Accepted by the faculty of the Department of Communication, College of Communication Arts, Michigan State University, in partial fulfillment of the requirements for the Doctor of PhiIOSOphy degree. mfélmz DirefiEr of Thesis Guidance Committee: M f M , Chairman 0% Z - W , Co-Chairman ABSTRACT DYNAMIC MATHEMATICAL MODELS OF DYADIC INTERACTION BASED ON INFORMATION PROCESSING ASSUMPTIONS By Joseph N. Cappella In order to treat the dynamic behavior of dyads as attitudes are being negotiated through the process of mutual influence, mathematical models are necessary. The reason is that even for dyadic influence the 'dynamics of Change is too complex to be handled with purely verbal models. The models develOped for mutual influence in this thesis orig- inate from Newcomb's structuring of dyads and, therefore, include vari- ables for each person's attitude, each person's perception of the other's attitude, and eaCh person's attraction to the other. In addition to these six variables, we consider two aSpects of the communicative under— change: the rate of transmission of messages and the content of gen- erated messages. In the case of content, two alternative models are considered: a veridical model in which the speaker's message always reflects his attitudes and a shi§t_model in which the speaker's message is shifted a fraction of the distance toward the speaker's perception of the other's attitude. Because Newcomb's paradigm for dyadic situations does not specify the £929:°f the change equations for attitudes, perceptions, and attrac- tions, two well-known theories of attitude change were invoked to Joseph N. Cappella specify the form of the change equations for attraction (Social Judg— ment Theory) and for attitudes and perceptions (InfOrmation Processing or ReinfOrcement Theory). ‘While Social Judgment Theory has not generally been applied to attraction change, there are sound reasons for doing so. In order to solve the system.of six non-linear differential equations for the veridical and shift message cases (that is, deter- mine stability characteristics and "direction" of movement) generated from the assumptions of Information Processing Theory, several simpli- fications were made. First, with attraction and message transmission held constant, the shift and veridical message models always converged to a point of equilibrium although the points differed between the models. Second, with attraction constant but transmission varying, both message models tended to converge to a point at which one individual perceived no discrepancy from the other and was silent while the other perceived discrepancy and was transmitting. Third, with attraction varying but transmdssion constant, both message content models produced infinite dislike with actual and perceived attitudinal differences only when both persons' initial messages to the other were well outside his region of acceptance. In all other cases, infinite liking with no actual or perceived differences obtained. Finally, with both variable attraction and transmission results which were a combination of simpli— fications two and three above were obtained. It was concluded that the method of analysis and framework for IHJtual influence had promise for future model building, theory construc- ‘tirnm, and research. However, that promise could be realized only by Joseph N. Cappella comparing other models with alternative psychological assumptions (for example, congruity and dissonance) and with alternative message content assumptions to one another. DYNAMIC MATHEMATICAL MODELS OF DYADIC INTERACTION BASED ON INFORMATION PROCESSING.ASSUMPTIONS By Joseph N. Cappella A.DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Communication 1974 DEDICATION To Professor Robert O. Brennan, friend and mentor, who was first to confuse me with mathematics. ii ACKNOWLEIEMENTS I hope that before this is read each person that I wish to thank has been thanked in a more personal way. Nonetheless these intellectual and personal debts should be made public. Of course, my entire committee is deserving of my deepest gratitutde for all of their time, energy, and understanding in the completion of this work. John Hunter who directed and dissected this dissertation deserves as much credit for its comple- tion as I do. His mark, both on this particular work and on the char- acter of my future work, will be unmistakable. I must also express an admiration for his ability to combine a sharp critical attitude toward intellectual pursuits but tempered with enough personal warmth and support to keep one moving ahead. Gerald R. Miller in his role as adviser offered personal support and an intellectual Openness without whiCh this dissertation would never have been undertaken, let alone completed. Donald Cushman has been both friend and sounding board for the entirety of my graduate education. His influence on my orientation to theory and types of theory construction can be seen by any who have studied with him. Joseph Wbelfel has offered by his example and personal discussions support for the aims of theory develOpment. R. Vincent Farace's contacts with the Environmental Design and.Management group headed by William Cooper and Herman Koenig made possible financial support for me in an environment free of time constraints and commitments. For his efforts in this regard, iii I am deeply indebted. The Department of Communication deserves mention since it has provided an Open, flexible, and stimulating intellectual environment in whidh to pursue interests which would have been unacceptable in more constrained academic units. Elena, Jeffrey and Elise Cappella managed to maintain their own high spirits which often buoyed my own through the weeks and.mohths of frustration, disappointment, and failure which preceded the completion of this work. Finally, the manuscript was professionally prepared and typed by Ruth Langenbacher. iv TABLE OF CONTENTS Chapter I INTRODUCTION II DEVELOPING MATHEMATICAL MODELS OF CHANGE FOR IJX SITUATIONS . . Information.Frocessing Model . . The Generation of Message Content . . The Transmission of Messages . . Internal Changes According to Information Processing Theory . Simplifying the Dyamics of IJX Situations III ANALYZING THE DYNAMICS OF CHANGE IN IJX SITUATIONS: INFORMATION PROCESSING THEORY Information Processing with Constant Attraction . . . . Internal Changes Only . . External IP with Shift Message and Constant Attraction and Transmission . . . IP with Veridical Messages: Constant Attraction and Transmission . IP with Constant Attraction and Variable Transmission: Veridical and Shift Messages . . . . General Discussion . . . The Case of Equal Parameters: a = b The Case of Unequal Parameters . The Unlikelihood of a Common Limit fOr the Unequal Parameters Case InfOrmation Processing with Varying .Attraction . . . Internal Changes Only Page 17 18 23 30 32 3H 3H 39 55 57 57 59 61 63 72 73 Chapter IP with.Message Shift and Variable Attraction: Transmission Constant . IP with Veridical Messages and Varying Attraction: Transmission Constant . . IP with Varying.Attraction and Transmission: Veridical and Shift Models . . IV CONCLUSIONS The Expected Results . . . . . The Unexpected Results . . . . An Experimental Test . . . . . . Future work . . . . . . . . APPENDICES . . . . . . . . . APPENDIX.A Internal Changes Only. . . . IP with Shift: Constant Attraction and Transmission . . . . . IP with Veridical Messages: Constant Amtraction and Transmission . . . IP with Constant Attraction and Variable Transmission Shift and Veridical . . APPENDIX B Internal Changes Only. . . IP with Message Shift and Varying Amtraction: Transmission Constant IP with Veridical Messages and Varying Attraction: Transmission Constant IP with Varying Attraction and Varying Transmission: Both.Message Models . BIBLIOGRAPHY . . vi Page 78 86 88 102 103 105 108 110 111 111 112 121 122 127 128 130 131 135 LIST OF TABLES Table Page 1 Change Equations fer Internal and External Changes Based upon Information Processing Theory . . . . 30 2 Predictions of Relative Transmission between I and J as a Function of Relative Attraction and Initial Perceived Agreement (PA) and Perceived Disagreement (PD) 0 O O O I O O O O O O O O ”3 3 Summary of Rating Scheme for the Degree of Fit between Attraction Parameters, Initial Values and Predictions from Transmission.Model . . . . . nu u The Possible Combination of the Attraction Subsystem ‘with.Attitude-Perception Subsystem fer IP with Message Shift . . . . . . . . . . . 91 5 Tentative Final States for IP with.Message Shift, Varying Attraction and Transmission, as a Function of the Initial Values . . . . . . . . . 99 V11 Figure LIST OF FIGURES Changes in Attitude versus Discrepancy between Message and Attitude with Attraction = O (a) and versus Attraction with Discrepancy between Message and Attitude = +l and —l (b): Varying Levels of Transmission . . . . . . . . . . Changes in Perception of the Other versus Dis- crepancy between Message and Perception of the Other with.Attraction = O (a) and versus Attraction with Discrepancy between Message and Perception = +1 and -l (b): Varying Levels of Transmission Changes in.Attraction versus Discrepancy between Message and Attitude for Varying Levels of Transmission . . . . . . . Transmission versus Perceived Discrepancy for Various Levels of Attraction . . . . Transmission to the Slider versus Perceived Dis— crepancy for High and Low Attraction (Schachter, 1951, Table 8). . . . . . . . . The Change in P. as a Function of Perceived Discrepancy with Attraction held Constant (a) and as a Function of Attraction with Discrepancy held Constant (b) fOr Internal Forces . . Regions of Acceptance and Rejection as Related to Changes in Attraction . . . . . The Change in Attraction as a Function of Perceived Discrepancy According to a Social Judgment Model for Internal Forces . . . . . Internal Change Trajectories for IP with Positive Attraction (+3) (a) and Negative.Attraction =-3) (b) fOr Initial Perceived Disagreement viii Page 11 13 16 20 22 25 28 28 36 Figure Page 10 Phase Plane Trajectories for Internal Changes with Constant Attraction: (a) Line of Critical Points only, (b) Critical Points plus Trajectories for q/r = 3, (c) for q/r = l, and (d) q/r = -.33 . . . 37 ll Trajectories for IP with Message Shift and Constant Attraction: Equal Attraction (3, 3) and Equal Thwemfismn(l,l) . . . . . . . . . H2 12 Trajectories fer IP with Message Shift and Constant Attraction: Equal Attraction but Unequal Transmission . . . . . . . . . . . H5 13 Trajectories for IP with.Message Shift and Constant Attraction: Unequal Attraction and Equal Transmission . . . . . . . . . . . H7 14 Trajectories fer IP with Message Shift and Constant Attraction: Unequal Attraction and Unequal Transmission-Discrepant from Transmission.Model . . M9 15 Trajectories for IP with.Message Shift and Constant Attraction: Unequal Attraction and Unequal Transmission-Consistent with Transmission.Model . . 50 16 Phase Plane Plots of I's versus J's Perceived Discrepancy fOr IP’with Message Shift and Varying Transmission: Attraction = (0, l) fOr (a) and Attraction = (l, 0) for (b) . . . . . . . 60 17 IP with Constant Attraction and Varying / Transmission: Unequal Parameters . . . . . . 62 18 IP with Shift Message and Constant Attraction: Transmission Variable . . . . . . . . . 65 19 IP with Shift Message and Constant Attraction: Transmission Variable . . . . . . . . . 66 20 IP with Message Shift: Varying Transmission and Constant Attraction . . . . . . . . 67 21 IP with Message Shift and Constant.Attraction: Transmission Variable . . . . . . . . . 69 22 IP with Shift Message and Constant Attraction: Transmission Variable . . . . . . . . . 70 1X Figure Page 23 IP with Shift Messages: Constant Attraction and Variable Transmission . . . . . . . . 71 2M Phase Plane Trajectories for Internal Changes with Perceived Discrepancy versus Attraction . . . 76 25 Time Trajectories for Internal Changes with IP and Varying Attraction: Initial Perceived Disagreement . . . . . . . . . . . 77 26 Time Trajectories for IP with.Message Shift and Varying.Attraction: Initial Messages both Inside the Other's Acceptance Region . . . . . . . 80 27 Time Trajectories for IP with.Message Shift and Varying Attraction: Initial Messages well Outside Other's Acceptance Region . . . . . . . . 82 28 Time Trajectories fer IP with Message Shift and Varying Attraction: Initial Messages both Outside the Other's.Acceptance Region . . . . . . . 83 29 Time Trajectories for IP with.Message Shift and Varying Attraction: Initial Message from J within I's Acceptance Region but from J Outside I's Acceptance Region . . . . . . . . . . 85 30 Time Trajectories for IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages well Outside the Other's Acceptance Region . . . . . . . . . . 93 31 Time Trajectories for IP with Message Shift, Varying.Attraction and Transmdssion: Both Initial.Messages within the Other's Acceptance Regions . . . . . . . . . . . . 9H 32 Time Trajectories for IP Message Shift, Varying Attraction and Transmission: Both Initial Messages within the Other's Acceptance Region . . . . . . . . . . . . 95 33 Time Trajectories for IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages just Outside the Other's Acceptance Region . . . . . . . . . . 96 Figure 3H 35 Page Trajectories fer IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages just Outside the Other's Acceptance Region . . . . . . . . . . . . . 97 Time Trajectories fOr IP with Message Shift, Varying Attraction and Transmission: J's Initial Message within I's Acceptance Region while I's Message is Outside J's Region . . . . . . . 98 xi CHAPTERI INTRODUCTION The purpose of this work is to discuss and develop mathematical models of small group interaction, particularly where the group is task- oriented and communicating about some issue of'mmtual importance and mutual relevance. Although the literature pertinent to small group processes is voluminous, there are at least two areas which have been sufficiently well—developed theoretically and researched empirically to warrant treatment in a mathematical framework. These include the baLance-related theories of interpersonal situations (Heider, 19MB; Newcomb, 1953, 1961, 1968; Osgood and Tannenbaum, 1955) and the attitude change theories of the passive communication situation (Festinger, 1957; Aronson, TUrner, and Carlsmith, 1963; HOVland, Janis, and Kelly, 1953; Sherif, Sherif, and Nebergall, 1965; Hunter and Cohen, 197a). Both of these points of View have their advantages and limita- tions. These can best be seen if we first adopt a structural and termi- nological framework for discussing the basic dyadic process. The framework is that supplied by Newcomb's.ABX model (1953; 1961), which we shall refer to as the IJX model for reasons of uniformity of notation. Newcomb assumes that the components of the interpersonal situation consist of two persons, I and J, and an object of mutual concern, X. 1 2 The relations among the components are attraction between I and J and the orientations or attitudes of I to X and of J to X. In addition, each individual is presumed to have a perception of what the other's orientation toward X is. With the addition of these two perception relations, Newcomb can divide the IJX situation into parts on the basis of which of the six relations are relevant. These parts are the two individual (or intrapersonal systems) and the collective (or objective) system. The individual system is constructed from cognitions available to the fecal individual. These include I's attraction to J, I's atti- tude toward X, and I's perception of J's attitude toward X. .A similar set of relations constitute J's individual system. The collective system is constructed from.two individual systems and, hence, is con- structed from information which at_any;pgint_in_time_is unavailable to either of the individuals in the interpersonal situation. The collec- tive system consists of four relations: I's attraction to J, J's attraction to I, I's attitude toward X, and J's attitude toward X. The beauty of Newcomb's structuring of IJX situations is found in the types of actions which an individual may undertake when he experi— ences individual strain. That is, when an individual acts to alter the IJX situation, his actions may be inner-directed toward changes in the individual system, or they may be outer-directed toward inducing changes in the collective system. Changes in the individual system would involve changes in attitude, attraction, or perception of the other. Actions directed toward the collective system would take the form of communicative acts, or persuasive efforts. UnfOrtunately, Newcomb's model does not go so far as to suggest the fOrms of the change equations 3 for internal changes, or the processes by which attitudes, perceptions, and attraction are altered as a result of communication. We must turn to other models to provide answers to these questions. But Newcomb's model provides the framework for analysis Which goes beyond either pure balance—related positions or pure attitude change theories. But why build mathematical models rather than verbal models? The answer is two fold: complexity and precision. In accepting the Newcomb paradigm for IJX situations we have already focused upon three variables fOr each person: attitude, attraction, and perception of the other person. With two people this means six variables which are all changing as non-linear functions of one another. They are not merely structurally related to one another but dynamically related to one another. Even ignoring the type of messages that are generated or the rate at which they are transmitted, this is already a very complex system to analyze without some mathematically SOphisticated tools. Secondly, verbal models of complex processes often have difficulty drawing correct and complete deductions from initial assumptions. In mathematical modeling, not only are assumptions quite explicit but the results of those assumptions constitute a complete and logically consistent set of conclusions. Thus, in terms of their ability to treat the dynamics of complex situa— tions and their ability to completely and explicitly array the alter- native assumptions and their implications, mathematical models are to be preferred to other forms of theory construction and model building. Balance processes are presumed to be primarily cognitive and intrapersonal. The role of interpersonal processes of information trans— mission is minimized, obscured or left absent. In the most careful and m thorough review of balance in small groups to date, Taylor (1970, p. 41) discusses the role of communication as fellows: "Through communication with the other, the focal person discovers the other's attitude toward X. In this respect, communication allows the tension mechanism.tg_02erate in_the balance process" (emphasis in the original). Thus, the only role that communication plays is that of informing the individual's perception of the other so that it becomes accurate. On the other hand, attitude change theories have been primarily interactive and interpersonal by involving explicitly the effects of messages. But this emphasis has thereby excluded processes whereby intrapersonal changes can occur in the absence of message input. Secondly, changes in perception of the other have not been treated in detail by either balance or attitude change researchers. Its role in the passive communication context is minimal since there is usually no assumption of continued interaction with the speaker beyond a single message. However, in IJX situations where continued interaction is the fecus, the role of perception of the other is crucial in understanding whether an individual's perceptions are accurate or inaccurate and, hence, whether individual and collective systems differ or are inter— changeable. Thirdly, balance-related theories usually begin with definitions of balance which are static. That is, at balance the system is presumed to be at rest. Deviations from.the static balanced state define imr balance and point out the apprOpriate Changes toward balance. However, --and this is our third criticisme—the emphasis in balance research for IJX situations has ignored the dynamics of change (see Hunter, 197” for 5 an exception) and has focused primarily upon verifying the definition of balance. Attitude change theories, on the contrary, have been con- cerned in a fundamental way with the dynamics of attitude change (Hunter and Cohen, 197a) but have been less concerned with the dynamics of attraction change (this is not true of congruity theory or dissonance theory). .As noted above, none of the passive communication models have been concerned with changes in perception of the other. Finally, attitude change theories, because they have been cast in the passive paradigm, do not consider alternative processes fOr the generation of messages. In the passive paradigmu messages may be treated as a constant input from some source but in the interactive IJX situation messages are an output from.an individual system whose content should change as the individual system Changes. With the above criticisms in mind, the present work seeks to develop alternate mathematical models of the dynamics of IJX situations based upon passive attitude change models (Hunter and Cohen, 1974) extended to fit more closely the Newcomb structuring of IJX situations. The extensions will include (1) processes of individual, intrapersonal change as well as interpersonal change due to message transmission, (2) processes of the generation of message content and of message trans— mission as a function of individual system states, and (3) a considera- tion of the process of change in the perception of the other as parallel to but independent of attitude change. The model to be considered is the information processing model (Hovland, Janis, and Kelly, 1953; Hunter and Cohen, 197“, pp. 28-39). This is deve10ped in the following chapter. CHAPTER II DEVELOPING MATHEMATICAI.MODELS OF CHANGE FOR IJX SITUATIONS As indicated in the previous chapter, Newcomb' s structuring of IJX situations offers a useful framework for the development of dynamic models of dyadic processes. Following Newcomb's discussion, any dynamic model must consider processes of internal as well as external change. That is, individuals would be expected to alter their cognitions in the direction of minimizing internal strain independently g the messages that they receive from the other and to alter their cognitions as a function of the messages that they receive from the other. Also following Newcomb, the cognit ions which are altered are the relationships which constitute the individual system: I' s attitude toward X, Pi’ I' s perception of J's attitude toward X, Qij’ and I's attraction to J, Aij' These constitute the three state variables of I's individual system. Together with Pj , Qj variables are the state variables for the collective system. Because i’ and Aji from J's individual system, the six Newcomb's paradigm allows the individual to act on the collective system (that is , the other individual) through the process of communication, then any dynamic model of the IJX situation must also account for the generation of message content and the transmission of those messages. 6 7 As we also noted in the previous chapter, while Newcomb's IJX system is suggestive for structuring the dynamics of interpersonal processes, it does not go beyond suggestion to the specifics of change. To do so we shall be forced to invoke the assumptions of other, more deveIOped, models and to extend them where necessary. In addition to presenting models of message transmdssion and the generation of message content, we shall invoke a model of attitude change which has been thoroughly researched and developed for the passive communication paradigmr-the information processing model (Hovland, Janis, and Kelly, 1953; Hunter and Cohen, 197”, pp. 28—39). As we take up these models in turn, we shall first consider external changes in the six state variables as a function of the other's message, and then the comparable process of internal spontaneous change occurring independently of the other's message. Finally, we shall consider the message transmission and generation processes which link the output of each individual system to the input of the other individual system. The key differences between the interactive models of this chapter and the passive models upon which they are based are feund (l) in treating the perception of the other's attitude toward X, Qij and jS as a relevant system variable, and (2) in considering the generation of message content and its transmission as the mathematical and substantive link between individual systems. Information Processing Model As Hunter and Cohen (197”, p. 30) point out, the fundamental tenets of the information processing models of attitude change are 8 . . . that (l) the magnitude of change is pro- portional to the discrepancy between the receiver's attitude and the position advocated by the message and (2) the change is always in the direction advocated by the message. These Changes arise from.the internal comparison processes which individuals undergo when the incoming message is compared to their own attitudinal position. The greater the difference between the incoming message and the individual's attitude, the greater the expected change in attitude. That is, change in P. = P.(t) - P.(t—l) 1 1 1 = AP. 1 = aCMo 0-PI) ji 1 where Mji is the message sent by J to I and a is a constant of prOpor— tionality which is greater than zero but less than one. As the dis— crepancy or distance between I's attitude and J's message increases, so does the expected amount of attitude change. That is, the amount of attitude change is a linear function of the amount of discrepancy betweenMji and Pi' Since this is the basic change characteristic of the information processing model, we Shall also refer to it as the linear discrepancy (LD) model. The above Change equation has a simple verbal interpretation. I's attitude toward X at time t is given by I's attitude toward X at the previous time (t-l) plus an increment in the direction of I's perception of J's attitude toward X. The fractional amount of that increment is given by a. As Hunter and Cohen (197”, p. 3M) point out, information pro- cessing theorists have given a great deal of attention to the effects of source credibility in inducing the desired amount of attitude change. 9 In purely interpersonal situations where the credibility Of the source can be identified almost completely with.his attractiveness, then the amount of attitude change depends upon the attractiveness of the source. That is, the more attractive the source, the greater the attitude change at least when the attraction is positive. However, when the source is thoroughly disliked so that his attractiveness is negative, then the attitude change can either (1) go to zero so that credibility is always positive, or (2) actually go negative thus causing the attitude to change Opposite to the direction advocated by the message. We shall Opt fer the first alternative fOr two reasons: First, the research related to balance in.IJX situations suggests that there exists a.strong positive balance tendency in the amount of tension or strain produced in IJX situations Zajonc, 1968; Newcomb, 1968). That is, the change of per— ceptions in IJX situations does not arise when the other is negatively evaluated but only when he is positively evaluated and there is dis— agreement. Second, the experimental production Of a boomerang effect is very difficult to achieve (Cohen, 1962; Cohen, 1964; Whittaker, 1967) and as a result should not be postulated as the primary mechanism of attitude change. Thus, the credibility factor should increase from zero fOr an infinitely incredible source to one for an infinitely credible source. .A function which achieves this is e ij / (l + e ij) so that AP. =a e13 (M.. -P.). 1 -————7§———— 31 1 l + e ij Our change model fOr attitudes will be complete when the factor Of transmission from J, Nji’ is included. In the passive communication 10 context the difference equation above can be applied again and again fOr each message that is generated. However, in the interactive mode, we desire to have the transmission process included explicitly so that once the process of interaction is begun, it will be terminated by_tge interactants with a termination Of transmission. Also, as the rate of transmission increases, the more messages J sends to I and, hence, the faster I should change toward the message. When transmission is zero, then the change in attitudes should also be zero. This suggests that transmission, like credibility, is a multiplicative factor in the change equation for attitudes: APi = a e ij (Mji - Pi) N... (1) 7;. 3i 1 + e l] Figure 1 shows the effects of discrepancy (Mji - Pi), attraction,.Aij, and transmission, Nji’ on the change in attitude. In Figure la, the more positive Mji is than Pi’ the greater will be the positive change in Pi (that is, in the direction of the message). For the same amount Of discrepancy, the greater the transmission from J. the greater the amount of Change in I's attitude. In frame b Of the same figure, we see that fOr a fixed amount of discrepancy, the greater the attraction, the greater the attitude change. For a fixed discrepancy and fixed level of attraction to the source, the greater the transmission the greater the change in the direction advocated by the message. Having laid the ground work fOr a change in I's attitudes as a function Of J's messages according to information processing theory, developing the change equation fOr Qij is an easy matter. The change 11 .cowmmwamcmpe mo mam>mq mafizsm> "on an new H+ u moovflPH< new mmMmmmzncmmzymo mocmmmhomfio rvfl3.:0flvomnpu<.msmfim> ecu Amv o n coweogppu< eves wospweH< new mmwmmmz.cmm3pmo mocmamgomfio msmgm> mozppr< ca mmwcmno .H wasmwm .1”- Nn..Z/ Hm, N\Hu..z mun- / H Hm. - ..< . “.ml..: '1 \ Hm \\N\HH..Z Hm N\H"uo Hm, \ HH..Z Hm a .o .m 922 .na 3 ma 12 in I's perception of J's attitude should be exactly analogous to the change in 1's attitude as a function of J's message. That is, the more discrepant J's message is from I's position, the more change in Qij in the direction Of the message should be observed. Also the more attrac- tive J is to I, the more change in Qij that should be realized. And as the transmission from.J increases, the rate of change of Qij in the direction Of the message should increase as well. Thus, AQ.. = b eij (M..-Q..)N... (2) l] A.. 31 l] 31 l + e l] Figure 2 presents the graphical.ftmmlof equation (2) for the cases of constant attraction and constant discrepancy with varying levels of transmission. Comparing Figures 1 and 2 for equations (1) and (2) shows in a striking manner the similarity of the two change equations. The only differences between equations (1) and (2) are found in the para- meters a and b. Although they both have the same function, a is not necessarily equal to b and their ratio indicates whether a given message elicits more change in Pi (a > b) or in Qij (b > a). Based upon a study by wackman and Beatty (1971), cited in wackman (1973), we shall always assume in our examples that perceptions of the other are less resistant to change than are the attitudes that one holds. This means that fOr the same discrepancy, I's perception of J's attitude will change more in the direction of J's message than will I's own attitude, hence b > a. At this point, it is interesting to note that Mji plays a dual role in equations (1) and (2). In equation (1) its effect is that of persuasion and in equation (2) its effect is that of informing I Of J's position. 13 .cowmmdzmcmgw MO £923 mcfi>9m> 25 HI ecu .T. n couflmoocwm one mmMmmmz 5953 zoqmomcdmwa 5pm.”: coauromcflfla 2699.. 98 A3 0 u soapomfiuk spas 9930 out. mo counvmmgmm ecu mmmmmoz apogee moqmomoomwm moms? amnvo may mo cowvmmoamm Cw mmmcmso .m gem md «"22 ..oa 1L: This duality is not unreasonable since in attempting to persuade another Of his position, I is simultaneously Offering him.infOrmation on the exact nature of his position. UnfOrtunately, deriving a change equation for attraction form infOrmation processing assumptions is not as easy as it was for percep- tions and attitudes. The reason, as Hunter and Cohen (1974, p. 38) note, is that the information processing theorists were not interested in change in the attraction of the source as a function of his message. In fact, attitude change theorists did not seriously begin to think about the effect that source change could have on the processes of attitude change until Aronson, Turner, and Carlsmith's famous attempt (1963) to explain nonlinear changes in attitude by invoking changes in the attrac- tiveness of the source. Since the information processing researCh pre- dated the Aronson, Turner, and Carlsmith piece, the information processing model takes no explicit stand on sOurce change. However, certain requirements fOr the change in attraction can be stipulated: First, based upon the work of Byrne (1969) and his colleagues, we expect that changes in the attractiveness Of the other will depend upon the degree of similarity that is perceived by the focal individual. Second, changes in attraction should be both positive and negative so that attraction is capable of either increasing or decreasing. Obviously, if attraction can gnly_increase or only_decrease, then the patterns of attraction whiCh can emerge from such a model will be less than interesting. we believe that any model allowing gnly_increases or only_ decreases in attraction lacks face validity and conflicts with everyday acquaintance processes. Newcomb's famous field study of the acquaintance 15 process (1961) observed and measured both increases and decreases in attractiveness between members of a housing unit and related those changes to initial socioeconomic and religious similarities Of the subjects at least at the early stages Of acquaintance formation. The reason that this point is being emphasized is that dissonance theory of source change (as Hunter and Cohen, 1974, show) is one of pure source derogation. we feel that such a model of attraction change is too limited to apply to dyadic processes. Similarly, a straighthrward extension of Byrne's so—called "law Of attraction" would posit AA.. = nA IM..—P.|. 13 31 1 HOwever, this model has the peculiar characteristic that if I and J initially hate one another but upon interaction find that they agree (that is Mji: Pi), then they will remain unattracted to one another despite being in agreement. we find this implausible and at Odds with the evidence presented in Newcomb (1961). Rather, we shall posit a model of attraction change Which is basically social judgmental in character. That is, we assume that when Mji: Pi’ the change in attraction is positive and maximumu WhenMji and Pi are discrepant, then whether the change in attraction is positive or negative depends upon what amount of discrepancy the focal individual is willing to accept. That is, if person I is willing to accept a certain amount of disagreement but no more before his attraction to J begins to decrease, then that amount defines the boundaries Of his acceptance region. The Change equation which will describe the above process is (see Hunter and Cohen, 1979): 2 _ _ 2 AAij = C (tij (Mji Pi) ) Nji . (3) 1 + t%. 13 This equation is graphed in Figure 3. 2tij is the width of the AA.. 13 N .=2 '1 =1/2 ji\ / “(13' iii \ ‘Mja'fPij Figure 3. Changes in.Attraction versus Discrepancy between Message and Attitude fOr Varying Levels Of Transmission. r-il acceptance region centered at Pi. When M.. - P. > t.. or'M.. - P. < 31 1 13 31 1 “tij’ then the change in attraction decreases. When -tij < Mji _ Pi < tij’ then the change in attraction is positive and is a.maximum.for Mji - Pi = 0 or perfect perceived agreement. WhenMji - Pi falls exactly on the border between the acceptance and rejection regions, then the Change in attraction is zero. Obviously, the behavior Of equation (3) depends upon the value of tij' we shall assume with Hunter and Cohen (1979, p. H9) and Sherif, Sherif, and Nebergall (1965, p. 189) that the width Of the acceptance region depends at least in part upon the attractiveness of the other. The more attractive the other, the wider the acceptance region and the less attractive the other, the narrower the acceptance region. Specif- ii ically, it is assumed that ti. = e In this way, the more positive 17 the attraction, the more likely that discrepancies will fall within the acceptance region and produce even greater attraction. Also, the more negative the attraction, the more likely that discrepancies will fall outside the acceptance region and produce further decreases in attraction. In sum, the information processing model for external changes is summarized by equations (1), (2), and (3). The key problem in the infor— mation processing model surrounds the choice of an attraction change equation which is consistent with the information processing position. Because the information processing point Of view has failed to consider source change along with attitude change, we introduced independent criteria fOr attraction change and were led to a model which best fits the social judgment position. However, the change equations for Pi and Qij are ngt_social judgment equations. we next turn to the development of models of the generation Of message content and its transmission in order to complete the input and output characteristics fOr the informa: tion processing model of external changes. The Generation of Message Content we shall consider two models of the generation of message content. Each of these models will be highly speculative since the question of what_is said in interaction has not been well researched. First, suppose the subject always speaks his mind. That is, the message will just be his attitude or M..: P. . (M) This prediction constitutes the first model of message generation and has been the one most commonly adopted in interactive models of atti- tude change (Abelson, 1969; Taylor, 1968). It will be called the 18 "veridical" model. In the veridical model, the speaker says the same thing regard- less Of who the listener might be. But suppose that he seeks to ingratiate himself with J by shifting his message in the direction of his perception of J's attitude. That is, where p is a weighting factor between 0 and 1. If we presume that indi- viduals are more ingratiating for more attractive others and less ingratiating fOr less attractive others, then the weighting factor p would be a function of attractiOn. Furthermore, the more attractive the other, the closer p should be to unity. This implies that p could be chosen to be p= l/(l + e ij). Rewriting the above equation with this new expression fOr p, we have M..: P. + e lj (Q..-P.). (5) l] l _—T._ l] 1. When I thoroughly dislikes J, then he speaks his mind (that is, is veridical) and does not seek interpersonal rewards from J by ingrati- ating him. When I likes J a great deal, then he seeks to further the favors and good graces from J by saying what he thinks J wishes to hear. we shall call this the "shift" model because of the cynicism associated with an "ingratiation" model. The Transmission of.Messages Recall that Newcomb's discussion of message transmission was as a possible response to individual system strain. He presumes that the reaction to individual system.strain through communication to the other 19 actually took the fOrm.of attempts to influence the other's point Of View concerning X. Consequently, influence attempts directed toward the other should arise from.forces created by perceived discrepancies on X. That is, we assume that transmission is intended to alter the other's attitudes toward X and not to alter I's attraction tO J. If N.. is the number of influence attempts generated by I toward J, then 13 we assume that N..: d | Q..-P.| A13 13 ___3£J___l__ 1 + e (5) — z 0. V 1+ (Qij P15 1. + eAlJ where d is a positive constant. Equation (6) is the product Of a discrepancy termland an attrac- tion term. It was chosen to yield the fOllowing specifications: (1) For constant attraction, the greater the discrepancy perceived by I, the greater the transmission from I. (2) For constant perceived dis- crepancy, the greater the attraction which I has fOr J, the greater the transmission from I (see Figure u). (3) For large negative attraction, transmission is still positive and depends upon the amount of perceived disagreement (see Aij: -w in Figure u). The empirical evidence relevant to the evaluation Of equation (6) is both limited and relatively Old. In the early 1950's Festinger and his colleagues at the University Of Michigan undertook a program of field and experimental search related to the question of communication and attitude change in the small group context. In summarizing the results of this research program Festinger (1951) states two prOpositions also predicted by our equation 6: 2O ij ij Figure H. Transmission versus Perceived Discrepancy for Various Levels of Attraction. The pressure on members to communicate to others in the group concerning "item.x" increases mono- tonically with increase in the perceived discrepancy in Opinion concerning "item x" among members of the group (p. 274). and The pressure . . . to communicate . . . concerning "item.x" increases montonically with increase in the cohesiveness of the group (p. 27”). we note that "cOhesiveness" was generally Operationalized as attractive- ness: "Cohesiveness is the attraction of membership in a group for its members" (Back, 1951, p. 9). Specifically, research by BaCk (1951) fOund that increases in group cohesiveness resulted in greater total discussion as well as a greater number of influence attempts. This supports Festinger's second proposition. Research by Festinger and Thibaut (1951) fOund that the weighted number Of communications to dis— crepant others decreased as the other's attitude moved from.an extreme position toward a pre—established group norm. This supports Festinger's first prOposition, and our model. 21 In the most explicit and well-known discussion Of communication and rejection in small groups, Schachter (1951) presents and tests a model in Which the effects of discrepancy and attraction on transmission were presumed independent and additive. This view agrees with that of Festinger (1951) but disagrees with our equation (6) where an inter— action between attraction and perceived discrepancy is assumed. While SchaChter's data are by no means unequivocal (see Berkowitz, 1971 for a discussion of interpretation difficulties), they do tend to support the attraction-transmission hypothesis. No interaction hypothesis is tested. Hewever, certain Of SchaChter's data tend to support an inter- action between discrepancy and attraction for predicting transmission. In eaCh experimental group, Schachter planted a "deviate" who took an Opposite position to that of the group and maintained it, a "mode" who adopted the position most frequently chosen by the group members, and a "slider" who initially took an extreme position and over time converged toward the group norm. If the number Of communications addressed to the slider over time is graphed, then an interpretation Of the graph as the number of messages versus discrepancy is possible since the slider's position is systemically changed toward that of the group over time. This data is presented in Figure 5 for the high and low attraction conditions. Only the data fOr the groups discussing topics :relevant to their purpose rather than irrelevant is presented. The ireason fOr this omission is that SchaChter did not measure the number Of influence attempts or even the relevant messages but rather measured the EIYXSS number Of communications. Berkowitz (1971, p. 238) has criticized this; measure as a possible explanation of the equivocality of 22 hi Attraction lo Attraction / lo lo— hi— hi med med Discrepancy Figure 5. Transmission to the Slider versus Perceived Discrepancy fOr High and Low Attraction (SChaChter, 1951, Table 8). SChachter's results. In addition, a recent study by Rosenfeld and Sullwold (1969) fOund a large increase in irrelevant discussion as indi— viduals who had little use for eaCh other's information interacted over time. Thus, to increase the validity of SChachter's measure as an indicator of influence attempts, only the data from groups interacting over relevant tOpics is considered. This is especially true if the "lo attraction" condition of Figure 5 can be viewed as the high dislike state fOr the experiment. In this case, the high dislike case still produces transmission with low discrepancy rather than yielding no transmission. Clearly, Figure 5 suggests that in predicting transmission, attraction and perceived discrepancy interact thus supporting the impli— cations Of equation (6). With the completion of the message transmission model and the content generation models, we now have the input and output linkages between individual systems through the process of interaction. But befOre summarizing the external Change model according to information 23 processing theory, we consider an infOrmation processing model of internal changes. Internal Changes According to Information Processing Theory Before discussing the application of information processing theory to internal, spontaneous Changes in the individual's cognitions in his intrapersonal system, it deserves mention that most information pro— cessing theorists would argue that spontaneous Changes in perceptions and attitudes do not occur. Rather, changes in perceptions and attitudes occur as a function Of the rational consideration of one's position rela- tive to the available infOrmation and arguments to which one attends. However, there is one rationale which infOrmation processing theorists might find acceptable. Even when the other, J, is not present, I may think about him.and may think what J would say about X. That is, I imagines J giving messages and those messages should have some impact. There are then two possibilities: I remembers J's actual messages in which case attitudes and perceptions change as some function of the actual message. This function would presumably reflect such effects as fOrgetting and selective retention. Or I can create messages fOr J based upon his perception of J in which case attitudes Change as a func— tion of his perception of J's position. we have Opted fOr the latter in the models below. In the individual system, it is not discrepancies between incoming messages and attitudes, or messages and perceptions which are evaluated relative to one another to determine change, but the discrepancies between the internal system.variables Pi and Qij' That is, the change 24 in Pi fOr internal information processing is exactly analogous to equation (1) fOr external information processing, except that the infor- mation being processed relative to the focal person's attitude is infor— mation about the other whiCh is internal to one's own cognitive system, that is Qij' Thus, rather than evaluating his attitude relative to the "hard" information provided by the other's message, the fecal individual evaluates his attitude relative to the "soft" information about the other which is already in storage. That is, A.. AP. = r e 13 (Q.. — P.) , (7) 1 13 1 1 + eAij which is graphed in Figure 6. Notice that this equation, like equation (1), has a credibility multiplier so that more attractive others produce greater changes in attitudes in the direction of the perception Of the other. Unlike equation (1), no transmission term is involved since transmission is relevant only in the interaction or external change processes. Notice in Figure 6a that the greater the perceived discrepancy, the greater the change in Pi fOr constant attraction. For the same amount of perceived discrepancy, the greater the attraction the greater the change in Pi. Figure 6b shows that the change in Pi goes to zero asAij goes to negative infinity if (Qij-Pi) stays finite. .As Aij goes to positive infinity the change in Pi becomes a simple linear function O:mamsom..5 £33 coapomfibt mo couflogm m mm. US... 23 955:8 oagrcoufiogflc. fies monumgmflo om>amosom mo cowpoqom m mm .m a...” omqmfiu m5. lumflU .mlnlmfipn/ .. was: Hum..HHunI..I.l..........\\\\\\\\ 26 he will change his attitude. However, if I dislikes J a great deal, then perceived discrepancies on X will produce little or no spontaneous changes in attitude. An equation to represent internal changes in Qij is also simple to develop since it is exactly the same as equation (7) with the roles of Pi and Qij reversed. That is, in evaluating his perception of the other's position I compares it to his own position and changes his per— ception in proportion to the discrepancy between Pi and Qij' Thus, we may write an equation for Qij directly: AQij = q e 1: (Pi — Qij) . (8) l + e lj The equation for the change in Qij due to internal fOrces has exactly the same qualitative description and the same graphical representation as those fOr Pi except that Pi and Qij have been interchanged. That is, Qij is changing in the direction of Pi by an amount Which is a function of I's attraction to J and the constant Q. Like the constant r in equation (7), q is positive and less than or equal to l. q is not necessarily equal to r and their ratio would indicate whether Pi is easier to change than Qij (r > q) or Qij easier to change than Pi (q > r) for internal change. Finally, given the validity of equation (3) fOr changes in attrac- tion due tO messages from the other, the comparable equation fOr internal changes in attraction merely replaces the external informationMji with the internal information Qij’ Thus, _ 2 _ _ 2 AAij - (tij (Qij Pi) ) a (9) 1 + t%. 11 27 where we assume as before that tij = e ij. Figure 7 shows the critical points for the change in attraction at differing amounts of perceived disagreement. When Qij—Pi> tij or Qij—Pi < _tij’ then AAij is less than zero. When -tij < Qij-Pi < tij’ then AAij is greater than zero and should be maximum for Qij—Pi = 0 or perfect perceived agreement. When Qij—Pi falls exactly on the border between the acceptance and rejection regions, then AAij is zero. Figure 8 graphs the cases fOr tij = m and tij = 0. When tij = m, the acceptance region is infinitely wide so that any finite discrepancy is perceived as "near zero" and the change in attraction is always positive. When tij = O, the acceptance region is infinitesmal so that any non—zero discrepancy no matter how small produces decreases in attraction. This latter case is similar to the dissonance model of source derogation (Hunter and Cohen, 197”, p. 81). Obviously, then, the behavior of equation (9) depends upon the value Of tij' we shall assume, as we did in the case fOr external changes, that the width of the acceptance region depends at least in part upon the attractiveness Of the other. The more attractive the other, the wider the acceptance region and the less attractive the other, the narrower the acceptance region. Specifically, it is assumed that tij = e ij. In this way, the more positive the attraction, the more likely that discrepancies will fall within the acceptance region and produce even greater attraction. .Also, the more negative the attraction, the more likely that discrepancies will fall outside the acceptance region and produce further decreases in attraction. 28 .mmosom HmcsmPcH sow Hmooz Homemoob HMHOOm m Op mcaosooo< >Ocmmmgomam om>flmooom mo coavoeom m we coweomsep<.ea wmemno wee .m madman m h.nwl..b.v ..u./ l.\...flu.nu 8”. .U. ma ..<< .eoapomepp< CH mmmcmno OP ompmaom mm eoapooflom ecu moememmoo<.mo meOHmmm .5 mQDMflm scammm :oawmm coawmm coapommom occupamou< COflaommmm ma ma ma OH..<.< XMOEH <4 OH..<< ma . ma . ma - ma - o v ..¢a o A ..¢d o A ..<< 0 v ..<< L)!, /N, _ /K\\ 1 ma. A.. ma. .p+ .m ..Pl 29 It is important to recognize what we have done in equations (1) through (3) and (7)-(9) in the light of Newcomb's paradigm. Recall that Newcomb carefully distinguishes between an individual and a col— lective systemxand argues that changes in an individual's attitudes, perceptions, and attractions can arise both through internal, Spontan— eous changes and through the communicative and persuasive acts Of the other. But as we also noted, Newcomb's model does not Offer the specif- ics as to how internal and external changes occur nor how they may differ. What we have done thus far, is to bring the assumptions of information processing theory to describe external changes due to messages from.the other and to extend the theory to describe internal changes as well. The change equations which have been developed are summarized in Table 1. Notice that in the external change equations, it is the information provided by the other—-an outside source-—Which is compared to the internal reference points Pi and Qij' This gives rise to the difference terms in the external change equations. On the other hand, fOr the internal changes it is information which is directly available from.the fOcal person's cognitive space which is compared. Thus, the difference terms in the internal change equations arise from comparisons Of internal reference points to one another. In a sense, internal infOrmation processing is an "irrational" process which seeks to make compatible, information which is incompatible. By altering attitudes, perceptions, and attraction to the other on the basis of internal information alone, these alterations may or may not have a basis in fact. External infOrmation processing is "rational" at least in the sense that attitudes, perceptions, and attractions change on the 30 Table 1. Change Equations for Internal and External Changes Based upon InfOrmation Processing Theory. Internal External Aij Aij APi: r e A, (Qij—Pi) a e A (Mji-Pimji 1 + e 13 1 + e 13 A13“ Aij A ..: e (P.— ..) b e (M..— ..)N.. Q13 q A.. 1 Q13 A.. 31 Q13 31 1 + e 13 1 + e 13 . 2 _ _ 2 M13" 8 tij (Qij Pi) c ti]. — (Mji-Pi)2 Nji 1 + t?. 1 + t%. 13 l] A. . where t.. = e 13 1 Input-Output Transmission Content _ A.. Nij _ d iQii-Pil 1 + e U Mi]. -.- Pi ' . A.. _ 2 /'1 + (Qij Pi) 1 + e 1] Mij = Pi + ti. (Qij-Pi) 1 + tij basis Of additional external infOrmation rather than through a "ration— alizing" process Of internal changes. Simplifying thegDynamics of IJX Situations Our general procedure fOr building an understanding of the complex dynamics embodied in the equations of this chapter will be to introduce simplifying assumptions and reductions of complexity initially and then to relax those assumptions. First, the equations of spontaneous change 31 will be analyzed independently of the equations for induced Change. This corresponds to the assumption that the internal and external pro- cesses do not Operate at the same time. This would fellow from.our assumption that the autonomous fOrces arise fromlimagined messages from J to I whiCh take place when I is thinking about J and J is not in fact present. Second, models in which the rate of transmission is held constant will be considered separately. In fact some investigators have found equal transmission rates to be the norm if only two people are talking. Thus, our equal, fixed transmission rate models may ultimately prove to be more realistic for dyads than our fancy model Which.was derived from small group studies. we will also consider separate models in which the attraction which I has fOr J will be assumed constant. These serve to build some understanding of the changes in Pi and Qij befOre considering the more complex case. The assumption of constant attraction will both reduce the number Of equations to be considered and remove some pesky non- linearities. Substantively, constancy of attraction is associated with long-term friendships which are unlikely to change or with disagreement over relatively unimportant topics. Also we restrict our analysis to the interaction of two individual systems. CHAPTER III ANALYZING THE DYNAMICS OF CHANGE IN IJX SITUATIONS: INFORMATION PROCESSING THEORY The previous chapter develOped static descriptions of the change in state variables, Pi’ Qij"Aij’ and.Mij. These change equations are summarized in Table l. The task of this chapter is to develOp an understanding of the behavior of each model over time so that compar- isons between models can be made. In achieving this goal, we shall find it advantageous to move from a difference equation format to a differential equation fOrmat. In this way, the mathematics will be facilitated without any conceptual or substantive changes. Understanding the dynamics Of IJX situations will be develOped through standard mathematical analysis fOr systems of differential equations when that is possible and through numerical analyses of the system when analytical procedures break down. Most of the mathematical analyses will be relegated to the two appendices.A and B. The numer- ical results were generated on a CDC 6500 computer using a standard Runge-Kutta method. The program.was developed at the NOrthwestern University Vogelback Computing Center by John Michelson and adapted for local use by the author. Most of the over time trajectories presented below were generated numerically. 32 33 Our general method Of proceeding is as fellows: The chapter is divided into two:major sections. The first Of these sections will take up the information processing (IP) model under the assumption of ESE? stant_attraction. Within that section, we shall discuss the internal change model, the two message models with transmission constant, and then the two message models under conditions of varying transmission. The second major section will take up the IP model with variable attrac- tion. Once again, within that section, we discuss the internal change model with varying attraction, the two message models with varying attraction but constant transmission, and then the two message models with varying attraction and variable transmission. Of primary interest in each model is the presence or absence Of equilibria or critical points and the stability of those critical points. Since the critical point defines a point fOr which there are no changes in the state vari— ables, those points (if there is more than one) define the balance points for the system. If there are no critical points, then we shall be interested in the direction that the state variables are moving as time t becomes infinite. That is, if the system is not going to a balance point where is it going? Most mathematical discussions and derivations will be relegated to the appendices with the text reserved for graphical and verbal reports. we hope that such a format will facilitate comprehension with— out sacrificing mathematical rigor. 3” Information Processing with Constant Attraction The information processing model with constant attraction consists of two parts: an internal change process due to Spontaneous fOrces toward change and an external Change process due to interaction between the individual systems. The reader is reminded that in keeping attraction and transmission constant, and separating internal and external Change processes, certain strict substantive assumptions are presumed valid. Namely, attraction is deep—seated and long—termu the amount of discussion is rather evenly spaced throughout the period of interaction, and either internal or external processes dominate as a result Of exogeneous factors enhancing or limiting discussion Of the topic. With these strong assumptions the mathematical analysis of the IP case becomes somewhat simplified. Internal Changes Only - The change equations governing Spontaneous individual changes toward.ba1ance are the same fOr both individuals and are given by: d P. = r k.. (Q..—P.) (10a) 1 13 13 1 dt dt where kij replaces e ij/(l + e ij) because Aij is presumed constant. The analysis of this pair of linear equations is quite simple. From.an intuitive point of view, since r, q, and kij are positive, equation (10a) says that the change in Pi is always toward Qij and (10b) says that Qij is always Changing toward Pi' In a manner Of speaking, Qij and Pi are "chasing each other" over time and, hence, converging toward 35 one another. The bigger the value Of kij’ the faster that they will converge. If the ratio Of‘P to q is large, then Pi will be changing more toward Qij than Qij will be changing toward Pi. Empirical studies by Kogan and Tagiuri (1958), Newcomb (1961), and Curry and Emerson (1970) support the view that it is the perception of the other that is changed while attitudes remain relatively stable (that is, the q to r ratio is large). TherefOre, in our numerical examples we will take q/r to be 3 to l. The graphs of Figure 9 show the over time behavior of Pi and Qij when there is initial perceived agreement and initial perceived disagreement for two levels Of attraction. Notice that in both graphs there is an asymmetry in that Qij changes more than Pi changes. This is due to setting the parameters r and q in a ratio of l to 3. An interesting case in Figure 9 is the over time behavior Of the system when there is perceived agreement, Pi = Qij' In this case, the vari— ables do not change over time at all. The system.starts out "at rest" and remains there. Such a point is called an equilibrium.point or critical point of the system. These points can always be fOund (assuming that they exist) by setting the equations for example (10a) and (10b) to zero and solving for those values of Pi and Qij which sat- isfy the equations. In the case at hand any pair in which the value of Pi equals Qij is a critical point. That is, we have an infinity of critical points which lie along the line Pi = Qij in a plane of Pi, Qij points (see Figure 10a). As we shall see in this and upcoming sections, models which emphasize discrepancies often have such an infinity of critical points (which in non—linear cases produce certain other 36 13 ijj ii =.3 q=.9 / "/ t Qij r=.3 q=.9 Figure 9. Internal Change Trajectories for IP with Positive Attraction (+3) (a) and Negative Attraction (=-3) (b) for Initial Perceived Disagreement. mathematical difficulties). Now that we have seen that the solutions to our equations con- verge toward one another and what the critical points Of the systemlare, we are easily led to conclude that any set of initial values will con- verge toward a critical point. This result is shown mathematically in Appendix A. Figure 10 also shows this convergence for different values 37 .mm... u QB 90m 33 use .H u «Co pom on .m u EU 90% mmHQOPommmfiH. mode 393m Hooavasu A5 .38 356m Hmoafiflo mo mead Amv “coauaowufia. Pcmpmeoo spun; mmmqmso HMEMHEH pom mOHQOPOO muse Osman mmmem . 3” 93mm // \\ \\\\ v/ // r a , 30/ a 2 .M// A o E 38 of the parameters. These graphs are called phase—plane trajectories and indicate possible movements of the pair of points (Pi’ Qij) over time. The arrows on each line indicate the direction toward which the pair of values is moving. .As can be seen, fOr all values Of the para— meters the trajectories terminate on the line Pi = Qij which is the line of critical points. Thus, fOr any set of initial conditions and any set of parameter values (as long as they are positive) the system will converge toward a critical point. What conclusions can be reached about individual system balance as a result of considering only the internal forces with attraction held constant? First, the system.is unchanging when the perceived discrep- ancy is zero, Pi = Qij’ regardless of;how_I_feels toward 3. This would mean, fOr example, that if I disliked J a great deal but perceived no differences between himself and J regarding X, then no spontaneous changes would ensue at all. Such a position is clearly predicted by Newcomb's positive balance model and by dissonance theory. In addition, the above description would be balanced in both Heider's and Festinger's descriptions Of unchanging IJX situations. This principle also implies that if I likes J but perceives no differences between himself and J regarding X, then no spontaneous changes would result. This prediction agrees with Newcomb's, Festinger's, and Heider's models. Second, when there is initial perceived disagreement, then rates of change toward perceived agreement depend upon the attractiveness Of the other. .As Figure 9 illustrates, the less attractive the other the slower the rate of convergence toward perceived agreement. In the limit as attractiveness goes toward negative infinity,kij goes to zero 39 and Pi and Qij remain constant. In other words, if I hates J and per- ceives that they disagree, he will not undergo any spontaneous changes toward perceived agreement. This is more in keeping with Newcomb's positive balance model since extreme dislike should not "engage" I in the IJX situation enough to result in spontaneous changes toward balance. On the other hand, in Heider's model only our predictions for positive attraction would yield "balanced" states. For finite negative attraction, the limiting states of this model are imbalanced. For infinite negative attraction the final states are balanced only if the initial states happen to have opposite signs. Thus, the key point of differentiation fOr the internal model is with regard to the level of attraction. When the attraction is in the vicinity of neutrality or is positive, the ultimate states are balanced according to Heider and Newcomb. When attraction is highly negative, the changes follow Newcomb's predictions from positive balance. However, for moderate negative attractions, the equilibrium of Pi = Qij is reached. Such a point wouid be imbalanced according to Heider's view but non- balanced (or vacuously balanced) according to Newcomb's positive balance model. That is, Newcomb would not have predicted any change. External IP with Shift Message and Constant Attraction and Transmission When I and J are interacting so that the internal processes are not Operative, then the change equations governing induced changes due to the messages transmitted are: dPi : a. kn. (Moo — P0) N00 (1h) -d-t— lj 31 l 31 dQ..= bk..(M..—Q..)N.. (11b) 1.! l] 31 l] j]. ”O dP. = a k.. (M..—P.) N.. (11c) J 31- l] J 1] dt dQ.i = b kji (Mij-jS) Nij (11d) dt where N.. and N.. are assumed to be constant and k.. and k.. are the 13 31 13 31 constant attraction parameters defined as before . Also, according to the shift model, M.. = P. + k..(Q..—P.) and M.. = P. + k..(Oj..—P.). In 13 1 13 13 1 31 3 31 1 3 this interactive case, both pairs Of individual system variables are "chasing" the message values which the other is generating. But since the message values are themselves dependent upon individual system vari- ables, then I's state variables Pi and Qij are chasing a weighted sum Of J's state variables (as a message) and vice versa. Since a and b are positive constants less than one, and the attraction and transmission terms, k.., k.. l] , and N. . and N. . respectivel , are positive, then we 31 13 31 y might expect the system of equations (11) to converge upon one another so that eventually Pi: Qij: sz jS. This is exactly what happens, as is proven in the second section Of Appendix A. In fact, Appendix A presents the general solution for the common limit Pi*= Pj*= Qijig: Q3. 1* as a function of the initial values (for the present case Of fixed attrac— tion and fixed transmission). The conclusions found below by looking at examples are all borne out by examining that general solution. The simplest way to begin analyzing the system of equations (11) is to consider the case in which I and J are equally attracted and trans— mit an equal number of messages. As we noted earlier, this model may best represent the interaction of natural dyads since there is some evidence to indicate both an equality in Speaking time (Jaffe and ”1 Feldstein, 1970) and in attraction (Willis and Burgess, 197”) fOr the dyadic case. Also, this case can be completely solved mathematically (Appendix.A, section 2). Throughout our discussion of this model, it will be assumed that the parameter b > a and that the ratio is 3 to 1. .A study by wackman and Beatty (1971), reported also in wackman.(l973), supports the view that changes in perception of the other's position occur muCh more quickly in interaction than do changes in one's own position. Figure 11 presents fOur sets of trajectories for the equal attraction, equal transmission case. Obviously, there are an infinity of initial values and we may choose but a few substantively interesting ones to discuss. The fOur trajectories differ in their initial values: (1) In frame a, I and J initially disagree with J perceiving disagreement but I perceiving agreement; (2) in frame b, I and J agree but both per— ceive disagreement; (3) in frame c, I and J initially disagree but both perceive agreement; and (”) in frame d, I and J disagree and both per- ceive disagreement. The parameters, a and b, and the attraction and transmission terms all play important roles in determining the speed of convergence of each of the variables. First, we assume that b/a = 3. TherefOre, with attraction and transmission equal we should Observe more rapid Changes in Qij and jS toward the point of convergence than Pi and Pj. This is exactly what we find in all frames Of Figure 11. Notice that in frames a and b, the attitude variable is changing morg_in.magnif tug§_than the perception variable. This agrees with the general result fOr final state derived in Appendix A, section 2. However, even in frames a and b, the perception variables reach the equilibrium point well befOre the attitude variables. This is more visible in frames c and d where the ”2 .3 .8 88853 9:8 as... S .8 8335.5. H93 Eoflowfiaa 2388 a6 tea 888: at“: an 8m mmflopommme .2 85mm % equilibrium point (= 0) is equidistant from the initial values of all variables. Clearly, perceptions converge much more quickly than attitudes . Thus far, we have been assuming that transmission is constant and, hence, can be assigned any values that we wish. However, some Of these values are inconsistent with our model of varying transmission which stipulates that if I and J are equally attracted and both perceive disagreement, then transmission should be equal. If they are equally attracted but both perceive agreement, then transmission is equal and zero. When I and J are equally attracted, but I perceives agreement and J perceives disagreement, then J's transmission should be greater than I's which is zero. These cases and the eight other possible cases are summarized in Table 2. The table presents a comprehensive categorization Table 2. Predictions of Relative Transmission between I and J as a Function of Relative Attraction and Initial Perceived Agreement (PA) and Perceived Disagreement (PD). PD PA PD(I) PD(J) IgJ ISJ RA(J) PA(I) ”=A” m.=m. o N >0=NH m.>0=NH 11 31 13 11 11 31 11 11 A >A” m.>m. 0 N >0=NH m.>o=NH 1] 11 11 31 11 31 11 11 A >Au m.>m. 0 N >0=NH m.>0=NH j]. l] j]. l] l] 31 31 l] of predictions about transmission based upon choice of attraction para- meters and initial values on perceived agreement (PA) or perceived dis- agreement (PD). Each graph in Figure 11 is rated according to its consistency (C) with or degree of discrepancy (Dl—DS) from the 1+” predictions of the transmission model. That is, if a graph receives a rating of D5, its initial values and attraction values are highly dis— crepant from the predictions that would be made based upon the trans— mission model. The rating scheme is summarized in Table 3 and can be used as a quick reference to ascertain the fit between attraction para- meters, initial values and the predictions from.the transmission model. Table 3. Summary of Rating SCheme fOr the Degree Of Fit between Attraction Parameters, Initial Values and Predictions from Transmission Model. Symbol _ Meaning Dl If Nij or Nji is predicted to be zero, but one is greater than zero. D2 If Nij §E§.Nji are predicted to be zero, but both are set greater than zero. D3 If Nij > Nji’ or Nji > N13’ or Nij = Nji is predicted, but the set values are different. D” If D1 and D3 hold. D5 If D2 and D3 hold. C The predicted values of Nij’ Nji are the ones chosen. The ratings are presented fOr the graphs Of Figure 11 in each frame. Frames b and d are consistent with the transmission assumptions while d is someWhat discrepant and a is very discrepant. Consider the equal attraction but unequal transmission case. Figure 12 presents this case for the same set of initial conditions presented in the same order as for Figure 11. The most direct way of understanding Figure 12 is by comparison with Figure 11 since the two ”5 "coapomswe< Hoopmcoo out Pwaem OMMmmmz eyes CH sow mmHQOpommmsH .coammdemcmsa amoeba: poo coavomsve< Hmoom .NH mamas Am .HV u coawmaamcmga Am .Hv u coammaEmcmsH Am .mV n coauomHHa< N Am .mV n coavomspp< p U. NHO NHO Hm /u So. 8 Na mm .o .0 Am .HV u coammHEmcmsH Am .Hv u coammHEmcmsH Am .mV n COflwomppH< Am .mV n coapomsee< P P . \ 8 a H 8 Na K A e ”6 differ only in the inequality of transmission. There are two Observa— tions to note: (1) the convergence points are not the same and (2) the rates of change Of Pi and Pj can differ. The easiest case to see with— out perfOrming a lot of calculation is in frame c. In Figure 11c the point of convergence was 0. But in Figure 12c, the convergence point is much closer to J's initial values. As we Show With equation (A6) in Appendix.A, fOr equal attraction, when J out-transmits I then I will do most of the changing toward a weighted sum Of J's initial values. Frame c is such a case and the change in convergence point can literally be "seen" by comparison with Figure 11c. Secondly, in Figures 11b and 11c the convergence rates fOr Pi and Pj and fOr Qij and jS were equal. In Figures 12b and 12c they are "distored" in the expected direction. That is, both Pi and Qij are changing more per unit time than their counterparts Pj and jS precisely because J is transmitting more to I. In Figure 13 we present the equal transmission, unequal attrac- tion case with the same set of initial values as fOr Figures 11 and 12. The most striking aSpect Of the trajectories of this figure, compared to those of Figure 11, are the convergence points. In all cases, the fact that J dislikes I so much while I likes J yields the strong result that J's attitudes and perceptions change very little or very slowly while I's change a lot and rapidly. Furthermore, since J dislikes I so much, his message to I is for all intents and purposes just his attitude, which.does not change much. .As a result, all the other state variables converge to Pj which stays close to Pj(0), just his attitude. That is, he does not shift his message so as to ingratiate I. Since his attitude does not change much, both I's perception of J and I's attitude converge .coammHEmcmnH Human ecu cOHPOMHPH< Hmzomco "coapomsve< 3.5.866 on... tum 888: .33 H 8m mmHBBmmme .mH 85mm S .3 u Emmanuel S .m; n 83883 p Hm Hm NH 0 - mm 0 mm [MHHHHHW1\\11\\\ O m 7 U m U u! r H mm m 1/// Am .mV n scammaawcmse Am .mV n coammaamcmoe mac Hm .muv u cOHpomgpea HH_- Hm .muv u aOHpomeH< H /r . - m 1 / . ma so Hmo Hma me to J's message value (Which is still almost identical to his initial attitude). At this point I's message equals I's attitude which equals J's attitude. Hence, J's perception of I converges to 1'8 message Which is I's own attitude. As a result, all other state variables converge to Pj which stays close to Pj(0). Figures 1” and 15 can be discussed together since they represent the two fOrms of the unequal transmission and unequal attraction cases. Figure 1” offers the case whiCh is at Odds with our transmission pre- dictions: namely, that the less attracted Speaker will be the greater transmitter (see the ratings for Figure 1”). Figure 15 presents the more plausible situation (Collins and Raven, 1968, p. 123) where attrac- tion and transmission are positively correlated. As the ratings show, this figure has parameters and initial values which are more consistent with our own transmission assumptions. Let us compare the convergence points in Figures 1” and 15. In both figures these points are very close to one another for the same set Of initial conditions. That is, 1”b and 15a have convergence points which are very similar, as do 1”b and 15b, and so forth. The only dif- ference between Figure 1” and Figure 15 is that in the former J out- transmits I by a ratio of 5 to 1 while in the latter I out—transmits J by the same ratio. Thus, it is the attraction parameter which (3, -3) which almost completely determines the final state of the system.given the same set Of initial conditions. But the reason fOr this dominance has to do with the choice of parameters in this case. For attraction (3, -3), we have kij = .95 and kji = .05. With these values fOr attrac- tion I would have to out—transmit J by a ratio of 19 to l to achieve an .Hmoozncoammaamcmsy seem HemomsOmHQTcowmmaamcmsH Hmsomco_o:m . cowpomsye< Hooves: “coawomsvv<.pQMPmcoo ocm pmaem mwmmw z Spaz.mH mom mmH90900mmnB .:H mesmam ”9 cm «mex Am .Hv u cowmmaawCMQB Am .HV u scammaamcmsa Am: .mv n coapomseH< Hm Am: .mV n coapomoea< P # mo Hmo .U ma Na manu1“\\\\\\‘ Am .HV u conmHewamge HHo Am- .mV n coHpoman< HNO HNO ma Am .Hv n coammasmcmsy Am- .mV n cOHpoman< :Q -.-..4-a)#— A. 7.37: a. ..Q. 50 soavowbpa. Hugues .Hmooz coammaamemfiw new... vaVmflmcooncoflmmHEmg Home: out .:0Hpomnpp<.pcmpmcoo cam yuaam mmmmmmz eves AH pom mmHaopowmmge .mH mgamHm NW \ NH0 Q .3 u coammafimfimfia Am- .mv cOHpomppe< H .p Hm H - mo .,Na a «Ho 3H .3 scammgg Am- .mV n cOHpompppa p. Ho 0 H .8 n COHmmdamsmHe Hm- .mV n cOHpom9pe< .3 AH .mV n Am- .mv COflmmHEmfié :OHpomnpea 51 equal weighting fOr I's and J's initial values. When I out-transmits J by a ratio of 5 to 1 there will be somewhat more shifting toward I's initial values than if J out-transmits I by the same ratio. This is exactly what we see by comparing the two figures 1” and 15 frame by frame. For example, in frame a the point of convergence is somewhat less negative for Figure 15 than 1”. There is also somewhat less change in Pi and Qij in Figure 15 than 1” and somewhat more change in Pj and jS. Both of these characteristics are due to I's transmission being greater than J's in Figure 15 while the reverse is true in Figure 1”. Graphical representations of the over time behavior Of a complex system such as that of equations (11) does not carry as much informa- tion about the overall pattern of trajectories as does a phase-plane graph (cf. Figure 10). However, when there are more than two variables to be considered, then a 3—, ”-, . . . , N—dimensional phase spage_is required. While there is no conceptual or mathematical limit to the dimensionality of a phase space, its heuristic value in portraying the set of possible trajectories for even three variables is essentially negligible. Thus, we are restricted primarily to representing only over time trajectories. The disadvantage of using the time trajectories is that the trajectories hold fOr only a single set of initial values. This is the reason that four different initial values were graphed in the Figures 11 through 15. In the IP case with constant attraction, the choice Of initial conditions is not crucial since, as we show in Appendix A, all initial values ultimately converge to one Of the critical points, Pi = Q.. : P. = Q... In other words, the convergence 1] J 31 or non—convergence of the state variables is independent of the initial 52 conditions. Clearly, this does not mean that the initial conditions are not important since they do determine the p9int_of convergence (along with the parameters) fOr the system with constant attraction. BefOre considering some interesting Special cases where con- vergence of the state variables to a common limit is not Observed, a few remarks on the general character and properties of equations (11a) throngh (11d) will be made. As is shown in Appendix A, the equations (11) are a system of four linear differential equations with constant coefficients. It may also be written: its = yes, (12) where S'is a ” x 1 vector of the state variables and WLis a ” x ” matrix Of coefficients. The interesting and useful result from linear systems theory is that the critical points of the system and their stability depend upon the matrix W. As is discussed in Appendix A, systems of equations like those of (11a) through (11d) yield matrices which allow one to conclude immediately that each solution converges to some critical point, and that there exist an infinity of critical points such 1] linear models built upon discrepancy assumptions Often will have the that Pi = Q.. = P3. = jS. The interesting general Observation is that characteristics necessary to conclude convergence to one of the critical points of the system. Because linear discrepancy models have played an important role in social psychological theory, it is exciting to dis— cover this linkage to mathematical conditions for stability. Returning to the IP model Of equations (11a) through (11d), thus far we know that for any fixed finite values fOr attraction, Aij and Aji’ 53 and for non-zero transmission, Nij and Nji’ both individual systems and the collective systemnwill converge over time toward perceived agreement and actual agreement. That is, one Of the infinite set Of critical points will be reached. Even for highly negative attractions the above result holds. Of course, the more negative the attraction, the slower the change toward convergence. Convergence is also guaranteed even for tiny transmission rates although the change is slower. Thus, the con- vergence point is the point Where both individuals perceive no dis— crepancy between their own position and that of the other and their perceptions of the other are accurate. If we do not insist that attraction be a finite parameter or that transmission be non-zero, then two interesting cases develOp. If J likes I so that A.ji > 0 and kji > 0, and if I hates J so thatAij is appro- aching negative infinity and kij is zero in the limit, then d Pi/dt = 0 and d Qij/dt = 0. This only leaves P3. and jS as changeable. .As we show in section 2 Of Appendix A, the critical value for Pj and jS is just the message whiCh I initially sends and does not deviate fromn That is, I does not change at all but J changes toward the position that I advocates. This means that in J's individual system there is perceived agreement and that in I's individual system there may or may not be perceived discrepancy depending upon the initial values of Pi and Qij' In the collective system.there would be actual agreement. The reason is that since I hates J the message being sent by J is not shifted at all in the direction of J and, hence, is just I's constant attitude. On the other hand if Nij is zero while Nji is non—zero, then J receives no messages from I and de/dt = 0 and dei /dt. Only person I su changes. Now if both individuals are moderately attracted to one another, then (see Appendix A), the critical point for I is just J's constant message. This message is Shifted part of the distance toward Q ultimately has no perceived discrepancy. But there is actual discrepancy ..(0). Since both P. and Q.. converge to M.., I's individual system 31 1 13 31 in the collective system; Pi will eventually be Mji’ not Pj' Since Pj remains constant at Pj(0), the limit Of Pi will always differ fromPj by the shift in J's message toward his initial perception Of 1. Thus, the absence of communication from 1 leads to a stable critical point which is also a point of collective system discrepancy in attitudes. Such a result could never have been anticipated by any of the balance theorists because the process of interaction is never explicitly included in their models. The reason for the discrepancy in attitudes in this case is twofold: (1) there is no communication from I thus leading J to retain his inaccurcies about I, and (2) the message content is shifted in the direction of the other. were there no shift at all (that is if the veridical message model was operative), thenMji = Pj(0) and eventually Pi(t) = Pj(0). That is, had J's message been veridical, then ultimately there would have been no discrepancy in attitude at the collective level. To summarize, the IP model with constant attraction was shown to have an infinity Of critical points satisfying Pi = Pj = jS = jS when the parameter of attraction is finite and that of transmission is finite and non-zero. Furthermore, it was shown graphically and through some powerful theorems in Appendix.A that the system always converges to a point of equality. That is, regardless of the initial values of the state variables, they will always converge toward a critical point of 55 no discrepancy at either the individual level or the collective level. We also saw graphically that the role of attraction and transmission was to speed up or slow down the rates of convergence depending upon whether they were larger or smaller values. Finally, in taking up the Special case of no transmission by one of the fecal individuals, we saw a dramatic demonstration of the differences between collective system dis— crepancies and individual system discrepancies when the role of com— munication is explicitly included. IP with Veridical Messages: Constant Attraction and Transmission In the veridical message case, equations (11a) through (11d) still describe the behavior of the IJX system.except that each person speaks his mind, or M.. = P. and M.. = P.. In this way, P. and Q.. are 13 1 31. 3 1 13 "chasing" Pj and Pj and jS are chasing Pi’ and we might expect, as before, that the system will converge to the point of equality Pi = P3. = Qij = jS. This is exactly the case (see Appendix B). Furthermore, the peie£_of convergence is also shown to be a weighted sum.of 1's and J's initial messages which in this case means their attitudes. That is, the common limiting value Pi* = Pj* = Qij* = jS* is entirely independent of their perceptions of one another. The weights given eaCh initial attitude in the equation for the limiting values are kiiji fOr Pj(0) and kjiNij fOr Pi(0)' That is, when the product of the credibility factor and transmission factor is greater for I than for J, kiiji > kjiNij’ then the final state is closer to Pj(0) and I is doing most Of the changing. When k..N.. and k..N.. are equal, then the final state 13 31 31 1] merely "splits the difference" between I's and J's initial attitudes. Notice that the veridical message model differs from.the shift model in 56 that the final state does not depend upon Qij and jS at all! The perception of the other is extraneous to the final state of the system. In fact the attitudes do not depend upon the perceptions at all. The equations for Pi and Pj in the veridical message model are 9P1 —dt— = a kiiji(Pj_Pi) dP. —d—l = a k..N..(P.-P.) . t 31 l] l 3 And if the transmission rates are assumed constant, then Qij and jS appear nowhere in the quations for Pi and Pj' we should also note, that the IP shift and veridical models also undergo no change if Nij and Nji are zero, or if in the limit Ai' and ‘A'i go to negative infinity. If we are to be consistent with our~model J of transmission, then N.. and N.. should be zero only When Q.. — P. = 0 13 31. 13 1 and jS — P. = 0 or when both individuals perceive no disagreement. 3 Recall that transmission is §e£_terminated byAij andAji going tO negative infinity. But if they hate each other so muCh that Aij = Aji = —w, then both I and J view the other as infinitely incredible and stop changing in his direction. That is, ifAij =Aji = —m, there is no change in the system. Like the shift model, the veridical model converges to an equilibrium.point with all equal values fOr finite attraction and non— zero transmission. Unlike the shift model, the veridical model con— verges tO a final state whiCh depends only upon I's and J's initial attitudes and not upon a weighted sum of I's and J's initial attitudes and perceptions. 57 IP with Constant Attraction and Variable Transmission: Veridical and Shift Messages General Discussion - In the next several sections we will consider the infOrmation processing models in which transmission is allowed to vary. In each case we use the basic equations dP. l - — 'TTE‘ ' a kij Nji (Mji Pi) inj = b k.. N.. (M.. - Q..) '71E“' in combination with the variable transmission equation .. = Qij" Pi <1 + k..) 13 l] / l + (Q.. - P.)2 13 1 with similar equations for Pj, jS, and Nji' Two distinct models of this type are defined by the two models Of message generation. In discussing the models in which transmission was assumed to be fixed, we noted that in eaCh case the basic results fell into one of three categories: First there was the "typical" case in which neither attraction was negative infinity and in which neither transmission rate was taken to be 0. Second, there was the Special case obtained if one or the other attraction was allowed to be negative infinity (and this will still be a special case below). Third, there was the special case in which one or the other Of the transmission rates was taken to be zero. In the "typical" case, both people kept transmitting messages to eaCh other until all four system variables were driven to a common limit of 58 The brunt of the analysis of each model then consisted of the determi- nation Of the relationship between the limiting value of the system and the fOur initial values. In the "special" case in which one or the other Of the trans— mission rates was assumed zero, the variables did not converge to a common limit. If for example Nij were 0, then person I never trans- mitted a message to person J and hence person J never Changed. Thus person J had his attitude and his perception fixed at whatever value they were to begin with. And Since person J had a fixed attitude and perception Of person I, he always sent the same message to person I. Person I then responded to these messages by having both Of his values converge to that constant message value transmitted by J. For the variable transmission models to be considered below, it turns out that the "typical" case is the case in which one or the other of the transmission rates is 0. To see this, we need merely look at the implications of the typical case for fixed transmission. If all fOur state variables were converging to the common asymptotic value P*, then in particular Pi and Qij would be converging to the same value P*. But if Pi and Qij are converging to the same value, then they are con— verging toward each other. That is, the discrepancy between Pi and Qij must be going to zero. But, if the discrepancy between Pi and Qij reaches 0, then _S_(_)_ too does N ... That is, since N.. = 0 whenever 1] 1] lPi — Qijl = 0, the assumption Of the typical case for fixed transmission implies the special case in which one or the other of the transmission rates becomes 0. 59 The critical question for the variable transmission models is this: Can one of the discrepancies Pi — Qij or Pj — jS reach 0 befOre the other one does? If so, then the one that reaches 0 causes a cessa— tion of messages to the other person and hence brings the other's dis- crepancy to a halt befOre it reaches 0. The answer to that question is: Yes. In most cases, one discrepancy will reach 0 first. To establish the plausibility Of this fact we will first consider the two models under the admittedly unlikely assumption that the parameters a and b are equal. In this case, the mathematical results are Simple and stark. .After that we will give the only slightly more complicated conditions required to Show that our claim is true if a and b are not equal. The Case of qual Parameters: a = b — Let us assign variable names to the two attitude-perception discrepancies. Let x 2 Pi - Qij and let y = Pj - jS. Then in Appendix A.we show that regardless of the message generation model, we have the fOllowing differential equations for x and y wherever a = b: dx- avg-'9 __L.X_L x 94% ll H, E ‘< where e and f are complicated symmetric functions of the two constant attractionsAij and Aji' That is, if the parameters a and b are equal, then the two variables x and y Obey a bivariate pair of differential equations that are each functions eely_of x and y. Thus x and y can be related to one another in a two dimensional phase plane. Two such phase planes are Shown in Figure 16. Ans 90% 8 .5 u coapomfiut 9.6 .3 90m 3 .8 u coapogfia. "scammdumsmfiw grams pom ermem wwmmmoz fia\3\wH 80m .mocmmwhomfim 8389mm wB 9.699, m_H ..HO EOE THE ummfim .3 9.53m \ \S/Q \% \6 d X 61 The important point about the phase planes in Figure 16 is that every trajectory converges to a point on either the x—axis or on the y—axis. Thus every trajectory converges to a point for which y = 0 or converges to a point fOr Which x = 0. Thus it is always the case that at least one of the discrepancies converges to 0. Where do we look in this phase plane fOr the possible common limit Pi = Qij = P? = jS? If all four were equal, then in particular P. = Q.. or x = 0 and in particular P. = Q.. 0. Thus the case 1 13 3 31 or y of the common limit is represented by the point x y = 0 or the origin. That is, the only trajectories which represent all fOur system.variables converging to a common limit are the trajectories in Figure 16 which converge to the origin. And that is clearly a very uncommon special case. Thus if the parameters a and b are equal, we have proved that in all but unlikely Special cases, either the discrepancy Pi — Qij hits zero before the discrepancy for person J, or else the discrepancy Pj - jS hits 0 before the discrepancy for person I. The Case of Unequal Parameters - If the parameters a and b are not equal, then the preceding development breaks down. Basically the problem.is this: If Pi - Qij = 0, then at that moment in time Nij = 0 and person J is not changing. However, the fact that Pi - Qij = 0 at that point in time need §e£_imply that Pi — Qij will e:ey_equal to 0. This problem is illustrated by the trajectory in Figure 17. The trajectory in Figure 17 starts out with Pi = Qij and indeed at time 0, Nij = 0. HOwever since Qij changes three times as fast as does Pi’ Qij decreases more rapidly toward Mji than does Pi. Thus a 62 jSw. t Jl__, Qij/ ///// Attraction = (3, 3) Figure 17. IP with Constant Attraction and Varying Transmission: Unequal Parameters. gap Opens between Pi and Qij and hence Nij ceases to be zero. Mathematically the correSponding problem in the case that a f b is the fact that the differential equations fOr the discrepancies x and y are not functions of only x and y, but are functions of the other system variables as well. Thus if a ¢ b, then there is no two dimen- sional phase plane for the two discrepancies. If it is to be the case that one discrepancy hits 0 befOre the other does, then what is the additional condition that must be met beyond the condition Pi — Qij = O? The problem is that although Pi = Qij fOr one instant in time, they may not e:ey_equal. In order that Pi stay equal to Qij it is necessary that in addition to equality of the vari- ables we must have equality of the derivatives. If Pi = Qij’ then we have 63 only if a(M.. - P.) = b(M.. - Q..) = b(M.. - P.) . 31 1 31 13 31 1 That is, only if M.. . M.. .. 31 1 31 13 Thus we are lead to consider the condition Pi = Qij = Mji’ And it requires only minimal checking to establish the fact that this determines a critical point. How different is the condition Pi = Qij = Mji from the condition Pi = Qij? Not as different as might appear. If the parameters a = b, then Pi = Qij implies that Pi stays equal to Qij' But that doesn't mean that either Pi or Qij remains constant. In point of fact it turns out that once P. = Q.., then P. and Q.. converge together to M... So 1. 13 1 13 31 even in the case of equal parameters, the critical point is Obtained at Pi = Qij = Mji’ its just that the trajectory reaches Pi = Qij first. The Unlikelihood of a Common Limit for the Unequal Parameters Case - Thusfar, we have found out that the entire system stops changing when P. = Q.. = M.. fOr a not equal to b. In other words, P. = Q.. = M.. 1 13 31 1 13 31 are a set of equilibria for the IP equations with constant attraction. However, these are not the only set of equilibria. If at the same time that Pi and Qij were converg1ng to Mji’ Pj and jS were converg1ng to Mij’ then a p0581b1e outcome would be Pi = Qij = Pj = jS. In th1s case, both I's perceived discrepancy and J'S perceived discrepancy would be zero and Nij = Nji = 0. The system would stOp changing because both I and J would stOp transmitting. Also I and J would have reached agreement (Pi = Pj) and both would accurately perceive this agreement. This 6” Situation constitutes a second set of equilibria for this model. What we wish to do fOr the remainder Of this section is to make some educated guesses as to whiCh equilibria is most likely to be reached: Pi = Qij = Mji with Pj and jS arbitrary or Pi = Qij = Pj = jS? Once again our remarks will be applicable to both the veridical and shift models although the trajectories to be presented are those Of the shift model. Figures 18 and 19 Show the only trajectories which converge to a common limit. In each case, the initial values for J are all perfectly symmetric (in either a positive or negative sense, see Appendix B) to the initial values for I. Moreover, in each case both attraction values and transmission rays are equal. The least deviation from all of these highly unlikely conditions produces a trajectory fOr which one discrep- ancy hits 0 befOre the other one does. That is, the ee£_of equilibria Pi* = Qijk = Pj* = jS* is unstable. Figure 20 shows two trajectories fOr which attraction between I and J is equal but I's initial transmission is greater than J's. Although the same value Of attraction is set fOr both I and J the system does not converge to a point Of equality fOr all fOur state variables. Observe that the initial value of J's perceived discrepancy is 0 while I's is 2. Thus, I initially out—transmits J by .91 to 0. Of course, J's perceived discrepancy does not remain at zero since a message from I changes jS more than Pj' But the damage has been done. That is, with I out—transmitting J initially, J converges to I's message con— siderably faster than I converges to J's message. The result is that P3. = jS = Mij EEESE§.P1 and Qij reach the same point. The transmission from J is shut Off so that I stops Changing. But I continues to transmit 65 Attraction = (3, 3) Attraction = (3, 3) Attraction = (3, 3) Figure 18. IP with Shift Message and Constant Attraction: Transmission Variable. 66 Attraction = (3, 3) . Q.- .\pi 11 \ Attraction = (-3, -3) P. _ 3 Figure 19. IP with Shift Message and Constant Attraction: Transmission Variable. 67 P . l \ \Fj \_ Qj t Qij Attraction = (3, 3) ji Attraction = (—3, -3) //"’” [ Qij Figure 20. IP with Message Shift: Varying Transmission and Constant Attract ion . messages to J which are identical with J's attitudes and perceptions. Thus, even the slight asymmetry in the system due to differences in initial perceived agreement produce an equilibrium other than that of complete equality. 68 Figures 21 and 22 present trajectories which are symmetric in their initial conditions (that is, IP. - Q..| = [P — Q..|, IM — Pl 1 13 3 31 1 ji = [M — le, andlMji - Qijl = IMij — jSl) but are asymmetric in ii attraction. In Figure 21 I is much more attracted to J than J is to I while in Figure 22, the reverse attraction pattern is present. In Figure 21, the system converges to Pi = Qij = Mji W1th Pj and jS con— stant and in Figure 22 the system converges to Pj = Q.. = Mij with Pi 31 and Qij constant. This difference in which of the two individuals stops transmitting is due to the reversal in attraction patterns between the two figures. Let us fOcus upon Figure 21. Since I likes J much more than J likes I, then Pi and Qij will converge upon J's message very rapidly. .As Pi and Qij both rush toward J's message, then they are also changing toward one another. Since Nij depends upon the difference between Pi and Qij (whiCh is getting smaller), then Nij is getting smaller. As Nij decreases then so does the rate of change of Pj and jS since they depend directly upon how many messages are being received. Now, in the examples of Figures 21 and 22 we have deliberately chosen Aij failure of the system.to converge to a common limit. However, there is andAji very discrepant in order to represent in a dramatic way the no reason why muCh smaller asymmetries in attraction Should not produce the same results, although the final differences among Pi’Qij’ Pj’ and jS would be much smaller. Thus, even if the initial conditions on attitudes, perceptions, and messages e§e_symmetrical, asymmetries in 1's attraction to J and J's attraction to I will produce convergence to the ° ' : .. : .. . = .* .. = ..*. more likely f1nal state Pi Q13 M31, P3 P3 ,le Q31 .mHomst> scammaamcmsa 8033.33. finance 6:... t..:a mwmmmmz as... H H. Pamfl a NH; Am mlv u aoapomspa<. m- ...o - .m P r. H ...... Am .mlv u cospomspe< Hm ..O 6 mm .m Am .muv u coapompppt .manmPHme/ coammfiamgfiw 605083.33. 2398 new mwmmmmz tea fie. H .mm name... m .i A .W///// .m. .mV n COHnoannna / .. HH .mH H .m .m. .mV n COHpomnnaa 70 .H ll .m/ m / ma ..0 .i .m. .m+o n coaaomnnea H “an ju 71 Finally, we consider the dual asymmetry in attraction and in initial transmission. Two example trajectories for this case are pre— sented in Figure 23. Based upon our remarks above, we expect and find P. a. \ l Pja jS I t Attraction = (—3, 3) Qij b P, P -‘\ T J jS t Attraction = (3, -3) Qij__————————-—— Figure 23. IP with Shift Messages: Constant Attraction and Variable Transmission. that the system converges to P. = Q.. = M.. in frame a and P. — Q.. = , 1 13 31 3 31 Mij in frame b. That is, that there is convergence to the more likely equilibrium and failure to converge to the common limit. 72 What we find, then, for the IP model with message shift or with veridical messages is that when the initial conditions are symmetric the variable Pi’ Qij’ P3’ and jS converge to a common limit Pi = P. = 3 Q.. = Q... If any of the symmetry conditions are violated, then it 13 31 becomes possible fOr either Pi and Qij or Pj and jS to converge to the other's message PEESEE the other converges to his message. In this case either the system will converge to Pi = Qij = Mji or to Pj = jS = Mij' If our analysis Of the shift and veridical models for the case of vari— able transmission is correct, then convergence to a common limit is a very special and unstable result dependent upon some very unusual initial conditions. The more common result would be that one or the other of the persons stops transmitting while the other continues sending messages identical to his silent audience's attitudes and perceptions. If trajectories fOr the veridical case had been presented, they would be qualitatively similar to those for the shift model. That is, the same conclusions about the final state of the system as a function of initial values could be supported. The chief difference between the two models would be fOund in the point at whiCh the system reached equilibrium. Of course, this fact is traceable directly to the dif- ferences in message assumptions between the two models. Information Processingywith Varying Attraction In this section, we consider the same set of models in the same order as the previous section but now permit attraction to vary both for the internal and external processes. Unfortunately, the powerful mathe— matical tools which were invoked in the previous section cannot be 73 invoked with the IP models of varying attraction. First, ell_of the models in this section are nonlinear so that the mathematics of linear systems is precluded. Second, none of the models in this section have any critical points (except fOr the varying transmission case) thus pre- cluding the usual techniques of analyzing the stability of critical points with linear approximations. .As a result, we are fOrced to focus primarily upon the numerical solutions to the variable attraction models. On the positive side, we have a useful set of mathematical results to build upon from the previous section. Internal Changes Only - The model for internal, spontaneous changes is summarized in the equations (7), (8), and (9) of Chapter II. The internal changes fOr I and J are independent of each other as we noted befOre. The key difference between the IP model with constant attraction Of the previous section and the IP model with variable attrac- tion is that attraction can now become infinite either in the positive or negative direction. As I's attraction to J becomes very positive, then (1) the changes Of Pi and Qij in eaCh other's direction becomes faster, and (2) the acceptance region fOr I becomes larger. AS I's attraction to J becomes extremely negative, then (1) Changes in Pi and Q.. 13 toward.each slow down and eventually stOp in the limit as attraction approaches negative infinity and (2) 1'3 acceptance region approaches zero. Although there are no critical points in this model (or any of this section), we still can note that as attraction goes to positive infinity or as discrepancy goes to zero the experienced internal force would be zero. .Although this is not a peie£_in the sense that the number 6.3” is a point on the real line, it does indicate the direction that 71+ the IJX situation is moving and would be considered balanced by Newcomb and Heider. That which determines whether the system.will tend toward zero discrepancy with infinite attraction or finite discrepancy with negatively infinite attraction is where the individual system begins. First, suppose that I thinks that the discrepancy between himself and J is within his acceptance region. Then I's attraction will increase and his acceptance region will get wider. Hence, the discrepancy will be smaller and the acceptance region larger producing;more positive increases in attraction and so fOrth. For the second case suppose that I thinks J is very dis- crepant so that he is fe§_outside I's acceptance region. Then there will be little convergence of Pi and Qij’ I will derogate J a great deal and I's acceptance region will shrink appreciably. Although the dis- crepancy will be slightly smaller, the acceptance region will be muCh smaller and the change in attraction will be very negative. This cycle will produce attraction going to negative infinity SO fast that the discrepancy never reaches zero, but stops at a non-zero asymptote. Third, we consider the difficult case: When I perceives J to be only slightly outside the acceptance region. Now attraction will decrease a small amount and the convergence of Pi and Qij will be slowed. However, if Pi and Qij change enough to move back into the acceptance region, then attraction will increase again. The cycle toward positive infinity and zero discrepancy will have begun. If the convergence of Pi and Qij is tOO slow, then attraction will decrease again and the Spiral toward negative attraction will have begun. 75 The intuitive arguments above can best be seen in the phase plane graph of attraction versus perceived discrepancy in Figure 2”. The equations used to derive the integral curves are discussed in Appendix B. The arrows which indicate a flow to the right (toward positive attraction) represent the trajectories tending toward infinite, positive attraction and zero discrepancy. The arrows which indicate flows to the left represent trajectories tending toward infinite, negative attraction and non-zero discrepancies. The dotted line Which separates trajectories flowing to the right from those flowing to the left is known as the separatrix. The separatrix is the dividing line between the cases whose initial values will yield a "right—flowing" trajectory from those whose initial values will yield a "left-flowing" trajectory. In other terms, the separatrix divides those individual systems with Aij = + w, Pi = Qij Which are balanced and those with Aij = -m, Pi’ Qij arbitrary which are not balanced. In general, the separatrix must be numerically determined based upon the particular parameters of the equations being modeled. TO round out our discussion of internal changes, Figure 25 pre— sents four over—time trajectories, two Of which are balanced (a and b) and two which are not balanced (c and d). The fOur figures represent the same pair Of initial conditions (discrepancy = 1 in all cases) but in the one case I is attracted to J initially and in the other I finds J unattractive enough initially SO as to fail to converge back toward liking J. BefOre taking up the external model, it will be useful to compare our results thus far with the results of the previous section. In the . .a. £030.3th. moms? mosawHOmaQ om>amosmi CHE: mquteo HmEoPcH cHOm mmasouroo muse oqum mmmfim ..a gamer .vcmamwsmcmaa cox/amoeba HapflcH .coflbmfiok wfi§> one LH :33 mowcmfio anaemia sow mmasovom muse mafia .mm magmam .HHo / a /1\\\\\l / K . I} o \ 1} ||II‘I\|\II\I\|\IA / H . .i S :0 .HH \ 78 previous section, convergence toward zero discrepancy was always the case as long as attraction remained finite. This leads to zero discrep- ancy, negative attraction as well as zero discrepancy positive attrac- tion final states. SuCh a result differs from the Heider approach to balance fOr the negative case. But when attraction is allowed to vary we find that positive infinite attraction accompanies zero discrepancy in support of positive balance and Heider's model. Also negative, infinite attraction accompanies finite discrepancy as Heider's view Of IJX situations predicts. IP with Message Shift and Variable Attraction: Transmission Constant Equations (1), (2), and (3) and their counterparts fOr J con— stitute the mathematical system.fOr IP with message shift and variable attraction. Based upon the results of the immediately preceding section we might expect that the behavior of the IJX system.under equations (1) through (3) would depend upon the initial conditions for attraction and message values. That is, the direction which the system.tends may depend upon Whether the initially generated messages are within the other's latitude of acceptance or not. This is exactly What we shall conclude. Let us discuss three cases: (1) both I's and J's initial messages within the other's acceptance region, (2) both 1'8 and J's initial messages exterior to the other's acceptance region, and (3) 1'8 initial message within J's acceptance region, but J's initial message outside I's acceptnace region. Since conclusions about I and J are symmetric, these above three cases cover all possible combinations of initial conditions. If I's initial message is within J's acceptance 79 region and vice versa, then both I's attraction for J and J's attraction fOr I will increase. This mutual increase in attraction will simultan- eously increase the width of both I's and J's acceptance regions and increase the rate of convergence of Pi’ Qij’ and Pj’ jS. Thus the "next round" of message interchanges will be even closer to the other person's attitude than the "first round" of message interchanges and will be with- in an even wider acceptance region. Hence, attraction will increase mutually once again and the cycle will continue. The result quite simply is that When both I and J extend messages whiCh are initially within the latitude of acceptance of the other, then those IJX situations will produce infinite attraction fOr both individuals and zero discrep— ancy in the collective system.and both individual systems. Figure 26 Shows fOur numerically generated trajectories fOr the case Where both I and J send initial messages within the other person's acceptance region. Notice that the attraction trajectories show positive changes across all values Of time and that the variables Pi’ Pj’ Q.. 13 and jS Show a rapid convergence toward zero discrepancy. If both I's and J's initial messages fall outside the other person's region of acceptance, then two situations need to be considered: (1) when the messages are both jee£_outside the acceptance region of the other and (2) when the messages are both well outside the acceptance region. In the latter case, both I's attraction to J and J's to I will decrease by a large amount (the change in attraction recall is quadratic) which will shrink both regions Of acceptance by a large amount (in fact at an exponential rate) and.wi11 slow down the convergence rate fOr Pi’ Qij and Pj, jS. With a large mutual decrease in attraction, the next .coammm mocmecmoo< n.9mnpo one moncH Lyon momemmz HmapacH ”coawomepe< wcabsu> ecu pwaem mMMmmm2_£pH3 mH mom mmaooycmmmsfi mafia .mm mssmam 80 Hm Hm < \\\\lll. p < H K Am .Hv u scammaewcuoH Am .HV n scammaemcmea mH- ml .... Hm 7.. a .4. . .. 2a .... ... a H o m- m .m H .m r o .0 p . u COHmmHEmcu AH mv . . 99 AH .mV n coammflameusH nH ..< ...... .... 81 round Of messages (which will not have changed much since Pi’ Qij’ Pj, and jS will not have changed much) will be (relatively) even further outside the shrunken acceptance regions, thus producing greater decreases in attraction. This cycle will continue, giving rise to trajectories such as those in Figure 27. In both frames, the attraction goes toward negative infinity very rapidly and changes in Pi, Qij’ Pj and jS toward one another then cease. Note that it is only correct to say that changes in Pi, Qij’ Pj and jS crease in the limit as attraction becomes nega- tively infinite. Large negative (but finite) values Of attraction may slow the process of change to a negligible level but only in the limit as attraction becomes infinitely negative does convergence stOp. In the case where the initial messages generated by I and J are just outside each other's acceptance region, then attraction for both will decrease slightly. .At the same time, both acceptance regions will Shrink slightly and the convergence process will be slowed by a small amount. But convergence for Pi’ Qij’ Pj, and jS is still taking place. If the rate of decrease in attraction is slow enough, then I's messages and J's messages can "catch" Pj and Pi respectively and produce changes in attraction which are positive. Once this change occurs fOr both I and J, then the spiral toward positive infinity and convergence of Pi’ Pj, Qij’ and jS begins again. Figure 28 presents one of the few observed trajectories in which both messages were initially outside the other's acceptance region, and after slight initial decreases in attrac- tion, the pattern of attraction change became positive. Substantively, the situation of Figure 28 represents the results of continuing inter- action between two individuals Whose differences are resolved by the 82 "coapomnpe< mcmmsu> ecu pmmem mwmmmmz_£pmz mH sow mmHoOpommmsH mama IL .3 S .H Hm TH V ..O .coawmm mocupcmoo<_m.so£po moampso mmmz mmmummoz HMHPHCH E H COHmmflEmCMfiB .2 answer S .3 mH ..O coammHEmcmoH 83 Transmission = (l, 5) ii Figure 28. Time Trajectories fOr IP with Message Shift and Varying Attraction: Initial Messages both Outside the Other's Acceptance Region. process Of interaction. Perhaps the most interesting set of results from the IP shift model with variable attraction is fOund fOr the case in which the initial message from I is outside J's acceptance region while the initial message from.J is within I's acceptance region. With J's message inside I's acceptance region, I's attraction to J will increase and, hence, the width of the acceptance region will increase. More importantly, however, the rate of convergence of I's attitude and of 1's perception Of J toward J's message will increase. On the other hand, J's acceptance region will shrink and the rate of convergence of jS and Pj toward J's message will decrease. The two key competing processes in this case are the shrinking of J's acceptance region and the convergence 8” of 1's attitude and perception Of J toward J's message. As Pi and Qij converge toward J's message, then I's message will ultimately be identical with J's attitude. When this occurs Mij - Pj = 0 and I's message will be within J's acceptance region even though J's acceptance region becomes very narrow. In other words, the ultimate result of an initial case where J's message is within I's acceptance region but I's message is outside J's region, is that I will Change his position on.X and his perception Of J's position until they are coincident with J's actual position precisely because I likes J so much. When this occurs, J will begin to Change his Opinion of I and like him.more and more. In a way this result represents the phenomenon of ingratiation on the part of I. Figure 29 presents four of the approximately 30 trajectories for the ingratiation model that were generated. In each frame we see that aimost all of the change in attitude and perception variables is done by the person whose attraction to the other is more positive. Most importantly, however, eaCh frame shows that fOr the person receiving the "extreme" message, his attraction initially decreases but then reverses itself and changes in the positive direction. The most striking case of this is fOund in frame d where large initial decreases in attraction are fOllowed by very small positive increments. Despite the lack Of the strong mathematical conclusions Which characterized the IP model with constant attraction we still may Offer some conclusions. First, whenever 1'8 and J's initial messages are within the other's region of acceptance, or both are slightly outside the region, or when one message is within the other's region and one is .coawmm moqmyawooa. m .H $333.0 . m. 609m “Sn coammm moqmpamog m .H 6.5va h. EosuH www.mwmz .mmwficm "coercing/5. mfi§> US... vam www.mmmz eyes .3 now mmHQOPOo mg mama. . mm 85me L llama. is Hfl- \hfl con .... j. Am .3 u :33ng Am .3 u commmmfimqmfim / IF 1 in m... Hm- ..< ..O Am .C u commmdamcmfim 3 .C n scanning I .m 86 outside, then the IJX situation will change toward increasing mutual positive attraction and toward the absence of discrepancy at the collective system level and for both individual systems. All these trajectories end in a state that is "balanced" according to positive balance theory as well as Heider's model. They are also consistent with dissonance theory. Second, whenever both I's and J's initial messages are well out- side the other's acceptance region, then the IJX situation will change toward.increasing mutual dislike and toward non-zero discrepancies at the individual level, the collective level or both levels. These results would be consistent with dissonance theory or with Newcomb's model, but not with Heider. The dissatisfying aspect of the IP shift model with varying attraction is our inability to Obtain a more precise estimate of where the separatrix fOr the six variable model would lie in the six—dimens- ional space. Even reducing this Space to four dimensions by treating discrepancies rather than the raw attitudes and perceptions, does not allow the clear graphic representation Of the separatrix as did Figure 2” for the internal changes case. Rather we were forced to rely almost exclusively on the sample of trajectories which have been numerically generated. IP with Veridical Messages and Varying Attraction: Transmission Constant With the results from.the shift model under our belt, analyzing the veridical model is quite easy although its implications are prO- fOundly different. For the Shift model, the final states of the system 87 were dependent upon the location of 1's and J's messages relative to the other's acceptance region. Since the fOrm of the change equations has not altered by considering the new message model but only the messages have a new form, then the final states of the IP model with varying attraction should depend upon the location of each person's initial messages relative to the other's acceptance region. In the veridical case, the initial messages are just the initial attitudes of the person sending them. AS a result, changes toward mutual positive attraction and zero discrepancy will depend upon either (1) I's and J's attitudes initially being within the other's acceptance region or (2) their initial attitudes being only Slightly outside the other's region. On the other hand, change toward increasing mutual dislike results if I's and J's initial attitudes are well outside eaCh other's acceptance region. In other words, the same relationship between initial messages and accep— tance regions predicting the final states of the system.are found in the shift and veridica1.models given our rough, qualitative results. In the veridical case the messages would be unshifted from the speaker's position. In the Shift case, the initial messages are shifted away from the Speaker's attitude and toward the speaker's perception Of his receiver's position. Let us explore the implications briefly. Suppose first that I and J disagree so that Pi # Pj and both are well outside each other's acceptance region. If even one of the two persons in the IJX situation is accurate, let us say Qij = Pj’ then under the Shift and sequential models I'S initial message will be shifted in the direction of J's actual position and the likelihood that I's initial message will fall into J'S acceptance region is increased. If the shift 88 message does fall into the acceptance region, then the IJX situation will tend toward a qualitatively different state (mutual positive attraction and zero discrepancy) than the veridical model would predict (mutual dislike with non—zero discrepancy). On the other~hand, if I and J initially agreed but were both vastly inaccurate, then the verid— ical model would predict change toward mutual positive attraction and zero discrepancy and the shift and sequential models would predict mutual hostility and finite discrepancy. Of course, this does not cover all possibilities. But the two cases cited do serve to point out that the choice of message model is crucial in predicting where the IJX situation is tending under IP with varying attraction and constant transmission. IP with VagyingpAttraction and Transmission: Veridical and Shift Models Finally we take up the most complex of the IP models incorporating both variable attraction and variable transmission. Our discussion of this case will consider both the veridical and shift models simultaneously since the initial message values will be one Of the crucial determinants of the system's behavior regardless of how those messages are determined. With both transmission and attraction variable, there are two Pj, zero transmission by both I and J and (2) mutual infinite dislike by factors whiCh can bring changes in Pi, Q. ., and Q.. to a.halt: (l) 13 31 both I and J. If the transmission from I and the transmission for J is zero, then all six state variables will stop changing. This mutual cessation ef_transmission can occur only if Pi = Qij and Pj = jS. If A.ij andAji go toward negative infinity, the transmission Of information 89 is 29: terminated but both I and J find one another totally without credibility and, hence, Pi, Qij’ Pj’ jS will stop changing. Aij and Aji will continue to change. Thus, the two sets of crucial points are Pi = Qij = Pj = jS andAij = Aji equal to negative infinity. But there is another, more important, set of points determining the final states for the system. When we considered the varying transmission.model with constant attraction, we fOund that Pi = Qij = Mji would shut the system down. This resulted since I stopped transmitting to J while J continued transmitting to I. In the variable attraction case, this same point is ee:_an equilibrium point (see Appendix B) since if the system.hits this point J will continue to transmit messages to I which are exactly equal to 1'8 attitudes. Thus, I's attraction to J will be continually rein- fOrced and will continue to increase toward positive infinity. Thus, we do not have an equilibrium point for Pi = Qij = Mji but only have an equilibrium point fOr Pi = Qij and PJ. 2 jS. Notice that if the system converges to Pi = Qij = Pj = jS, then both I and J stop transmitting and the system of equations Shuts down. However, we will Show that this latter equilibrium is both unstable and ye§y_unlikely to occur. Further, the more likely final states will be shown to be Pi = Qij = Mji with P3. and jS arbitrary,Aji going to positive infinity and Aij constant QEnAij and A.. going to negative infinity with Pi’ Pj’ Qij’ and 31 Q the complexities of the variable attraction, variable transmission models ji approaching different asymptotic values. The best way Of discussing with both veridical and shift messages is to review the results Obtained previously for the constant transmission-variable attraction and variable attraction—constant transmission cases. The final states fOr both the 9O veridical and Shift models under constant transmission—variable attrac- tion assumptions were seen to be dependent upon whether both I's and J's initial messages were (1) both within the other's acceptance region, (2) both well outside the other's acceptance region, (3) both jee£_out— side the other's acceptance region, or (”) whether one message was with- in while the other was outside the initial acceptance region. Only in condition 2 did the system move toward.Ai. and A. 3 31 at negative infinity with P1’ Qij’ Pj and jS discrepant from one another. More simply, for fixed (and non-zero) transmission the final state of the 1P system for both veridical and shift messages depends upon the location Of 1'8 and J's initial messages relative to the other's acceptance region. Under the assumption of constant attraction and variable transmission, we con— cluded that the final states for both the veridical and shift models depended upon the symmetric or asymmetric configuration of the initial values. The initial values were called symmetric whenAij =Aji (¢ negative infinity), IPi(0) - Qij(0)l = IPj(0) - jS(0)|, IMji(0) - Pi(0)l = IMij(0) - Pj(0)l and IMji(0) - Qij(0)l = IMij(0) - jS<0>| and were called asymmetric otherwise. Under the symmetric conditions, the system would converge to the common limit Pi = Qij = P3. = jS and under the (muCh more likely) asymmetric conditions the system would converge to Pi = Qij = Mji with Pj and jS arbitrary (if I converged onMji before J converged on Mij)' In simpler terms, the final states of the constant attraction-variable transmission IP model depends upon the symmetry or asymmetry Of the initial conditions fOr both the veridical and Shift models . 91 With Ee£§_attraction and transmission varying the final states of the system should depend upon the location Of the initial messages relative to the other's acceptance region eee_whether the initial con— ditions on attraction, attitudes, perceptions, and messages are sym— metric Or asymmetric. Table ” summarizes all possible categories of initial conditions which could lead to different final states of the system. Cell IV—S in Table ” is actually an empty cell since it is logically contradictory to require all initial conditions to be symr metric but to have one message outside and the other message inside the initial acceptance region. That is, with.IMji(0) - Pi(0)l = IMij(0) — Pj(0)l and Aij(0) = Aji(0)’ either both messages are within or both are Table ”. The Possible Combination of the Attraction Subsystem.with Attitude-Perception Subsystem for IP with.Message Shift. Attitude-Perception Subsystem Symmetric Asymmetric . Both Messages Attraction . . Subsystem. W1th1n I-S IHA Both.Messages well Outside II—S IIHA Both Messages Just Outside III-S III—A One Message Within and One Outside IV—S IVHA outside the other's acceptance region. we also note that if both I's and J's initial messages are well outside the other's acceptance region, then Aij andAji will tend toward negative infinity quickly whether the other initial values are symmetric or not. Thus, in both the symmetric 92 and asymmetric cases with 1'8 and J's initial messages well outside the other's acceptance region, Aij and A.ji will go toward negative infinity and perceptions and attitudes will fail to converge to a common point. Figure 30 presents trajectories for both of these cases fOr Shift messages. Frame a has both 1's and J's well outside the other's accep- tance region initially and also is asymmetric in that J's initial trans— mission is greater than I's. Frame b and c of the same figure are symmetric in every reSpect fOr I and J with both initial messages well outside the other's acceptance region. Clearly, all three cases result inAij and Aji going to negative infinity with either I or J both per- ceiving some discrepancy. Figure 31 presents three trajectories (also for shift messages) in which the initial messages are well within the other's acceptance regions and the initial values are completely symmetric in I and J. Clearly all three trajectories show Pi’ Q.., Pj, and jS converging to l] a common point and Aij and A.. going off to positive infinity. The rate 31 of change of'Aij andAji will become constant in this case since both 1's and J's transmission will cease. Figure 32, on the other hand, pre- sents two trajectories (Shift messages) in which both 1's and J's initial messages are within the other's acceptance regions Ee:_the initial values on attitudes and perceptions are not symmetric. Notice that in both cases attitudes and attractions fail to converge to a common point despite the fact that attraction is always increasing or is at least constant and positive. In frame a, Pj and jS converge to I's message (essentially equal to jS), thus shutting off transmission to I. This causes Aij to slow down and eventually just be a positive 93 t P. P. “13 ‘l C.- 931 g Q13 A.., A.. 13 31 P. J Figure 30. Time Trajectories for IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages well Outside the Other's Acceptance Region. 91+ ji Figure 31. Time Trajectories for IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages within the Other's Acceptance Regions. 95 A.. ii J . Figure 32. Time Trajectories for IP Message Shift, Varying Attraction and Transmission: Both Initial Messages within the Other' 8 Acceptance Region . constant. In frame b, it is I which converges to J's message SO that Aji becomes constant with the cessation of transmission by 1. Notice that J does not stOp transmitting and, hence, I' s attraction to J con- tinues to grow toward positive infinity. Thus, when both I's and J '5 initial conditions are symmetric the system tends toward constant positive attraction and Pi = P]. = Qij = jS. When both messages are within the other' S region but there is an asymmetry, the system tends 96 toward positive attraction with perceived discrepancy by at least one of the persons. Of course it is the asymmetric case which would arise in fact; symmetric initial values are highly unlikely. The symmetric and asymmetric cases for both messages initially just outside the other's acceptance region are presented in Figures 33 and 3” respectively. In Figure 33, with symmetric initial values, the attitudes and perceptions are clearly converging toward a common point and the attractions have reversed themselves to change in a positive direction. Since 1'5 and J's attitudes are converging, then trans— mission from.both I and J is decreasing. .As a result,.Aij and A.ji will increase more and more slowly until, at the point of convergence Of attitudes and perceptions, they become constant. For the asymmetric Figure 33. Time Trajectories for IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages just Outside the Other's Acceptance Region. 97 ji ii Figure 3”. Trajectories for IP with Message Shift, Varying Attraction and Transmission: Both Initial Messages just Outside the Other's Acceptance Region. case of Figure 3”, the direction Of movement of the trajectories is not clear. It is possible that Pj and jS are close to convergence on I's message but the amount of change in Qij is puzzling. The likelihood is that jS and Pj are converging on Mji and the rate of change of Qij is decreasing. The problem here is that‘withflAij and A.ji negative, the change in attitudes and perceptions is very Slow. Therefore, after a much longer time interval than is graphed, Pj = jS = Mij with Pi and Qij equal to their equilibrium values, Aij constant and A.ji tending to positive infinity. Thus, the symmetric case once again produces con— vergence of attitudes and perceptions with.Aij and jS constant. The asymmetric case produces perceived discrepancy fOr one of the individuals and perceived agreement fOr the other. HOwever, we must caution the reader that the results fOr either case are only tentative. 98 Finally, in Figure 35 we consider three trajectories for the situation where J's message is within I's acceptance region but I's message is outside J'S acceptance region. Since this situation is already asymmetric then we might expect that attitudes and perceptions Figure 35. Time Trajectories fOr IP with Message Shift, Varying Attraction and Transmission: J's Initial Message within I's Acceptance Region while I's Message is Outside J's Region. 99 would fail to converge to a common point. This is exactly what happens in these numerical results. In each case I converges to J's message be— fOre J converges to 1'3 message. I's transmission to J terminates and Aji becomes a constant. Since J still perceives discrepancy, he continues to transmit to I causingAij to increase toward positive infinity. While the varying transmission and varying attraction model is the most complex and least tractable of all the models considered thus far, it is also the most interesting. The symmetry Of the initial values and the location of the initial messages relative to the other's acceptance region determine the final state of the model. These are summarized in Table 5. Of course these results are based upon a crude Table 5. Tentative Final States for IP with Message Shift, Varying .Attraction and Transmission, as a Function of the Initial Values. Symmetric Asymmetric Both A..,.A.. + +, constant A.. +-+cn, A.. +-+ constant Inside 1] 31 l] 31 Pi‘Qij ‘Pj ‘jS Pi‘Qij‘Mji P = * .. = * 1 1 ’ Q11 Q11 Both Well Aij’ Aji + — w Aij"Aji + — w OUtSlde P.* ¢ Q..* ¢ P.* ¢ Q..* P.* ¢ Q * ¢ Q..* ¢ P i 1 13 3 31 1 13 31 3 Both Just A.., A.. + +/-, constant A. +~++a2_m¥AM1+m+am_kn18§J 2 which can be shown to be real and negative. Thus fOr differences we have r t r t + Pj a3e 3 aqe H r t r t Qd b3e 3 + bue H . To get back to the original variables we note that Pi = (P8 + Pd)/2 Pj 2 (PS - Pd)/2 Qij = (Q8 + Qd)/2 jS = (Q8 - Qd)/2 . Thus, the state variables are just linear combinations of the solutions for Ps’ P Qs’ and Qd which are themselves exponentially decreasing. d, That is, Pi = 1/2 (a1 + a2er2t + aser3t + auerut) Pj = 1/2 (a1 + a2er2t — aseust - auerut) Qij = 1/2 (bl + b2er2t + b3er3t + buerht) jS = 1/2 (bl + 132er‘2t — 13333t — bqerut) . It can be shown that a1 = bl whiCh equals 2 2 (r2 + ak N)PS(0) — ak N QS(0) I" 2 llS TherefOre, l/2a1 is the value toward which P., P. Q.., and Q.. converge 3- ]: l] 31 in the limit as t goes to infinity. For the sake of completeness we also note that a = ak2N (QS(O) —ps(0) 2 I’2 a3 = 1 [(ak(k - 2) N — r”) Pd(0) - de(0)] I" -r 3 u a” = 1 [(333 - ak(k - 2)N) Pd(0) + de(0)] I’ ~11“ 3 u _ 2 b2 - r2 + ak N [QS(0) - PS(0)] I‘2 b = ak(k - 2)N - r (rB—rh)k 3 (ak(k — 2) N — r”) Pd(0) - de(0)] b = ak(k - 2)N - r r3-ru)k H [(r3 - ak(k - 2) N) Pd(0) + de(0)] This gives us a complete solution for equations (Al) - (AH). TherefOre, we may generate the trajectories of Figure 9 directly. we note that if the parameters a and b were equal, then r2 = wakN and al/2 would be the average of the initial message values. That is, if'a = b, then Pi, Qij’ Pj’ jS would converge to the average of their initial messages. Now, let us turn to the more general case fOr which both trans- mission and attraction are still constant but unequal. If the expressions fOr Mij andM[ji (that is, equation (5)) from the shift model are substituted into equations (lla) through (11d), then the matrix representation of this system.can be denoted 116 g§= m (m) m where S is the column vector: S : (Pi, Pj, Qij, jS> and w is the matrix: "ak N” ale-kJN” 0 ak.k.N. 1] 31 13 31 31 JJ 31 31 ak 0(l — k0.) No. -a k.. N.. aka. k.. N.. 0 33- 13 l] 31 1] 1] 31 1:] _b kc. No. b kc. k0. No. 0 b kij(1 - k'i) Nji 13 31 13 31 31 b k..(l - k..) N.. 0 b k.- ko. No. —b kc. No- 31 l] l] 31 l] 13 :11 l] The matrix w has four important characteristics: 1. All diagonal elements are less than zero. 2. All off-diagonal elements are geater than zero. 3. The sum of the elements of each row is zero. 1+. The matrix is compact (Abelson, 1964, p. IRS) in that each variable is at least an indirect cause of every other variable . Matrices with the above characteristics permit some direct and powerful inferences concerning the dynamic character of the system of linear equations. It is well—known in linear system theory that if the eigenvalues of the system (equation A5) can be determined, then the asymptotic characteristics are immediately known. If one or more of the eigenvalues is positive, the system is unstable. If all of the non-zero eigenvalues have negative real parts , then the trajectory for each set of initial values will converge to some critical point. A theorem which Abelson (1969, p. 145) invokes (see also, McKenzie, 1960) states that if the above four conditions are met, then all the non—zero 117 eigenvalues will have negative real parts and at least one of the eigen- values will be zero. For such a linear system, every trajectory con— verges to a critical point and every critical point is a right eigenvector of w fOr eigenvalue 0. If the matrix w is compact, then every right eigenvector is a scalar multiple of the column vector (1, l, l, 1). Thus, in our case we can use Abelson's theorem to conclude immediately that each of the variables converges to some common limit, i.e., Pi’ Qij’ Pj’ jS goes to L as t goes to w. Hewever, this limit is not "stable": if S is randomly jarred fromlthe critical point 8* = (L, L, L, L), it will then converge to a new nearby critical point (L L L L ). The distance from L to L1 is always less than or 1’ 1’ 1’ 1 equal to the size of the random movement. That is, random.events not accounted for’by the model produce a.random.motion from.one equilibrium point to another nearby. Thus, given any set of initial conditions the system (A5) will converge to a point at which there are no longer any changes. SuCh a point is by definition a critical point. The specific critical point toward Which the system is converging is determined by the initial conditions. we know that the system of equations converges to a point of equality but we do not know what that point is as a function of the initial conditions. However, this can be fOund in a straighthrward but tedious way by determining the left eigenvector, V, for the system Vw = 0 and then setting the initial dot product v. 8 equal to the asymptotic dot product V.S*. That is, we write V 1x + v2y + v z + v W'= v x* + v2y* + v 2* + v w*. 3 4 1 3 4 But since all the limiting values are the same, we have 118 k _ v x + v2y + v z'+ v w 3 H v1 + v2 + V3 + Vu X:'€:y7'€:z :w* l Thus, we see that the final values all converge to a weighted average of the initial values. The appropriate left eigenvector is v = (bkji (1 — kij)Nij, bkij(l — k..)N >. .., ak..k..N.., ak..k..N.. 31 31 13 31 13 13 31 31 The first and third.components of this vector correspond to Pi and Qij’ while the second and fOurth correspond to Pj and jS. Since these pairs each have a common factor, it is instructive to break up the final value by persons. If we let the sum of the weights be denoted s, i.e. , s = bk..(l - k..) N.. + bk.., then the contribution to the final value 31» l] l] 1] made by person I is divided by s, while the contribution made by person J is N k.(b(l—k..)P.+ak Q .. . .. ..) Jl 1] 31- J 31 31 divided by s. we see then that the contribution to the final value made by person I is proportional to Ni' 3 made by person I is prOportional to the rate at which I transmits to J and kji' That is, the contribution and logistically proportional to how much I is liked by J. Similarly, the contribution to the final value made by person J is proportional to the rate at which J transmits to I and logistically proportional to how much J is liked by I. If we look at the contribution made by each person separately, then we can assess the relative weight given to perceptions as compared to attitudes. For person I we have weight attitude _ weight perception _ ak.. ' .A. ae Thus, I's perception of J is weighted by parameter a, while I's attitude toward the issue is weighted by parameter b. Thus, according to wackman's study (1973), this model would predict that the person's perb ception of the other is ultimately three times as important as his atti— tude toward.the Object or issue. If this is compared to the veridical model, then we see that the assumption of ingratiation is an extremely strong assumption. Of course, there is a second term to the relative weight of per— ception to attitude, namely eAij. If I likes J initially, then e ij > 1, and perception will ultimately be weighted even.more than three times as much as the pzrson's attitude. On the other hand, if I initially dislikes J, then e l] < l, and his attitude has more weight in the final analysis. ForAij attitude become equal, While as Aij goes to —w, the model approaches —l.lO, the weights fOr perception and the assumption of veridicality. Before leaving the shift model let us consider two special cases: (1) the limit as A.ij approaches negative infinity fOr Aji finite and (2) the case of Nij = 0 fOr Nji > 0. Notice that both of these cases drastically change (A5) and, in particular, the vital characteristics of w. .As.Aij_gpes to negative infinity kij goes to zero and d Pi/dt = O and d Qij/dt = 0. This implies that Pi and Qij are constant which we take to be their initial values Pi(0) and Qij(0). Since kij is zero, the message sent by I is not shifted so that Mij = Pi(0). TherefOre, 120 d P' = a k.. N.. (p.(0) — p.) dt 31 13 1 3 dQ.: bk.N.@Xm-QHL 31 31 JJ 1 31 dt The critical value for Pj and jS, then, is just Pi(0). This is stable critical point since the eigenvalue for each equation separately is negative and Pi(0) is fixed. Thus, Pj and jS will ultimately converge toward Pi(0) and I will not change from.the initial values, Pi(0) and For N.. = O, we also have d P./dt = O and d Qj./dt = 0 which 13 3 1 imply as above that Pj = Pj(0) and jS = jS(0) for all time. Now the message which J generates is shifted toward his perception of I M.. = P.(O) + k.. (Q..(O) - P.(O)). But, M..(O) is fixed over 31 3 31 31 3 31 time and so, dPi= ak.M.mum)-R) _—dt' 13 31 31. 1 dt which has the same critical value: Pi = M (0) and Qij = M (0). That is, both converge to J's initial message. This critical point is stable as befOre. Thus Pi and Qij will ultimately converge toward Mji(0) = Pj(0) + k.. (Qj.(0) — P.(O)) While P. and Q.. remain constant at their initial 31 l J J 31 values. It is interesting to note at this juncture that if the model of equation (A5) had been extended to 3, H, or more individuals, the expanded matrix would still satisfy the conditions dictated by Abelson. 121 As a result, the conclusions about critical points and their convergence readhed fOr the dyadic case would hold identically for a larger group of persons. It is also interesting to note that the symmetries Which give rise to the special case of w arise fromlthe basic discrepancy principle which treats change as a function of the difference between a state variable (for example, I's attitude) and a target variable (for example, J's message). When presented in the language of target and state vari— ables, the link to a general cybernetic fOrmat is implied. IP with Veridical Messages: Constant Attraction and Transmission The system.of equations (lla) through (lld) fOrMij = Pi and Mji = Pj also satisfies Abelson's Theorem as can be seen by inspection of the coefficient matrix: ‘a]<.. No. ak.. No- 0 O T l] 3]. l] 31 so No. _akoo No. 0 0 11 13 31 1] bk N.. 0 -bk 0 l] 31 l] 31 0 bk.. N.. O -bk.. N.. j]. l] 31 l] L _ Therefore, in the veridical case I and J always converge to Pi = Pj = Qij = jS. As before, the point of convergence can be related to the initial values. For the veridical case, I's and J's attitudes and perceptions converge to k .. N P.(O) + k.. N.. P.(O) 11} 31 3 ij 1 13 k.. N.. + k.. N.. 31 13 l] 31 122 This is the simpler final value of the two message models since it depends neither on Qi' j’ Qj i or the parameters a and b. Clearly, if k.. N.. > k.. N.., 31 JJ 1] 31 attitude than to J's which means that J will have done more changing then the final value will be closer to 1'8 initial than I. In the case where I and J transmit equally and kji > kij’ then J's attitude changes kji/kij as muCh as I's attitude. Similarly, when I and J are equally attracted to each other, then J's attitude changes Nij/Nji as much as I's attitude. In general, the ratio of the weights k. . N. /k .. indicates how much change J undergoes relative to I. 31 ll 11 N31- IP with Constant Attraction and Variable Transmission Shift and Veridical If we consider the situation Where Nij and Nji are variable but attraction is constant, then we can discuss both message models simul- taneously. If we assume that a = b, then subtracting equation (11a) from (llc) and (llb) from (lld) regardless of What M'i. and M. i are, we "—"31 have a(PiC-lQij) : akile-jSl (Qij_ t . /’l + P. - .. 1 ( 3 Q31) d (P. - ..) a k. P. - .. ( .. — P.) ,1 Q31 = 3i 1 Q11] Q31 3 dt _ z /' l + (Pi Qij) where k. = k.. (l + k. i) and k. = k.. (l + k..). Let P. - Q.. = S and 1 13 3 31 13 1 13 l Pj - jS = 82' This yields dS a k. S _d%' = l A 2 ("81) (A10) /1 + S ‘— 2 EEE_ = ak ,j [SI (-82) (All) dt /1+87 1 123 we note immediately that S1 = O or 82 = O are equilibrium.points for the variable transmission case. That is, when a = b, if either I or J or both perceives no disagreement, then Pi - Qij and Pj - jS will remain constant fromxthe point of zero perceived disagreement on. For 81, 82 > 0 or 81’ 82 < O, the equations (A10) and (All) are separable yielding + S dSl : ki 1 ; dsz k./ 1 + Si 3 2 which integrates to kj 1n (Sl +V 1 + 812) + C = ki 1n (S2 +V 1 + SE) where C is an arbitrary integration constant. These constitute the integral curves of quadrants l and 3 of Figure 16. For Sl > O, 82 < 0 or Sl < 0, 82 > O the integral curves are k. In (S + V1 + S 2) + c = —k. In (S + 1 + S 3 1 l 1 2 2 thus yielding the integral curves for quadrants 2 and H of the same figure. Note that when S1 = O, the slope dSZ/dsl = (kj/ki) #1 + S22 Which increases as 82 increases in absolute value. Similarly when 82 = O the Slope dSZ/dsl is (kj/ki) (l/Vl + 812). As Sl increases in absolute value, then the lepe of the phase plane trajectories approaches 0. These results suggest the nature of the trajectories in Figure 16 whiCh are near to the S1 and S2 axes respectively. we remark that the above analysis holds fer the veridical model and the Shift model. ____ ___,.__.——- 124 The unequal parameters case (a ¢ b) behaves somewhat differently. First consider the situation in whichAij is large and positive andAji is large and negative. For the Shift message model, this implies that M.. .. and M.. P. and that k.. is near uni while k..is near zero. 11 ” Q11 11” J 11 W 11 The equations for this case have the form: dP. aN.. (P.-P.);dQ.. bN.. (P.-Q..) 1 z 31 3 1 13 = 31 3 13 dt dt dP. ..N..(..-P.)'d.. bk..N..(..- ..) —l = 31 13 Q13 3 ’31:: 11 11 Q1] Q11 dt dt a; O 2% 0 Since the variables fer J are changing very slowly, Pj is almost constant. But the variables for I are moving rapidly toward Pj so that in a very short time Pi = Qij = Pj' In fact the greater the initial discrepancy, Pj - jS, toward Pj' The convergence of Pi and Qij stOps the transmission from the more rapid the convergence of Pi’ and Qij I to J and, hence, "freezes" jS at its current value. Although J continues to transmit to I, he is transmitting a message equal to Pj so that I does not deviate fromPi = Qij = Pj' These behaviors are depicted in the trajectories of Figures 21 and 22 in the text. In the more general case, where messages are Mij and Mji’ then if for any reason Pi and Qij converge to M.ji befOre Pj and jS converge to Mij’ then dPi/dt = inj/dt = 0 Since Pi = Qij = Mji and de/dt = dei/dt = 0 since Nij will be zero due to I's perception of agreement. The key question, however, whiCh we have been unable to answer with any precision is What conditions will necessarily produce such a convergence. Suppose Nij(0) = Nji(0)’ [M.. —P.| = IM.. 31 1 - P.| initially, and 11 J 125 IM — Qijl = [M . — jSI initially. .Actually, we only need to assume ii i1 two of the three of these conditions since the other follows immediately given any two. With these initial values and kij > kji’ then Pi and Qij will always converge faster to Mji than Pj and jS converge to Mij' With Pi and Qij converging faster, then Nij will be decreasing faster than N.. is decreasing. As a.resu1t, P. = Q.. = M.. with P. and Q.. 31 1 13 31 3 31 arbitrary constants whenever the initial values of the state variables meet the conditions above and kij > kji' This result also suggests sufficient conditions for convergence of the vary1ng;transm1ss1on systemlto Pi = Qij = Pj = jS. Suppose IPi(O) - Qij(0)| = IPj(O) - jS(0)| or that I and J transmit equally at time t 0. Also suppose that [Mij(0)l = IMji(0)I or that initial messages are of the same magnitude. If we fUrther assume thatAij =.Aji and is not infinitely negative then we can show that the system of equations (11a) through (lld) converges to Pi = Qij = Pj = jS as follows: Consider the first increment in attitudes, APi and APj. We have AP. 1 a N..(O) k.. (M..(0) - P.(O)) 31 13 31 1 AP. a N..(O) k.. (M..(0) — P.(O)). J 13 33- l] 3 But since N..(O) = N..(0), and k.. = k.., the difference between AP. 13 31 13 31 1 and AP. depends only upon M..(0) — P.(O) and M..(0) - P.(O). But these 3 31 1 13 3 two differences are equal in.maghitude (but not necessarily in sign) since 2 IPj(O) + kji (jS(0) — Pj(0))l according to the shift model. Therefore, IAPiI = IAPjI in the first time increment. Similarly, we can Show that IAQijl = IAjSI. However, it is n93 the case that IAQijI = IAPiI since 126 we assume that b > a. Since attitudes are changing by the same amount and perceptions are changing by the same amount, this implies that Nij (1) = Nji(l), IMij(1)I = IMji(l)I and further that the magnitude of the change in attitudes will be the same in the next time increment; similarly fOr the change in I's and J's perceptions. The overall impli- cation of our "proof" is that Pi and Pj will converge on eaCh other's messages by changing equal amounts and Qij and jS will converge on eaCh other's messages faster than Pi and Pj do but also by changing equal amounts. This means that Pi = P3. = Qij = jS when the restrictive symmetry assumptions detailed above hold. we note that the same proof can be made fOr the veridical model. While the symmetric fOrm.of the model of equations (11a) through (lld) for varying transmission clearly will yield convergence to Pi = Pj = Qij = jS (that is, the symmetry conditions are sufficient), it is not clear that these restrictive conditions are also necessary. If they could be shown to be necessary, the result would be a strong one. If it is false, then we would be back to the issue that plagued our search for a separatrix in the other 3, 4, . . . variable models. Namely, how.much asymmetry is possible before Pi’ Qij’ Pj, jS fail to converge? APPENDD< B APPENDIX B In this appendix the few mathematical results pertaining to the IP model with variable attraction are discussed. As noted in the text, this model consists of six first-order, nonlinear differential equations which have no critical points (the varying transmission case is an exception). The nonlinearities make the mathematics impossible to treat analytically while the absence of critical points makes the usual approx- imation procedures also inapplicable. The main procedures of this appendix are to discuss special cases of the models. Internal Chages Only If we let the parameters r and q be equal in equations (7) and (8) of Chapter II, then we may subtract to obtain A.. d(Pi Qij) - 2r e (Q.. - Pi) 0 dt ’ ""711" . 13 l+elj Together with equation (9) 2A.. dA.. (e 13 — (Q.. — p.)2> 13 :3 13 1 dt 2A.. we have the internal change model with varying attraction. Dividing the former equation into the latter does not yield a separable form and so we generate the trajectories of Figure 21+ numerically. Notice that 127 128 along the.A.. axis, Q.. — P. = O and 13 13 1 dA.. ij 13 = s e dt ZAi. l + e j which is always positive. This suggests that the separatrix (the dotted line in Figure 24) is asymptotic to theAij axis for large negative Aij’ Q.. - P. = O. 13 1 IP with.Message Shift and Varying Attraction: Transmission Constant Let us consider some special cases of the IP message shift model with varying attraction so that the asymptotic behaviors can be under- stood. First, consider the case in whichA.j andAji are large and A.. .A.. .. .. positive so that e 13/ (l + e 13) and e 31/ (l + e 31) are both approxe A”. A». imately one, and e l] and e 31 are extremely large. In this case the model reduces to dPi in. ——ldt _ b Nji (jS — Qij) (132) dAi. dt—l': c Nji (B3) with equivalent equations for Pj’ jS, and Aji' .After attraction becomes large, it simply increases in a linear fashion with increasing time. Also, the change equations for attitudes and perceptions become linear discrepancy equations converging rapidly to Pi = Qij = Pj = jS. Results Which indicate such behavior may be fOund in Figure 26 eSpecially frames a and b. Note that if N.. ¢ N.. such that N.. > N.. then, 31. 13 31 13 129 dAij/dt > dAji/dt while both are increasing at a constant rate. In Figure 26a Nij > Nji so that J's increase in attraction while constant is much greater than I's constant increase. Next, consider the situation fOr whichAij and A31 are large negative values (note thatAij = -5, yields a‘s = .0067) so that e 13/ (l + e l3) and e 31/(l + e 31) are approximately zero, and so are e l] A.. and e 31. In this case the model reduces to dP. dP. d .. d .. __1 = 1= Q11 = Q31 2 0 dt d1: (11: dt dAij 2 2 =-CN (P.- P.) =-CN..II1 dt 1 1 31 dA'i 2 2 —l— =—cN.. (P. —P.) =-cN..m dt 13 1 3 13 where m.is a constant since Pi and Pj are not changing. Note that for large negative values ofAij and Aji’ attraction decreases in a linear fashion and the variables Pi’ Q ., and P. remain fixed. The tra— ij’ Q11 J jectories of Figure 27 approximate such behavior. Once again, notice that Nji > Nij will yield dAij/dt as the more negative. Figure 27a shows the effect of transmission asymmetry on the slopes of dAij/dt and dA../d‘t. 31 Finally consider the situation in which the variables Pi’ Qij’ P., jS have converged to a common limit. In this case we have that J P., Q.., P., and jS are unchanging as before but 1 13 3 dA.. 13 _.]_'J_ : c N e dt 31 l + e 13 dA.. 2Aji 1 c N e dt ° 130 both of these equations are directly integrable but this would be unnecessarily complicating. Rather, given the convergence of attitudes and perceptions, the rate at which attraction increases depends upon the value of attraction. For large positive values, the change is approximately constant, fOr large negative values it is very slow, and as attraction increases from.negative to positive the rate of change of attraction also increases. The attraction trajectories in Figures 28 and 29 show each of these cases. IP with Veridical Messages and Varying Attraction: Transmission Constant The equations fOr the veridical case have a particularly simple fOrm since the change in attitudes and attraction are independent of changes in perception. That is, for the veridical case, we have dPi e l] a? 3 a Nji (Pj — Pl) 1 + e 1] DP. Aji 4a. = a Nij ..:—F (Pi - 13-) l + e 31 dt = C 2A. 1 + e l] dA.. 31 (P. _ P )2 1 = c e - #41 dt 2A.. 1 + e 31 which does not depend upon Qij or jS. The change in Qij and jS on the other hand, does depend upon the behavior of Pi’ Pj’ Aij’ and Aji: in. e 13 dt ‘ iji—T (Pj "Qij' l + e i] 131 A.. dQ.i e 31 '7?%‘ z b Nij ""'ZIT' (Pi ’ jS)° l + e 31 If Pi and Pj are within the other's acceptance region, then Pi and Pj converge toward one another and Aij andAji go toward negative infinity. As this happens Qij and jS converge toward Pj and Pi respectively and, hence, toward one another. If Pi and Pj are well outside the other's acceptance region, thenAij andAji will decrease toward negative infinity. As this occurs, Qij and jS will stop changing. In any case, the behavior of the attitude-attraction subsystemlcan be studied inde- pendently of the perception subsystemh And the behavior of the attitude- attraction subsystem.determines the behavior of the perception subsystem while the reverse is not true. IP with Varying Attraction and Varying Transmission: Both Message Models Perhaps the simplest way to come to some understanding of the varying attraction, varying transmission model is to consider the Pi’ Pj’ Qij’ jS subsystem and its interaction with the attraction subsystem. The equations for the attitude-perception subsystem are dPi e 1] dt a Nji 1 (Mil - P ) l + e 3 in. éAij dt _ b N31 A1 (M31 - Qij) l + e 3 where '- 1 ji z Qij ' P1 1 + e 13 fi A O o ‘/ __ Z 132 and Mji depends upon the particular message model chosen. The com- parable equations for J complete the attitude-perception subsystem. There are two characteristics of this subsystem that are crucial: (1) As long as both values for Aij and Aji are not large and negative, Pi’ Qij , Pj , Qj i will always change toward each other. (2) For fixed values of attraction, Pi = Qij = M. . is an equilibrium value for the subsystem. 3 1 But note that for the attraction subsystem .. e 13 — (M.. — P.)2 —ll = c N . J1 1 dt 31 2A.. 1 + e 13 2A'i 2 dA.. e 3 - (M — P.) i : c N.. ll 3 dt 13 2A.. 1 + e 31 that P. = Q.. only insures that N.. 1 13 1 0. Thus dA../dt = 0, A.. =.A..*. ] 31 l J 31 ButP.=P.* ..: ..*whenP.= ..=M..andJ'sattitudeand r— 1 1’ Q11 Q11 1 Q11 11 pe ception will not in general be equal. Therefore at Pi = Qij = Mji dA.. JPJ" _ Q..i:l 2Aij __3.LJ_-_-(1+k,.*)1 j _e_____ dt 31 /1 + (P **— Q.T*)Z 2Aij j 31 1 + e 2A.. = N '" e l] 31 ""‘7ZKTT 1 + e l] A. A. "‘ where kji* = e ji*/ (1 + e 31 ). In other words if the attitude-percep- tion subsystem equilibrates, the attraction subsystem does not as Aij will approach positive infinity at a rate equal to Nj i*‘ On the other hand, Aji will be frozen at Aj 1* which can be positive or negative depending upon the rate of convergence to Pi = Q. . = M'i and A. .(0), the 13 3 31 initial attraction of J to I. The graphs of Figure 32 exemplify the 133 above behaviors. The infinite but unstable set of equilibria which shuts down both subsystems is Pi = Qij = Pj = jS. If such a point is reached = N.. = 0 and all changes are zero with A.. = A,.*,.A.. = Ai.*. 31 13 13 31 31 ii As we showed in the constant attraction, varying transmission case, the equilibria Pi = Qij = Pj = jS will be reaChed only if the initial values are completely symmetric. In the varying attraction case, this means.Aij(0) = Aji(0), IPi(0) - Qij(0)l = IPj(0) - jS(0)|, and the distance between the initial incoming messages and eaCh person's atti- tudes are the same, and the distance between the initial incoming messages and eaCh person's initial perception of the other is the same. When these restrictive initial conditions are met, we say that the system.is symmetric. If the initial messages are in or not too far from the acceptance regions, then the system will converge to Pi = Qij = P. = Q.. withAij = Aji' As in the constant attraction, varying trans- ] 31 mission case, it is not clear that the above conditions are necessary to have convergence to a common limit although they are clearly sufficient with one qualification: If 1'3 and J's initial messages are well out— side the other's acceptance regions, thenAij andAji will go toward negative infinity very rapidly thus shutting off the attitude—perception = .* ..: ..* .= .* ..: ..*. ' subsystem at Pi Pl , Q1] Q1] , P3 P3 , and Q31 Q31 In thls case bothAij andAji will go to negative infinity with the rate 2 3'6- ': 1 + (Pi Qij’) for A.ji and a comparable rate fOr.Aij. Figures 30b and 30c exhibit 134 trajectories with the above behavior since initial messages were well outside the other's acceptance region. Figure 31 shows three symmetric sets of initial conditions with initial messages within the other's acceptance region so that there is convergence of attitudes and percep- tions to a.crmmon point with the change inAij andAji decreasing to a constant value. Finally, we remark that the symmetric set of initial conditions described above is a very unlikely set to find. 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