative) which obtains a majority in a pairwise race with each other candidate (or alternative), now called the Condorcet winner, has the highest social ranking and ought to be chosen. Condorcet showed, however, that pairwise comparisons would not necessarily give a social preference order which was internally consistent, foreshadowing Arrow's work. However, he suggested that the propositions be taken in successive order with the size of the majority, and "as soon as these propositions produce a result, it should be taken as the result, without regard for the less probable decisions that follow."3 The third part of his work discusses the probability of obtaining an inconsistent social ordering and represents the first attempt to estimate the frequency of the paradox of voting. Given a set of n candidates, there are n! sets of consistent social rankings. If each candidate is paired 9 with each other candidate, then there are (1/2)n(n:1) pairs, i.e. candidate A vs. candidate B, candidate A vs. candidate C., etc., which is equal to the number of combinations of n things taken two at a time. In each of these pairings, a choice must be made between the two candidates. Therefore, 2(1/2)n(n-1) gives the number of possible social ‘preference profiles'. This minus n!, the set of internally consistent pairings, is the number of inconsistent preference orderings, and the limit of the percentage of inconsistent social orderings, 2(1/2)n(n-1) - nl 2(1/2)n(n-1) is equal to one as n-+m. Condorcet's work does not make any obvious assumptions about individual voter preferences, except that given two candidates, any voter is equally likely to vote for either. He does not require that an individual’s vote be consistent with a preference ordering. In 1795, LaPlace essentially duplicated Borda's method of ranking using a different line of reasoning. He assumed that the "merit“ attributed on average to candidates was linear, similar to Borda, and that the candidate who ought to be elected is the one to whom the most merit is attributed by the entire group of voters. Interestingly, the merit attributed on average to candidates will be linear, as will individual expected utilities for candidates by rank, if all voter utilities are drawn from an identical uniform distribution. The “merit“ discussed by LaPlace is 10 the voter's marginal rate of substitution or ratio of exchange of the candidate for money. Other early work was produced by Hare, Nanson, Galton, and Dodgson, and contained the same types of arguments. The most extensive review of this work is contained in Black [7]. A more rigorous approach did not appear until Hotelling's work. 1.2 W Preferences 1.2.1 Impassihility—Theereme Work in this area has focused on the incompatibility of specific characteristics in an aggregation procedure. The seminal work, Arrow's General Possibility Theorem [1], showed the incompatibility of 1) unrestricted domain on (ordinal) preferences; 2) consistency with the Pareto principle; 3) independence of irrelevant alternatives; and 4) nondictatorship. Zeckhauser's [145] explanation of these conditions is clear and concise. "(1) The procedure must include all logically possible combinations of individuals' orderings. (2) It must lead to Pareto-optimal outcomes. (3) The choice between any two alternatives cannot be influenced by the presence or nonpresence of a third alternative. (4) No individual can always secure his choice regardless of the presence of others."7 Arrow proved that there is no aggregation procedure (social welfare function) which simultaneously fulfills these conditions. Condition 2 is simply that if all individuals prefer an alternative x to 11 an alternative y, or are indifferent between them, with at least one individual strictly preferring x to y, then x is socially preferred to (Pareto dominates) y. Any alternative y for which an alternative x can be found which fulfills this condition is not an acceptable outcome. Condition 3, independence of irrelevant alternatives, is the requirement that the social ranking between any two alternatives be independent of any other alternative. In essence, this limits us to pairwise comparisons of alternatives, as in Condorcet’s method, and implicitly accepts the Condorcet criterion. However, the General Possibility Theorem shows that if we limit ourselves to using pairwise comparisons, then any aggregation procedure which is to be used for all preference profiles (unrestricted domain) is either inconsistent with the Pareto principle (some outcomes will be Pareto-dominated), or dictatorial. Arrow's work was followed by many attempts at relaxing his requirements in order to find a set of compatible conditions with little success. Expansion and comment (e.g. Sen [124],[125],[127], Plott [107],[108]) provided insight into Arrow's result, but no progress in solving the problem of social choice. To clarify the issue, the problem needed to be stated in a different form. Gibbard [58] did just that: instead of referring to a social welfare function, he looked at the problem in terms of a game form. A game form, in Gibbard's terms, is "...any scheme which makes an outcome depend on individual actions of some 12 specified sort...strategies. A voting scheme, then, is a game form in which a strategy is a profession of preferences..."8 He also makes use of the term ‘straightforward’ to mean a game form for which all players, for every preference profile, have a dominant strategy. A strategy is dominant for an individual player if, given any set of strategies of the other players, no other strategy available to the player will produce an outcome preferable to him. Using these definitions, Gibbard proved that every straightforward game form with at least three possible outcomes is dictatorial, and every voting scheme with at least three outcomes is either dictatorial, or can be manipulated by an individual.9 Satterthwaite [120] independently made the same contribution, although his terminology differs somewhat. Instead of straightforwardness, he looks at strategy— proofness, which in his work corresponds to Arrow's independence of irrelevant alternatives and Pareto conditions for social welfare functions. He showed that all strategy-proof voting procedures are dictatorial. Interestingly, these results break down if lotteries over alternatives are allowed as outcomes of a social choice function (Gibbard [59]). However, Gibbard proved that all strategy-proof decision schemes are either random dictatorships, pairwise majority rule over a random pair, or a system which chooses randomly between the first two. Unfortunately, either method violates one of Arrow’s CO fi Wh CO di: So] of to 0V4 Pl": the Pl" le. Er Se in fr Ch ra 13 conditions, which is where the alteration in terminology, at first appearance eminently useful, comes back to haunt us. When randomness is introduced, strategy-proofness no longer corresponds to Arrow’s second and third conditions. A final work in this area is discussed because of the direct relevance it bears on this work. Postlewaite and Schmeidler [109] considered social choice functions in terms of (first-degree) stochastic dominance. "A person is said to prefer in the stochastic dominance sense one lottery- over-outcomes over another lottery-over—outcomes if the probability of his (at least) first choice being selected in the first lottery is greater than or equal to the analogous probability in the second lottery, the probability of his at least second choice being selected in the first lottery is greater than or equal to the analogous probability in the second lottery, and so on, with at least one strict inequality."1° Individuals, assumed to know the relative frequency of (ordinal) preference profiles for two social choice functions (which may include an element of randomness) can compare the social choice functions in terms of stochastic dominance. If a social choice function F stochastically dominates a social choice function G for all individuals in a society, F stochastically dominates G socially. This implies ex ante Pareto efficiency of F over G. Postlewaite and Schmeidler comment that "Arrow's Pareto principle, which is ex post, should be implied by a reasonable notion of ex ante efficiency in a model which rn C: Q: 14 admits such evaluations."11 Their main result is that for more than 3 voters and alternatives, there does not exist a social choice function which is simultaneously Pareto undominated (ex ante efficient) and straightforward.12 That is, a social choice function which is ex ante efficient in the stochastic dominance sense will present individuals in the society with situations in which misrepresenting their preferences (as a strategy) dominates their sincere strategy of truthful revelation of preferences. These major contributions to the social choice literature provide a background for comparisons of voting systems, but do not provide any positive criteria which can be used for comparison because of the incompatibility of desired characteristics. If these characteristics were compatible, a social welfare function could be constructed that would specify the “correct“ choice for every social choice situation. 1.2.2 lneenfixe_flemnatihilitx The concern with strategy-proofness or manipulability has been addressed from another viewpoint, that of incentive compatibility. In this line of research, attempts have been made to construct voting systems which are incentive— compatible: truthful revelation of preferences is a dominant strategy in an incentive-compatible mechanism. This emphasis on incentive compatibility is due in large part to the ‘free-rider problem' which is a consequence of the existence of pure public goods (the 15 classic example is national defense). The main characteristics of a pure public good (Samuelson [119]) are joint consumption and nonexcludability. Joint consumption is the property that all members of the consuming body for this good benefit from its production (although not necessarily equally), without preventing other consumers from benefiting or reducing the benefits available to them. Nonexcludability is just that: individuals cannot be prevented from enjoying these benefits. The problem is to determine the Pareto-optimal level of a pure public good to be produced. The condition for Pareto-optimal production of a good is that marginal benefit be equal to marginal cost. Since marginal benefit is distributed across the consuming body, the marginal benefit for one unit of a pure public good is the sum of marginal benefits for all consumers. The level of the pure public good should be chosen such that the sum of marginal benefits across consumers is equal to the marginal cost of production. The difficulty lies in determining what the sum of marginal benefits across consumers is for different levels of production. Generally, individuals would be asked to provide their marginal benefit curve. However, the method of financing production of the pure public good influences the information provided. If individual marginal cost (the marginal tax rate) is zero over the level of production of the good (total cost is constant), each individual has an incentive to overstate his marginal benefits at each level of the pure public good, 16 which will lead to overproduction of the good and a misallocation of resources. If, on the other hand, individual marginal cost is set to correspond with stated marginal benefit, individuals have an incentive to understate marginal benefits in order to reduce their marginal cost, which leads to underproduction of the good. Because of this difficulty, attention focused on the formulation of a direct mechanism which would induce truthful revelation as a dominant strategy. Dasgupta, Hammond, and Maskin [33] review the major results of this approach. They discuss general results on incentive compatibility in the implementation of social choice rules. Their discussion involves the use of a "planner" to implement the social choice rule; however, a “planner" is not necessary to their discussion except as a pedagogical tool. The general problem is approached as follows: A social choice mechanism depends on signals from the individual agents to implement the social choice rule. It is assumed that each individual agent sends his own signal. The mechanism is then a rule which specifies a social state for each list of signals sent by the individual agents. It is assumed that each agent knows the precise form of the mechanism being used. Then each agent realizes that he is involved in a game, because the outcome of the mechanism depends on the signals which he and all the other agents send. More precisely, this is a "game form,” in which there is a fixed set of strategies, consisting of signals, and in 17 which the outcomes of these strategies are known to all “players." It is then assumed that the players in this game form, who are the individuals in the society, reach some kind of equilibrium which depends on their true characteristics - in particular, their preferences. The mechanism generates a particular social state given these equilibrium signals. "Presumably, one wants this social state to be in the social choice set given the agents' true characteristics - i.e. to be something the planner might have chosen had he known these characteristics right from the start. ...The basic problem, then, is to devise a game form which always has at least one equilibrium, and whose possible outcomes in equilibrium all belong to the appropriate social choice set for the individuals' true characteristics. A mechanism (or game form) with this property is said to implemenl.the social choice rule."13 Dasgupta, Hammond and Maskin discuss mechanisms which are individually incentive compatible, both direct and indirect. A direct mechanism is one where the agent's signal is a characteristic: preferences, endowments, etc., relevant to the economic decision to be made. In contrast, with an indirect mechanism, agents' signals "may be quite arbitrary, without any obvious economic significance."14 Such mechanisms can be and have been found, such as the Clarke tax [24]. However, as Dasgupta, Hammond, and Maskin point out, "the papers which find straightforward mechanisms restrict themselves to rather special economic environments. 18 Either the preferences are special, (Clarke [24], Green and Laffont [61], Groves and Loeb [65]) or there is a large economy in which no one individual's lie can significantly affect the overall outcome (Hammond [66], Roberts and Postlewaite [116])."15 They then present their versions of impossibility theorems, which extend Arrow's work. First, in any "rich economic environment"13 (e.g. unrestricted domain of ordinal preferences), any Pareto optimal single valued social choice rule which can be truthfully implemented in dominant strategies is dictatorial. Secondly, in any "rich economic environment," any Pareto-optimal single—valued social choice rule which can be implemented in Nash strategies is dictatorial. ,This follows naturally from their proof that in a rich economic environment, a single-valued social choice function which is implementable in Nash strategies is truthfully implementable in dominant strategies. However, this does not mean that the task is hopeless. All that this implies is that a non-dictatorial Pareto optimal single-valued social choice rule cannot be ‘implemented' in Nash strategies. This means that the use of the Nash equilibrium concept implies that all possible outcomes in equilibrium do not belong to the appropriate social choice set for the individuals' true characteristics. However, recall from the previous section that straightforwardness (truthful implementation in dominant strategies) is inconsistent with ex ante efficiency in the 19 stochastic dominance sense. Postlewaite and Schmeidler's result is that without restricting preferences, ex ante efficiency comes at the cost of straightforwardness. 1.3 WWWM A different approach to voting systems is to look at specific parts of a system. How are voter preferences formed? What are admissible strategies? Finally, what is (are) the equilibrium outcome(s)? 1.3.1 Wiring: One branch of this literature concerns itself with the equilibrium outcome(s) of specific voting systems. Different assumptions about the restrictions on formation of voter preferences account for the differences in outcomes, but the models are set up in essentially the same way. The most famous of these is the median voter model. 1.3.1.1 UnmimeneimaLfinatialJedeuMediaLVeteLMedeu The spatial theory of voting has a long and distinguished history. Black [7] states that "Galton (1907) notices the property of the median optimum when the variable under consideration is measurable (provided the voters' preference curves can be taken as single-peaked)."17 However, a close reading of his citation from Galton reveals that what Galton noted was the equilibrium property of the median.18 The impetus to the approach must lie with Hotelling [74] and Smithies [135], who showed the existence of a spatial location equilibrium in a model where producers of goods must choose a location given the existence of posi‘ Galtc of t] majO' exte: voti pref if, vote dire "Pea Sing POin COnt this SYSt. maxil impl. aVai; to be indix PrgdL “oulc rate SUE 20 positive transportation costs. Their work, along with Galton's, inspired Black to prove the equilibrium properties of the median position in pairwise majority voting. Black [6] essentially limited his analysis to pairwise majority voting, although in a related work he includes an extensive discussion of the literature including alternative voting methods. He first defines single-peakedness of preferences. Preferences of a society are single-peaked, if, for some arrangement (order) of alternatives, each voter's utility curve over alternatives "changes its direction at most once, from up to down.“19 In this case, the highest point on an individual's utility curve is his "peak preference." It is important to point out that single-peakedness of preferences does not imply a ‘satiation point.’ The median voter model is ordinarily used in the context of decisions on the production of public goods. In this context, given a method of financing production (tax system), the individual is solving a constrained maximization problem based on his resources (income). This implies an optimal level of consumption of each good available, including the public good on which a decision is to be made. It is not unreasonable that a graph of the individual's total utility as a function of the level of production of the public good would be single peaked (as it would be, for example, if there were a constant marginal tax rate for increments of the public good and the usual assumptions on individual utility were made). pref of t voti have will pref: doma: eqUil conte or $1 Outco leVel Under 21 The median voter is the individual with the median peak preference. Black's main result is that the peak preference of the median voter, in this case, is the pairwise majority voting equilibrium. As Galton deduced, anything less will have a majority in favor of increasing it, and anything more will have a majority in favor of decreasing it. However, preferences must be single-peaked, and the unrestricted domain used by Arrow will cause nonexistence of an equilibrium point for some cases in this model. Bowen [15] extended Black's result to an economic context. He showed that under certain conditions plurality or simple majority voting would produce a Pareto optimal outcome in equilibrium, when the decision to be made is the level of production of a pure public good. The conditions under which this holds are: (1) There is complete and sincere participation of the voting population; all voters in the voting population do vote, and they vote sincerely, i.e. in correspondence with their true preferences. (2) The cost curves for production of the public good are known. (3) The public good is produced under conditions of (eventually) nondecreasing marginal cost. (4) The cost of the public good is divided equally across the population, or there are equal tax shares. (5) The marginal rate of substitution of the public good for money is normally distributed across the population at any level of the public good. (6) The public good is nonexcludable and equally ava def pea} publ marl are the Prel acrc dist and out; OCCu has max: Pref that COSt marg bene COSt eQui Dubl Blae 22 available to all voters, corresponding to Samuelson's definition of a pure public good. Conditions 2, 3, 4, and 5 imply that there are single— peaked preferences for all members of the population for the public good, if it is a normal good with a decreasing marginal rate of substitution for money. Since preferences are single-peaked, each voter has a most preferred level of the public good, and condition 5 implies that the most preferred level then has a continuous normal distribution across the population. The point of maximum density of this distribution would be the simple plurality voting winner, and the outcome under a simple plurality system would be the output of the public good for which this maximum density occurs. Since the most preferred level of the public good has a continuous symmetric distribution, the point of maximum density coincides with both the median and mean most preferred level. Since each voter's most preferred level is that at which his marginal benefit is equal to his marginal cost, this implies that mean marginal benefit equals mean marginal cost, and therefore the sum of the marginal benefits across the population will be equal to the marginal cost of production of the public good. In other words, the equilibrium point of the simple plurality system is Pareto-optimal. With simple majority voting over increments of the public goods, the outcome will be the same. As shown by Black, the median most preferred level (median peak 23 preference) is the equilibrium point. However, in Bowen's model, the median coincides with the mean, and Pareto- optimality results. Therefore simple plurality voting or simple majority voting will produce the optimal level of the public good if the conditions postulated by Bowen are fulfilled. It should be noted here that any continuous symmetric distribution of peak preferences for which the point of maximum density is both the mean and the median will produce this same result. 1.3.1.2 Mulfidimensiena1_Snafia1_Medel The multidimensional spatial model, developed by Enelow and Hinich [40], is a simple extension of Bowen and Black's median voter model. The major difference is that one dimension is no longer thought sufficient to describe how individuals' preferences are formed. An issue may have more than one dimension, and each dimension in this model is a dimension in the “issue space."‘ The justification for this assumption is the prevalence of ‘package votes,’ such as a decision on the level of two or more public goods at once. The peak preference level of the unidimensional model is described here as a voter's ideal point in the issue space. However, preferences are again assumed to be single-peaked. "The key element of spatial models is the relationship between preference and distance. ...The weighted Euclidean distance between y and z is defined to be "y - zNa = [a11(y1-zi)2 + 2a12(y1-21)(y2-zz) + azz(y2-zz)2]1/2, where a11>0, azz>0, and (a12)20 24 for all yiz. ...Weighted Euclidean distance defines a symmetric preference rule...the closer (in weighted Euclidean distance) an alternative is to his ideal point, the more he prefers it.“2° a12=0 implies separability of preferences; that is, the most preferred level in one dimension is independent of the most preferred level in all other dimensions. Given this mechanism for formation of preferences, and again assuming, with Black and Bowen, complete and sincere participation of the voting population, determination of the equilibrium is made. In the classic spatial model, it makes a great deal of difference whether ‘dimensions' are voted on sequentially or simultaneously. Unless all voters' preferences are separable, the equilibrium outcome will differ. Separability of preferences along with sequential voting implies Pareto- optimality of the equilibrium outcome, just as in the unidimensional model. If preferences are not separable, however, sequential voting produces differing outcomes depending on the order in which dimensions are voted on. In essence, this is because in all but the first election, voters take the values of public goods decided on in previous elections as given. In any case, results of ‘secondary' elections may be Pareto-optimal given the result of the first election, but the converse does not hold, as shown in Figure 1.1. This in turn implies that the overall results of the system are not Pareto-optimal. 25 median net marginal benefit median net marginal benefit of x1 given x2 median net marginal benefit of x: given x1 x? & / Pareto-optimal production: x1=xv, x2=xp 1 2 Figure 1.1. Sequential Voting Equilibrium with Reintroduction of Issues. If voting on x1 takes place first, the level selected will be xi. A vote on x2 then selects xi. If x1 were reintroduced, the level chosen would be xi', etc. 26 Separability or nonseparability of preferences does not matter with respect to Pareto-optimality of outcomes if a dimension can be re-introduced into the process. In Figure 1.1, reintroduction of the first dimension after the second has been decided on will move the outcome towards the Pareto-optimal point, and if this process is continued, the limiting equilibrium point is indeed Pareto-optimal. However, if preferences are separable but voting on dimensions is simultaneous, the outcome is not necessarily the peak preference point, which corresponds to the median ideal point on each dimension. "...once both issues are voted on simultaneously,...xmed can be beaten in a majority contest, and furthermore there may exist no proposal that cannot be beaten...this result is a general problem for the multidimensional spatial model."21 A dominant point only exists if there is a point in the multidimensional space which is a median in all directions. "If a dominant point exists, all that we are guaranteed is that no other point can beat it in a pairwise contest. This does not mean that a dominant point beats all others.“22 A dominant point receives at least as many votes as any other point in a pairwise contest. In other words, some other point may tie with the dominant point in a pairwise contest. However, if a point y is closer to Xmed than 2, then y beats z in a majority contest. This suggests that the limiting equilibrium point is the dominant point xmed. 27 In the absence of a dominant point, the outcome of a sequence of pairwise votes depends upon the agenda. "It is possible to reach literally any point in the space through same sequence of votes, pairing each previously winning proposal with some new proposal that a majority prefers until the chosen point is finally reached.“23 Thus the spatial model, in the absence of a dominant point, has no implications for outcomes without a model of agenda control, which is beyond the scope of this work. 1.3.2 WW2. "Much theorizing about the utility of voting concludes that voting is an irrational act in that it usually costs more to vote than one can expect to get in return."24 This includes the work of Downs [37] and Tullock [141]. If we are to apply a rational choice perspective, the expected return from voting should be at least equal to the cost or expected cost of voting in order to induce voters to participate. The expected return is the difference in utility between the voter's preferred alternative and another alternative, times the probability that the voter is decisive (the probability that his action in voting causes the change in outcome). If expected return exceeds expected cost, it is rational to vote; if not, voting is an irrational act. Since in any election where the voting population is large, as in the 0.8., the probability of being decisive is very small (Riker cites 10-8), the difference in utility must be extremely large in order to 28 compensate for a relatively low cost of voting. The general conclusion is that voting is not a rational act. Some attempts to modify this conclusion have postulated direct benefits from voting as opposed to its expected return. Palfrey and Rosenthal [102] critique this approach, commenting that "...many observations are inconsistent with the proposition that an individual's net cost of voting...is anywhere near constant. The greater turnout in presidential than in off-year elections and the greater turnout in contested than in uncontested elections belie any simple citizen-duty story. Of course, citizen duty could be rescued by arguing that there is a greater sense of duty in presidential and contested elections, but such logic is difficult if not impossible to test."25 Another approach is Ferejohn and Fiorina's [42] minimax regret model. They contrast voting as decision-making under risk, which is the conventional analysis, with voting as decision-making under uncertainty. "Under risk, probabilities can be assigned to the states of nature; under uncertainty, state probabilities are unknown or unknowable."28 They analyze voting under Savage's minimax regret criterion, and come to two interesting conclusions. First, voting for one's second choice is never minimax regret optimal. This implies that strategic voting never occurs, which would make the current work irrelevant if believed. Secondly, minimax regret decision makers find it rational to vote for their most-preferred alternative rather than thus analy enpir signi regre or kn Parti model Proba can r Pr0ba Citiz tUrnc Dartj Spatj POpu] his c PrefG ProbE the86 rela. Prov! Mode‘ 29 than abstain under relatively weak conditions. This model thus avoids the difficulties that the expected utility analysis runs into. Unfortunately, a great deal of empirical evidence indicates that probabilities have a significant effect on voter participation.27 The minimax regret framework denies that these probabilities are known or knowable. An alternative approach to the problem of voter participation is suggested by Palfrey and Rosenthal, who model simultaneous determination of participation and the probability of being decisive. "If everyone else votes, p can readily be very small. But if no one else votes, the probability of being decisive would be 1. Clearly, if citizens are rational, the voting probabilities and the turnout decisions are simultaneously determined.“28 Ledyard [85],[86] modeled simultaneity of voting participation and the probability of being decisive in the spatial model. Each voter knows the size of the voting population, the spatial positions of the alternatives, and his own preferences. His information on other voters' preferences is limited to knowledge of the continuous probability distribution from which they are drawn. Under these conditions, if expected return is sufficiently large relative to the cost of voting, turnout is positive, and he proves existence of a symmetric equilibrium. Palfrey and Rosenthal take a similar approach, but model only two types ("teams") of citizens, each with 30 identical preferences. Voting in this model is over two fixed alternatives "as in a two-candidate election or in a referendum or initiative vote between a proposal and a status quo."29 They find the possibility of substantial voter turnout in equilibrium, although depending on the size of the electorate, multiple equilibria are common. Thus for small numbers of voters "there are not strong predictions about the size of voter turnout."30 For a large voting population, they find only two types of equilibria: one in which turnout approaches zero, and one in which percentage turnout approaches twice the ‘minority' side's percentage of the electorate. Table 1.1 below presents percentage turnout for the 1972, 1976, and 1980 presidential elections along with the percentage of voters registered under the ‘minority' party (Republican or Democrat only).31 Percentage turnout can only roughly be described as double the minority side's percentage of the electorate, but Palfrey and Rosenthal's conditions are not strictly complied with. There are more than two ‘types' of citizens, and it is improbable that all citizens of a specified type have identical preferences. Certainly, this can be considered as some support for Palfrey and Rosenthal's model. However, the important point is that even with a cost of voting, substantial turnout can be an equilibrium outcome for rational voters. Tel 1972 1976 1980 1.3.3 Se Probabi and 0rd these m abstain However strateg arbitra two. H alterna Voting two alt fulfill 8Ystem’ Model . use a d Probabi 31 Table 1.1 Voter Turnout as Percentage of Minority Registration Registered Total % Turnout % Registered Voters Votes as ‘Minority' (thousands) Cast (R or D only) 1972 92,702 77,719 83.84 37.5 1976 105,837 81,556 77.06 48.0 1980 112,945 86,515 76.60 41.0 1.3.3 RandemnesLinJefinLMedele Several models have introduced randomness via probabilistic voting. These have included Hinich, Ledyard, and Ordeshook [69], and Fishburn and Gehrlein [56],[57]. In these models, there is a probability that an individual will abstain as opposed to voting his (sincere) preferences. However, if we think of sincere voting as one possible strategy and abstention as another, this type of model arbitrarily restricts voters' possible strategies to these two. Hinich, Ledyard, and Ordeshook model a two- alternative system which makes this plausible, since sincere voting is the unique optimal strategy when there are only two alternatives. However, a social welfare function fulfilling Arrow's conditions exists for a two-alternative system, casting some doubt on the applicability of this model. Intriligator [79] and Coughlin and Nitzan [31],[32] use a different type of model, in which each voter has a probabilistic density function fi(x), and for any subset A of the set of feasible social alternatives X, la fi(x) is the probability that individual i chooses some member of A, given than he can unilaterally determine the social choice. An ind his st model ‘ Pareto contra: density and usi probabi Probabi candida analee equilib: The all utij implYing lowest_I mania specifle into an SenSe. understa] generatil 1.3.4 Farg voting 8y iscuSSed procedure 32 An individual's choice probabilities are "proportional to his strength of preferences.“32 Intriligator develops this model to extend standard systems (Borda, majority rule, Pareto rule, etc.) into a probabilistic framework. In contrast, Coughlin and Nitzan assume that each individual's density function is also his differential utility function, and using two candidates, develop a model based on the probability of voting for each candidate. These probabilities are determined by the alternatives each candidate proposes to enact if elected. They then go on to analyze candidate behavior in the sense that electoral equilibrium depends on proposed policies. The major drawback to Intriligator's model is that if all utilities are positive, all probabilities are positive, implying that in some case an individual would choose his lowest-ranked alternative, gi1en_that_he_cguld_unilaterallx. determine_the_sggial_ghgige, Unless a framework is specified in which the welfare of other individuals enters into an individual's utility function, this doesn't make sense. In the current literature, the only readily understandable context for randomness in voting models is in generating preferences or utilities, or in tie-breaking. 1.3.4 Strafefiic_letini. Farquaharson [41] was the first author to approach voting systems from a game-theoretic point of view. He discussed only binary procedures; at any point within the procedure, voters have only two choices. This is distingui outcomes . bill may 1 decision 1 The possi‘: ____—_—— passage of the proces (voting in optimal. "vulnerabl "A si A set of 8' equilibriu: Strategy Rel ParquaharSC equilibrium W111 not al games in wh Pareto Prin this Case. voter fOr S GiVen . elncere s t 33 distinguished from pairwise voting because all possible outcomes are not paired with each other. In Congress, a bill may be amended or not, but if it is amended, the decision to be made is to pass or fail the amended bill. The possible outcomes of passage of the amended bill and passage of the original bill are not directly compared in the process. In one-stage binary processes, sincere voting (voting in accordance with one’s preferences) is always optimal. In contrast, multistage binary processes are "vulnerable"33 to strategic voting. "A situation is vulnerable if another situation i) can be obtained from the first by substituting a strategy of at least one voter; ii) is preferred to it by that voter or those voters."34 A set of strategies is "invulnerable" if it is a Nash equilibrium, one in which "each voter can say ‘no other strategy would have given a better outcome.’"35 As Farquaharson points out, sincere voting may or may not be an equilibrium. In fact, it is certain that sincere voting will not always be an equilibrium strategy in multistage games in which Gibbard’s conditions on unrestricted domain, Pareto principle, and nondictatorship are fulfilled. In this case, another strategy will be used by at least one voter for some social preference profile. Given this, is it reasonable to assume that voters use sincere strategies, or is it possible that voters actually calculate optimal strategies? ‘Sophisticated’ voting as developed by Farquaharson (his terminology for the use of 34 optimal strategies) received theoretical attention from McKelvey and Niemi [89], focusing on legislative voting games characterized by a finite sequence of two-alternative issues. This theory is examined by Enelow and Koehler [39], who look specifically at two amendment strategies: (1) amend to save a losing bill; (2) amend to "kill a winning bill.“ In either case, the amendment is voted on first (amended bill ab vs. bill b), followed by the vote on final passage, with each voter voting either yes or no on each. The game tree for this is shown in Figure 1.2. amended bill ab vs. bill b amended bill vs. 0 bill vs. 0 ab 9 b 9 Figure 1.2. The Game Tree for Pairwise Majority Voting with Amendment. If the first strategy is being employed, then the original bill is expected to lose, and the amended bill is expected to win. Therefore, "...the sophisticated voter realizes that while the nominal contest on the amendment vote is ab vs. b, the expected fate of ab and b, respectively, on final passage indicates that the actual contest on the amendment vote is between ab and 0. Therefore, the sophisticated voter votes for the amendment if he prefers ab to 0 and against the amendment if he prefers 0 to ab.”36 35 Table 1.2. Possible Preference Orders, Sincere Votes, and "Sophisticated" Votes on a Saving Amendment and Final Passage (Amendment Expected to Pass) preference order b>ab>¢ b>g>ab ab>b>¢ ab>g>b fi>b>ab g>ab>b sincere votes N,Y N,N Y,Y Y,Y N,N Y,N sophisticated Y,Y N,N Y,Y Y,Y N,N N,N Votes on passage of a "saving amendment" (the Sarasin amendment on House bill 4250) were compared to predicted votes.. Actual voting patterns were: Y,Y - 204 or 48.5%; N,N - 177 or 42.0%; Y,N - 40 or 9.5%; N,Y - 0 or 0%. 90.5% of these vote patterns used were predicted by the theory. A more in-depth analysis of how the vote patterns support the theory is presented in the article. An analysis of a killer amendment is also presented. Enelow [38] subsequently extended this paper to conform to an "expected utility theory of sophisticated voting."37 In this case, comparison of the ‘lotteries’ described by the left hand and right hand second branches determines voting on the amendment for an individual voter. In order to test this model, "...group rating scores were used to distinguish among congressmen by preference types. It was then shown that the aggregate voting patterns on a well—known example of a saving amendment and a well-known example of a killer amendment were consistent with the predictions of the E08 (expected utility sophisticated) voting model for each preference type."38 Thus these articles indicate that there is empirical support for the notion of ‘sophisticated’ (strategic) voting. CHAPTER 2 LITERATURE REVIEW: EXPECTED OUTCOMES 2.1 W Given the different outcomes of voting systems, an explanation of the criteria that can be used to compare them is necessary for any comparisons to be meaningful. Two measures have been used in comparing voting systems: Condorcet efficiency, and social utility of voting systems. 2.1.1 GendemeLEffieienel In order to understand the idea of Condorcet efficiency, it is necessary to define the Condorcet winner. Given a set of alternatives, the Condorcet winner is that alternative which would achieve a majority in a pairwise race with any other alternative. For example, if there are three alternatives A, B, and C, there are three pairwise races possible: A vs. B, A vs. C, and B vs. C. Let A>B indicate that alternative A achieves a majority over B in a pairwise race. Then A is the Condorcet winner if and only if A>B and A>C. Similarly, in a four-alternative election, A is the Condorcet winner if and only if A>B, A>C, and A>D. Condorcet efficiency is a measure of the extent to which a voting system complies with the Condorcet criterion: "...a candidate who receives a majority as against each other candidate should be elected.“1 As Arrow points out, this criterion implicitly accepts that there should be independence of irrelevant alternatives. Since pairwise majority choice may lead to intransitivity of social 36 37 preferences, only those cases where a Condorcet winner exists are used in the construction of Condorcet efficiency. Explicitly, Condorcet efficiency is the percentage of Condorcet winners expected to be elected by a voting system, when they exist. By this measure, a voting system which is more likely to elect Condorcet winners (i.e. has a higher expected percentage of Condorcet winners) is judged to be a “better“ voting system. 2.1.1.1 WW One difficulty with Condorcet efficiency is that a Condorcet winner may not exist. Existence of a Condorcet winner is not precluded by the presence of majority voting cycles; however, all of the alternatives in any cycle must be beaten in a pairwise contest by another alternative (which is the Condorcet winner) to avoid this problem. What is the frequency of existence of a Condorcet winner? It should be substantial if Condorcet efficiency is to be used as a comparison measure, since it is undesirable to compare voting systems on the basis of a minority of cases. Fortunately, probabilities of a social preference profile with no Condorcet winner have been calculated by Niemi and Weisberg [99] for an infinite voting population where all preference orders are equally likely. The probabilities are shown in Table 2.1 below. These are limiting probabilities for an infinite population; however, for small numbers of voters, probabilities for existence of a Condorcet winner are slightly higher. Until the number of alternatives 38 exceeds ten, the majority of social preference profiles do have a Condorcet winner. Table 2.1 Probabilities of the Existence of a Condorcet Winner for Various Numbers of Alternatives # of alternatives P(no Condorcet P(Condorcet winner) winner) 2 0 1 3 .0877 .9123 4 .1755 .8245 5 .2513 .7487 6 .3152 .6848 7 .3692 .6308 8 .4151 .5849 9 .4545 .5455 10 .4887 .5113 11 .5187 .4813 2.1.1.2 CendenceLEffieieneLandJiaimieLMeieritLlefing As mentioned previously, the Condorcet winner, when it exists, is the pairwise majority voting equilibrium. This is true regardless of whether voters use sincere or "sophisticated" strategies, since the Condorcet winner is a pairwise majority voting equilibrium in either case. A simple example should make this clear. Suppose there are three alternatives: A, B, and C. The game trees below diagram possible outcomes of a pairwise majority voting game, depending on the agenda. A a. b. c. Figure 2.1. Possible Agendas and Outcomes for Pairwise Majority Voting 39 Let C be the Condorcet winner. Then in Figure 2.1a, at the first branch of the tree, individuals ranking C last cannot prevent C from being considered as an alternative, and at the second branch, cannot prevent it from being chosen since a majority of the voting population sincerely prefers C and has no incentive to vote other than sincerely. In Figure 2.1b and 2.1c, these individuals could prevent the choice of C if they could influence the game by moving down the left branch of the tree. However, again they are working against a majority of the voting population which has no incentive to vote other than sincerely. Clearly, whether voters are assumed to vote sincerely or strategically, the Condorcet winner remains a pairwise majority voting equilibrium. Current legislative voting systems are characterized by a sequence of pairwise votes. Thus, when a Condorcet winner exists, it is the unique equilibrium outcome. Condorcet efficiency is therefore one measure of how closely different voting systems would correspond to current legislative methods’ equilibria in those cases where a Condorcet winner exists. 2.1.2 " ' " Another way of looking at the problem of comparing voting systems is to use a social welfare function even though we know this cannot fulfill all of Arrow’s conditions. Specifically, if individuals have cardinal utilities for the alternatives in the choice set, then a social welfare function of the form 40 {2: [uiJJTlllT T S 1; T ? 0. where T is a constant reflecting society’s aversion to inequality, is often used to measure the social utility of each of these alternatives. If T = 1, one way to interpret this efficiency measure is as the a 221921 expected utility of the outcome of a voting system, given the stated assumptions about individual utility. It is equally likely than an individual voter will have any of the possible preference orderings. Thus his expected utility for the outcome is 1/n times the expected social utility of the outcome as measured by a utilitarian social welfare function. Given a distribution from which utilities are drawn and a method of determining voters’ strategies, an expected value for social utility can be determined for each voting system. As an example, Weber’s derivation of "effectiveness" for the standard voting system with two alternatives is reproduced here.2 In this work, individual utilities are independent identically distributed random variables drawn from a uniform [0,1] distribution. Given no specific information about other voters’ strategies and a ‘large’ voting population, an individual voter’s optimal strategy in the standard voting system is to cast his vote for his most-preferred alternative.3 Since the winner is the alternative with the most votes, the expected social utility of the elected alternative is: n n Zk=o (1/2n) (R) (2/3 max(k, n-k) + 1/3 min(k, n-k)), 41 n where (1/2“) (k) describes the probability of a certain pattern of votes occurring, max(k, n-k) is the number of votes cast for the winning candidate and min(k, n-k) is the number of votes cast for the losing alternative, and 2/3 and 1/3 are expected values for the utility of an alternative ranked first and second, respectively, since the expected values of the maximum and minimum of two independent [0,1] uniform random variables are 2/3 and 1/3. Using Stirling’s factorial approximation, this expression simplifies to n/2 + W. Weber uses a transformation of this to make social utility measures more comprehensible. He defines the effectiveness of a voting system as follows: ElelecfeMLnandeml. E(maximal) - E(random) where E(*) is the expected social utility of the elected, maximal, or random alternative. Values for ‘effectiveness’ of course will vary according to the scaling factor used, which is E(random) in this transformation, but relative effectiveness of any two systems (in terms of ranking) will remain the same regardless of the scaling factor used. This is a particularly nice transformation since E(random) = n/2 and E(maximal) is asymptotic to n/2 + IH7T2'Normmax (m). Normmax (m) is the expected value of the maximum of m unit normal random variables, and Normmax (2) = l/ffi, simplifying the expression considerably. Effectiveness of the two alternative standard voting system is $273 : .8165. 42 This method can be used to determine the theoretic effectiveness (hereafter referred to as social utility efficiency) for the Borda system. The theoretic social utility efficiency as derived by Weber, of the standard voting system and the Borda voting system for m-alternative elections is: Standard voting system: 1357(m+1) Borda system: m m+ Weber was not able to derive a formula in terms of m for the approval voting system; however, he did derive social utility efficiency for a 3-alternative election: 87.5%. 2.2 RelatienehiLofJemeariseLMeasuree For voting populations which are assumed to use sincere strategies, both comparison measures generally have given the same rankings of voting systems, indicating some overlap in criteria. Indeed, it is easily verified that when 1 individual utilities are i.i.d. random variables of a given distribution, when a Condorcet winner exists, it has maximum expected social utility over all alternatives. Since at least a majority of voters prefer the Condorcet winner to any other alternative, the expected social utility of the Condorcet winner is greater than or equal to (int[n/2]+1)(E[distmax(2)l) + (n-int[n/2]-1)(E[distmin(2)]), where int[n/Z] is the largest integer smaller than or equal to n/2 and E[distmax(2)] and E[distmin(2)] are the expected values of the maximum and minimum of two independent random variables of the given distribution. For at least a majorit Condorc contras alterna (n-int[ This im Condorc utility ranking be due Voting exist, which h. Utility Condorc. Smaller diSPrOp. v°tins : F0 compari 2.3 Kg Th 100k lng [45]. expecte 43 majority of the voting population, the utility of the Condorcet winner exceeds that of the other alternative. In contrast, the expected social utility of the other alternative does not exceed (n-int[n/2]-1)(E[distmax(2)]) + (int[n/2]+1)(E[distmin(2)]). This implies that a voting system which always chooses the Condorcet winner when it exists maximizes expected social utility in these situations. Therefore, differences in rankings which occur given the two efficiency measures may be due to statistical variation or to the outcomes of the voting systems in cases where the Condorcet winner does not exist. An additional possibility is that a voting system which has a lower Condorcet efficiency but higher social utility efficiency chooses another alternative than the Condorcet winner in precisely those situations in which a smaller than majority group of voters benefit disproportionately. This would be expected to occur in voting systems with greater scope for strategic voting. For the interested reader, optimality properties of comparison measures are discussed in chapter 5. 2.3 Wes The more modern approach of comparing voting systems by looking at their expected outcomes was pioneered by Fishburn [45]. Rather than analyzing the characteristics of the process or mechanism, he analyzed the characteristics of expected or mean outcomes. I_1 F previo with. (unres that all pox to whic logical equally assumpt Partici voter m behavio Uni Which t] fail to Borda i: pairwis. he "as ‘ method < the COm CYcliCa: voting a this is would Cc 44 Fishburn’s approach was designed to fulfill many of the previously discussed conditions on the process. To begin with, he allowed all logically possible preference orderings (unrestricted domain), in keeping with Arrow’s justification that "...the decision making process should be applicable to all possible profiles since when we choose it, we don’t know to which profiles it will be applied.“4 In addition, all logically possible preferences orderings are taken as equally likely (since termed the ‘impartial culture’ assumption). He assumed there would be complete and sincere 'participation of the voting population, as in the median voter model, and that other voters’ preferences and voting behavior are independent of each other. Under these conditions, Fishburn analyzed the degree to which the Borda and Copeland extension of Borda give, or fail to give, the same selection. The Copeland extension of Borda is a Condorcet completion method, consisting of pairwise comparisons of all alternatives, so in essence what he was doing was determining the degree to which the Borda method of ranking would produce the Condorcet winner. Since the Condorcet winner is the result (in the absence of cyclical majorities) which is chosen by a pairwise majority voting system such as is used in Congress or in Parliament, this is one way of comparing how closely the Borda system would correspond to equilibrium outcomes of voting systems currently used. Add Fishburn along the their lib exists. Gehrlein 4 They con: restrict Their su: effiCien. 45 Additional work by Fishburn and Brams [51],[52] and Fishburn and Gehrlein [53],[54],[55],[56],[57] proceeds along the same lines, comparing voting systems in terms of their likelihood of choosing the Condorcet winner when it exists. In an article summarizing their work, Fishburn and Gehrlein [55] present the findings of their earlier studies. They consider the cases of 3, 4, and 5 alternatives, but restrict their summarization to ‘large numbers’ of voters. Their summary of ‘simple majority efficiencies’ (Condorcet efficiencies) for one stage procedures is presented below. Table 2.2. Condorcet Efficiencies for Various Voting System35 (%) Profile Generating Method random model 1 model 2 MAX n=101 power 1 power 1 n=101 Procedures n=101 n=101 vote for 1 77 76 81 78 vote for 2 74 72 73 76 vote for 52 75-79 w=(2,1,0) 91 vote for 1 66 67 69 63 vote for 2 74 76 72 77 vote for 3 61 61 62 65 vote for 32 70-76 vote for $3 64-70 w=(3,2,1,0) 87 87 w=(2,1,0,0) 82 79 vote for 1 58 58 76 58 vote for 2 70 71 64 68 vote for 3 68 67 54 71 vote for 4 53 50 38 54 vote for 32 61-72 vote for 13 63-69 vote for $4 59-64 w=(4,3,2,1,0) 85 87 87 w=(3,2,1,0,0) 84 w=(2,1,0,0,0) 73 73 72 46 Fishburn and Gehrlein used several methods to generate preference profiles. These include (a) random: each of the voters is independently and randomly assigned one of the m! linear orders on the m candidates; (b) model 1, power 1: same as random but recorded differently (power 2 squares the number of voters with each preference order); (c) model 2, power 1: each of the linear orders is selected randomly. Each order is then sequentially assigned voters, with the probability that n1 voters have this order assigned according to a binomial distribution. The second order is then taken and n2 voters assigned it, etc., until all voters have been assigned a preference order or until the last preference order is reached, in which case all remaining voters are assigned it; (d) MAX: each of the preference orders is randomly assigned an integer in {1,2,...,101} as the number of voters who have that preference order (the number of voters varies between m! and 101(m!)). All of these methods have the expectation of producing the same number of voters for each preference order, but the variance of methods (b), (c), and (d) differs. The methods used do "tend to generate ‘close elections’ among the m;3 contenders."6 Fishburn and Gehrlein see this as a drawback because "the efficiency percentages...may represent only a small proportion of relevant multicandidate elections, and the ‘correct’ efficiency figures could well be much higher than those given in the tables."7 However, there is no reason to believe that rankings of voting systems would 47 change by adding in ‘non-close’ elections. In these cases, the result is pretty much a foregone conclusion regardless of the voting system used. In fact, the relevant cases for a comparison of voting systems are precisely those in which the outcome would differ depending on the system used. Table 2.2 clearly shows that the Borda weighted ranking system achieves higher Condorcet efficiency than the standard voting system. The approval voting system (vote for $(m-1)) generates a range of Condorcet efficiency numbers that in 2 out of 3 cases contains the estimated Condorcet efficiency for the standard voting system and in one case exceeds it. Fishburn and Gehrlein’s work produces the following ranking: (1) Borda system; (2) approval voting system; (3) standard voting system. Although the work assumed sincere voting, Fishburn and Gehrlein do discuss the possible effect of strategic voting on Condorcet efficiencies. They argue that "approval voting is more immune to strategic voting than any of the other =k or 5k procedures...its efficiency estimates may compare more favorably to the efficiencies of other procedures when strategic voting is taken into account."8 They do not predict the effect of strategic voting on Condorcet efficiency of the Borda system, but do not “count its apparent sensitivity to strategic misrepresentation of preferences in its favor."9 Weber [143] also compared voting systems from the point of view of their outcomes or expected outcomes. He did not 48 use Condorcet efficiency as his comparison measure; instead he used social utility efficiency. Social utility can be considered the expected utility of a given alternative to a randomly chosen voter. Weber, assuming equally likely preference orders and complete and sincere participation, performed his analysis to determine the efficiency of a voting system in terms of social utility. Individual utilities were drawn from a uniform [0,1] distribution, and as previously noted, social utility was defined as the sum of individual utilities over all voters. Weber then determined what the expected social utility of the elected candidate for particular voting systems would be, and by comparing this with the expected social utility of the alternative with maximum social utility, scaled by the social utility of a randomly chosen alternative, developed the social utility efficiency measure: the percentage of maximum social utility a voting system is expected to produce. Using statistical tools for expected value, Weber computed the theoretical values of this efficiency measure for an infinite population of voters. Weber [143] showed that the Borda system, the approval voting system and the standard voting system could be ranked in the order given. The social utility efficiencies of the systems for a 3-alternative race are, respectively, 87.5%, 86.6%, and 75%. Weber also showed that the Borda system increases in efficiency as the number of candidates is increased, whereas the standard voting system decreases 49 in efficiency as the number of alternatives increases. He also proved that sincere voting is an Optimal strategy asymptotically, and produces a unique symmetric Nash equilibrium. Sincere strategies are also sophisticated optimal strategies, given no information about the preferences of other voters. In a subsequent article, Weber [143] first defined essentially equivalent voting systems as voting systems whose weights are positive affine transformations of each other; if a positive affine transformation of an optimal strategy under one system will yield the optimal strategy under the other system, this implies that these voting systems are essentially equivalent. He also showed that every nontrivial voting system is essentially equivalent to a unique minimal 0-1 normalized voting system ; the voting system weights are 0-1 normalized and the voting system is minimal in the sense that for every weight set of the system, there is at least one vector of utilities for which the weight set must be used in the corresponding optimal strategy. Using this definition, it is clear that all two- alternative voting systems are essentially equivalent to the standard voting system, which in the previous article was shown to have a social utility efficiency of {2/3 = 81.65%. Following Weber’s analyses, voting systems were compared by Chamberlin and Cohen [22]. Chamberlin and Cohen used the comparison method of the expected percentage of Condorcet winners (Condorcet efficiency), but also compared 50 the multidimensional spatial model with the unidimensional impartial culture model. Spatial theory assumes that there are dimensions to an election corresponding to salient issues, and that every voter has a preferred ideal position in the voting space. The voter is assumed to cast his vote in the standard voting system for the alternative or candidate closest to him in the space that describes the factors that are of concern to the voter. They perceive the use of the spatial model as a generalization of the impartial culture assumption. This is not strictly correct, since, as noted earlier, the classic spatial model, with individual utility being a function of weighted Euclidean distance, gives all voters single-peaked preferences. The standard assumptions on complete and sincere participation continue to apply. The voting systems which Chamberlin and Cohen compare include the standard voting system, the Borda system, and two multistage systems, the Hare and Coombs voting systems, which will not be discussed here. Their impartial culture results as presented below do not differ significantly from previous results. Table 2.3. Chamberlin and Cohen: Proportion of Condorcet Winners Selected - Impartial Culture 21 voters 1000 voters Borda system 86% 89% standard system 69% 69% 51 In contrast, their spatial model simulations produce varying results. All voters are represented by their ideal points in a four-dimensional space. The four numbers are generated as follows: voter j’s position on the first dimension is chosen from a standard normal distribution; his position on the second dimension is generated from the first dimension position by perturbing it with normal noise; the third position is produced from the second with fresh noise, and the fourth from the third likewise. All values are then normalized to have variance 1. However, this produces an electorate characterized by the correlation matrix shown in Table 2.4. Table 2.4. Expected Correlations Among Voter Dimensions Dimension 1 2 3 4 1 — .45 .33 .28 2 .45 - .75 .68 3 .33 .75 - .83 4 .28 .68 .83 - Candidate or alternative positions are generated in the same way, but three variances are used: low (.04), medium (1.0), and high (1.5). Given this structure, Chamberlin and Cohen find that the existence of a Condorcet winner is more likely .in the spatial model than the impartial culture assumption. Depending on candidate (alternative) dispersion and the number of voters, the probability ranges from 92 to 100%, as opposed to 84-85% for the impartial culture assumption for 4 alternatives. The arbitrary correlation used in assigning utilities to voters may have some influence on this result. 52 However, as shown below, the ranking of the Borda and standard voting system does not change. Because of the arbitrary nature of dimensional correlation, Chamberlin and Cohen’s results do not generalize well for the spatial model. Table 2.5. Chamberlin and Cohen: Proportion of Condorcet Winners Selected - Spatial Model 21 voters 1000 voters cand variance: low med high low med high Borda 83 83 92 85 86 97 standard 59 53 77 27 33 70 Following Chamberlin and Cohen, Merrill [93] compared voting systems using both Condorcet efficiency and social utility efficiency. He also varied the candidate dispersion in space in the spatial model relative to voters. Merrill’s results for the impartial culture model bear a striking similarity to Fishburn and Gehrlein’s. His spatial model results differ from Chamberlin and Cohen’s, but he used a multivariate normal distribution to generate voter and candidate positions, with a variety of correlation structures. Table 2.6. Merrill: Proportion of Condorcet Winners Selected (%) Impartial Culture (25 voters) # of candidates: 2 3 41 5 7 10 standard 100 79.1 69.4 62.1 52.0 42.6 approval 100 76.0 69.8 67.1 63.7 61.3 Borda 100 90.8 87.3 86.2 85.3 84.3 53 Table 2.6 (cont’d.) Spatial Model (201 voters, 5 candidates) dispersion 1.0 .5 4 of dimensions 2 4 2 4 standard 61 81 27 42 approval 81 84 75 82 Borda 89 92 86 88 % with Condorcet 99+ 99+ 98 99 winner Merrill’s dispersion is the ratio of standard deviations of the marginal distributions for candidates and voters. Thus if dispersion is greater than 1, there is more variance in candidate positions than in voter positions and vice versa. His results do indicate that as dispersion increases, Condorcet efficiency increases for all voting systems. If candidate dispersion is high relative to voter dispersion, the median has a greater probability of winning, whereas if candidate dispersion is low, extreme candidates or alternatives have a greater probability of winning. Thus there should exist an equilibrium level of relative dispersion under which all distances from the center or median of the vOting space are equally attractive to candidates. This nonconvergent equilibrium is in strong contrast to the median voter result of the unidimensional model, but is due to the discrete choice set. The same result occurs in the unidimensional model when a discrete choice set is used. Also, the nonconvergent equilibrium depends on the voting system. For the Borda and approval systems of voting, the advantage of the centrist candidates is little affected by the relative dispersion of voters and 54 candidates because they are systems in which either approximately half or all but one of the candidates receive votes from voters. Another interesting point is that Chamberlin and Cohen’s assertion that existence of a Condorcet winner is more likely under spatial model assumptions is borne out. In their development of the spatial model, Enelow and Hinich show that when more than one dimension is used, the existence of a Condorcet winner, or a median in all directions (dominant point), becomes less and less likely as the number of dimensions is increased. However, any point in the issue space may be introduced as an alternative in their model. They are essentially working with a continuous choice set. In contrast, the discrete choice set may have an equilibrium where the continuous one does not, and based on Merrill’s results, an increase in the number of dimensions increases the likelihood of an equilibrium point (Condorcet winner) when the size of the choice set (number of alternatives) remains constant. Merrill’s social utility efficiency results for the impartial culture assumption differ from his Condorcet results only in ranking the approval voting system above the standard voting system for a 3 alternative election. Otherwise all rankings remain the same. His results for the spatial model also parallel his Condorcet results. [\1 Bc 55 Table 2.7. Merrill: Social Utility Efficiency Impartial Culture (25 voters) # of candidates 2 3 4 .5 7 10 standard 100 83.0 75.0 69.2 62.8 53.3 approval 100 95.4 91.1 89.1 87.8 87.0 Borda 100 94.8 94.1 94.4 95.4 95.9 Spatial Model (201 voters, 5 candidates) dispersion 1.0 .5 it of dimensions L 4 2 4 standard 74 93 22 52 approval 97 98 95 98 Borda 98 99 96 99 Note the close correspondence between social utility efficiency and Condorcet efficiency numbers between Tables 2.6 and 2.7. The distinct relationship between the two efficiency measures as discussed earlier is apparent here. Merrill’s social utility efficiencies for the two and three alternative races are appreciably larger than the asymptotic limits calculated by Weber. He cannot be using the same exact formulation, since Weber calculates that expected social utility of a two alternative election for all voting systems is 81.65%. A final work using the expected outcomes approach to comparisons was written by Bordley [12]. He used both the spatial model and impartial culture assumptions to simulate the effect of various changes in the model on social utility efficiency. The variables analyzed included the number of alternatives and the number of voters. Generally, regardless of these values, rankings of the systems were (1) Borda; (2) approval voting; and (3) the standard voting fl) (0 56 system. However, social utility efficiency estimates for the approval voting system approached those of the Borda system as the ratio of the number of voters to the number of alternatives increased. 2.4 W 2.4.1 We In a 1974 article, Fishburn [49] took a different approach. In this article, he analyzed how many candidates should be voted for, as a parameter of the voting system, in order to maximize the efficiency of a voting system, in terms of agreement with the Condorcet criterion. He looked at both the simple voting system (vote for k of m), and the rank ordering system in which k are rank-ordered of m. He determined that a simple voting system reaches maximum efficiency by this criterion when as close to half of the candidates as possible are voted for. He also determined that weighted ranking systems, such as the Borda system, are most efficient when all candidates are ranked (kzm). Evidence about the efficiency of various values for the k parameter, which is the number of alternatives about which information is provided, is presented in Table 2.2. For the standard voting system and the approval voting system, voting for as close to half as possible of the alternatives is seen to increase Condorcet efficiency; for the Borda system, ranking less than all alternatives decreases efficiency. tn ff) 57 Weber also analyzed how the weight sets used (admissible strategies) affect the efficiency of the approval voting system and the Borda system. First it must be clarified that use of a different weight set may not produce an essentially equivalent system, which would have identical social utility efficiency to the original system. Although the Borda system with weights (m,m-1,...,1) is essentially equivalent to the Borda system with weights (m- 1,m-2,...,0), the 3 alternative system with weights (4,3,0) is not essentially equivalent to the one with weights (2,1,0). His analysis does show that alternative weight sets can increase the social utility efficiency of a voting system. Weber [143] also directly compared three voting systems with different parameter values: a. vote for k of m voting system, the family in which k e (1,...,m-1). The standard system with which we are all acquainted has k=1. b. the weighted ranking voting system with a single weight set (W1,...,Wm), of which the Borda system is representative with the weight set (m,m-1,...1). This is in contrast to the original Borda system with weight set (m-1,...,0). c. the vote for-or-against k system, with weight sets (w1,...wm) and (wi,...,wm), where the first set, with w1 through Wk = 1, wk+1 through wm : 0 corresponds to voting for k candidates, and the second set, with wi 58 through wfl-k = 1 and wfi-k+1 through we = 0 corresponds to voting against k candidates. Where m=3, this is the approval voting system with k=1. With more than three candidates, however, the approval voting system does not fit this model because in approval voting k is a choice variable for each voter. Examining these three systems, Weber determined which k would maximize the effectiveness of each voting system according to his social utility efficiency measure. Looking at system a, in which one votes for k of m, he determined that its efficiency measure was: 1/(m+1) x [W] 1/2 m-l Taking the derivative of this with respect to k and setting it equal to zero gives the result that when m/2 = k, the social utility efficiency of the vote for k system is maximized. Therefore, in the vote for k system, as close to half as possible of the candidates should be voted for. It is easily verified that social utility efficiency is symmetric about m/2 and that k = m/2 - 1 and k = m/2 + 1 are equivalent. Interestingly, this result corresponds to Fishburn’s earlier work showing that Condorcet efficiency is also maximized when k = m/2. It can be shown that if k is iset to be m/2, then as the number of candidates increases (ms’m), then the effectiveness of the vote for k system approaches a constant equal to [372 (z .866 or 86.6% social utility efficiency). This is in contrast to any fixed k as n increa decrease In utility (3,7 Any weig utility opposed Con. showed t; k) are e: efficient VX Where w ; betWeen a Voled for 81 - b1 - in the se most‘like do not re f0r~0rgag efficienc maximum 8 limiting 1 against k 59 m increases, in which case social utility efficiency decreases and the limiting value is zero. In the weighted ranking voting system, the social utility efficiency of the voting system is: m m+ Any weighted ranking voting system has maximum social utility efficiency when all candidates are ranked, as opposed to any k < m. Considering the vote for-or-against k system, Weber showed that vote for k and vote for n-k (i.e., vote against k) are essentially equivalent. The social utility efficiency of the vote for-or-against k system is: w x [leLmzkl 1/2 m(m-l) where w : the expected value of the difference in utility between a1 and b1, where [a] is the set of alternatives voted for and [b] is the set of alternatives not voted for. a1 - b1 is the difference in utility between the most-liked in the set of alternatives which receive votes and the most-liked (or least-hated) in the set of alternatives which do not receive votes. It is easily verified that the vote for-or-against k system has strictly greater social utility efficiency than the vote for k system. This system has maximum social utility efficiency when k = .368m. The limiting social utility efficiency of the vote for-or- against k system is 92.25%. However, the limiting social utility efficiency of the Borda system is 100%. AsymPt effici 2.4.2 F. of dif: In a 15 concorc concord the deg Condorc Concord Varianc. POPulat. Candida: concord. Substan. POpulat Showed and the either acrOSs Select 60 Asymptotically, the Borda system has social utility efficiency as great as any voting system possible. 2.4.2 Vefenfieneemamnmrrelatien Fishburn extended his work to allow for the possibility of different ‘levels of agreement’ in the voting population. In a 1973 article [48], he examined the effect of voter concordance as measured by the Kendall-Smith coefficient of concordance, W, on the existence of a Condorcet winner and the degree to which the Borda system agrees with the Condorcet winner. The Kendall-Smith coefficient of concordance, developed in 1939, is a transformation of the variance of the rank of candidates across the voting population, adjusted for the numbers of voters and candidates. If variance in rank is high, there is little concordance, whereas if variance in rank is low, there is a substantial amount of agreement among voters in the population as to what is a desirable outcome. The analysis showed that there is more agreement between the Borda system and the Condorcet winner when W is extreme. If there is either very little voter concordance or extreme agreement across the voting population, then the systems tend to select the same outcome. That is, the likelihood of the Borda system selecting the Condorcet winner is greater at the extremes of W. Bordley [12] examined the correlation coefficient r, with a range of -1 to 1. For the correlation coefficient, his assumption was of two equally sized groups in the voting populat' the two opposed the two changes, rz-l, di voting : voting : excepti contras Coeffic Flshbu; eiflCie Concor. aSSump decree If 31; Varia. by 23 diffs BQrQ' 2.4. time “C}1 61 population, with the correlation being between utilities in the two groups. When r=-1, there are two diametrically opposed groups in the voting population, whereas when r=1, the two groups are identical. Bordley showed that as r changes, the best voting system will change radically. When r=-1, dictatorship may be a preferable alternative to any voting system, whereas when correlation is perfect, the voting system used is of little importance with the exception of the approval voting system. This is in contrast to Fishburn’s results on the Kendall-Smith coefficient of concordance and Condorcet efficiency. Fishburn’s results showed an increase in Condorcet efficiency for the Borda system when there was little concordance. Presumably the difference is due to Bordley’s assumption of diametrically opposed groups, which would decrease the variance in rank across the voting population. If all preference orders occurred in equal numbers, this variance would increase; the ‘extreme disagreement’ implied by a high Kendall-Smith coefficient is qualitatively different from the extreme disagreement produced in Bordley’s model by a correlation coefficient of -1. 2.4.3 cheniaramefere Bordley’s work indicates that the effect of altering the standard deviation of utilities is negligible. "Changing the standard deviation only changes the scale of utilities and does not affect results.“1° Normal or uniform distributions for utilities of alternatives to voters were also co. idea th. is used not chaJ 62 also compared. Bordley provided evidence to support the idea that whether a normal or uniform utility distribution is used, the results in terms of ranking voting systems do not change. A voting closel be meal previo: be incc of indi common] A Prefere Structu and inf. ChooSes OUtCOme of Ca] inc thl dis Dre 111:. C911 CHAPTER 3 A MODEL FOR SIMULATION OF VOTING SYSTEMS WITH STRATEGIC VOTING A model which investigates the results of strategic voting as opposed to sincere voting should correspond as closely as possible to previous work for the comparisons to be meaningful. Therefore, the standard assumptions of previous work, with the exception of sincere voting should be incorporated into the model. In particular, generation of individual preference profiles is identical to the most commonly used method. A more detailed explanation of how a voter’s preferences are formed, how the possible strategies and structure of the voting system along with these preferences and information about other voters determine the strategy he chooses, and how all voters’ strategies determine the outcome of the system is presented here. 3.1 Aesumntiens_Qf_the_Mede1 1. A voter’s preference ordering is based on the utility of various alternatives to him. Let Uij be the cardinal utility of alternative i to voter j, where i indexes alternatives 1 through m and j indexes voters 1 through n. All nij are independently and identically distributed uniformly on the interval [0,1]. This determination of cardinal utilities allows all ordinal preference orderings, and in fact makes them equally likely for any given individual - the ‘impartial culture’ assumption. Arrow does not require equal 63 64 probability for preference orderings, just admissibility of all preference orderings, or unrestricted domain. Voters’ possible strategies for a voting system include all weight sets W = [w1,...,wu] which conform to the requirements of the particular voting system. a) the Borda system strategy set includes all weight sets [w1,...,wn] for which each w: is an element of {0,1,...,m-1}, and w: 7 Wj for all i 1 j. Thus voters’ possible strategies for the Borda system include all permutations of [0,1,...,m-1]. b) the standard voting system strategy set includes all weight sets [w1,...wm] for which each Wi is an element of {0,1} and 2: W1 = 1. Therefore standard voting system strategies include exactly one weight of 1, with the remaining weights being 0. c) For the approval voting system, the strategy set includes all weight sets [w1,...,wm] for which each w: is an element of {0,1}. Approval voting system strategies may include from zero to m weights of 1, and correspondingly m to zero weights of 0. Voters’ strategies determine the outcome of a voting system. The weights assigned by the n voters are summed over alternatives. The outcome of a voting system is the alternative which receives the greatest 65 total weight over the voting population, or arg max 23:1 Wij. All ties are broken randomly. 1 4. Voters choose optimal strategies from their possible strategy sets by maximizing their expected utility based on their information about other voters’ strategies. 5. Let E{uj(W1,Hz,...,Wn)} be voter j’s expected utility as a function of all voters’ strategies including his own. An equilibrium point is a matrix of strategies (W1,E2,...fln) such that for each j=1,2,...,n, E{UJ(E1:E2:°--:Hn)} 2" max E{uj(fll’fl2!°'-tfln)} .1 Simply put, all outcomes of a voting system must be in the set of Nash equilibria for the associated voting game . 3.2 W 3.2.1 Wheaties A sincere strategy for an individual voter is the strategy he chooses based only on his own preferences. Thus for the Borda system, the sincere strategy is an assignment of weights [w1,...,wn], where W1 = m-rank(i), which corresponds to the voter’s true ranking of alternatives. Then if ul ; uz 1 ..._; um, the sincere strategy assigns weights so that w1 L wz L ... ; wm. For the standard voting system, the sincere strategy is to assign a weight of one to the most preferred alternative (Wi = 1 iff i = arg max ui, otherwise w: = 0.) For the approval voting system, the sincere strategy is to vote for every alternative of greater than average utility [143]. Let u,= (21 ui)/m. Then if (ui-u) the exp partici alterna 3.2.2 Sin informat importan assumed ‘ utilitie each ind Strategi key Piec individu aceruing level 01 his dete values ‘ his kno- than fu given h othOme 66 (ui-u) > 0, W1 = 1; otherwise w: = 0. Intuitively, u,is the expected utility of the election when the voter does not participate, and the individual voter "approves“ of any alternative which betters that. 3.2.2 W Since voters are maximizing expected utility, the information upon which they base their expectations is an important part of the model. In all cases voters are assumed to know the distribution from which all individual utilities are drawn (or, equivalently, the likelihood of each individual preference profile). They may know the strategies of voters other than themselves. However, the key piece of information that is used to determine an individual’s optimal strategy is his estimate of total votes accruing to each of the alternatives, and the confidence level of his estimates. If the voter has full information, his determination of optimal strategy is based on the actual values of total votes accruing to alternatives, 21 Wij, and his knowledge of his own strategy. If the voter has less than full information, it is based on his estimates E1. and given his confidence level, the probabilities of various outcomes occurring. The voter solves max 21 pi(flj,&:7j)u1j subjecEJto the constraints of the voting system. The use of maximizing behavior on the part of voters can make a great difference to the performance of a voting system as measured by either Condorcet efficiency or social utility efficiency. 3.3 Sup mm 3)v. and C. equally ‘represe assumpti corresPo sincere are Pres for the likelih the SYs Profile OUtCome for the Singer! (32.5/ 67 3.3 W Suppose that there is a committee of three (voters 1,2, and 3) which has to choose one of three alternatives A, B, and C. The model specified makes all preference orders equally likely. Consider then the problem of a ‘representative’ voter. This voter will, under the assumption of sincere voting, choose a strategy which corresponds to his true preference ordering. Outcomes of sincere strategies for the standard and Borda voting systems are presented in Table 3.1, along with the Condorcet winner for the given preference profile. Because of the equal likelihood of individual preference orders and symmetry of the system, the Condorcet efficiency of a system given a profile for voter 1 is the same as that for the system. The outcomes shown are used to determine CondOrcet efficiencies for the standard voting system and the Borda system with sincere voting, which are 88.24% (30/34) and 95.59% (32.5/34) respectively. Table 3.1. Outcomes for the Standard and Borda Voting Systems with Sincere Voting Voter 1: A>B>C 2 \a I 92329 I eeC>E I 32929 I BZC>9 a 92923 1 92326 I A>B>C : a/a/a : a/a/a : a/a/a :a/a,b/a: a/a/a : a/a/a : A>C>B : a/a/a : a/a/a : a/a/a : a/a/a : a/a/a : a/a/a : B>A>C : a/a/a : a/a/a : b/b/b : b/b/b : */a/a : */b/b : B>C>A :a/a,b/a: a/a/a : b/b/b : b/b/b : */*/X : */b/b : C>A>B : a/a/a : a/a/a : X/a/a : */*/X :c/a,c/c: c/c/c : C>B>A : a/a/a : a/a/a : x/b/b : */b/b : c/c/c : c/c/c : Entries are standard outcome/Borda outcome/Condorcet winner. A * indicates that a tie occurs among all alternatives, which is broken randomly. Where two alternatives are listed, only those two are tied. 68 When instead voters choose optimal strategies, a Nash equilibrium point will determine the outcome. Possible strategies for the standard and Borda systems and outcomes given total votes accrued are presented below. Table 3.2 Possible Strategies and Outcomes Given Others’ Strategies 3.2.1: Borda System n H: W115 2,1,0 2,0,1 1,2,0 1,0,2 0,2,1 0,1,2 4.2.0 a a a a a, a 4,0,2 a a a a fl 3,0 4.1.1 a a a a a a 3,3,0 a a b a b b 3,0,3 a a a c c 0 3,2,1 a a a,b a b * 3,1,2 a a a a,c * c 2,4,0 b a1h b b b b 2,0,4 a+g c c c c 0 2,3,1 a,b a b * b b 2,1,3 a a,c * c c 0 2,2,2 a a b c b 0 1,4,1 h h h h h h 1.1.4 1: c c e c e 1:332 x b. i b. C h b,c 1,2,3 X i c h c b,c c 0,4,2 h h h h h b,c 0,2,4 c c h+c c c c 0,3,3 h c h c h 0 3.2.2: Standard System n 35 W115 1,0,0 0,1,0 0,0,1 2,0,0 a a 3. 0,2,0 h h h 0,0,2 9. c Q 131:0 a b * 1,0,1 a * 0 0,1,1 X i h 0 ‘Optimal’ strategies for a voter with A>B>C are underlined. In the rows marked with an X, where more than one strategy is underlined, determination of the optimal strategy depends on whether the expected utility from a random choice exceeds the individual’s utility for his second-ranked alternative. If strategies are equivalent in terms of payoffs, the voter is assumed to maintain the current strategy (usually the sincere strategy). 69 If a preference profile is given, sincere strategies can be determined and then voters checked individually to see if an increase in expected utility can be obtained by changing strategy. When no voter can unilaterally increase his expected utility given the strategies of others, a Nash equilibrium has been reached. Given the classic majority cycle profile, either one, two or three Nash equilibria will be found for the standard voting system. Voter 1: A>B>C Voter 2: B>C>A Voter 3: C>A>B If all voters have expected utility for a random choice (EU(*)) exceeding the utility of their second-ranked alternative (uz), sincere voting will be the only Nash equilibrium found. If exactly one voter has u2>EU(*), that voter will vote for his second choice, which will be the Nash equilibrium outcome found. If more than one voter has uz>EU(*), then the number of voters with this characteristic is the number of equilibria this method of solving can find. The equilibria found are not equally probable for a given social preference profile. However, the probability of a particular equilibrium can be determined using the probability that voters’ cardinal utilities have specific characteristics, and the frequency with which this equilibrium is found will reflect this probability. The equilibrium outcomes of the 3-alternative 3 voter 70 election for the standard and Borda voting systems, assuming voters choose optimal strategies, are shown in Table 3.3. Table 3.3 Outcomes for the Standard and Borda Voting Systems with Strategic Voting Voter 1: A>B>C A>B>C : a/a/a : a/a/a : a/a/a :a/a,b/a: a/a/a : a/a/a : A>C>B : a/a/a : a/a/a : a/a/a : a/a/a : a/a/a : a/a/a : B>A>C : a/a/a : a/a/a : b/b/b : b/b/b : I/l/a : II/2/b: B>C>A :a/a,b/a: a/a/a : b/b/b : b/b/b : */*/X :III/3/b: C>A>B : a/a/a : a/a/a : I/1/a : */*/X :c/a,c/c: c/c/c : C>B>A : a/a/a : a/a/a : II/2/b:III/3/b: c/c/c : c/c/c : Entries are standard outcome/Borda outcome/Condorcet winner. Numbered outcomes have probabilities for each of the alternatives being an equilibrium outcome: I: 15/24 a + 8/24 b + 1/24 c II: 8/24 a + 15/24 b + 1/24 c III: 1/24 a + 15/24 b + 8/24 c 1: 19/24 a + 4/24 b + 1/24 c 2: 4/24 a + 19/24 b + 1/24 0 3: 1/24 a + 19/24 b + 4/24 c Condorcet efficiencies are now 93.38% for the standard voting system and 95.22% for the Borda system. Condorcet efficiencies clearly do change as the assumption of sincere voting is dropped. The Borda system suffers a slight decrease, while the standard voting system performs significantly better. 3.4 W Previous work has either assumed that voters use sincere strategies or, alternatively, that voting takes place under zero information conditions. Particularly for committee voting, zero information is not the most realistic assumption to make. Frequently committees have one or more ‘vocal’ members whose preferences are common knowledge. Th informa affects system. but now This of strateg: Our repr his card 502 Chan of his e this OCQ' indel’erld. uniforml‘ fanatic!) variable£ P[(4A+c)‘ S . peclfic. C 71 The example in the previous section assumed complete information. Let us see how ‘incomplete’ information affects Condorcet efficiency under the standard voting system. As before, voters may have any preference profile, but now know only the strategy of voter 3 beside their own. This of course implies that voter 3 knows only his own strategy. Table 3.4. Possible Strategies and Expected Utility for Voters 1 and 2, Given Voter 3’s Strategy EU(HJ) ‘ 11.: Voter 3 110:0 HA 4/9(uA uB)+1/9(uc) 4/9(uA uc)+l/9(uB) 0,1,0 4/9(uA + uB)+1/9(uc) uB 4/9(u3 + uc)+1/9(uA) 0,0,1 4/9(UA uc)+1/9(u8) 4/9(u8 + uc)+1/9(uA) uc +~+ + OOHDOHOOH CHOOHOOI—‘O HOOHOOHOO Our representative voter’s optimal strategy again depends on his cardinal utilities. However, where he previously had a 50% chance of preferring the insincere strategy on the basis of his expected utility, he now has only a 20% chance of this occurring (P[(4uA+uc)<5uB]). The median of three independent uniform random variables (A>B>C) is distributed uniformly on [C,A]. The conditional probability density function of the median of three independent uniform random variables on the same interval is 1/(A-C). Therefore P[(4A+C)<5B] = I?4A+C)/5 1/(A-C) dB = 1/5. With this specification of information structure, the equilibrium 72 outcomes of the standard voting system are as shown in Table 3.5. Table 3.5. Equilibrium Outcomes of the Standard Voting System When One Specific Voter’s Strategy is Known. Voter 1: A>B>C ED / v v v A>B>C A>C>B B>A>C B>C>A C>A>B C>B>A I I I I I I. I I I I I I IIIIII IIIIII O()OJ*OJW IIIII ' Numbered outcomes have probabilities for each of the alternatives being an equilibrium outcome: 1: .64 a + .36 b 2: .6933 a + .2533 b + .0533 c 3: .2667 a + .2667 b + .4667 c 4: .2667 a + .4667 b + .2667 c 5: .2133 a + .5733 b + .2133 c In this case, Condorcet efficiency for the standard voting system is 85.33%. If two voters’ strategies are known, the results change again. Suppose the strategies of voters 2 and 3 are known. Both of these voters calculate optimal strategies in accordance with Table 3.4, while voter 1 uses Table 3.2.2. Then results are as shown in Table 3.6. Table 3.6. Equilibrium Outcomes of the Standard Voting System When Two Voters’ Strategies are Known. Voter 1: A>B>C 2 \a 92329 EZCZB I B>ezc I 82929 I C>BZB I 92B>e I A>B>C : a l a : a : a : a : a : A>C>B : a l a : a : a i a : a : B>A>C : a : a : b : b i 1 : 2 : B>C>A : a : a : b I b : 3 : 4 : C>A>B : a : a : 1 : 3 : c : c : C>B>A : a : a : 2 i 4 : c : c : 73 Table 3.6 (cont’d.) Numbered outcomes have probabilities for each of the alternatives being an equilibrium outcome: 1: .1667 a + .6667 b + .1667 c 2: .1333 a + .7333 b + .1333 c 3: .1333 a + .5333 b + .3333 c 4: .1067 a + .6267 b + .2667 c Condorcet efficiency in this case is 91.33%. Interestingly, Condorcet efficiency does not follow a predictable pattern given the information level. Condorcet efficiencies when 0, 1, 2, and 3 voters’ strategies are known are 88.24%, 85.33%, 91.33%, and 93.38%. This is due in part to the asymmetry of information between voters. An important point here is that in the 3 voter, 3 alternative standard voting game, it is always in the voter’s interest to reveal his strategy. Presented below are the probabilities of first, second, and third choices being chosen by the system if the voter either does or does not reveal his strategy. Table 3.7. Probabilities of Voters’ lst, 2nd, and 3rd Choices Being Chosen by the Standard Voting System Given the Information Structure of the Game. zero information .6296 .1852 .1852 one voter known strategy revealed .6919 .1541 .1541 strategy unknown .5784 .2252 .1963 two voters known strategy revealed .6517 .1822 .1661 strategy unknown .5856 .2533 .1611 three voters known .6296 .2176 The first voter to reveal his strategy does so because this .1528 policy stochastically dominates that of concealing his 74 strategy (zero information). The same holds true for the second and third voters, who compare their previous strategies of one voter known, strategy unknown, and two voters known, strategy unknown, respectively. Interestingly, this implies that an incomplete information game, at least in this example, is not an equilibrium outcome, because it is in each individual’s interest to reveal his strategy. However, when the voting population becomes larger, it may in reality be difficult for each individual voter to communicate his strategy to all other voters unless there is systematic reporting, such as on support for various bills before Congress. An incomplete information game may therefore occur. 3.5 Sincere Strategies and Nash Equilibria It has been shown [58],[120] that every non-dictatorial voting system with at least three alternatives is manipulable. That is, there is always some social preference profile for which an individual can improve his utility by misrepresenting his preferences. In other words, there is always a case for which sincere strategies do not constitute a Nash equilibrium. Given the stated assumptions about voters’ behavior and an infinite voting population, Weber [139] showed that sincere strategies are asymptotically optimal if only the distribution from which cardinal utilities are drawn is known to voters besides their own utilities. To show this, he used the fact that the number of votes cast by one voter for a particular 75 candidate and the probability that this number of votes is critical (changes the outcome of the election) are asymptotically proportional. Then subjective expected gain from a vote vector [w1,...wm] is asymptotically proportional to chd (uc-ud) max {0, wc-wd} = m[2c wc(uc-u)]. An optimal strategy is then an assignment of weights which maximizes 2c wc(uc-u), and Weber demonstrates the optimality of sincere strategies for each voting system, showing that sincere strategies under these conditions produce a unique symmetric Nash equilibrium. It can be shown that either an infinite voting population or zero information conditions are sufficient for sincere strategies to constitute a Nash equilibrium, and that both are not needed. 3.5.1 An_1nfinife_!efing_Benuleiien A set of strategies is not a Nash equilibrium if for any voter j, there exists some strategy H for which EIuIOIIHJIlH > Efua'mgvmfljn. where W213 is the set of strategies for all other voters. Clearly, an individual must be able to change the outcome of the voting system by altering his strategy for (fl?#j,flj) to be excluded from the set of Nash equilibria. Ihegrem_1;_As the voting population becomes large, i.e. ne-m, the probability that sincere strategies constitute a Nash equilibrium approaches one. Erect (standard voting system): 76 For the individual voter, any Wij is a binomial random variable (either a vote is cast for it or not), with p = 1/m. Then W1 = 25 Wij is distributed approximately normally with mean np n/m and variance np(1-p) = n(m—1)/m2, and the W1 have an approximate multivariate normal distribution. In order for an individual voter to change the outcome of the system, there must be some [W1 - Wk] 3 1. That is, the voter’s maximum weight assignment of one can cause the ordering of two totals to change. Let Y = W: - Wk. Then Y z 2 2 has a mean uy = pw - pa = 0; and variance 0y : Ow + Ow + i k 20w w . Because of the relationship between the covariance i k and correlation coefficient this variance can be computed exactly; the correlation coefficient is —1/(m-1). Intuitively, when one of the W1 is above its mean, the others are expected to be slightly below the mean. Computing this, a variance of a: = 2n(m-2)/m2 is obtained. Obviously, as n-rm, the variance of Y becomes infinite. Therefore P{|Wi - Wk] 3 1} = P{—1 g Y s 1}, the probability that Y falls within the specified interval, approaches zero. Thus scope for strategic behavior diminishes asymptotically and the probability that sincere strategies constitute a Nash equilibrium approaches one. An analogous proof can be constructed for the Borda system and the approval voting system (see appendix A). 3.5.2 WWW Recall that voters choose optimal strategies based on their information about other voters’ strategies (assumption 77 4). I will assume that this information is obtained by sampling the voting population and that the information obtained is correct. As an individual voter’s sample size becomes smaller, his estimates of the total votes accruing to alternatives become less accurate, and their variances increase. Specifically, let W1 be representative voter j’s estimate of total votes accruing to alternative i and n: be the number of voters sampled, with W: being the sample total. Because of the independence of the uij, the voter’s best estimate W1 is w: = wI + (n - ns) E(Zi w:5)/m, where E(Zi Wij) is the expected total weight for an individual voter. For the standard voting system and the Borda system this can be calculated precisely since it is not random, but for the approval voting system it must be designated as an expectation. The variance of W1 is (n - ns) times var(wij). If n: = n, variance is zero and the voter has complete information. If n: = 0, then W: = A n E(Ej Wij)/m, and the variance of W1 is n times var(wij), which is the zero information condition used by Weber. Ihegrem_2; As individual voters’ estimates of other voters’ specific strategies become less accurate, i.e., their sample size becomes smaller, the probability that sincere strategies constitute a Nash equilibrium approaches one. Erggf; As shown above, as sample size diminishes, the limiting condition is the zero information condition used by Weber. It remains to be shown that with zero information, 78 the sincere strategy is the optimal strategy for an individual voter. Recall that Weber used the asymptotic proportionality of the number of votes cast by one voter for a particular candidate and the probability that this number of votes is critical (p1). With this, he shows that subjective expected gain from a vote vector [Wi,...,Wm] is asymptotically proportional to m[21 w1(u1-u)]. Asymptotic proportionality of pi and w1 is a sufficient but not necessary condition for this result. The necessary conditions are that P1, the probability that outcome 1 occurs, be a positive function of w1 (P1 = f(w1)), with 5P1/5w1 > 0; 53P1/5w12 L O; 21 P1 = 1 That is, the probability of a specified alternative 1 occurring is strictly positively related (increasing at an increasing rate) to the number of votes cast for alternative 1, w1, within the constraints of the voting system. This condition holds for the model employed here. Under zero information conditions, the probability of occurrence of a specified alternative increases at an increasing rate with w1, with a strict one-to-one correspondence of w1 and P1. Therefore, given that the sum of the w1 is a constant, a vote vector which maximizes 21 w1(u1-u) over W also maximizes 21 f(w1)u1 = 21 p1u1 over W, or expected utility under the constraints of the voting system, and is an optimal strategy for the voter. However, the vote vector which does this is simply the sincere strategy, as shown by Weber. Therefore, under zero 79 information conditions, sincere strategies constitute a Nash equilibrium. As an example for the standard voting system, consider the following three-alternative, three voter election. For representative voter j, possible values of Egg: are shown in the left-hand column, along with their probable occurrence in parentheses. Voter j’s possible strategies of voting for alternatives A, B, or C, and the possible outcomes of the strategy are shown in columns 2, 3, and 4. Table 3.8. Strategies and Possible Outcomes of a Three-Alternative,Three Voter Election. n HJ Hin‘ A =[1.0.01 B = [0.1.0] C = [0.0.1] [1.1.0] (2/9) a b * [1,0,1] (2/9) a X 0 [0.1.1] (2/9) * b 0 [2,0,0] (1/9) a a a [0,2,0] (1/9) b b b [0,0,2] (1/9) c c c *A tie occurs which will be broken randomly. It is easily verified that if voter j votes for alternative A, his expected utility is pAUA + pBuB + pcuc = .6926 uA + .1852 us + .1852 uc. For the standard and Borda voting systems, given the number of voters and alternatives, any vote vector has a corresponding probability vector, and a permutation of the vote vector corresponds to an analogous permutation of the probability vector. Although probabilities are not a linear function of the weights assigned for small voting populations, there is a strict mapping from vote vectors to 80 probability vectors (which is asymptotically linear). For the approval voting system, there is a strict mapping for any fixed number of total votes. Vote vectors and their corresponding probability vectors are as shown in Table 3.9. Table 3.9. Vote Vectors and Corresponding Probability Vectors for a 3 Alternative, 3 Voter Election Vote Vector Probability Vector (=[PA,PB,PC]) StandancLloizinLstfem [1,0,0] [.6926,.1852,.1852] [0,1,0] [.1852,.6926,.1852] [0,0,1] [.1852,.1852,.6926] BerdLSiLsiem [2,1,0] [.6162,.2689,.1159] [2,0,1] [.6162,.1159,.2689] [1,2,0] [.2689,.6162,.1159] [0,2,1] [.1159,.6162..2689] [1,0,2] [.2689,.1159,.6162] [0,1,2] [.1159,.2689,.6162] WW [1,0,0] [.6574,.1713,.1713] [0,1,0] [.1713,.6574,.1713] [0,0,1] [.1713,.1713,.6574] [1,1,0] [.4491,.4491,.1018] [1,0,1] [.4491,.1018..4491] [0,1,1] [.1018,.4491,.4491] 3.6 The_Simu1atien_Eregram_and_Selxins_AlsQrifhm The simulation program is set up in accordance with the model specified. (For specific programs, see appendix B.) A cardinal utility vector is generated for each voter. With these, the Condorcet winner, it if exists, and the alternative with maximum social utility are determined. Given the voting system, the sincere strategy corresponding to the utility vector for each voter is determined. Then for each voter, possible pure strategies are taken one at a time and the voter’s expected utility for the possible 81 strategy calculated. With complete information, expected utility is only an expectation in the case of a tie occurring (because the tie will be broken randomly). With incomplete information, expected utility depends on the calculation of probabilities of outcomes, which depend on the strategy chosen, as well as the voter’s sample size. Expected utility is calculated for all possible strategies. It is then compared with expected utility for the voter's sincere strategy. If expected utility from another strategy exceeds that of the voter’s sincere strategy (an alternative strategy strictly dominates the sincere strategy). the individual’s vote vector is changed accordingly. If more than one alternative strategy has the same (maximum) expected utility, one of these strategies is chosen randomly, and the individual's vote vector changed accordingly. The process continues, checking each possible strategy for an expected utility increase. If expected utility remains constant with a change of strategy, the original vote vector is kept; there is no reason to assume that a strategy will change unless a gain is expected. Each voter is checked in a similar fashion until a Nash equilibrium is reached, or a specified number of iterations checking strategies (40) have been done. If after 40 iterations no equilibrium has been found, voters are randomly reordered and the process repeated. When an equilibrium is found, the outcome of voters’ strategies is determined along with its social utility; it is compared 82 with the Condorcet winner when it exists to see if they are the same, and the results are used to estimate the efficiency measures. Efficiency measure estimates are based on 2000 repetitions of the voting system for a given number of voters and alternatives. Numbers of alternatives range from 3 to 6, and the size of the voting population ranges from 3 to 125. For the incomplete information game, the number of alternatives is set at three, and sample size (the number of voters used in determining total votes for a subset of the population) is taken as 2/3 of the voting population, rounded to the nearest integer (a = .6667). The probability of each outcome (given the sample) should be approximately equal for different electorate sizes, given essentially equivalent population profiles, inducing equivalent optimal strategy responses from voters. Differences are due to the reduced likelihood of a tie in the larger voting population, just as in the complete information simulations. Voting populations again range from 3 to 125. 3.7 ReaLn1QIiQn_1Q_Enze_fiiraI§81_EQnilihzia Voters’ possible strategies in the solving algorithm include only pure strategies. Models in which an element of randomness is introduced for voters (see p. 31) have in general introduced a probability of voting as opposed to probabilities for strategies. If we think of sincere voting as one possible strategy and abstention as another, this type of model arbitrarily restricts voters’ possible 83 strategies to these two. Also, I question whether it is reasonable to expect voters, even in committee voting, to choose a mixed strategy when an optimal pure strategy response can be found. Merrill [92] proved that all "potentially uniquely optimal strategies"1 are pure strategies. A potentially uniquely optimal strategy may be a unique best response to others’ strategies. A point Wj in a convex subset S of RI" is called extreme if it is not interior to any line segment contained in S. A sz (0.1.0) (Ofihl) w3j Figure 3.1. Possible Strategies in the Standard Voting System with 3 Alternatives: a Convex Subset of R“. If a voting system S is a convex subset of Rm, then “the potentially uniquely optimal strategies are extreme points of 8."2 Let E(i), the ‘strategic value’ of alternative i, be 2?:1 (u1-uj)p13, where p15 is the probability of being decisive between alternative i and j (p11=0). Merrill’s formulation of expected utility is 84 EU(W5) : 2?:1 E(i)v1, where v1 is the number of votes in W5 for alternative i. If W5 is a potentially uniquely optimal strategy, then there exists a total utility function such that EU(W5) > EU(W3) for all W3 in S other than W5. Because EU(W5) is a linear combination of the E(i)’s, W5 must be an extreme point. One significant conclusion can be drawn from Merrill’s work. A mixed strategy is a linear combination of pure strategies and therefore interior to a line segment contained in S. Therefore a mixed strategy is never potentially uniquely optimal. In other words, a mixed strategy can never be a unique best response in the game. All unique best responses are pure strategies. Additionally, in cases where a mixed strategy is a best response, there exists a pure strategy with equal expected utility. A mixed strategy is only optimal if the voter is indifferent between two or more pure strategies which in linear combination produce the mixed strategy. However, if this is the case, he is also indifferent between the pure strategies which produce the mixed strategy and the optimal mixed strategy itself. There is thus always a pure strategy response with equal expected utility to the optimal mixed strategy response. There are cases for which the solving algorithm does not find a pure strategy equilibrium given a fixed order of voters for checking strategies. An example of such a case 85 for the 3-alternative 5 voter Borda voting system will serve to illustrate the point. Table 3.10. An Example of Preferences for Which a Pure Strategy Equilibrium is Not Found when Voters are Taken in a Specified Order Expected Utility Matrix j \ 11. 1 2 3 1 4.967650E-002 9.129716E-001 3.133120E-001 2 8.345773E-001 7.409244E-001 7 170978E-001 3 2.628670E-001 6.382484E-003 2.704006E-001 4 6.233332E-001 3.598900E-001 6.352836E-001 5 3.980304E-001 6.974258E-001 6.352836E-001 Sincere Vote Matrix j \ Ii 1 2 3 1 0 2 1 2 2 1 0 3 1 0 2 4 1 O 2 5 O 2 1 Preferences Voter 1: 2 > 2,3 > 1,2 > 1,2,3 > 3 > 1,3 > 1 Voter 2: 1 > 1,2 > 1,3 > 1,2,3 > 2 > 2,3 > 3 Voter 3: 3 > 1,3 > 1 > 1,2,3 > 2,3 > 1,2 > 2 Voter 4: 3 > 1,3 > 1 > 1,2,3 > 2,3 > 1,2 > 2 Voter 5: 2 > 2,3 > 3 > 1,2,3 > 1,2 > 1,3 > 1 a,b denotes a tie which will be broken randomly. The solving algorithm produces the following sequence of strategy changes: Table 3.11. Sequence of Strategy Changes Produced by the Solving Algorithm Individual Strategies and Total Votes i1 1. 2 3 4 5 Total 0,2,1 2,1,0 1,0,2 1,0,2 0,2,1 4,5,6 H5 1,2,0 2,1,0 1,0,2 1,0,2 0,2,1 5,5,5 1,2,0 2,1,0 2,0,1 1,0,2 0,2,1 6,5,4 0,2,1 2,1,0 2,0,1 1,0,2 0,2,1 5,5,5 0,2,1 2,1,0 1,0,2 1,0,2 0,2,1 4,5,6 86 The pure strategy equilibria {[2,1,0], [2,1,0], [0,1,2], [1,0,2], [0,1,2]}, {[0,2,1], [2,1,0], [1,0,2], [1,0,2], [1,0,2]}, and {[0,2,1], [2,1,0], [2,0,1], [1,0,2], [0,1,2]} all exist for this preference profile but are not found by the solving algorithm because the voters are taken in a specified order. However, if voters are taken at random to have their strategies checked, there is no way of ensuring that all voters’ strategies are checked (verifying the existence of the Nash equilibrium). A random reordering of all voters and repeat of the process solves the problem, and an equilibrium is found in every case. 3.8 Wad Not all Nash equilibria are found by the solving algorithm. Because of its construction, if sincere strategies constitute a Nash equilibrium, then for that social preference profile the outcome of the voting system is the outcome of sincere voting. Only if sincere strategies do not constitute a Nash equilibrium is strategic voting taken into consideration. In the previous example (Tables 3.10 and 3.11) the strategy [2,1,0] for all voters is a pure strategy Nash equilibrium point; none of the voters can unilaterally increase his expected utility. However, the solving algorithm provides no motivation for individual voters to alter their strategies to reach this equilibrium. In fact, both voters 1 and 5 are strengthening their last-ranked alternative at the expense of their first and second choices. The strength of the solving algorithm 87 lies in the fact that any Nash equilibrium found can be reached via individual strategy changes (motivated by expected utility maximization) from the sincere strategy matrix. In this case, the equilibrium will correspond to a minimal B-coalition of the associated cooperative game. In cooperative games, the characteristic set V(s) delineates a set of payoff vectors for each possible coalition S which represent the worth or effectiveness of the coalition S. In beta theory, a vector of payoffs is included in the characteristic set V(s) if and only if it is non-preventable by players outside the coalition.3 In other words, if players outside the coalition have some strategy or set of strategies which could prevent this payoff vector from occurring, it is not included in the beta solution. A simple example using the standard voting system should clarify the idea of the beta solution. Table 3.12. Expected Utility, Preference Orderings, and Sincere Strategies of Voters Using the Standard Voting System. Voter Alternative 1 2 3 A .016 .365 .694 B .682 .482 .247 C .793 .218 .413 x .497 .355 .4513 Expected value of u15 Preference orderings Sincere strategies voter 1: C>B>*>A [0,0,1] voter 2: B>A>*>C [0,1,0] voter 3: A>*>C>B [1,0,0] 88 If voters 1 and 2 form a coalition, they can achieve any of the possible payoff vectors (rows of Table 3.12) for A, B, or C. They cannot guarantee the payoff vector for a random choice (*) because regardless of the strategies they choose, voter 3 has a strategy which can prevent it. If voters use sincere strategies, this final payoff vector is the outcome. Sincere strategies clearly do not constitute a Nash equilibrium in this case. If voter 1 votes for alternative B instead of his most-preferred alternative C, his expected utility increases. Additionally, if voter 1 does this, neither of the other voters can increase their expected utility by altering strategy and this set of strategies is a Nash equilibrium. 1: [0,1,0] 2: [0,1,0] 3: [1,0,0] However, the set of strategies 1: [0,0,1] 2: [1,0,0] 3: [1,0,0] is also a Nash equilibrium in this game. If the solving algorithm looks at voter 2 before voter 1, this is the equilibrium that will be found. Because of the randomness of individual utilities, the Monte Carlo techniques employed make it equally likely that the individual utilities will occur in either order, and a sufficient number of repetitions will find each equilibrium; furthermore, they will occur with equal probability (given that exactly 2 voters have u2>EU(*)). 89 The equilibria above correspond to minimal B-coalitions because the removal of one player from the coalition causes it to fall apart. If we looked at a standard voting system game with five players, a coalition of 4 would not be minimal because the removal of one player would still leave a decisive coalition of 3. The solving algorithm will not find an equilibrium in which individuals vote strategically corresponding to a non-minimal B—coalition in a game with complete information. Subsequent to the assignment of strategies corresponding to a minimal B-coalition, no voter outside the coalition can increase expected utility by altering his strategy so as to "join the coalition." All equilibria corresponding to non-minimal B-coalitions will be sincere strategy equilibria, and the coalitions will occur with probability determined by the approximate multivariate normal distribution. In contrast, in an incomplete information game, a non- minimal B—coalition equilibrium with strategic voting may occur because a player may have a positive probability that this minimal coalition does not exist, due to his uncertainty about voters’ strategies. Even though a minimal B-coalition already exists, a voter may have preferences such that either joining the coalition or voting strategically against it can increase his expected utility because of this positive probability. The equilibria found are also perfect equilibria in the sense of Selten [123]. Although his concept of a perfect 90 equilibrium was intended to apply to extensive games, the point of view which looks at “complete rationality as a limiting case of incomplete rationality"4 is useful in this model because of the difficulty of accepting the concept of the rational voter. Suppose voters are rational in the sense that they can evaluate different alternatives, compare strategies available to them, and estimate the effect of these strategies on the outcome of the system. However, this hypothetical rational voter is not perfect; he may make ‘mistakes.’ When he has had a ‘bad day’ with probability 8, he is equally likely to choose any of the strategies available to him, as he is no longer thinking straight. If all this happens to all voters, we have Selten’s perturbed game. The ‘rational’ part of the voter knows that this happens and uses it in his calculation of optimal strategy as far as he is able. Then if the strategies of the perturbed game approach the strategies of the original game ase:+'0, the Nash equilibrium of the original game is ‘perfect.’ The model as constructed is set up in exactly this way. The zero information game corresponds to a complete information game in which the rational voter assigns 5:: (Q-1)/Q (where Q is the number of admissible strategies) to every other voter and determines his optimal (sincere) strategy on that basis. As the original value of 6 gets smaller, the variance of estimates W295 decreases, exactly as if the voter had better information. Complete information (or perfect rationality) is the limiting case. 91 Conversely, a Nash equilibrium which cannot be reached from the sincere strategy matrix is not perfect, since the sincere strategy matrix is the unique equilibrium of a sufficiently perturbed game. Therefore if the number of repetitions is sufficiently large, the set of equilibria found will correspond to the set of perfect equilibria, and Nash equilibria which are not found will not be perfect equilibria. CHAPTER 4 RESULTS The results of the simulations are presented and analyzed here. Some of the questions examined are 1) the relationship between social utility efficiency estimates and Weber’s theoretical values; 2) how social utility efficiency estimates compare given the use of sincere strategies as compared to optimal strategies; 3) how Condorcet efficiency estimates compare given sincere and optimal strategies; 4) the effect of strategic voting on rankings of the systems using either social utility or Condorcet efficiency with strategic voting; 5) the relationship between Condorcet efficiency and social utility efficiency; and 6) the effect of the amount of information available to voters on efficiency estimates given optimal strategies. 4.1 Theorefiea1_la1uee 4.1.1 Sincerelefing Weber’s social utility efficiency values are asymptotic. It is therefore possible that social utility efficiencies may be significantly different for small voting populations. This possibility was investigated, but the differences were found to be insignificant for the most part. Using the student’s t distribution, t-tests indicated only three cases, all for the standard voting system, for which the differences were significant at the 90% level or better. In each of these cases, the number of voters differed from the number of alternatives by at most one, and 92 93 social utility efficiency was significantly greater than the theoretical value. However, in these cases, the standard voting system had appreciably lower social utility efficiency than either of the other systems considered, and rankings were not affected. Again, for 3 alternative elections, rankings according to Weber’s asymptotic social utility efficiencies are 1) approval voting system, 87.5%; 2) Borda system, 86.6%; and 3) standard voting system, 75%. The simulations tended to confirm this for 3 alternative elections, although there is difficulty in differentiating the efficiency of the Borda and approval systems. In fact, the Borda system ranked above the approval voting system 12/22 times, but a t-test detects no significant difference in means. Although Weber did not develop a formula for theoretical values of the approval voting system with more than three alternatives, simulation estimates of social utility efficiency for the approval voting system appear to indicate that asymptotic social utility efficiency is constant at 87.5%, regardless of the number of alternatives. Figures 4.5-4.9 (pages 96-97) show social utility efficiency estimates for the approval voting system and their approach to this limit. Deviations are greater for a smaller number of voters, and the size of the deviation is greater the larger the number of alternatives considered. For more than 3 alternatives, rankings were, without 94 exception: 1) Borda system; 2) approval voting system; and 3) standard voting system. 4.1.2 Wing When voters’ use of optimal strategies was incorporated, social utility efficiency estimates diverged markedly from theoretical values for the standard voting system. Differences are predictably greater for small electorates, and given the number of voters, greater for a larger number of alternatives. STANDARD VOTING SYSTEM: 3 ALTERNATIVES — STRATEGIC ..... LIMIT .....- SINCERE Figure 4.1 Social Utility Efficiency for the Standard Voting System with 3 Alternatives: Strategic, Sincere, and Limit Values 95 98 as. 88* 751 7.1 maxi-“~-‘bflx 1”\\ I’W”‘\\f'"’n~“‘. \ I ‘r ‘ I 65W! I I v I I I I I I 'fi STANDARD VOTING SYSTEM: 4 ALTERNATIVES _ STRATEGIC ..... LIMIT ...; SINCERE Figure 4.2 Social Utility Efficiency for the Standard Voting System with 4 Alternatives: Strategic, Sincere, and Limit Values 98 85 .t 88 73 . 703‘. 65 \’ ‘1 1‘s 1 60, j l I r f l7 V I .V I T . l 7 l I r Y I j STANDARD VOTING SYSTEM: 5 ALTERNATIVES __ srna'rsczc ...“ LIMIT ...- smczaz Figure 4.3 Social Utility Efficiency for the Standard Voting System with 5 Alternatives: Strategic, Sincere, and Limit Values 96 STANDARD VOTING SYSTEM: 6 ALTERNATIVES _STRATEGIC ,.... LIMIT ...- SINCERE Figure 4.4 Social Utility Efficiency for the Standard Voting System with 6 Alternatives: .Strategic, Sincere, and Limit Values This difference did not occur to such an extent for the approval voting system. In only two cases was the difference great enough to produce a t-statistic significant at the 80% level. However, an interesting pattern to social utility efficiency estimates appeared. For small voting populations, the estimates are very close to 87.5%; they decline as the number of voters increases and after a certain point begin to increase again toward 87 5%. This decline is more marked as the number of alternatives increases, as shown in Figures 4.5-4.8. 97 l ' I ' r ' l ' l ‘ I ‘ I ‘ I APPROVAL VOTING SYSTEM: 3 ALTERNATIVES ____STRATEGIC”«.LLMIT .... sxnczns Figure 4.5 Social Utility Efficiency for the Approval Voting System with 3 Alternatives: Strategic, Sincere, and Limit Values lffjl'l'l APPROVAL VOTING SYSTEM: 4 ALTERNATIVES __ swam-mm ..... LIMIT --- smcsaz Figure 4.6 Social Utility Efficiency for the Approval Voting System with 4 Alternatives: Strategic, Sincere, and Limit Values 98 Figure 4.7 f 1’! I ' I I F r APPROVAL VOTING SYSTEM: 5 ALTERNATIVES _ STRATEGIC...» LIMIT -_. SINCERE Social Utility Efficiency for the Approval Voting System with 5 Alternatives: Strategic, Sincere, and Limit Values 09 80 I? 86 85 “i ”I 82 81 80 I r l ‘ l APPROVAL VOTING SYSTEM: 6 ALTERNATIVES — STRATEGIC ..... LIMIT ...... SINCERE Figure 4.8 Social Utility Efficiency for the Approval Voting System with 6 Alternatives: Strategic, Sincere, and Limit Values 99 Strategic estimates again diverge for the Borda system; the same pattern is discernable as for the approval system. Once again the effect is greater where there is more scope for strategic voting. Efficiency measures for the 6 alternative system are predicted values using regression coefficients estimated (see page 115). A h 1‘ . / ~ I I: I \ , l‘ I \ 1 I n 1 I 1 ...” 1’ \ f I“; I i l’ ; MMJ M " r v ‘4‘; V 854 on. T I ' 7 ‘ I ' I ' I l I BORDA VOTING SYSTEM: .3 ALTERNATIVES -— STRATEGIC ..... LIMIT --- SINCERE Figure 4.9 Social Utility Efficiency for the Borda Voting System with 3 Alternatives: Strategic, Sincere, and Limit Values 100 82s- I Y I T I I l l: I If I ‘r 1 I I I r I I BORDA VOTING SYSTEM: 4 ALTERNATIVES _ STRATEGIC so... LIMIT ... SINCERE Figure 4.10 Social Utility Efficiency for the Borda Voting System with 4 Alternatives: Strategic, Sincere, and Limit Values BaLl'F‘I'I‘I'T'j'I‘I‘I BORDA VOTING SYSTEM: 5 ALTERNATIVES (ESTIMATED) _ STRATEGIC ..... LIMIT Figure 4.11 Social Utility Efficiency for the Borda Voting System with 5 Alternatives: Strategic and Limit Values 101 95.0 9.00‘ 87:51 flLI< BORDA VOTING SYSTEM: 6 ALTERNATIVES (ESTIMATED) ...—STRATEGIC ..... LIMIT Figure 4.12 Social Utility Efficiency for the Borda Voting System with 6 Alternatives: Strategic and Limit Values 4.2 Wm Wm Under the assumption of sincere voting, social utility efficiency rankings from the simulation estimates are compatible with the results of previous work. When voters are assumed to use optimal strategies, estimates of social utility efficiency are in many cases significantly different from their sincere voting estimates: Despite this, overall rankings of the systems do not change much. The approval voting system does rank above the Borda system for small electorates given more than 3 alternatives. As the voting population increases, this ranking is reversed. In all cases, the standard voting system is ranked below the other two systems, despite the pronounced increase in social 102 utility efficiency for the standard voting system and decrease for the Borda system. For the approval voting system, small voting population estimates of social utility efficiency are significantly greater than their sincere counterparts, while larger electorates tend to have strategic estimates below the sincere estimates. Given these changes, for small electorates the approval voting system moves up in ranking while the Borda system moves down to second place. The standard voting system, while having social utility estimates which are roughly comparable (for 4 alternatives and 3 voters, estimates are: approval, 89.1%; Borda, 88.0%; and standard, 87.9%), remains ranked in third place. As the size of the voting population increases, the ranking between the approval and Borda system is reversed, and the estimates for the standard voting system decrease steadily toward their limit. at 'o. 3...."IS-u-o5... oo- " oak! - ‘ no. 5.. ’ .. 5-4 \_’~-~ 5’ F". o " 88« SINCERE SOCIAL UTILITY EFFICIENCY: 3 ALTERNATIVES ...—swim» ..... APPROVAL -..- BORDA Figure 4.13 Sincere Social Utility Efficiency: 3 Alternatives 103 87 I 5 ‘\J’\ .5"): .-. «5 .. ... ... ......a" I o o a . ‘ .0 o. .0 .' 5 \ o o .0. -~. ‘ .. o. ‘ ....o~..~ .0 o“- ” \ A .... .0 ”bum“ . R -— ’ am - ‘-" ~ Rik-M: xx x , \z ".5 75.0 I'I'T‘I'T'Ifir'lfl‘lY STRATEGIC SOCIAL UTILITY EFFICIENCY: 3 ALTERNATIVES ~— _STANDARD ...... APPROVAL .-.. BORDA Figure 4.14 Strategic Social Utility Efficiency: 3 Alternatives 95 9" H f_~ I-h--’ -------‘ d .0... .- on... o~ u I. J." '00...."\.. ..'~o.. '.ouo...oo'°. . '50....‘a...’ ‘.. .0 ‘. .o‘ ..‘ooo’... '0. as 1 ......oo 0.. '0... ...... .... o w' 804 SINCERE SOCIAL UTILITY EFFICIENCY: 4 ALTERNATIVES Figure 4.15 Sincere Social Utility Efficiency: 4 Alternatives 104 90. I a? I s ;...~....”..~.'.\..ooc'o. " f-s ‘ 3' "" . ....«fi fan-i' 3' ”Q50" I an \.....---~, ":2: ,.-....r ‘4‘» oz 5 ‘w ‘--.-*~-: ._. . d 72.5fi_‘_—T——‘—-p—'-r l r if . 11f 1 T STRATEGIC SOCIAL UTILITY EFFICIENCY: 4 ALTERNATIVES P —_STANDARD.......APPROVAL -..- BORDA Figure 4.16 Strategic Social Utility Efficiency: 4 Alternatives ......aco‘0f0.000.00000.0000.. ..... .0 ... 0...... IO. 0' no. on! ..."00000009. SINCERE SOCIAL UTILITY EFFICIENCY: 5 ALTERNATIVES ._ STANDARD .... APPROVAL ...- BORDA (LIMIT) Figure 4.17 Sincere Social Utility Efficiency: 5 Alternatives 105 ‘ 35090500....- ’f- 85 k“; o.-....’.'...OO~~:-v #J- ...-5..‘......‘.. 7. ' I I I I I r I ' I ‘ I ' I ' I T I ‘ I ' STRATEGIC SOCIAL UTILITY EFFICIENCY: 5 ALTERNATIVES _STANDARD....... APPROVAL ...... BORDA (ESTIMATED) Figure 4.18 Strategic Social Utility Efficiency: 5 Alternatives 0" Q. O .OQ... ..ffihoooooooflhoo~.oooliCo. a...........oo‘0.. 5 #1 . 85 ‘ ”....o..o....oo..... 0.50.00... ...}..f 5"... f a..‘ C '0‘ 7'4 65 60 . I r I ' I ' I ' I ' l ' I ‘ I ‘ l ' I r SINCERE SOCIAL UTILITY EFFICIENCY: 6 ALTERNATIVES _ STANDARD u... APPROVAL ..- BORDA (LIMIT) Figure 4.19 Sincere Social Utility Efficiency: 6 Alternatives 106 \ ‘ - ..‘..~....... ... - 0' a. \ I‘ I I7 ‘ Ff! ‘ I ‘ I ' I ' I I STRATEGIC SOCIAL UTILITY EFFICIENCY: 6 ALTERNATIVES __ STANDARD ....... APPROVAL __- BORDA (ESTIMATED) Figure 4.20 Strategic Social Utility Efficiency: 6 Alternatives 4.3 QQndeeLEfficisncLRankizms. Condorcet efficiency rankings under sincere voting are, for ”small" electorates: 1) Borda system; 2) standard voting system; and 3) approval voting system. Given a specified number of alternatives, as the size of the electorate increases, the approval voting system reverses rank with the standard voting system, and as with social utility efficiency, we have the Borda system ranked first, followed by approval voting, followed by the standard voting system. Strategic voting produces a dramatic change in these rankings. Condorcet efficiency increases significantly for both the approval and standard voting systems, while in all but a few cases it decreases significantly for the Borda 107 system. For most small voting populations (committee size), the standard voting system is ranked first in Condorcet efficiency, followed by the Borda system, with approval voting ranked last. For any number of alternatives considered (3-6), Condorcet efficiency for the standard voting system with strategic voting peaks when there are five voters and decreases more or less consistently thereafter. In contrast, the approval voting system with strategic voting has maximum Condorcet efficiency with 3 voters and declines thereafter. Within the voting populations used in the simulation, there is no U-shaped curve as found for social utility efficiency; Condorcet efficiency does not reach some approximate minimum and begin to climb towards a limit. Instead Condorcet efficiency begins from a level above its “limiting value" and approaches the value in an approximate logarithmic curve. For the Borda system, with 3 or 4 alternatives, the U- shaped curve is again apparent. Estimates for Condorcet efficiency with 5 or 6 alternatives follow the same pattern. 108 ‘\ .e--— _ 85< no I. .‘m....~ so... 0 .50 7s .... 0. ...50000000.‘ ...-cocooiflfi..’. ‘0‘ ..‘ 0‘..- a“ .000"'.‘..'“~00000". ‘0. SINCERE CONDORCET EFFICIENCY: 3 ALTERNATIVES Figure 4.21 Sincere Condorcet Efficiency: 3 Alternatives .0. 0.. ... a. 80* ...... ~......¢..‘o. 753-T-w-T-n-T-w-q-w-q-w-1-—-1-—-q-—-f-—-7-w-f STRATEGIC CONDORCET EFFICIENCY: 3 ALTERNATIVES _STANDARD ....... APPROVAL ...- BORDA Figure 4.22 Strategic Condorcet Efficiency: 3 Alternatives 109 33K 5 J“ — r-fi - 00.....'-.. .ooocnooo'... SINCERE CONDORCET EFFICIENCY: 4 ALTERNATIVES __— __STANDARD ,.... APPROVAL ..- BORDA Figure 4.23 Sincere Condorcet Efficiency: 4 Alternatives I“ 95 99 83 « on ‘ d c 00.... 0.........IIOJ I. ?5-¢ .'-. . .~ ’ O ..~~ ... . a, 5...... o N 7o‘flf‘l'f'l‘l—‘f'l‘lfiI' STRATEGIC CONDORCET EFFICIENCY: 4 ALTERNATIVES -.——-—— .— __STANDARD ....... APPROVAL .....- BORDA Figure 4.24 Strategic Condorcet Efficiency: 4 Alternatives 110 '- ..o' q'°'ooooooo....-.. .- a. o “... a... o I ~.‘.oc’.... ~ ......... . . . ...~II..... SINCERE CONDORCET EFFICIENCY: 5 ALTERNATIVES — STANDARD ..... APPROVAL .... BORDA (ESTIMATED) Figure 4.25 Sincere Condorcet Efficiency: 5 Alternatives 65; I ' I ' I ‘ I ' I f I ‘ I ‘ I T I I I STRATEGIC CONDORCET EFFICIENCY: 5 ALTERNATIVES __ STANDARD ...... APPROVAL ___ BORDA (ESTIMATED) Figure 4.26 Strategic Condorcet Efficiency: 5 Alternatives 111 {.00'000000. fi ...'90. ..00. 0 .0 "boooooooooou . . 6L .roooooog...-' "a...IO' SINCERE CONDORCET EFFICIENCY: 6 ALTERNATIVES _ STANDARD ..... APPROVAL -..- BORDA (ESTIMATED) Figure 4.27 Sincere Condorcet Efficiency: 6 Alternatives 89+ 79+ so. .0... I O ...-o... I a...’.uc.~.‘..‘..o' I 00-” 684 SOW'I'I'I‘I‘I'I‘ STRATEGIC CONDORCET EFFICIENCY: 6 ALTERNATIVES _svmmmuu) ...... APPROVAL ___, BORDA (ESTIMATED) Figure 4.28 Strategic Condorcet Efficiency: 6 Alternatives 112 4.4 .... - ' '-. -.. . '- ' ’ ' ’-¢ The relationship between Condorcet efficiency and social utility efficiency was discussed in Chapter 2. An attempt to quantify this relationship more precisely was made by running simple linear regressions (OLS) of the form SCON = A + B1(SSU) + B2(ALTS) + BB(V), where SSU is strategic social utility efficiency, SCON is strategic Condorcet efficiency, ALTS is the number of alternatives, and V is the number of voters. This regression was run for each voting system. The results of the regressions are presented below, with Figures 4.29-4.31 showing estimated and actual strategic Condorcet efficiencies. Table 4.1 Regression Results for Strategic Condorcet Efficiency Borda_fixatem Dependent Variable: SCON Mean of Dependent Variable 86.898530 Standard Deviation 4.166679 Sum of Squared Residuals 159.787800 Standard Error of Regression 1.998673 Number of Observations 44 R2 .785960 Variable Estimate Std. Error T-Statistic Intercept -30.9682720 22.8332530 -1.3562795 ALTS -4.6555093 .6553843 —7.1034801 V -.0138721 .0078597 —1.7649777 SSU 1.5771413 .2558077 6.1653391 Standard_fiy§tem Dependent Variable: SCON Mean of Dependent Variable 81.270320 Standard Deviation 9.219158 Sum of Squared Residuals 472.696268 Standard Error of Regression 2.372201 Number of Observations 88 32 .936074 113 Variable Estimate Std. Error T-Statistic Intercept 29.9295465 11.3396821 2.6393638 ALTS -4.0469293 .3360611 -12.0422427 V -.0876380 .0101957 -8.5955649 SSU .9422485 .1261226 7.4708959 Annrnxal_fixstem. Dependent Variable: SCON Mean of Dependent Variable 76.712010 Standard Deviation 7.978904 Sum of Squared Residuals 699.152623 Standard Error of Regression 2.885004 Number of Observations 88 R2 .873769 Variable Estimate Std. Error T-Statistic Intercept -58.6598397 15.4701347 —3.7918118 ALTS -3.4314452 .3058458 -11.2195253 V -.0762712 .0083586 -9.1248403 SSU 1.8056124 .1722115 10.4848536 Note that in each of the regressions, strategic social utility efficiency has a fairly strong positive relationship with strategic Condorcet efficiency. In fact, strategic Condorcet efficiency can be predicted fairly well given the value of strategic social utility efficiency, as will be seen in Figures 4.29-4.31. Even so, values for strategic Condorcet efficiency estimated with regression coefficients are not too far off from the simulation values. Note also that the sign of the coefficients on both ALTS and V is negative in every case, as would be expected. The regressions do support the hypothesis of a strong relationship between the two efficiency measures. 114 3 alternatives 4 alternatives 5 alternatives 6 alternatives IIII‘ITUTIUI7I‘IIIYII IIII'YI'II'V‘I'II' IU'IIIIIIIII'I'I‘III VllT'VIITTYI'TIIIVIUI STRATEGIC CONDORCET EFFICIENCY: STANDARD VOTING SYSTEM ... _ACTUAL ....... ESTIMATED Figure 4.29 Actual and Estimated Condorcet Efficiency: Standard Voting System 6° 3 alternatives lo alternatives 5 alternatives IIIII‘IU'IIVIU'I'U IIIIIIVIIIUI'IIUU1li IIIU'I'I'IIIITIIIIUI' IIITITTTIIIU‘I'VIIII STRATEGIC CONDORCET EFFICIENCY: APPROVAL VOTING SYSTEM ___.ACTUAL 'ESTIMATED Figure 4.30 Actual and Estimated Condorcet Efficiency: Approval Voting System 115 Estimated with quation 2 Estimated with . equation 4 ' 7° 3 alternatives 4 alternatives 5 alternatives 6 alternatives TT‘IIIIYUVIIITII IIIIII'|"|'I"IIII'|" STRATEGIC CONDORCET‘EFFICIENCY: BORDA SYSTEM __ ACTUAL ....... ESTIMATED Figure 4.31 Actual and Estimated Condorcet Efficiency: Borda Voting System 4.5 Wines In addition, regressions were run to allow prediction of efficiency measures for these systems when the number of alternatives is greater than is feasible to simulate. The variables used for sincere efficiency measures were THEO, DIF, MEAN, and VAR. THEO is the theoretical social utility efficiency value. DIF is a measure of the difference between the actual distribution of total votes and the normal distribution which total votes approach as the number of voters increases. DIF is defined as the difference between the normal distribution standard deviation and the actual standard deviation divided by two times mean votes 116 for the voting system. MEAN and VAR are the mean and variance of total votes for the voting system. The variables used for strategic efficiency measures were THEO, DIF, l/P, Q“(ALTS/(2*(ALTS+V))), and four powers of V. The new variables are functions of P, the probability of a tie, and Q, the number of admissible strategies. P is defined as Q(lVK/SD)-Q(-IVK/SD), where V is the number of voters, SD is the standard deviation of (Wi-Wk), and K is a constant term equal to the maximum weight assignment of the voting system. Q has the above functional form to display the following characteristics: as V gets large, the effect of Q decreases, and as the number of admissible strategies Q increases, the damping effect of V decreases. The numerical results of these regressions are presented in appendix D. The results were used to forecast values for efficiency measures for 7 alternative elections, which are shown in Figures 4.32-4.35. 117 95 99- 85 ................. 89+ 78- 63- 55 I ' I ' I ' I ‘ I ' I ‘ I I I ‘ I I ‘ I. SINCERE SOCIAL UTILITY EFFICIENCY: 7 ALTERNATIVES (ESTIMATED) _ STANDARD ....... APPROVAL ..-- BORDA Figure 4.32 Sincere Social Utility Efficiency: 7 Alternatives (Estimated) 78 I ' I ' I ' I ‘ r ' I ' I ‘ I ‘ I I I STRATEGIC SOCIAL UTILITY EFFICIENCY: 7 ALTERNATIVES (ESTIMATED) —STANDARD ....... APPROVAL ..-- BORDA Figure 4.33 Strategic Social Utility Efficiency: 7 Alternatives (Estimated) 118 70‘ 65 . 5m , ............ .. ........... I ‘ I ' A I ' I I I I I ‘ I ’ I ‘ I ‘ r SINCERE CONDORCET EFFICIENCY: 7 ALTERNATIVES (ESTIMATED) —STANDARD .......APPROVAL -..- BORDA Figure 4.34 Sincere Condorcet Efficiency: 7 Alternatives (Estimated) ‘fl‘! so, ' .......... STRATEGIC CONDORCET EFFICIENCY: 7 ALTERNATIVES (ESTIMATED) _.STANDARD ....... APPROVAL ..-- BORDA Figure 4.35 Strategic Condorcet Efficiency: 7 Alternatives (Estimated) 119 4.6 Winn A variation of the simulations was run to determine the effect of less than full information on efficiency estimates. It was assumed that the total votes of 2/3 of the voting population (to the nearest integer) were known to all voters, who also knew whether or not they were included in the group. Unfortunately this provided no useful information because with more than approximately 7 voters, efficiency estimates were practically identical to those for sincere voting (the zero information case). This is due primarily to voters’ knowledge of the distribution from which individual utilities are drawn. Because of this, voters cannot treat the sample total vote vector as a random sample from population total votes and assign corresponding probabilities or expected values to the unknown votes. The unknown votes continue to have the known (approximately multivariate normal) distribution as under zero information, adjusted for sample size. Therefore, if 3 or more voters are not in the sample, the variance of population total votes is large enough to discourage most strategic voting (as in the analytical example, p. 69). The results that were obtained are presented here for the sake of completeness, although the simulations were aborted when it was apparent that the level of information was not large enough, given the structure of the model, to provide information on the movement of sincere estimates 120 toward strategic estimates as the information level increases. Table 4.2 Incomplete Information Efficiency Measures for 3 Alternatives and 3 Voters Wm Voters 80 SSU CON SCON 3 81.28015 85.64013 89.9151 92.43286 5 73.89849 74.64334 81.86399 81.4175 7 77.15541 77.15031 82.56971 82.80641 9 78.36027 79.40004 81.01244 81.60281 11 76.51844 75.74441 80.96372 80.25922 13 76.13150 75.44744 80.24274 80.33026 15 75.09851 75.92546 78.97618 80.12462 17 75.06600 73.67316 80.62215 80.56885 19 76.10229 76.65386 80.56049 80.69643 21 76.26308 75.64381 79.39276 78.56687 23 76.66461 76.86256 78.59663 78.81081 25 75.36212 75.63785 79.0997 79.17628 W Voters 80 SSU CON SCON 3 84.59656 84.38637 73.80964 74.48530 5 85.49946 84.88809 76.43347 75.29308 7 84.75676 85.13169 75.97089 75.95857 9 86.40394 86.43179 73.26244 73.43390 11 87.68660 87.51424 75.68910 76.54454 13 88.60281 87.75786 74.57857 74.97629 W Voters SU SSU CON SCON 3 87.47958 85.14944 97.28929 94.75961 5 86.7208 86.7208 93.3808 93.3808 7 85.8735 85.8735 92.6008 92.6008 9 86.7582 86.7582 92.7946 92.7946 11 86.8360 86.8360 93.3957 93.3957 13 85.4114 85.4114 90.2788 90.2788 15 88.1715 88.1715 91.4930 91.4930 CHAPTER 5 DISCUSSION AND SUGGESTIONS FOR FURTHER RESEARCH Perhaps the most pertinent question which can be addressed to this research is why it is of any interest to compare multi-alternative voting systems with strategic voting. After all, any voting system has 100% Condorcet efficiency with only two alternatives, regardless of whether sincere or optimal strategies are assumed. Additionally, most of the voting situations in which there are more than two alternatives occur with large electorates, where the possibility of strategic voting is more or less precluded. However, there are two points to keep in mind. First, a series of sequential pairwise votes on the same issue implies more than two alternatives, and this occurs frequently in committee voting. Second, we know that increasing the number of alternatives decreases the likelihood of a Condorcet winner, and as appealing as the Condorcet criterion is, that means that we disregard those cases where extreme conflict occurs (no Condorcet winner exists). We also know that maximum social utility efficiency of a two alternative election is 81.65%. Thus, we must expect social utility efficiency to decrease with every step in a sequence of pairwise votes. Multi-alternative elections are an option to be compared with a sequence of pairwise votes. Condorcet efficiency is the appropriate comparison measure for this purpose. However, different multi-alternative voting 121 122 systems can also be compared to each other using both Condorcet efficiency and social utility efficiency. Given this rationale, it is important to differentiate between sincere and strategic efficiency measures. Strategic efficiency measures are more appropriate because they recognize maximizing behavior on the part of individuals. 5.1 EffisisaCLMsasaathaasssJiiLSiaatssisLlctias The striking difference in the way social utility efficiencies change for a given voting system is not very difficult to explain. Recall that with the assumption of strategic voting, standard voting system social utility efficiency increased markedly, while for the approval and Borda systems it decreased, particularly for small electorates. However, in the standard voting system, for strategic voting to occur, some alternative must be ranked first by as large or nearly as large a percentage of the voting population as the winning or tied alternative. The individual who changes the outcome increases his utility by doing so; the voters who had ranked the strategic voter’s more preferred alternative as first gain, while those who had ranked his less-preferred alternative as first lose. The other voters' losses and gains essentially balance each other out, with the gain of the strategically voting individual being the predominant effect. In contrast, for both the approval and Borda systems, strategic voting can occur if there is an alternative which is ranked as high or 123 nearly as high Qn_a1erage as the winning or tied alternative. These characteristics are combined with the fact that you can't "go around in circles“ in the standard voting system. Strategic voting is an all or nothing proposition. Suppose two alternatives are vying for first place, and an individual changes his vote from his most preferred alternative to his more preferred of the two vying for first place. At that point, there is nothing more he can do to change the outcome, and he has reduced or eliminated the possibility of strategic voting on his most preferred alternative. In the Borda system he would have the option of ‘removing' votes from the less preferred alternative, which would increase the total of some 3rd alternative and the possibility of strategic voting on it. In the approval system he can either remove a vote from the less preferred alternative, or add one to the more preferred alternative, but this does not prevent yet another voter from adding or subtracting a vote without affecting his most preferred alternative. In other words, strategic voting in the Borda or approval system may entail changes in total votes which can cause other strategic (insincere) voters to change their minds. In the standard voting system, the total of the 3rd alternative can only decrease. It is easy to show that expected social utility of the standard voting system should increase and expected social utility of the Borda system should decrease with strategic _¢—- "-..—.4- 124 voting. Let Wk be the maximum of total votes with sincere voting, and W1 be within range of winning. Then [Wk-W1! ; k, where k is the maximum weight assignment of the system. Standazd_lgting_fiystem; The value of k is 1. Any strategic voter either makes or breaks a tie, and the adjusted total votes are such that {Wk-W1] ; 1. Let Ni>k be the number of voters who prefer i to k, and Nk>i be the number that prefer k to i. There continues to be an incentive for strategic voting until min(Wi,Wk) = min(Ni>k,Nk>i) and either WiiWk or Wi+Wk=N. However, for an odd number of voters, this implies that a majority of the voting population prefers the winning alternative after strategic voting to the contending alternative, and the change in expected social utility is positive if the outcome is different after strategic voting, and zero if the outcome remains the same. Borda_lgting_fiystem: The change in expected social utility from a change in outcome from i to k with strategic voting is E(Zj(uij-ukj). Let rj(i) be an individual voter's ranking of alternative i, and E(i) be the average rank across the voting population of alternative i. Given that individual utilities are i.i.d. uniform [0,1] variables, E(uij-uik)= (r5(k)-r5(i))/(m+1), and E(25(u15-uik))= n(z(i)‘n(k))/(m+1). However, we know that rj(i)=(m-Wij), where wij is the sincere vote. Using this information, we obtain E(Zj(uij-uik))= (Wi-Wk)/(m+1). But Wk 1 W1, so the change in expected social utility with strategic voting is negative or zero. 125 5.2 Innllcnlinnfi_9£_1h§_fls&nllfi The first major implication of the results is that strategic voting can increase efficiency measures of a voting system. Manipulability of a voting system is not necessarily an undesirable characteristic. It should be pointed out that the voting system which is least manipulable (the standard voting system), is the one which showed the most dramatic increase for both Condorcet efficiency and social utility efficiency. However, the fact remains that strategic voting can actually produce a “better" outcome. Unless a fairly high level of information is available to voters, rankings according to sincere efficiency estimates are correct. Without nearly complete information, the incentives for strategic voting disappear, and estimates approach their sincere counterparts. Similarly, with large electorates (>125 voters) the advantages of strategic voting disappear, although for the standard voting system, strategic Condorcet efficiency can still be significantly greater than sincere Condorcet efficiency. Second, when optimal strategies are used by voters, differences between voting systems are not as clear-cut for small electorates. For very small voting populations, efficiency estimates for all three voting systems fall within a very small range when the number of alternatives is 4 or less. The advantages of using the approval or Borda system as opposed to the standard voting system are not as 126 large as previous work has indicated for these situations. Again, however, for large electorates or less than nearly complete information, the conclusions of previous work hold. Third, multi-alternative voting decreases Condorcet efficiency, but fairly high efficiencies are still obtainable if strategic voting is assumed. The cost of repeated (sequential) pairwise votes may be large enough relative to a single multi-alternative election to justify multi-alternative voting in committees. 5.3 LiaiiatiaauajasLEauilihaiaflaad In estimating efficiency measures, the use of the first equilibrium found (sincere voting, if it is a Nash equilibrium) is based on two points. The model is designed to approximate as closely as possible to previous work, which has always assumed the use of sincere strategies by voters. The cases which differentiate the current research from previous work are those in which sincere voting is not a Nash equilibrium. A base vote matrix is necessary in solving for equilibria, and the sincere vote matrix is the simplest and most logical choice. Again, there is no reason to assume that individuals' strategies will change unless a gain in expected utility can be achieved. Therefore, if sincere voting is a Nash equilibrium, it is the equilibrium used. The algorithm does not go on to find all equilibria after the first both because of the number of equilibria that exist (regardless of the number of voters) and because 127 asymptotically this approach is incapable of differentiating between voting systems. Recall theorem 1, which says that asymptotically sincere strategies are a Nash equilibrium. The theorem implies that asymptotically, any set of strategies is a Nash equilibrium. If one assumes that equilibria are equally likely, then as the voting population increases, the voting system degenerates to a random choice. Some restriction of equilibria is necessary in order to differentiate between voting systems. For small electorates, efficiency estimates do differ when all equilibria are found. Table 5.1 presents these estimates for the 3 voter 3 alternative case (1000 repetitions). However, it is clear that efficiency estimates increase as the number of strategy profiles which are not equilibria increases. Table 5.1 Summary Statistics for the 3 Voter 3 Alternative System When All Nash Equilibria are Found Approval Borda Standard 3 possible strategy profiles 216 216 27 mean # equilibrium strategy profiles 18.297 29.544 6.111 % profiles which are equilibria 8.47 13.68 22.63 % profiles which are not equilibria 91.53 86.32 77.37 social utility efficiency (%) 97.7247 91.3310 55.5714 Condorcet efficiency (%) 75.7878 75.1140 58.7851 128 Finally, the use of sincere voting as the equilibrium each time it is a Nash equilibrium is supported by the concept of bounded rationality as expressed in perfect equilibria [123]. Each voter has some probability 61 for the breakdown of rationality. When this occurs, he will use each admissible strategy S1 with probability q , and Q Si 21:1 q = 1. 51 Theorem_3; If sincere voting is a Nash equilibrium, it is a perfect pure strategy equilibrium. Erggf; Let Q be the number of admissible pure strategies in the voting game. Qn is the set of admissible strategy profiles. Let 81 be a representative voter's sincere strategy, with 32 being any other admissible strategy. N(Si,Sz) E Qn is the subset of admissible strategy profiles for which the expected outcomes of the two strategies differ. For any profile k E N, p: is the probability that this profile occurs in the perturbed game. pi is a function of all voters' C1 and q vectors, and ZkeQn p: = 1. Then EU(81) - EU(Sz), the difference in expected utility of strategies 1 and 2, is equal to Zken p:(u1 - Uj) (1) As s-+O, all p: (k e Qn) approach either 0 or 1 (the degenerate distribution of the complete information case). Then there are two possible cases: either p: approaches one for a profile for which the expected outcomes of the two 2 strategies do not differ, or pk approaches one for k e N. a Case 1: pk +1, k i N. 129 For any sequence of s in which qs1 = qss = 1/Q for each voter, the maximand is a positive linear transformation of the zero information game maximand. As shown in Chapter 3, sincere strategies are optimal. Case 2: pic->1, k e N. (a) k 3: p:-+1 has u1 > us: Zken p:(u1-uj) > 0 :> strategy 61 is optimal. (b) k a: p§-+1 has u1 < UJ: Zken p:(u1-uj) < 0 :> Skew pk(u1-uj) < 0. However, this implies that strategy 81 was not optimal in the original game, and sincere strategies were not a Nash equilibrium, which is a contradiction. Therefore, if sincere voting is a Nash equilibrium, it is a perfect pure strategy equilibrium. It is also clear that for this sequence of e, the equilibrium is the only one that will be found. 5.4 QatimalitLEnaenisaMmaaaianaJlaaanna The optimality properties of the comparison measures used depend on the decisions being made by using a voting system. Two classic situations in which voting systems appear to be reasonable methods of choice are a) determination of the level of a pure public good to be produced; and b) choice of an allocation of "resources” along a Pareto-frontier. 5.4.1 Qhnina_Qf_Lhn_Lax§l_nfi_a_2nrs_2nhlin_finnd The condition for Pareto-optimal provision of a pure public good was derived by Samuelson in 1954 [119]. A pure 130 public good has the property that it is consumed simultaneously by all individuals in its entirety. The Samuelson condition is 21 MRSGX = MRTGx, where MRSCX is individual i's marginal rate of substitution of the private good X for the public good G, and MRTGX is the marginal rate of transformation of X for G. Intuitively, "...at the optimum, the marginal cost of supplying the last unit of G in terms of X foregone just equals the sum of the marginal benefits that all users of the increment G simultaneously obtain in terms of X.“1 Since individual marginal benefits are equally weighted, this is identical to maximizing social utility in terms of a utilitarian social welfare function. 5.4.1.1 Sgcial Utility Efficiency and Optimality in the E i . E E E 11' G I A voting system which maximizes social utility in terms of a utilitarian social welfare function will produce Pareto-optimal outcomes when used for decisions about the level of pure public goods to be produced. Social utility efficiency measures the "closeness" of outcomes of a voting system to maximum social utility, and is the ratio of the expected social utility of the outcome to the expected maximum social utility over the alternatives. If this ratio is equal to one, then the voting system being evaluated is expected to produce a Pareto-optimal outcome. Given the same variance, a voting system with lower social utility 131 efficiency will be expected to achieve a Pareto— optimal outcome less frequently. One difficulty is that the variance of social utility efficiency does not remain constant across voting systems. A better measure might be the frequency with which a voting system is expected to attain maximum social utility, but the same problem surfaces that occurs with Condorcet efficiency: there is no differentiation between social-utility outcomes which do not attain the maximum. Given this problem, the social utility efficiency measure used is a reasonable compromise. Because it does reflect to some extent the probability of Pareto-optimal provision of a pure public good, a voting system with greater social utility efficiency than another is in some sense "better." Because social utility efficiency reflects a random individual’s expected utility of a voting system’s outcome, a further insight into the optimality properties of this measure can be gained. Each alternative (level of the public good) X has a corresponding mean utility level across the population, u(x), which is the expected utility of that level to a randomly chosen voter. Conversely, an expected utility of the voting system's outcome implies one or more expected outcomes (the inverse function is not generally single-valued). If expected utility for the average (mean) voter is single peaked and symmetric about its maximum Xp, the Pareto-optimal level (Figure 5.1a), then as expected utility increases, the level of under- or over-provision of 132 the public good, :Xp - Xl, decreases, i.e. the level actually produced is closer to the Pareto-optimal level. Both single-peakedness and symmetry are necessary conditions for this conclusion, however. In Figures 5.1b and 5.10, an increase in expected utility does not necessarily move the level of provision of the public good closer to the Pareto-optimal level. expected utility fl, Xp output of public good X. 5.1a. Symmetric and single-peaked mean expected utility 133 expected utility r--——-_-—_ ~3> output of public good X. N H N O J‘ 5.1b. Symmetric and non-single-peaked mean expected utility IXp — X| may increase; |Xp - Xol < IXp - X1|. ex ected ut lity “1 ————— no -‘-‘-‘-- - n l a -.1....- u I I t I I I a- 0 x? x1 output of public good X. xs-—-—- 5.1c. Asymmetric and single-peaked mean expected utility. |Xp - X] may increase; IXp - X0] < lXp - X1]. Figure 5.1. Mean Expected Utility and Corresponding Levels of the Public Good (G) Produced. 134 5.4.1.2 QQadnzaa1_EiiiQiancx_and_QnLimalill_in_Lha We. Bowen [15] showed that the Condorcet winner (median voter equilibrium) is a Pareto-optimal outcome if the median voter is also the mean voter. Since the median voter is decisive in his model, and the equilibrium point is the median voter's most preferred level of the public good, if the median coincides with the mean then the mean voter also has a utility-maximizing outcome. In algebraic terms, Jim“ = t = has. N N or the average marginal rate of substitution of money for the public good G is equal to the marginal tax rate, which in his model is an equal share of the marginal cost of production of the public good. Under this condition, then, the Condorcet winner is a Pareto-optimal outcome. However, the existence of a Condorcet winner does not require single-peakedness of preferences, nor if preferences are single peaked does the mean peak preference necessarily coincide with the median. Without these assumptions, the Condorcet winner need not be a Pareto-optimal outcome in choosing the level of provision of a pure public good. 5.4.2 ChaiaLAlnaLahntnEnatiar The second situation in which voting systems are of interest to an economist is the situation of choice along a Pareto frontier. Using lump-sum taxes and transfers, the government can attain alternative points along the grand utility possibility frontier. When choosing an allocation 135 of resources along a Pareto-frontier, the general guideline is that the allocation which maximizes social welfare should be chosen. A widely-used formulation of the social welfare function [140] is {25 [uijT]}1/T T a 1; T i 0. If T = 1, then we are using a utilitarian social welfare function. 5.4.2.1 SQcial_fl1ilitx_Efficisncx_and_flhnice_Alnng_a W The social utility efficiency used by Weber and Bordley, among others, is a transformation of a utilitarian social welfare function. If indeed a society has a social welfare function for which T = 1, a voting system with higher social utility efficiency will be expected to produce outcomes of greater social welfare and will be in some sense a "better" voting system. Additionally, if in fact T ¢ 1, social utility efficiency measures can easily be constructed which use different values of T. If an estimate of T can be obtained, then a social utility efficiency measure can be constructed which will rank possible voting systems appropriately. 5.4.2.2. QnndorneI_Efficiencx_and_flhnice_AlQns_a Earetanrnntier As mentioned previously, when it exists, the Condorcet winner has maximum expected social utility. Therefore a voting system which is expected to choose the Condorcet winner with greater frequency when it exists might also have 136 greater expected social utility. Rankings of voting systems obtained by using Condorcet efficiency have agreed with those obtained with social utility efficiency when voters use sincere strategies. Unfortunately, because the Condorcet efficency measure does not differentiate between outcomes in cases where there is no Condorcet winner, no correspondence between the two measures can be shown unless preferences are restricted so that a Condorcet winner always exists. 5.4.3 MW Condorcet efficiency does have one implicit equity consideration. If the Condorcet winner is chosen, at least a majority of the voting population prefer it to any other alternative. Also, the Condorcet winner tends to have high social utility. The converse is not true. Social utility efficiency does not imply anything about equity. 5.5 W11 5.5.1 W A cost of voting is not included in the model used for the simulation. Tullock and Downs [141],[37] both concluded that "voting is an irrational act in that it costs more to vote than one can expect to get in return."2 An estimate of voting costs appropriate to a comparison of voting systems is presented below. The expected utility of voting is: EU = (ui-uj)pij - c 137 where (ui-Uj) is the gain in utility to the voter if alternative i defeats alternative j as a result of his vote; p15 is the probability of this occurring, and c is the cost to the individual of voting. Since pij approaches zero rapidly, and c is generally assummed to be positive, (ui—uj) must be of extreme magnitude for voting to be a rational act. Once again, let p be the probability that an individual voter is decisive. P depends upon the size of the voting population. Now, where a cost of voting is included, complete participation cannot be assumed. Voter participation will depend upon whether the expected utility of voting is positive, which in turn is based on the individual voter's estimate of p. The question of the ”rationality" of voting is not therefore as clear-cut as would appear on preliminary examination. Using a model in which p and n are determined simultaneously, Palfrey and Rosenthal find that substantial voter turnout can be consistent with the inclusion of a cost of voting. Their model uses only two alternatives; however, increasing the number of alternatives would, under the assumptions presented at the beginning of this chapter, only increase p, making substantial participation more likely. Thus the inclusion of a cost of voting is consistent with the rationality assumptions employed. However, because the purpose of this work is to compare voting systems, a determination of the possibly differential 138 costs of voting for different systems is necessary. Voting involves not only a fixed cost of taking the time to go to the polling place and vote, but the cost of determining which strategy (vote vector) to use. Strategy determination costs clearly vary with the level of information the individual has, since as discussed previously, under zero information conditions, sincere voting is the unique optimal strategy. However, even under zero information conditions this cost will vary across voting systems because of the amount of information ‘requested’ from the voter. The standard voting system asks only for the voter's most- preferred alternative; the approval voting system required identification of all alternatives with above-average utility; and the Borda voting system requires a full ranking of all alternatives. Let the individual cost of voting be approximated by c1 = ai + f(s[C],a) 0 g a g l where ai is some fixed cost to the individual voter i of taking the time to go to the polling place, s[C] is the number of possible strategies in the strategy set of the voting system or choice rule C, a is the information revel voters are assumed to have, and f(s[C],a) represents the cost of optimal strategy determination. Individual voting costs may differ due to ai, which may be modelled as a random variable. Given this determination of the individual cost of voting, an equilibrium in p and n can be determined. Palfrey and Rosenthal's model, however, finds multiple 139 equilibria in p and n, and there are no strong predictions about voter turnout. Their model used only two alternatives, so that an extension of this model would be necessary prior to drawing any conclusions about voter turnout. It is also highly likely that such an extension would produce multiple equilibria in p and n for small voting populations. However, the multiple equilibria problem could be handled as it has been here, with Monte Carlo techniques. At this point a pertinent consideration would be the administrative, or social costs of the voting system. Once individual strategies (including abstention) are determined, even if the equilibrium outcome of the election is known by the modeler, there is still the problem of "counting votes." Again there are differences between voting systems in this regard. The factor which immediately appears significant is the number of elements in the vote vector to be tallied. Let the social cost of the voting system be 03(8) 2 n(S) x k(S),where n(S) is expected participation in the voting system as determined above, i.e. the number of ballots completed, and k(S) is the number of positive elements in an individual vote vector. k(S) would of course be one for the standard voting system, m/2 for the approval voting system, and m-1 for the Borda system. Given this information, appropriate efficiency measures, based on the expected net social cost (= expected social utility of chosen alternative - 21 Ci - social cost) can be constructed for comparison of 140 these voting systems. Given the difficulties, extending the model to include a cost of voting at this time would probably not produce any useful results. 5.5.2 Qther_Ednilihriai_Ihs_flgmpstitixe_fiolntion The possibility of modeling voting systems as cooperative games has not been overlooked. “Cooperative game theory for the most part focuses on games with transferable utility, even though...this assumption excludes the possibility of modeling most interesting political coalition processes. For the more general case, though, standard solution concepts are inadequate because they are undefined or they fail to exist, and even if they do exist, they focus on predicting payoffs rather than the coalitions that are likely to form."3 Thus values such as the Shapley value or the Banzhaf-Coleman index of power, which have been widely used to estimate, for example, the "coalitional" value of states in a 0.8. presidential election game, cannot be used to compare different voting systems, as the only information which they can provide is on the "coalitional" value of the players and not on outcomes. McKelvey, Ordeshook and Winer [90] propose a different solution concept entirely, the competitive solution for games without transferable utility. The solution concept hypothesizes that "potential coalitions must bid for their members in a competitive environment via the proposals they offer. Given that several coalitions are attempting to form simultaneously, each coalition must if possible, bid 141 efficiently by appropriately rewarding its "critical“ members."4 Let A be the set of feasible outcomes. Then for any coalition C, v(C) = A if C is winning and v(C) = 9 if C is losing. Thus if there is a majority voting game and C is a majority coalition, v(C), in a repeated game, is "the set of all possible dispositions of all bills."5 A coalition's proposal is their policy platform; in their work a coalition's proposal is an ordered pair (u:C) such that u is an element of v(C) and u is an element of v(N). Then given two proposals, the coalition's proposal (u1:C1) is viable against the proposal (u2:Cz) if u1 ;,u2 for all individuals belonging to both coalitions (i 6 C1 n Cz). Let K be any set of proposals. (uzC) is viable in K if it is viable against all proposals in K. K is balanced if each coalition can have exactly one proposal, and all proposals in K are viable against each other. Of course, there may exist many distinct balanced sets of proposals. McKelvey, Ordeshook and Winer focus on the class of proposals in which the coalitions represented “make offers that are as attractive as possible to their respective critical members.”6 A proposal upsets a set of proposals K if it is a viable proposal in K and there is an alternative proposal (u':C’) in K for which u > u' for all individuals belonging to both coalitions. A set of proposals K is a competitive solution if K is balanced and there is no proposal (uzC) that upsets K. This ' 142 implies that the coalitions represented in K do indeed make offers that are as attractive as possible to their critical members. A stronger definition of "balanced" allows them to exclude coalitions greater than minimal winning size. K is strongly balanced if it is balanced and there are no two proposals (u1:C1) and (u2:C2) for which u1 ; uz, with strict inequality for at least one i, for all individuals belonging to both coalitions. If K is a competitive solution and strongly balanced, the authors refer to it as a "strong competitive solution." The competitive solution does predict vote trading; in one example the authors show that none of the proposals in the unique competitive solution correspond to the outcome of sincere voting. Additionally, a preliminary test or empirical validity found impressive correspondence between actual outcomes and the competitive solution's predictions. The predicted coalitions all formed at least once, and no other coalitions formed. As a solution concept this is very attractive. Not only do the conditions of the solution have intuitive appeal, but they can be placed in the familiar context of committee voting, as for example in Congress. Different voting systems such as the approval and Borda voting system can be analyzed, and the competitive solution predicts different size coalitions with each because of the different requirements for a winning coalition. However, some assumption about the likelihood of coalitions must be made 143 to get any prediction on expected outcomes in order to compare different voting systems. 5.5.3 MAW Because equity considerations are ignored in the utilitarian social welfare function, it would be useful to see if another formulation (T ¢ 1) would produce any changes in rankings of voting systems. Certainly if equity is important to the choice of a voting system, the utilitarian social welfare function is not the appropriate comparison measure to use. 5.6 Cnnnlusion It has been shown that the use of optimal strategies by voters as opposed to sincere strategies can significantly change both social utility and Condorcet efficiency estimates for multi-candidate voting systems. Furthermore, the changes in Condorcet efficiency estimates change the rankings of the voting systems when the voting population is small. The standard voting system is seen to achieve the highest Condorcet efficiency, followed by the Borda system, with approval voting ranked last. V. AL' 'I- APPENDICES APPENDIX A SINCERE VOTING AS A NASH EQUILIBRIUM WITH AN INFINITE VOTING POPULATION Theorem 1: As the voting population becomes large, i.e. n-+m, the probability that sincere strategies constitute a Nash equilibrium approaches one. A.1 Proof: Borda System For the individual voter, any Wij is a random variable with u = (m-1)/2, 02 = 22;: wZ/m - [(m-1)/2]2. Then W1 : Ej Wij is distributed approximately normally with mean n(m-1)/2 and variance n[2:;: wZ/m - ((m—1)/2)2], and the W1 have an approximate multivariate normal distribution. In order for an individual voter to change the outcome of the system, there must be some [W1 - Wk] 3 m-1. That is, the voter's maximum weight assignment of m-l can cause the ordering of two totals to change. Let Y = W1 - Wk. Then Y 2 2 2 has a mean py = pw - pu = O; and variance 0y = Ow + Cu + 1 K 1 k 20w w . Because of the relationship between the covariance andlczrrelation coefficient this variance can be computed exactly; the correlation coefficient is -1/(m-1). Intuitively, when one of the W1 is above its mean, the others are expected to be slightly below the mean. Computing this, a variance of a: = 2n([2:;: wz/m] - (m-2)(m-1)2/2m] - 2/(m—1) is obtained. Obviously, as n-1m, the variance of Y becomes infinite. Therefore P{'W1 — Wkl ; m-l} : P{-m+1 g Y ; m-1}, the probability that Y falls within the specified interval, 144 145 approaches zero. Thus scope for strategic behavior diminishes asymptotically and the probability that sincere strategies constitute a Nash equilibrium approaches one. A.2 Proof: Approval System For the individual voter, any Wij is a binomial random variable (either a vote is cast for it or not), with p = 1/2. Then W1 2 23 W15 is distributed approximately normally n/Z and variance np(1-p) = n/4, and the W1 with mean np have an approximate multivariate normal distribution. In order for an individual voter to change the outcome of the system, there must be some [W1 - Wk| ; 1. That is, the voter's maximum weight assignment of one can cause the ordering of two totals to change. Let Y 2 W1 - Wk. Then Y 2 2 2 has a mean py = pw - pw = 0; and variance 0y = Ow + 0w + 1 k 20w wk. Because of the relationship between the covariance andicorrelation coefficient this variance can be computed exactly; the correlation coefficient is -1/(m-1). Intuitively, when one of the W1 is above its mean, the others are expected to be slightly below the mean. Computing this, a variance of a: = [2n(m-1) - 8]/4(m-1) is obtained. Obviously, as n->m, the variance of Y becomes infinite. Therefore P{|W1 - Wkl g 1} = P{-1 g Y 3 1}, the probability that Y falls within the specified interval, approaches zero. Thus scope for strategic behavior diminishes asymptotically and the probability that sincere strategies constitute a Nash equilibrium approaches one. APPENDIX B SIMULATION PROGRAMS VARIABLES USED ALTS - number of alternatives used CHOOS - randomly chosen voter for the reordering COMMON - number of elections for which there is neither a Condorcet winner nor a pure strategy Nash equilibrium (always = 0) ; COMP - indices of alternatives within “reach" of the winner; [ those which need to be compared for strategic voting F CVOTES(6) - Condorcet votes CWINNE - Condorcet winner F - indicator of strategic voting G - number of tied alternatives G1(720) - vector of strategies with maximum expected utility G2 - number of tied strategies H - loop counter I - loop counter for alternatives J loop counter for voters K randomly chosen alternative for breaking ties L - loop counter LAST - loop counter for sorting by rank M - loOp counter for elections MONE - loop counter for random reorderings of voters N - loop counter for determining expected utility of strategies NCOND - number of elections for which the Condorcet winner is chosen by sincere voting NONASH - number of elections for which a pure strategy Nash equilibrium is not found (always = 0) P - loop counter for repetitions of 100 election simulations PVOTE(6,721) - matrix of admissible strategies within the voting system Q - number of admissible strategies for the voting system RANK(6) - the vector contains the index of the alternative in the specified rank for an individual voter SIN - number of elections for which sincere voting is not manipulable ' SNCOND - number of elections for which the Condorcet winner is chosen by strategic voting STRAT - strategy which maximizes expected utility for a given voter TEMPR - holding variable for sorting by rank TIED(6) - the vector contains the indices of the tied alternatives TOTAL(6) - total number of votes accruing to specified alternatives TRANK(6) - rank ordering of total votes 146 147 TVOTE - holding variable for random reordering of voters VOTE(6,125) - individual votes VOTERS - the number of voters VOTES(6) - total number of votes accruing to specified alternatives WINNER - the alternative chosen by the voting system WINS(6) - number of alternatives beaten in pairwise races by the specified alternative VMAX - maximum number of votes accruing to any alternative Z - 2,147,483,647: used in random number generation DSEED - current seed value for the random number generator EELECT - total social utility of all winners chosen by sincere voting for a voting system EFFIC - social utility efficiency with sincere voting EMAX - maximum social utility over alternatives EU(721) - expected utility of an admissible strategy NOCC - number of elections without a Condorcet winner _ NUM - number of elections with a Condorcet winner 5 RUTIL(6) - holding variable for sorting by rank SCEFFI - strategic Condorcet efficiency SEELEC - total social utility of all winners chosen by strategic voting for a voting system SEFFIC — strategic social utility efficiency SOCUT(6) - vector of social utilities of alternatives TEMPU - holding variable for random reordering of voters TOTUT - total utility of all alternatives in an election; divided by the number of alternatives, the expected utility of the election if the specified voter does not participate UTIL(6,125) - matrix of individual utilities UTMAX - sum over elections of maximum social utility M1 - mean Condorcet efficiency with sincere voting M2 - mean strategic Condorcet efficiency M3 - mean social utility efficiency with sincere voting M4 - mean strategic social utility efficiency SD1 - standard deviation of Condorcet efficiency with sincere voting SD2 - standard deviation of strategic Condorcet efficiency SD3 - standard deviation of social utility efficiency with sincere voting SD4 - standard deviation of strategic social utility efficiency X - multiplier for random number generation Y - double precision value of Z W115. PROGRAM STANDARD (BORDA, APPROVAL) COMMON/PICK/ALTS,VOTERS,I,J,TOTAL,VOTES,LAST, +TVOTE,TEMPR,COMP,WINNER,TIED.G.K,TRANK. +SOCUT.VOTE INTEGER*2 ALTS,CVOTES(6),CWINNE,F.G,H,I, WES CEFFIC - Condorcet efficiency with sincere voting . F (ICC? NH 148 +J,K,L,LAST,M.N.NCOND,RANK(6),COMP,TOTAL(6),MONE, +SNCOND,STRAT,COMMON,P,SIN.Q,PVOTE(6,721),GZ, +TEMPR,TIED(6).NONASH,TVOTE,G1(720),CHOOS,TRANK(6), +VOTE(6,125),VOTERS,VOTES(6),WINNER,WINS(6),VMAX REAL CEFFIC,EELECT.EFFIC,EMAX,EU(721),NOCC,NUM, +RUTIL(6),SCEFFI,SEELEC,SEFFIC,SOCUT(6),TEMPU, +TOTUT,UTIL(6,125),UTMAX,M1,M2,M3, +M4,SD1,SD2,SD3,SD4 INTEGER*4 Z REAL*8 DSEED.X.Y DATA X/1.6807D4/ Z=2147483647 Y=DBLE(Z) ALTS=5 Q=5 The value of Q (the number of admissible strategies) depends on the voting system being simulated. For the Borda system, Q=ALTS!, while forthe Approval alts-1 system, Q=(21=1 2i) ADMISSIBLE STRATEGIES ARE DETERMINED DO 2 N=1,Q DO 1 I=1,ALTS IF(I.EQ.N)THEN PVOTE(I,N)=1 ELSE PVOTE(I,N)=O ENDIF CONTINUE CONTINUE 1 2 For the Borda and Approval Systems, admissible strategies are read from a file. The above lines are replaced with the following: Open(5,File='STRAA',Statusz’Old') Do 2 N=1,Q Do 1 I=1,ALTS Read(5,*)PVOTE(I,N) Continue Continue OPEN(4,FILE='RESULT',STATUSz’OLD') OPEN(3,FILE='SEED',STATUS='OLD') READ(3,*)DSEED DO 295 VOTERS=3,25,2 GOO GOO 000 000 000 0000 A 149 INITIALIZE LOOP VALUES M1=0. M2=0. M3=0. M4=0. SD1=0. SD2=0. SD3=O. SD4=0. DO 20 REPETITIONS OF 100 ELECTIONS DO 294 P=1,20 INITIALIZE LOOP VALUES SIN=0 COMMON=0 NONASH=0 NCOND=0 SNCOND=0 EELECT20. SEELECzo. EMAX=0. NUM=100. DO 100 ELECTIONS DO 286 M=1,100 UTMAX=0 DO 4 I=1.ALTS SOCUT(I)=0 WINS(I)=0 ASSIGN UTILITIES TO VOTERS FOR EACH ALTERNATIVE DO 3 J=1,VOTERS DSEED=DMOD(DSEED*X,Y) UTIL(I,J)=SNGL(DSEED/Y) SOCUT(I)=SOCUT(I)+UTIL(I,J) CONTINUE IF(SOCUT(I).GT.UTMAX)THEN UTMAX=SOCUT(I) ENDIF CONTINUE EMAX=EMAX+UTMAX DETERMINE CONDORCET WINNER BY MAKING ALL PAIRWISE COMPARISONS CWINNE=O DO 11 I=1,ALTS 10 H H0000 00000—- N 13 14 00000 150 D0 10 H=I+1,ALTS CVOTES(I)=0 CVOTES(H)=O DO 9 J=1.VOTERS IF(UTIL(I.J).GT.UTIL(H.J))THEN CVOTES(I)=CVOTES(I)+1 ELSE CVOTES(H)=CVOTES(H)+1 ENDIF CONTINUE IF(CVOTES(I).GT.CVOTES(H))THEN WINS(I)=WINS(I)+1 ELSE IF(CVOTES(I).LT.CVOTES(H))THEN WINS(H)=WINS(H)+1 ENDIF CONTINUE IF(WINS(I).EQ.ALTS-1)THEN CWINNE=I GOTO 12 ENDIF CONTINUE NO CONDORCET WINNER: SUBTRACT 1 FROM NUMBER OF ELECTIONS WITH CONDORCET WINNER NUM=NUM-1. ORDER UTIL(I.J) AND RANK(I.J) SO WE HAVE UTILITIES IN ORDER AND CANDIDATES IN ORDER BY RANK DO 17 J=1,VOTERS DO 13 I=1,ALTS RANK(I)=I . RUTIL(I)=UTIL(I,J) CONTINUE DO 15 LAST:ALTS.2.-1 D0 14 I=1,LAST-1 IF(RUTIL(I).LT.RUTIL(I+1))THEN TENPU=RUTIL(I) RUTIL(I)=RUTIL(I+1) RUTIL(I+1)=TEMPU TEMPRzRANK(I) RANK(I)=RANK(I+1) RANK(I+1)=TEMPR ENDIF CONTINUE CONTINUE CANDIDATES ARE RANKED FROM HIGHEST TO LOWEST. RANK(I) GIVES NUMBER OF CANDIDATE IN RANK I FOR VOTER J. ASSIGN VOTES (SINCERE) FOR STANDARD SYSTEM. DO 16 I=1,ALTS K=RANK(I) 16 17 0000000 151 IF(I.EQ.1)THEN VOTE(K,J)=1 ELSE VOTE(K,J)=O ENDIF CONTINUE CONTINUE For the Borda and Approval systems, assignment of sincere votes differs slightly. The above lines are replaced with the following: Borda: Do 16 I=1,ALTS K=RANK(I) VOTE(K,J)=ALTS-I 16 Continue 17 Continue Approval: TOTUT=0 Do 16 I=1,ALTS TOTUT=TOTUT+UTIL(I,J) 16 Continue TOTUT=TOTUT/ALTS Do 18 I=1,ALTS If(UTIL(I.J).GT.TOTUT)Then VOTE(I.J)=1 Else VOTE(I,J)=O Endif 18 Continue 17 Continue CALL COUNT IF(WINNER.EQ.CWINNE)THEN NCOND=NCOND+1 ENDIF EELECT=EELECT+SOCUT(WINNER) ASSIGN VOTES (STRATEGIC) FOR STANDARD SYSTEM IF NUMBER OF ALTERNATIVES WITHIN "REACH" OF WINNING IS NOT EQUAL TO 1, THE ELECTION IS MANIPULABLE. OTHERWISE DO NOT NEED TO CHECK STRATEGIES. SKIP TO LINE 65, P. 159. IF(COMP.NE.1)THEN DO 37 L=1,40 DO 36 J=1,VOTERS N=Q+1 EU(N)=0 G=1 VMAX=MAX(VOTES(1),VOTES(2),VOTES(3),VOTES(4), 000 26 000 .4 00000N 00000 29 152 +VOTES(5)) DO 26 I=1,ALTS IF(VOTES(I).EQ.VMAX)THEN ASSIGN INDEX TO TIED ALTERNATIVE TIED(G)=I G=G+1 EU(N)=EU(N)+UTIL(I,J) ENDIF VOTES(I)=VOTES(I)-VOTE(I,J) PVOTE(I,Q+1)=VOTE(I,J) CONTINUE EU(N)=EU(N)/REAL(G-1) STRAT=Q+1 D0 31 N=1.Q IE(TOTAL(1).GT.TOTAL(2))THEN D0 27 I=1,ALTS IF STRATEGY CAN CHANGE OUTCOME OF ELECTION IF((TOTAL(1)-VOTE(TRANK(1),J)+PVOTE(TRANK(1),N) +)-(TOTAL(I)-VOTE(TRANK(I),J)+PVOTE(TRANK(I),N)).LE.0) +THEN GOTO 30 ENDIF CONTINUE OTHERWISE EXPECTED UTILITY OF STRATEGY IS EQUAL TO EU OF CURRENT STRATEGY. SKIP TO END OF LOOP AND GO TO NEXT STRATEGY. EU(N)=EU(Q+1) GOTO 31 ENDIF DETERMINE EXPECTED UTILITY OF STRATEGY DO 28 I=1,ALTS VOTES(I)=VOTES(I)+PVOTE(I,N) CONTINUE VMAX=MAX(VOTES(1),VOTES(2),VOTES(3),VOTES(4), +VOTES(5)) EU(N)=0 G21 D0 29 I=1,ALTS IF(VOTES(I).EQ.VMAX)THEN TIED(G)=I G:G+1 EU(N)=EU(N)+UTIL(I,J) ENDIF VOTES(I)=VOTES(I)-PVOTE(I,N) CONTINUE EU(N)=EU(N)/REAL(G-1) 0000 0000 33 0000 00000) 153 IF EXPECTED UTILITY EXCEEDS EU OF CURRENT STRATEGY, CHANGE STRATEGY IF(EU(N).GT.EU(STRAT))THEN STRAT=N ENDIF CONTINUE 62:1 IF(STRAT.NE.Q+1)THEN F=F+l DO 33 N=1,Q IF MORE THAN ONE STRATEGY HAS MAX EU. CHOOSE ONE RANDOMLY IF(EU(N).EQ.EU(STRAT))THEN G1(G2)=N G2=G2+1 ENDIF CONTINUE DSEED=DMOD(DSEED*X,Y) K=INT(((SNGL(DSEED/Y))*(REAL(G2-1)))+1.) STRAT=G1(K) ENDIF REASSIGN VOTES IN ACCORDANCE WITH CHOSEN STRATEGY DETERMINE NEW TOTALS DO 34 I=1,ALTS VOTE(I,J)=PVOTE(I,STRAT) VOTES(I)=VOTES(I)+VOTE(I,J) CONTINUE CONTINUE IF NO STRATEGY CHANGES HAVE OCCURRED, NASH EQUILIBRIUM HAS BEEN FOUND. DETERMINE WINNER AND GO TO CALCULATION OF STATISTICS. IF(F.EQ.O)THEN CALL COUNT IF(WINNER.EQ.CWINNE)THEN SNCOND=SNCOND+1 ENDIF SEELEC=SEELEC+SOCUT(WINNER) GOTO 285 ENDIF F=0 CONTINUE HERE WE HAVE NOT REACHED THE NASH EQUILIBRIUM WRITE UTILITIES AND VOTES TO A FILE 13 14 15 000.5 000% 47 48 154 REWIND 11 DO 44 J=1.VOTERS DO 13 I=1,ALTS RANK(I)=I RUTIL(I)=UTIL(I.J) CONTINUE DO 15 LAST=ALTS,2.-1 DO 14 I=1.LAST-l IF(RUTIL(I).LT.RUTIL(I+1))THEN TEMPU=RUTIL(I) RUTIL(I)=RUTIL(I+1) RUTIL(I+1)=TEMPU TEMPR=RANK(I) RANK(I)=RANK(I+1) RANK(I+1)=TEMPR ENDIF CONTINUE CONTINUE DO 43 I=1,ALTS K=RANK(I) VOTE(K,J)=ALTS-I CONTINUE WRITE(11,*)UTIL(1,J),UTIL(2,J),UTIL(3.J),UTIL(4,J) WRITE(11,*)VOTE(1,J),VOTE(2,J),VOTE(3,J),VOTE(4,J) CONTINUE DO LOOP FOR NUMBER OF REORDERINGS DO 63 MONE=1,40 REWIND 11 DO 46 J=1,VOTERS READ(11,*)UTIL(1,J),UTIL(2,J),UTIL(3,J),UTIL(4,J) READ(11,*)VOTE(1,J),VOTE(2,J),VOTE(3,J),VOTE(4,J) CONTINUE RANDOM REORDERING OF VOTERS DO 48 J=1,VOTERS DSEED=DMOD(DSEED*X,Y) CHOOS=INT(SNGL(DSEED/Y)*(VOTERS-J+1))+J IF(CHOOS.NE.J)THEN DO 47 I=1,ALTS TVOTE=VOTE(I,J) VOTE(I,J)=VOTE(I,CHOOS) VOTE(I,CHOOS)=TVOTE TEMPUzUTIL(I,J) UTIL(I,J)=UTIL(I,CHOOS) UTIL(I,CHOOS)=TEMPU CONTINUE ENDIF CONTINUE WRITE(*,*)’SEARCH ',MONE ‘0000 50 52 53 55 57 155 AFTER REORDERING, REPEAT PROCESS OF SEARCHING FOR NASH EQUILIBRIUM DO 62 L=1.40 DO 60 J=l.VOTERS N:Q+1 EU(N)=O G=1 VMAX=MAX(VOTES(1),VOTESOZ),VOTES(3),VOTES(4), +VOTES(5)) DO 50 I=1,ALTS IE(VOTES(I).EQ.VMAX)THEN TIED(G)=I G:G+l EU(N)=EU(N)+UTIL(I.J) ENDIF VOTES(I)=VOTES(I)-VOTE(I,J) PVOTE(I,Q+1)=VOTE(I,J) CONTINUE EU(N)=EU(N)/REAL(G-1) STRAT=Q+1 DO 55 N=1.Q DO 52 I=1,ALTS VOTES(I):VOTES(I)+PVOTE(I,N) CONTINUE VMAX=MAX(VOTES(1),VOTES(2),VOTES(3),VOTES(4), +VOTES(5)) EU(N)=0 G=1 DO 53 I=1,ALTS IF(VOTES(I).EQ.VMAX)THEN TIED(G)=I G:G+l EU(N)=EU(N)+UTIL(I,J) ENDIF VOTES(I)=VOTES(I)-PVOTE(I,N) CONTINUE EU(N)=EU(N)/REAL(G-1) IE(EU(N).GT.EU(STRAT))THEN STRAT=N ENDIF CONTINUE G2=1 IF(STRAT.NE.Q+1)THEN F:F+1 DO 57 N=l,Q IF(EU(N).EQ.EU(STRAT))THEN G1(G2)=N G2=G2+1 ENDIF CONTINUE DSEED=DMOD(DSEED*X.Y) K=INT(((SNGL(DSEED/Y))*(REAL(G2-1)))+1.) 156 STRAT=G1(K) ENDIF DO 58 I=1,ALTS VOTE(I,J)=PVOTE(I,STRAT) VOTES(I)=VOTES(I)+VOTE(I.J) 58 CONTINUE 60 CONTINUE IF(F.EQ.O)THEN CALL COUNT IF(WINNER.EQ.CWINNE)THEN SNCOND=SNCOND+1 ENDIF SEELEC=SEELEC+SOCUT(WINNER) GOTO 285 ENDIF F=0 61 CONTINUE 62 CONTINUE 63 CONTINUE IF AFTER 40 RANDOM REORDERINGS OF VOTERS, AN EQUILIBRIUM STILL HAS NOT BEEN FOUND, 0000 NONASH=NONASH+1 IF(CWINNE.EQ.0)THEN COMMON=COMMON+1 ENDIF WRITE(10,*)VOTERS.ALTS.P 5 ELSE SINCERE VOTING IS A NASH EQUILIBRIUM 0000) SIN=SIN+1 IF(WINNER.EQ.CWINNE)THEN SNCOND=SNCOND+1 ENDIF SEELEC=SEELEC+SOCUT(WINNER) ENDIF 285 WRITE(*,*)VOTERS,P,M 286 CONTINUE CALCULATE STATISTICS FOR 100 ELECTION SIMULATION AND WRITE TO RESULT FILE 0000 EFFIC=((EELECT/100.)-(VOTERS/2.))/ +((EMAX/100.)-(VOTERS/2.)) SEFFIC=((SEELEC/(REAL(100-NONASH)))‘(VOTERS/Z-))/ +((EMAX/100.)-(VOTERS/2.)) CEFFIC=(REAL(NCOND))/NUM SCEFFI=(REAL(SNCOND))/(NUM-REAL(NONASH)+REAL(COMMON)) NOCC=100.-NUM WRITE(4,*)ALTS,VOTERS WRITE(4,*)NONASH,NOCC,COMMON,SIN WRITE(4,*)EFFIC,SEFFIC,CEFFIC,SCEFFI 294 295 298 157 IF(NONASH.GT.0)THEN WRITE(12,*)EELECT.SEELEC.EMAX WRITE(12,*)NCOND,SNCOND,NUM ENDIF M1=M1+EFFIC M2=M2+SEFFIC M3=M3+CEFFIC M4=M4+SCEFFI SDl=SD1+(EFFIC**2) SD2=SD2+(SEFFIC**2) SD3=SD3+(CEFFIC**2) SD4=SD4+(SCEFFI**2) CONTINUE CALCULATE STATISTICS FOR 20 REPETITIONS OF 100 ELECTION SIMULATION AND WRITE TO RESULT FILE M1=M1/20. N2=M2/20. M3=M3/20. M42M4/20. SD1=((SD1-(20.x(M1**2)))/19.)**0.5 SD2=((SD2-(20.*(M2**2)))/19.)**0.5 SD3=((SD3-(20.*(N3**2)))/19.)**0.5 SD4=((SD4-(20.*(M4**2)))/19.)**0.5 WRITE(4,*)M1,M2.M3,M4 WRITE(4;*)SD1,SD2,SD3,SD4 REWIND 3 WRITE(3,*)DSEED CONTINUE STOP END SUBROUTINE COUNT COMMON/PICK/ALTS,VOTERS,I,J,TOTAL,VOTES,LAST, +TVOTE,TEMPR,COMP,WINNER,TIED,G,K,TRANK, +SOCUT.VOTE INTEGER*2 ALTS,CVOTES(6),CWINNE,F,G,H,I, +J,K,L,LAST,M,N,NCOND,RANK(6),COMP,TOTAL(6),MONE, +SNCOND,STRAT,COMMON,P,SIN.Q.PVOTE(6,721),G2, +TEMPR,TIED(6),NONASH,TVOTE,G1(720),CHOOS,TRANK(6), +VOTE(6,125),VOTERS,VOTES(6),WINNER,WINS(6),VMAX REAL CEFFIC,EELECT.EFFIC.EMAX.EU(721).NOCC.NUM. +RUTIL(6),SCEFFI,SEELEC,SEFFIC,SOCUT(6),TEMPU, +UTIL(6,125),UTMAX,M1,M2,M3, +M4,SD1,SD2,SD3,SD4 INTEGER¥4 Z REAL*8 DSEED.X.Y DATA X/1.6807D4/ Z=2147483647 Y=DBLE(Z) CALL ADD COMP=0 DO 23 I=2.ALTS IF((TOTAL(1)-TOTAL(I)).GT.2)THEN 23 24 25 000:4 158 For the Borda system, the difference between totals must be (ALTS-1)x2 for the totals to be comparable. COMP=I-l GOTO 24 ENDIF CONTINUE COMP=ALTS G=1 WINNER=TRANK(1) DO 25 I=1,COMP IF(TOTAL(I).EQ.TOTAL(1))THEN TIED(G)=TRANK(I) G:G+1 ENDIF CONTINUE DSEED=DMOD(DSEED*X.Y) K=INT(((SNGL(DSEED/Y))*REAL(G-l))+1.) WINNER=TIED(K) END SUBROUTINE ADD COMMON/PICK/ALTS,VOTERS,I,J,TOTAL,VOTES,LAST, +TVOTE,TEMPR,COMP,WINNER,TIED,G,K,TRANK, +SOCUT,VOTE INTEGER*2 ALTS,CVOTES(C),CWINNE,F,G,H,I, +J,K,L,LAST,M,N,NCOND,RANK(6),COMP,TOTAL(6),MONE, +SNCOND,STRAT,COMMON.P,SIN1Q1PVOTE(6,721),G2, +TEMPR,TIED(6),NONASH,TVOTE,GI(720),CHOOS,TRANK(6), +VOTE(6,125),VOTERS,VOTES(6),WINNER,WINS(6),VMAX REAL CEFFIC,EELECT,EFFIC,EMAX,EU(721),NOCC,NUM, +RUTIL(6),SCEFFI,SEELEC,SEFFIC,SOCUT(6),TEMPU, +UTIL(6,125),UTMAX,M1,M2,M3, +M4,SD1,SD2,SD3,SD4 INTEGER¥4 Z REAL*8 DSEED.X.Y DO 19 I=1,ALTS VOTES(I)=O DO 18 J=1,VOTERS VOTES(I)=VOTES(I)+VOTE(I,J) CONTINUE TOTAL(I)=VOTES(I) TRANK(I)=I CONTINUE SORT TOTALS FROM HIGHEST TO LOWEST DO 22 LAST=ALTS,2,-1 DO 21 I=1,LAST-l IF(TOTAL(I).LT.TOTAL(I+1))THEN TVOTE=TOTAL(I) TOTAL(I)=TOTAL(I+1) TOTAL(I+1)=TVOTE TEMPR=TRANK(I) 159 TRANK(I)=TRANK(I+1) TRANK(I+1)=TEMPR ENDIF 21 CONTINUE 22 CONTINUE END STAN APP BOR APPENDIX C NUMERICAL EFFICIENCY ESTIMATES number of alternatives number of voters SU = sincere social utility efficiency estimate strategic social utility efficiency estimate sincere Condorcet efficiency estimate strategic Condorcet efficiency estimate standard voting system Approval voting system Borda voting system SYSTEM M AAhphha»pump»wwwwwwwwmwwwwwwwwwwwww Table 0.1 Numerical Efficiency Estimates 161 Table 0.1 (cont’d.) SYSTEM M V SU SSU CON SCON STAN 4 35 68 69625 76 43157 67 31848 82 23264 STAN 4 45 70 75148 77 81714 67 28976 80 31257 STAN 4 55 69 56684 76 58865 68 50451 79 62701 STAN 4 65 71 68784 76 79339 68 73268 79 57056 STAN 4 75 69 76920 75 28825 65 53801 74 82826 STAN 4 85 70 28945 77 55321 66 03845 78 15042 STAN 4 95 71 34871 77 18910 67 95308 78 11021 STAN 4 105 71 99607 77 21820 66 63617 75 37265 STAN 4 115 70 39350 75 21690 68 25966 76 30755 STAN 4 125 69 60443 75 41456 65 07824 73 55512 STAN 5 3 69 30213 85 41085 73 50789 85 04213 STAN 5 5 72 01346 81 82657 71 19073 92 17719 STAN 5 7 70 53767 80 76190 67 97408 87 65211 STAN 5 9 68 56567 78 66927 65 11762 85 37357 STAN 5 11 70 35698 79 95623 67 46361 85 56151 STAN 5 13 68 58490 79 37249 63 63131 82 46771 STAN 5 15 68 84727 79 47679 65 33905 83 63716 STAN 5 17 66 64464 76 05931 61 66896 79 90193 STAN 5 19 67 48163 77 99164 61 80851 81 43187 STAN 5 21 65 22192 77 77954 58 44027 78 62954 STAN 5 23 67 13834 76 69456 61 56055 77 48210 STAN 5 25 68 53179 78 24449 63 45115 79 38334 STAN 5 35 66 07011 75 17092 58 49493 74 21334 STAN 5 45 66 22192 75 15441 59 75353 73 99453 STAN 5 55 68 44178 76 39528 60 68997 74 27016 STAN 5 65 64 36284 72 59757 60 49124 72 53312 STAN 5 75 65 96298 74 23602 60 45285 71 83448 STAN 5 85 62 85065 72 40006 57 94659 68 78442 STAN 5 95 69 14971 74 65492 61 47445 71 10004 STAN 5 105 68 48776 74 97199 60 55389 71 30886 STAN 5 115 66 05291 72 51652 58 55390 68 91547 STAN 5 125 66 53121 72 02755 59 26700 68 81782 STAN 6 3 65 05819 84 54540 68 88606 82 76113 STAN 6 5 70 21628 80 06387 68 45763 89 08249 STAN 6 7 67 70756 80 83061 63 49053 84 24884 STAN 6 9 67 32233 78 70402 61 83347 82 75975 STAN 6 11 64 75424 76 42666 58 51207 80 27396 STAN 6 13 67 02317 78 15276 59 82922 79 05948 STAN 6 15 64 16193 75 80341 57 94467 77 83272 STAN 6 17 67 08984 77 70395 58 05382 77 74515 STAN 6 19 64 30106 74 64041 55 95757 75 27487 STAN 6 21 63 41932 75 27146 55 99650 75 10687 STAN 6 23 64 62094 75 43137 55 87289 73 15285 STAN 6 25 64 84376 74 44135 55 40077 72 46808 STAN 6 35 63 34317 73 88823 54 30782 69 83503 STAN 6 45 62 89053 72 27750 54 37664 69 03371 STAN 6 55 61 16387 71 43884 51 11140 64 65038 STAN 6 65 63 93198 72 13074 54 54405 66 66238 STAN 6 75 62 34725 71 82034 51 38808 63 97619 Table 0.1 (cont'd.) SYSTEM M ----—-----’-----------------------_--—------—---------_. -_----------------------q--------_-----------—-_--_----- APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP has.»pump»«spa-«Napm-panda-AAant-x»pwwwwmwwmwwwwwwwwwwwwwmmmmmm V SU 162 Table C.1 (cont'd.) SYSTEM M 163 APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP APP BOR BOR BOR BOR BOR cocowwwmmmmmmmmmmmmmmmmmmmmo:mmmmwmmmmmmmmmmmmmwmmmm 164 Table C.1 (cont'd.) SYSTEM M V SU SSU CON SCON BOR 3 13 85 6521 86 0514 91 0840 90 3618 BOR 3 15 86 0413 85 1241 91 4254 89 3947 BOR 3 17 86 5786 86 0729 91 4962 89 9234 BOR 3 19 87 4681 85 8277 91 0791 89 1793 BOR 3 21 86 9231 85 7577 90 8613 88 8679 BOR 3 23 86 0169 85 5753 90 4614 87 0584 BOR 3 25 85 9809 84 8370 91 0876 89 3036 BOR 3 35 86 9052 86 0876 90 8584 88 7467 BOR 3 45 85 9639 84 1891 90 9682 88 4536 BOR 3 55 87 2799 86 2050 89 4846 88 3376 BOR 3 65 85 7878 84 6726 90 2689 90 4917 BOR 3 75 86 1616 85 6402 89 1739 88 4144 BOR 3 85 85 9609 85 6280 89 6652 88 9880 BOR 3 95 85 6847 84 3600 90 0702 89 1871 BOR 3 105 87 8802 86 7592 89 7204 89 3822 BOR 3 115 85 7873 84 9270 89 8063 89 2558 BOR 3 125 87 5652 86 4840 90 8998 90 1502 BOR 4 3 90 2885 87 5803 91 0800 95 5689 BOR 4 5 88 9593 85 8698 87 7640 90 8517 BOR 4 7 89 8753 84 8071 88 8192 84 8081 BOR 4 9 89 9198 84 6571 87 7144 82 8791 BOR 4 11 88 9953 85 2660 88 8147 81 8023 BOR 4 13 89 6839 83 8101 87 0150 80 7521 BOR 4 15 89 5930 83 4478 87 9289 80 5604 BOR 4 17 89 5385 84 8454 88 0877 81 9271 BOR 4 19 89 6547 83 4629 87 7410 80 7406 BOR 4 21 88 7063 83 3710 88 5702 82 1670 BOR 4 23 88 1805 82 9895 88 7625 83 2401 BOR 4 25 88 6720 83 7541 87 7058 80 8639 BOR 4 35 88 7871 83 1956 87 1246 80 7768 BOR 4 45 88 8816 85 0088 87 8488 84 0707 BOR 4 55 89 4796 84 7656 86 7280 82 7086 BOR 4 65 89 5331 85 8105 87 5145 83 1699 BOR 4 75 89 8240 85 5151 88 3134 84 3461 BOR 4 85 90 2245 86 5894 88 0251 84 6901 BOR 4 95 88 5688 86 2847 86 0952 83 4126 BOR 4 105 88 3059 85 4874 87 2052 83 8800 BOR 4 115 89 0266 84 7904 87 8618 84 9852 BOR 4 125 90 2115 87 5790 86 7301 84 9834 Values estimated with regression coefficients: BOR 5 3 91.28709 93.62172 90.37112 92.07680 BOR 5 5 91.28709 90.17734 89.51751 86.85212 BOR 5 7 91.28709 88.38428 89.06592 84.12371 BOR 5 9 91.28709 87.30332 88.75550 82.47178 BOR 5 11 91.28709 86.59463 88.51114 81.38258 BOR 5 13 91.28709 86.10698 88.30303 80.62754 BOR 5 15 91.28709 85.76300 88.11699 80.08968 Table C.1 (cont'd.) SYSTEM M BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR BOR mmmmmm02050503090)0:050:05030303mmmmmmmmmmmmmmmmmm V SU 165 APPENDIX D REGRESSION RESULTS Equation 1: C(4)*VAR Table D.1. Sincere Social Utility Efficiency Regression Results Willem Coefficient Std. Error C(1) 0.9660104 0.0134421 C(2) 12.5755110 2.0373614 C(3) -0.0908287 0.0509604 C(4) 0.1744424 0.0523831 32 Standard Error of Regression Sum of Squared Residuals WWW Coefficient Std. Error C(1) 0.9834290 0.0049852 C(2) -14.4611890 2.6824025 C(3) 3.6120629 0.5574343 C(4) -3.6261399 0.5612308 32 Standard Error of Regression Sum of Squared Residuals Wm . Coefficient Std. Error C(1) 0.9988894 0.0028917 C(2) -1.4911846 5.7122250 C(3) 0.0056901 0.0148756 C(4) -0.0036556 0.0094704 R2 Standard Error of Regression Sum of Squared Residuals 166 T-Stat. 71.8642880 6.1724490 -1.7823393 3.3301283 T-Stat. 197.2711100 -5.3911334 6.4797997 —6.4610496 T-Stat. 345.4374500 -O.2610514 0.3825100 -0.3860020 SU = C(1)*THEO + C(2)*DIF + C(3)*MEAN + 0.862424 1.717966 247.9183 0.432635 0.995403 83.22952 0.834664 0.663750 17.62255 "7"“! 167 Equation 2: CC = C(1)*THEO + C(2)*DIF + C(3)*MEAN + C(4)*VAR Table D.2. Sincere Condorcet Efficiency Regression Results WWII; Coefficient Std. Error T-Stat. C(1) 0.9347468 0.0225973 41.3654810 C(2) 17.4107530 3.4249585 5.0848752 C(3) 0.7782854 0.0856683 9.0848752 C(4) -0.8640046 0.0880599 -9.8115514 R2 0.908270 Standard Error of Regression 2.888031 Sum of Squared Residuals 700.6208 Annmallatinmm Coefficient Std. Error T-Stat. C(1) 0.6636912 0.0092258 71.9385700 C(2) -18.8665650 4.9641934 -3.8005298 C(3) 20.5012300 1.0316170 19.8729090 C(4) -20.4961870 1.0386428 -19.7336240 R2 0.854252 Standard Error of Regression 1.842145 . Sum of Squared Residuals 285.0538 WW Coefficient Std. Error T-Stat. C(1) 0.9702868 0.0032940 294.5577000 C(2) 136.2546000 6.5070940 20.9393930 C(3) 0.1357765 0.0169456 8.0125011 C(4) -0.0832901 0.0107883 —7.7204378 Rz 0.887074 Standard Error of Regression 0.756112 Sum of Squared Residuals 22.86822 168 SSU = C(1)*THEO + C(2)*DIF + C(3)/P + C(4)*(Q‘(ALTS/(2*(ALTS+V)))) + C(5)*V + C(6)*V‘2 + C(7)*V“3 + C(8)*V‘4 Equation 3: Table D.3. Strategic Social Utility Efficiency Regression Results Standardistimflsm Coefficient Std. Error T-Stat. C(1) 0.5207019 0.0542326 9.6012750 C(2) 18.1723560 17.5191300 1.0372864 C(3) 22.8116840 4.8082186 4.7443110 C(4) 15.1048090 2.0875383 7.2357038 C(5) 0.0750898 0.1612042 0.4658056 C(6) -0.0026245 0.0039308 -0.6676666 C(7) 3.237D-05 4.058D-05 0.7978145 C(8) -1.343D-07 1.459D—07 -0.9206889 R2 0.921995 Standard Error of Regression 1.116703 Sum of Squared Residuals 99.76201 WWII Coefficient Std. Error T-Stat. C(1) -1.4149151 0.2701843 -5.2368524 C(2) 92.3039750 10.0521490 9.1825119 C(3) 169.3911200 19.1548330 8.8432624 C(4) 1.3284882 0.5041483 2.6351140 C(5) -0.2029406 0.1278940 -1.5867872 C(6) 0.0051127 0.0033746 1.5154758 C(7) -3.835D-05 3.624D-05 -1.0583883 C(8) 9.423D-08 1.333D-07 0.7066768 R2 0.741881 Standard Error of Regression 1.097526 Sum of Squared Residuals 96.36510 W Coefficient Std. Error T-Stat. C(1) 0.7174308 0.0998939 7.1819298 C(2) 262.6485400 54.7449520 4.7976760 C(3) 15.0476340 10.2901170 1.4623384 C(4) 2.6690842 1.3929331 1.9161611 C(5) -0.0878516 0.1415911 -0.6204598 C(6) 0.0044036 0.0036143 1.2183975 C(7) -5.407D-05 3.806D-05 -1.4209206 C(8) 2.095D-07 1.383D-07 1.5148517 R2 0.709214 Standard Error of Regression 0.766167 Sum of Squared Residuals 21.13243 169 Equation 4: SCC 2 C(1) + C(2)*SSU + C(3)*ALTS + C(4)*V See Chapter 4, page 112. APPENDIX E NOTES TO TEXT Introduction and Chapter 1 lBlack [7], p. 180. 2Condorcet, in Rosenstein [118], p. 36. 3Condorcet, in Rosenstein [118], p. 46-47. 4Condorcet, in Rosenstein [118], p. 53. 5Condorcet, in Rosenstein [118], p. 51. 6Condorcet, in Rosenstein [118], p. 56. 7Zeckhauser [145], p. 935. 8Gibbard [58], p. 587. 9Gibbard [58], p. 595. 10Postlewaite and Schmeidler [109], p. 37. 11Postlewaite and Schmeidler [109], p. 38. 12Postlewaite and Schmeidler [109], p. 37. 13Dasgupta, Hammond, and Maskin [33], p 186. 14Dasgupta, Hammond, and Maskin [33], p. 186. 15Dasgupta, Hammond, and Maskin [33], p. 188. 16Dasgupta, Hammond, and Maskin [33], p 189. 1'7Black [7],‘p. 188. 13Black [7], p. 188. 19Black [7], p. 7. 20Enelow and Hinich [40], p. 16. 2lEnelow and Hinich [40], p. 30. 22Enelow and Hinich [40], p. 30. 23Enelow and Hinich [40], pp. 30-31. 2“Riker and Ordeshook [114], pp. 25-26. 25Palfrey and Rosenthal [102], p. 9. 26Ferejohn and Fiorina [42], p. 527. 27Palfrey and Rosenthal [102], p. 9. 28Palfrey and Rosenthal [102], p. 10. 29Palfrey and Rosenthal [102], p. 8. 30Palfrey and Rosenthal [102], p. 31Data are taken from the ”tr ‘ t .~ ' Statesg_1982;83, 103rd Edition. U.S. Bureau of the Census, 1982. 32Intriligator [79], p. 553. 33Farquaharson [41], p. 24. 34Farquaharson [41], p. 24. 35Farquaharson [41], p. 25. 36Enelow and Koehler [39], p. 399. 37Enelow [38], p. 1062. 38Enelow [38], pp. 1088-1089. ’30) Chapter 2 1Arrow [1], pp. 94-95. 2Weber [143], pp. 7-8. 3Weber [143], pp. 9-11. 4Postlewaite and Schmeidler [109], p. 38. 170 5Fishburn 6Fishburn 7Fishburn 8Fishburn 9Fishburn 10Bordley Chapter 3 1Merrill [92], 2Merrill [92], 3Shubik [130], 4Selten [123], Chapter 5 1Boadway, Co., 1979, 2Riker and 3McKelvey, 4McKelvey, 5McKelvey, 6McKelvey, 171 and Gehrlein [55], p. 143. and Gehrlein [55], p. 149. and Gehrlein [55], p. 149. and Gehrlein [55], p. 151. and Gehrlein [55], p. 151. [12], p. 129. p. 119. p. 119. p 136. p 35. R.W.. WW Little. Brown, and pp. 71-72. Ordeshook [114], pp. 25-26. Ordeshook, and Winer [90], p. 599. Ordeshook, and Winer [90], p. 605. Ordeshook, and Winer [90], p. 602. Ordeshook, and Winer [90], p. 606. 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