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Mental accounting in a sequential Prisoner's Dilemma game
ELSEVIER
Journal
of Economic
Psychology
15 (1994) 351-374
Mental accounting in a sequential Prisoner’s Dilemma game * Gerrit
Antonides
**
Dept. of Economic Sociology and Psychology, Erasmus Utzinillersity, P. 0. Box 1738, 3000 DR Rotter-dam, The Netherlands Received
November
19,1992;
accepted
February
28, 1994
Abstract The two-person Prisoner’s Dilemma (PD) game is decomposed in three different ways. The games differ with respect to positive and negative outcomes, whether they are played simultaneously or sequentially and whether one or the other person is allowed to play the sequential game first. The observed strategies of 5? participants playing these games in a computer experiment are explained by the values of the payoffs, the expectations regarding the other person’s moves and by personal information. It appears that a subjective expected utility model can be used to test different psychological models of choice. A model assuming segregated evaluation of the payoffs in the sequential games gives a better explanation of the choices than a model assuming integration of payoffs. This implies some rules for giving rewards for behavior in the social dilemma situations concerned.
1. Introduction Choices in social dilemmas are predicted by game theory from the normative rational model. In the Prisoner’s Dilemma (PD) game, this results in collectively inefficient solutions. In psychology, many factors have
* Bouke den Hoed is gratefully acknowledged for creating the dataset. I appreciate the computational assistance by Edwin Jansen van Rosendaal. I thank Henk Elffers, Werner Giith, Charles Noussair and two anonymous referees for their helpful comments. ** Fax: +31 10 452 2646, E-mail: [email protected] 0167-4870/94/$07.00 0 1994 Elsevier SSDI 0167-4870(94)00006-V
Science
B.V. All rights reserved
352
G. Antonides /Joumal of Economic kychology 15 (1993) 351-374
been investigated to explain deviations from the game-theoretic solution. These include, inter a&, altruism, conscience and norms, communication, group size, noticeability and expectations (see, for example, Dawes, 1980; Liebrand et al., 1992). These factors may prevent society from disasters such as, for example, overpopulation or depletion of natural resources. Also, they influence the provision of public goods. Here, we shall not deal with large scale consequences of choices in social dilemmas but concentrate on the mental processes involved in making a choice. In particular, we shall consider situations in which the payoff is not obtained instantaneousty but depends on the choice made by another person at a later point in time. These situations are common in dynamic work environments (e.g. streetworkers who help each other at different points in time; see Pruitt, 1967) and in other situations where reactions of other people to a person’s behavior are delayed (e.g. in competitive markets, agency problems, venture capital markets, etc.). A number of experiments have been run to estimate the effects of the psychological factors mentioned above (cf. Dawes, 1980; Colman, 1982; Liebrand et at., 1992). Typically, in these experiments, the factors are manipulated and the behavior is observed. Although this procedure obtains reliable results, the mental choice process is barely considered. In the present paper, choice models are derived based on different mental processes taking place in decision making. In experimental studies of the Prisoner’s Dilemma, assumptions about the utility of the choice alternatives are made but utilities usually are not estimated. However, knowledge of the utility function of choice alternatives is important in understanding the choice process. Expectations regarding the opponent’s choice are considered to be among the determinants of choice (Dawes, 1980). Here, expectations will be investigated systematically in relation to the choices made. The explanation of these expectations will also be considered. Utilities of alternatives and expectations concerning the outcomes are combined in subjective expected utility models to explain the choices of the participants in an experiment including Prisoner’s Dilemma games. Rather than asking participants about the reasons for their choices (e.g. Pruitt, 19701 subjective information has been obtained during the process in order to test hypotheses regarding the choices made. Our concern will be choices in one-shot games rather than choices in a dynamic context. Hence, the results may be applicable in situations where the players do not build relationships.
G. Antonides /Journal
2. The sequential
Prisoner’s
353
of Economic Psychology 15 (1994) 351-374
Dilemma game
All the games considered here are symmetric, that is, the distinction between the players is arbitrary - both face the same payoff matrix. Although there is no compelling need to use symmetric games, we decided to base our experiments on Pruitt’s games (1967) for reasons of comparability. The symmetric simultaneous PD game is shown in Fig. 1 in its strategic form representation. In the PD game, the condition a > b > c > d applies by definition. Individuals i and j both choose simultaneously from the strategies C (cooperative) and D (defecting). The individual payoffs are known directly after choices have been made. The SPD game in Fig. 1 is run in two stages. First, Individual 1 chooses C or D, getting the corresponding i (self) payoff for him/herself and giving the j (other) payoff to Individual 2. Second, after being informed about l’s choice, Individual 2 makes a choice, getting the corresponding i payoff for him/herself and giving the j payoff to Individual 1. The summed payoffs resulting from the two choices equal the payoffs in the simultaneous PD. It follows that the symmetric SPD is subject to the conditions p + q = b, r+s=c,p+s=dandr+q=a,andhencethata+d=b+cisanecessary condition for a PD to be decomposed symmetrically. However, an infinite number of different values of p, q, r and s will generate the same PD. Since different decompositions constitute the same PD, game theory would predict the same strategy (D) in each decomposition. If the sign of the payoffs in Fig. 1 is reversed, the dominant strategy of the players is changed. After transposing the rows and the columns of the payoff matrix, the cooperative and defecting strategies appear as usual. However, Pruitt
individual other
j’s choice
Simultaneous
self
PD
Fig. 1. Payoff matrices of simultaneous and sequential j = 1, 2; i # j.) (C = cooperate, D = defect.)
Sequential symmetric
Prisoner’s
PD
Dilemma
games.
(i = 1, 2;
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of Economic Psychology 15 (1994) 351-374
(1970) obtained different proportions of defecting strategies on presenting three different SPD’s to his participants. From the reports given by the participants in retrospect, their behavior was inspired by different motives, i.e. tactical altruism, fair division and greed. Since the assignment of participants to the SPD’s was random, the framing of the SPD’s should be responsible for eliciting the different motives. The next sections deal with the question of how different strategies can be explained by the different frames. 2. I. Utilities of payoffs Usually, social dilemma games are investigated empirically by providing the participants with payoffs resulting from their choices. Since the payoffs are of only moderate financial value, one could argue that the choices would not matter to the participants very much. However, as Dawes et al. (1977) report, participants may show hostile behavior to non-cooperative persons even if moderate payoffs are at stake. This suggests that the payoffs should not be considered as marginal increments of their wealth but as isolated outcomes in their own right. Kahneman and Tversky (1979) also notice the isolation effect in two-stage games. This implies that utilities should be defined on the payoffs of the game, rather than on end states of wealth including the payoffs. As an alternative to the economic utility function, Kahneman and Tversky (1979) propose a value function which is concave for gains and convex for losses. The inflection point is located at the point of reference, i.e. the present state of affairs. In essence, they propose their value function as an alternative to the economic utility function which is defined over total assets. Alternatively, one could say that utility theory assumes only one utility function for the evaluation of payoffs, whereas prospect theory assumes two different functions depending on whether the prospect is framed positively or negatively. This suggests that the utilities associated with positive and negative payoffs in PD games may not be described by the same utility function. 2.2. Subjective
expected utilities of payoffs
Choice in a PD game is associated with uncertainty regarding the opponent’s strategy. It is hypothesized here that the probability of making a cooperative choice depends both on the utilities of the payoffs and the
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subjective expectation that the opponent chooses a particular strategy. It appears that the explanation of behavior in a PD calls for a subjective expected utility model (cf. Schoemaker, 1982). More specifically, the subjective expected utility of a C choice may be compared with the subjective expected utility of a D choice. The outcome of the comparison may be associated with the decision to make a C choice. Let the value of a C-C outcome for an individual be V(b) and the value of a C-D outcome V(d). Analogously, the value of a D-C outcome is V(a) and the value of a D-D outcome is V(c). If an individual is 100% sure that the opponent will choose C, the subjective expected utility (SEU) of a C choice is V(b) and the SEU of a D choice is V(a). We assume that the higher V(b) relative to V(a), the more likely a C choice by the individual will be. Furthermore, we assume that a vector of other variables (X), such as psychological factors, will affect the probability of choosing C. Usually, individuals are poor at predicting the choices made by the other player in the PD game (Dawes, 1980). For this reason, perceived probabilities of outcomes, i.e. expectations about the choices to be made by the other player, become important in explaining the choice behavior. If 7~ is the perceived probability that the other player will choose C, the SEU of a C choice is TV(~) + (1 - ~)l/(d) and the SEU of a D choice is VI’(~) + (1 - r).)I/(c). We assume that the difference between the two SEU’s, together with personal factors X, is related to the probability of a C choice, P(C). Applying the probit model to the discrete choice between C and D (Maddala, 1983) we have:
P(C) =N((dqb) -(7TV(a)
+ (1 - %-)I+)) + (1 - 7T)V(c)) +x’p),
(1)
where N is the cumulative standard normal distribution function and p is a vector of parameters. Eq. (1) assumes that the probability of a C choice depends on the difference between the two SEU’s and on the value of X’p (one of the X’s equals one to allow for a constant term). We observe that a high expectation of the other player’s C choice does not necessarily induce a high probability of a C choice by the individual. Eq. (1) implies that cooperative choice depends on the interactions between the subjective probabilities that the other player chooses cooperatively and the payoff values. Without knowledge of the value function, one cannot predict whether expectations positively or negatively affect cooperative choice.
ml
G. Antonides /Journal
of Economic Psychology 15 (1994) 351-374
Despite conclusions from specific studies regarding the effect of expectations (e.g. Dawes et al., 1977; Marwell and Ames, 1981) it appears that no general rules can be given on this matter. Eq. (1) shows the basic model for evaluating choice probabilities. The same type of modeling is applied to the sequential PD games in Appendix A.
3. Mental accounting
of payoffs
In the PD, integrated payoffs are evaluated. However, since the payoffs in the sequential PD are given separately, they may be accounted separately. As Thaler (1985) has argued, the mental accounting of separate payoffs may proceed differently from neo-classical computation of utilities. Mental accounting (or mental arithmetic) is a psychological mechanism for evaluating the balances of different accounts. Money values are frequently not cleared within the same mental account (cf. Antonides, 1991). In particular, Thaler (1985) assumes that V(x) + V(y) differs from V(x + y) generally. This implies that the summed evaluations of the two different stages of the SPD may be different from the evaluation of the integrated payoffs. For example, in the SPD, the equation p + q = b must hold. If payoffs are integrated, V(p + q) = V(b) should also hold. However, if the payoffs in the two stages of the game are evaluated separately and the function V is nonlinear, then in general V(p) + V(q) # V(b). If decision makers in an SPD integrate the payoffs, the choice model is different than if they evaluate the payoffs separately. This is shown more fully in Appendix A. We need to consider two variants of the SPD, depending on whether we are focusing on the first or the second player. The second player makes his/her choice after the other player has moved. In this case, the payoffs resulting from the other person’s behavior are known and expectations do not come into play. This game is referred to here as SPD2. The choice models that apply to it are also derived in Appendix A. We characterize the games where the different models apply: - The choice model for the simultaneous PD involves the same expectations about the other player’s behavior, whether one chooses C or D. - The choice models for the SPD include different expectations about the other player’s behavior after a C choice than after a D choice, because the other player moves second.
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- The choice models for integrated or segregated - The choice models for first, differ according to evaluated by integration
4. Personal
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357
the SPD game differ according to whether evaluation of payoffs is assumed. the SPD game, where the other player moves whether payoffs in the first round are ignored, or evaluated by segregation.
factors in the Prisoner’s
Dilemma game
Since the game-theoretic solution of the PD is frequently violated, other factors should be added to explain cooperative choice. Dawes (1980) describes a number of factors influencing choices in social dilemmas, including personal factors, X above. A personal characteristic influencing choices is strategy preference (this is similar to the concept of social value orientation in Liebrand et al., 1992, p. 17). A number of basic preferences can be distinguished, four of which are of practical relevance: altruism, cooperation, competition and individualism (cf. McClintock et al., 1984). Altruism is characterized by maximizing the payoffs of the other player, regardless of one’s own payoff. Cooperation is characterized by maximizing the players’ joint payoffs, competition by maximizing the difference between the players’ payoffs, and individualism by maximizing the individual payoff regardless of the other player’s payoff. Alternatively, concern for the other player may be modeled by including the second player’s outcomes in the utility function of the first player (cf. Becker, 1981; Antonides and Hagenaars, 1989; Sawyer, 1965). If the payoffs of the second player are valued within the utility function of the first player, the expected utility of the second player (as considered by the first player) can be added to the models considered before, weighted by a coefficient, 4, indicating the extent of concern for the second player (see Appendix A). Locus of Control refers to the disposition to attribute the results of one’s behavior to one’s own actions (internal control) or to the environment (external control) (cf. Rotter, 1966; Phares, 1978). Internal control may increase cooperative choice because it assumes that one is able to influence the other player’s strategy. External control may increase defecting choice because one believes that one cannot influence the other player. Frequently, in experimental games men have been found to be more cooperative than women (Colman, 1982, p. 124). Income may have a positive effect on cooperation (cf. Schneider and Pommerehne, 1981),
358
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possibly because at higher incomes the risk of a loss counts less. Education might have a positive effect on cooperation because it may help to overcome the inefficient equilibrium of the PD by insight. Finally, we assume that married people show more cooperation than singles because of selection effects (cooperative people marry more frequently) and learning effects (married people may learn to cooperate).
5. Experimental
procedures
In March 1992, an experiment was run with individuals from a neighborhood near Erasmus University. In total, 300 people were invited to participate by putting a letter in their mailbox. The letter said that on average Dfl.lO.- could be earned by playing a computer game with other people. A few days later, recipients to the letter were asked in person to make an appointment for the experiment. Failures to make appointments occurred because participants were not at home on two different occasions (100 cases), unable to make an appointment on the two scheduled dates or refused (133). Thus 67 people agreed to participate. Groups of four participants were scheduled on one of two consecutive Saturdays each for about one hour. Each participant was seated in a separate room in front of a personal computer. All the instructions, games and questionnaires were administered by computer. Participants were instructed that they were matched with another individual, who acted as the other player. Because we wanted to standardize the behavior of the other player, they actually played with the computer which made cooperative and defecting choices at mndom, each with 50% probability. A 50% probability is comparable with the average cooperation rates in first trials reported by Pruitt (1967, 1970). Participants received instructions for playing the PD and the SPD games and performed at least three practice trials in each type of game until they understood the game. Each point in the payoff matrices earned lOc, so the payoffs in each experimental trial ranged from -Dfl.1.80 to +Dfl.l.SO. They were told that a minimum of Dfl.5.- plus the total payoffs exceeding this amount could be taken home. The experiment lasted for about 45 minutes on average after which the participants were paid. Since the participants were likely to communicate with each other because they live in the same neighborhood, we could not reveal that they actually played
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with a computer. We did not keep records of the identities of the participants, so it was impossible to contact them after the experiment for debriefing or to ask further questions. Before each trial, participants were asked to state their expectations regarding the other person’s choices and the subjective probabilities that these expectations would come true. In the SPD, separate expectations were recorded for the cases where one would oneself choose a cooperative or defecting strategy, respectively. After each trial participants were informed about the total amount won or lost. In the SPD’s in which the participant made the second move, no expectations were elicited since the other player’s choice was already known. The participants played a number of different games; they were told that each game consisted of three trials. At the start of each game, a change of the other player was announced. The participants were distributed almost evenly over 12 conditions of the experimental design shown in Appendix C. PD I and SPD’s III, IV and V in Appendix B were used by Pruitt (19671. PD II, SPD’s VI, VII and VIII are similar, except for the sign of the payoffs. SPD’s IX-XIV are similar to SPD’s III-VIII, except that the player is allowed to move after the other player. All games involved payoffs that were multiples of 6, which facilitates the analysis of the value function. After the games, the participants completed several questionnaires regarding their socio-demographic situation, their strategy preference and their Locus of Control by means of the personal computer. To measure the strategy preferences, Van Lange et al. (1990) used a scale consisting of nine items, each with three different answers. One answer typically implies high joint payoffs, another answer implies a high difference of payoffs and the remaining item implies high individual payoff. Sub-scale scores associated with the strategy preferences range from zero to nine and the sum of scores equals 27, so two scales are sufficient to identify the strategy preferences. The motivation for cooperation is measured by counting the number of answers maximizing joint payoffs (MOTCOOP), whereas the motivation for competition is measured by counting the number of answers maximizing the difference between the players’ payoffs (MOTCOMP). The LOC scale was adapted from an abbreviated version by Phares (1978) consisting of five questions (see Appendix D). The fifth question of this scale deals with student grades and was not relevant for our sample. A question concerning the points earned with the games was included instead.
360
Pruitt’s outcomes Since our elicitation and all of
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/Journal
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Psycho&y
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questionnaires between trials (1967, 1970) did not result in different from those in a control group without questionnaires. games are similar to Pruitt’s experiment, it is assumed that the of expectations does not interfere with the participants’ behavior the participants were treated as explained above.
6. Results The experiment unintentionally took place when the weather was fine and as a result the number of participants was further reduced to 52. The sample composition was as follows: male: 54%; married: 31%; age 40 years or over: 38%; higher education: 54%; reported after-tax monthly household income over Dfl.2000: 56%. The sample included 16% less married people than the Dutch population (cf. NCBS, 19921, possibly due to family obligations on Saturday. Also, the sample included about 20% less people with an after-tax household income over Dfl.2000, possibly due to underreporting of income to some extent. 6.1. Effect of game order Since the dynamics of experimental games is not the object of study, we first tested for the presence of dynamic effects. If dynamic effects are present, we expect the position of a game within the session to affect the probability of a cooperative move. Table 1 shows the results of a probit analysis on the observed strategies (0 = defecting choice, 1 = cooperative choice). The analysis included 1248 choices (52 individuals times 8 different games times 3 repetitions). Dummy variables associated with the type of the game were included and the logarithm of the sequential position of the game (ranging from 1 to 24) should capture dynamic effects. Several dummies had significant effects on the probability of cooperation, notably those associated with the negative simultaneous game and the third SPD. The effect of the positive simultaneous PD game was included in the constant term. Apparently, the order of the game in the experiment did not significantly affect cooperation. A quadratic order function obtained similar results. It appears that on average the participants acted as if each game was played with a different person, unlike experiments where players were allowed to play a number of games with each other (e.g.
G. Antonides /Journal Table 1 Probit analysis
of dynamic
effects
Constant PD(l -) SPDfl + ) SPD(2 +) SPD(3 + ) SPDfl -) SPD(2 - ) SPD(3 - 1 SPD2f 1 + ) SPD2(2 + ) SPD2f3 +) SPD2( 1 - ) SPD2(2 - ) SPD2(3 - ) Inforder of the game)
of Economic Psycholow 15 (1994) 351-374
in cooperative
choice (t-values
361
in parentheses)
- 1.19 (-7.62) 0.38 (2.46) 0.11 (0.65) - 0.08 ( - 0.44) 0.50 (2.90) 0.20 (1.16) 0.13 (0.71) 0.57 (3.38) 0.16 (0.72) 0.27 (1.19) 0.32 (1.46) 0.04 (0.16) - 0.23 (- 0.90) 0.27 (1.19) -0.02(-0.41)
PD( .) refers to simultaneous PD’s. SPD( .) refers to sequential PD’s where the individual moves first. SPD2( .) refers to sequential PD’s where the individual moves second.
Pruitt, 1967, 1970). In what follows, it is assumed that the order of the games did not affect the choice of strategy. 6.2. Cooper-ation rate Since the number of trials in each game is known and finite, game theory would predict non-cooperative choice. However, the average cooperation rate was 26.2% overall. The average cooperation rates in the first, second and third trials of the games were 26.0%, 26.2% and 26.4%, respectively. These cooperation rates are not significantly different from each other. The cooperation rate was lower than cooperation rates on the first trial obtained by Pruitt (1967, 1970) with undergraduate students (ranging 45%-80% in different games). Pruitt’s participants (1970) were not told the exact number of trials, which may account for the higher cooperation rate at least in part. Cooperation rates different from zero have also been found by Dawes (19801, Dawes et al. (19771, Stroebe and Frey (1982) and Pruitt (1967), among others. The cooperation rates differed across the games to some extent (see Table 2). The average cooperation rate of the SPD (27.6%) was not significantly different, statistically, from the SPD2 where the player moves
362
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Table 2 Cooperation parentheses)
PD(+) PDt-) Average
rates
PD
and
expectations
of Economic Psychology 15 (1994) 351-374
of the
other
player’s
cooperative
choice
(standard
Cooperation rate
Expectation of the other player’s cooperative choice
Total number of trials
19.9% (3.2) 32.1% (3.7) 26.0% (2.5)
39.6% (2.2) 47.1% (2.5) 43.4% (1.7)
156 156
SPD(l+) SPD(2 + ) SPD(3 +) SPDtl -) SPD(2 - ) SPD(3 -) Average SPD
23.2% 17.7% 36.3% 25.9% 23.5% 39.2% 27.6%
(4.1) (3.8) (4.8) (4.2) (4.2) (4.8) (1.8)
SPDU 1 + ) SPD2(2 + 1 SPD2(3 + 1 SPD2t 1 - ) SPD2(2 - ) SPD2(3 - ) Average SPD2
24.1% 20.4% 27.5% 13.7% 29.4% 27.5% 24.4%
(5.8) (5.6) (6.3) (4.7) (6.4) (6.3) (2.4)
Expectation in case of choosing C
Expectation in case of choosing D
40.2% 41.1% 48.4% 40.2% 45.8% 44.0% 43.2%
34.5% 39.1% 41.9% 36.6% 38.7% 37.7% 38.0%
(3.0) (3.1) (3.4) (3.9) (3.3) (3.3) (1.3)
(3.3) (3.3) (3.3) (3.4) (3.3) (3.6) (1.4)
errors
in
108 102 102 108 102 102
54 51 51 54 51 51
PD( . ) refers to simultaneous PD’s. SPD( .) refers to sequential PD’s where the individual moves first. SPD2( .) refers to sequential PD’s where the individual moves second.
second (24.4%). On average, it does not seem to matter if the players move first or second. This result is not expected from game theory, although it is consistent with the results obtained by Bolle and Ockenfels (1990). 6.3. Expectations Table 2 shows the average expectations of the players regarding the other player’s move for each type of game. The expectation is the perceived probability r, that the other player will choose cooperatively. The average expectations were lower than the actual percentage of cooperative moves made by the computer (51.0%), i.e. the true percentage was underestimated. On the other hand, the observed cooperation rates were lower than the expectations regarding the other player’s cooperative
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strategy, indicating exploitative motives to some extent. In the SPD, the average expectation of cooperative choice was 5% higher when participants had themselves made a C choice than when they had made a D choice (t = 2.72). The average difference of 5% applies to each individual’s choices, since at each move the individual was asked to state an expectation conditional on a cooperative choice and an expectation conditional on a defecting choice. The fact that expectations differ according to the choice made indicates a belief that the other player will react to one’s own move (i.e. that the other player does not follow the game-theoretic solution). 6.4. Estimation of the choice models The models in Appendix A were estimated by means of probit analysis. In this analysis, the probabilities of the choices were related to the observed expectations of the other players’ choices and to personal factors. The evaluations of payoffs were estimated as sample constants. In each model, each of the choices made by an individual in 24 games were analyzed for the sample of 52 individuals (1068 observations in total). Note that the effects of personal factors were assumed to be constant over all games and individuals. Table 3 shows the results of three model estimations. Model 1 assumes segregation of payoffs in the sequential games, according to Eqs. (A.31 and (A.41 in Appendix A. Model 2 assumes integration of payoffs in all of the games. Model 3 adds the values of the other player’s payoffs to Model 1 (including Eqs. (A.la), (A.3a) and (A.4a) in Appendix A). In Games III and V, a negative payoff ( - 6) occurs in combination with positive payoffs, and it might be that this context would give it a different value from the one it took in Games VI-VIII, where the other payoffs were also negative. Similarly Games VI and VIII involve a positive payoff ( + 6) in a context of negative payoffs. To test for context effects, the value differences v(O) - V( - 6) and V(6) - V(0) were estimated separately for positive and negative contexts. If we arbitrarily set T/(O) equal to zero, these differences give the values of 1/(6) and V( - 6) directly. The x2 values reported in the table equal - 2 times the log likelihood ratio resulting from the specified model relative to a model including a constant term only, and can be used to assess the overall significance of the explanatory variables included in the model (see Maddala, 1983, p. 40). Model 1 yielded a very significant x2 and a pseudo-R2 of 0.126.
364 Table 3 Results of probit
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analyses
of cooperative
of Economic Psychology 15 (1994) 351-374
choice (absolute
Model 1 Segregation
t-values in parentheses)
Model 2 Integration
Model 3 Segregation + weight of other’s
2.05
(2.28)
0.43
(2.34)
-0.05 -0.03 0.01 - 0.00 0.30 0.61 -0.04 - 0.46
(1.21) (2.10) (0.69) (0.01) (2.68) (5.11) (2.12) (4.33)
2.25 0.61 0.35 0.33 0.66 0.39 0.36 0.11 0.10
Constant V(- 12)- V(- 18) V(-6)I’-12) I’(O)- I’-6) V ” (6)-V ’ (0) V”(O)-I’“(-6) V(6) - V(O) V(12)- V(6) W18)- V(12) 4 (weight of other’s payoff) LOC MOTCOMP MOTCOOP Male/Female Age Married/Unmarried Education Income
2.40 0.49 0.43 0.37 0.09 0.13 0.64 0.43 0.36
(2.63) (2.59) (3.24) (2.72) (0.65) (0.92) (4.71) (3.1 I) (1.62)
- 0.05 - 0.03 0.01 0.04 0.29 0.63 -0.04 - 0.46
(1.14) (2.14) (1.02) (0.46) (2.62) (5.25) (2.11) (4.30)
log(likelihood)
- 666.1
- 676.4
- 666.0
103.4 15 0.126
82.8 9 0.102
103.5 16 0.116
payoff
(2.34) (2.14) (1.60) (1.74) (4.70) (2.46) (1.61) (0.70) (0.74)
0.18 (0.48) - 0.04 (1.06) - 0.03 (2.08) 0.01 (1.09) 0.04 (0.46) 0.29 (2.53) 0.63 (5.23) - 0.04 (2.12) - 0.46 (4.26)
2
$f Pseudo-R’ a The sign of the payoff
differs
from the other
payoffs
in the games
concerned.
Model 2 assumes integrated payoffs in all the games, so that, for example, in SPD(l + > V(6) + V(6) = 1/(12), V(6) + V( - 6) = V(O), V(12) + V(6) = T/(18) and V(12) + V( -6) = V(6). That is, a linear value function is used. It was therefore only necessary to estimate one segment of the linear value function, V(6) - V(0). The x2 of this model was also very significant. Since Model 2 is a restricted version of Model 1, a x2 concerning the difference between Model 1 and Model 2 can be computed. The x2 equals 20.6 (df= 6) and is significant at the 1% level in favor of Model 1. The result indicates segregation of payoffs, i.e. mental accounting in the SPD games, rather than integration. Model 3 is a non-linear probit model because the weight given to the other player’s outcome, 4 is multiplied by the expected values of the other person’s payoffs (see Appendix A). The coefficient 4 was not significantly
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different from zero, so on average concern for the other player’s payoffs seems to be absent, once the motivation for cooperation (MOTCOOP) and motivation for competition (MOTCOMP) were taken into account. Above, a linear value function was rejected statistically. This seems to be due mainly to different evaluations of payoffs of + 6 and -6. If these payoffs occurred in games otherwise including payoffs with a different sign (indicated by a superscript ‘a’ in Table 31, the value was different from that found in games where all the payoffs had the same sign. It appears that the context of the payoff influences its value. In Model 1, the estimate of V(6) - V(O) is almost twice as large as V(O) - V( - 61, so the same difference of payoffs was evaluated differently in the positive area than in the negative area. According to prospect theory, the value function for losses is steeper than for gains (Kahneman and Tversky, 1979). Our result seems to contradict this assumption. Although the result was not reproduced in Model 3, in that case the two slopes are equal. According to all models, value differences act as coefficients of expectations. Therefore, significant t-values in the models indicate that expectations significantly affect the probability of cooperation. Although this is consistent with the theory of cooperative choice, our result shows that the effect of expectations is modified by the values of the payoffs. The Locus of Control did not significantly influence cooperative choice. The motivation to compete (MOTCOMP) with the other player significantly decreased cooperation, whereas the motivation to cooperate (MOTCOOP) and gender did not have significant effects. A higher age and a married status both increased cooperation and both education and income diminished cooperation significantly. Although Model 1 is favored over Model 2, one might object that it actually consists of three different models describing behavior in three different types of games. To deal with this objection, we considered the three game types separately (PD, SPD and SPD2, respectively). In each sequential game type, the restricted model fitted significantly worse than the unrestricted model, confirming that segregation was more likely than integration of payoffs in the SPD games. 6.5. Expectations and previous choices Finally, we consider the subjective probabilities regarding the other player’s cooperative move, excluding games where the individual moves
G. Antonides /Joumul
366 Table 4 Regressions
of expected
Constant Previous computer move minus previous expectation Next-to-previous computer move minus next-to-previous expectation R’ (adjusted) N
cooperation
of Economic Psychology I5 (1994) 351-374
(t-values
in parentheses)
PD
on previous PD
behavior
SPD own C choice
of the other
SPD own C choice
SPD own D choice
SPD own D choice
0.09 (2.65) 0.44 (21.55)
0.43 (14.95) 0.11 (2.35)
0.45 (28.17) 0.12 (4.89)
0.45 (19.66) 0.07 (1.70)
0.41 (25.25) 0.19 (7.41)
0.42 (18.23) 0.13 (3.29)
0.08 (1.57)
0.03 208
0.06 104
player
0.13 (3.46)
0.05 416
0.08 208
0.14 (3.69)
0.11 416
0.13 208
second. We tested the hypothesis that expectations are revised if the actual moves of the computer deviate from expectations in former trials. Table 4 shows the results of regressions of the expectations in trial t (E,) on the differences between actual strategies played by the computer and expectations in the former (two) move(s) (S,pl - E,_1 and S,_, - El_,, respectively). In case of the SPD, the participants stated two expectations, one conditional on a C choice and one conditional on a D choice. The ‘previous expectation’ in the regression is the one associated with the strategy one has actually played. For example, if S, _, is cooperative (or defecting) E, _ , is the expectation associated with one’s own cooperative (or defecting) choice. Since the participants were told that in each game they were matched with another person, only the second and third moves in each game could be explained by the preceding events, whereas only the third move in a game could be explained by the two former events. The table shows that expectations were significantly affected by the deviations of the previous move(s) of the other player from previous expectations, though the amount of variance accounted for in this way was small.
7. Discussion The choices in the Prisoner’s Dilemma games have been modeled assuming different hypotheses concerning the choice process. The hypothe-
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367
sis of evaluating integrated payoffs has been rejected in favor of segregated evaluation. In general, this implies that the evaluation of two payoffs of the same magnitude is different from the evaluation of their sum. In the example of the two streetworkers, this means that it is better if both obtain half of a positive payoff on each of two occasions than if they obtain the total payoff on one occasion and zero payoff on another occasion (besides discounting the advancement or postponement of a reward). It appears that the choice models discriminate between the different mental choice processes concerned. Also, by estimating them, estimates of the value function were obtained. In our experiment, we did not find a convex value function for losses, as postulated by Kahneman and Tversky (1979). Since we only experimented with small payoffs, the hypothesis of a convex value function for losses may still hold for large negative payoffs. This would imply an inflection of the value function in this area. An inflection of the value function would be consistent with the lognormal utility function of income proposed by Van Praag (1968). For very low incomes (comparable with a very low payoff) the lognormal utility function is convex and for high incomes it is concave. However, this function applies to incomes rather than (small) wealth changes as used in our experiment. The value of a particular payoff seems to depend partly on the other payoffs included in the game. A positive payoff in a game otherwise including negative payoffs is evaluated as smaller than the same payoff in a game with otherwise positive payoffs and vice versa. This indicates a context or framing effect, resulting in underestimation of payoffs that are inconsistent with the frame. This result seems interesting enough to be studied in future research. We assumed a constant value function for all participants, with only additional effects of personal characteristics. Since much variation is captured by the personal information in the models, we think that this assumption is justified to some extent. In any case, it does not seem worse than utility functions in microeconomics where personal information is almost excluded. Since the participants each played eight different games, a different value function could in principle be estimated for each individual. However, since this would require many more parameters to be estimated, this is left for future research. The expectations regarding the other person’s strategy explain a significant part of the choice behavior (otherwise the estimated utility intervals would not have been statistically significant). However, expectations of a C
368
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/Journal
of Economic
Psychology
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351-374
choice by the other player do not directly increase the probability of a C choice by the individual; their effects are mediated by the value function in the SEU model. Only in the simultaneous PD does it seem that the higher cooperation rate in games with negative payoffs is due to a higher expectation that the other player will play cooperatively. The expectations in trial t can be explained to some extent by the differences between the actual outcomes and expectations in former trials (t - 1 or t - 2). This suggests a learning process despite the finite number of trials (three in each game). The non-zero cooperation rates in the experiment show that the strong form of the game-theoretic hypothesis of playing a defecting strategy in the PD has to be rejected. Even in the SPD where the individual moves second, the cooperation rate is not significantly different from SPD’s where the individual moves first. On average, however, exploitation occurs, because the cooperation rate is about half of the expectation that the other player chooses cooperatively. Although the computer chose C in 51% of the cases, the average expectation that the other player chooses C was 43% in the PD and the SPD. The objective probability of 51% was underestimated by the participants. This is in agreement with the subjective weighting function of probabilities in Kahneman and Tversky (1979), who also find that objective probabilities around 50% are underestimated. Several personal factors influenced cooperative choice in the experiment. The motivation to compete with the other player has a negative effect, whereas the motivation to cooperate does not have a significant effect. The weight associated with the other player’s payoffs is not significantly different from zero. The latter result suggests a lack of altruistic motivation on the average. Locus of Control did not have a significant effect on choice. Possibly, the need to exert power (McClelland, 1975) should be distinguished from the attribution of control. Age and married status both affected cooperative choice positively and income and education both had a negative effect.
8. Appendix A: Choice models in simultaneous Dilemma games
and sequential
Prisoner’s
Let the subjective probability that the opponent chooses C in the PD be r,(P,(C)>, with pj(C> being the probability that individual j chooses C and
G. Antonides /Journal
T~( *> the subjective estimate of this probability 1). For the ease of presentation rj(Pj(C)) Furthermore, the individual values of payoffs refers to a vector of personal factors. Simultaneous PD The probability specified as:
that an individual
P(C) = F{7TV(b) -TV(a) with F some probability
369
of Economic Psychology 15 (1994) 351-374
chooses
by individual i (0 I V& - ) I will be abbreviated to r. will be denoted by I’( *>. X
C in the PD, P(C) has been
+ (1 - $7)V(d) - (1 - ~)l/(c) function
+X’/3},
and p a vector
(Al) of coefficients.
Sequential PD with integration of payoffs Eq. (A.l) deals with choice in a simultaneous PD in which the payoffs resulting from the choices of both players are integrated. According to expected utility theory, integration of payoffs should occur in the SPD, too. However, in the SPD the expectations regarding the other person’s strategy may differ according to the player’s own choice of strategy. In particular, it may be assumed that the expectation that the other person chooses C is higher if one chooses C oneself than if one chooses D. The probability that an individual chooses C in an SPD, P(C) has been specified as: P(C) = F{rcV(b) -(I
+ (1 - n,)V(d)
- %)J+)
- r,V(a)
+m),
(A4
with rc the subjective expectation that the other player chooses C (0 I rc( - ) I 1) if one chooses C oneself and 7r,, if one chooses D oneself. Eq. (A.21 is similar to Eq. (A.11 except that r differs according to whether one chooses C or D oneself. Sequential PD with segregation of payoffs Given the assumption of segregated valuation of the payoffs mental accounting hypothesis, the probability that an individual in the SPD, P(C) has been specified as follows: P(C)
=FMW) -Q(W)
Inspection restrictions
+ %))
+ (1 - %)(I+4
+ V(s))
+ I’(q))
- (1 -Q)(W)
+ I+))
under the chooses C
+X’P}.
(A.3)
shows that Eq. (A.21 is a restricted version of Eq. (A.31, the concerning the integration of payoffs. The probabilities of
G. Antonides /Journal
370
of Economic Psychology 15 (1994) 351-374
cooperative choice, P(C) in Eqs. (A.2) and (A.31 may be different for a particular individual because of the different decompositions of the game. Sequential PD, the individual moves second
Assuming that the payoffs in the first round are neglected, the probability that an individual chooses C in these games, P(C) is derived from Equation (A.3) with the expectations replaced by the other player’s actual behavior: P(C) =F{V(p)
- V(r) +x’p}.
(A4
In this case, the individual does not include the payoffs resulting from the other player’s behavior in making a choice. Sequential PD, the individual moves second and values payoffs from the first move
Assuming integration of payoffs, Eq. (A.2) is modified, resulting in Eq. (AS). Eq. (AS) is similar to Eq. (A.21, except that now the expectations rr are substituted by facts, reflected by D. D equals 1 if the other player has moved cooperatively and 0 otherwise. P(C) = F{DV(b) + (1 - D)V(d) - DV(a) -(l - D)V(c) +X’p}. (A-5) Evaluation of the other players payoffs
In Eqs. (A.l)-(AS), no weight is given to the payoffs of the other player. However, if cooperation is assumed, the payoffs of the other player should be evaluated besides the effects of motivation for cooperation and motivation for competition. The equations corresponding to Eqs. (A.11, (A.31 and (A.41, including the value of the other player’s payoffs are, respectively: P(C) = F{~l/(b) + (1 - r)V(d) - TV(a) - (1 - z-)V(c) +4(rV(6) + (1 - r)V(a) - TV(d) (A.la) -(l - 7r)V(c)) +x’p}. P(C) = WQMP) -TOYr) +Vq))
+ V(q))
+ (I-
%)(VP)
+ V(s))
+ V(q))
- (I-
TAG
+ V(s)) + +4%)
+ (I-
- (1 - r&v(r)
P(C)
=F(~(P)
-v(r)
Tz)(V(q)
+ V(r))
+ v(s))) +4(v(s)
- T#Ys)
+ VP))
(A.3a)
+X’PL -
v(s)
+ V(P)
-
v(r)>
+X’P}.
(A.4a)
371
9. Appendix B: Games used in the experiments
Game (I)
Your Gains
Other’s Gains
Your Gains
c
c
D
D Game (III)
Your Gains
Other’s Gains
Game (II)
Your Gains
Other’s Gains
Other’s Gains
~~~ Game (IV)
Your Gains
Game (V)
Your Gains
Other’s Gains
Other’s Gains
C ~~1
D Game (VI)
Game (VII)
Games IX-XIV are similar to Games III-VIII moves aster the other player.
Game (VIII)
except that the player
372
G. Antonides /Journal
10. Appendix
C: Experimental
(Rows refer to conditions,
11. Appendix
(Adapted
of Ecorzomic I?yw/dogy 15 (1994) 351-374
design
cells refer to the games in Appendix B.)
D: Short scale of internal-external
Locus of Control
from Phares, 1978.)
I more strongly believe that: 1. a. Many people can be described as victims of circumstances. b. What happens to other people is pretty much of their own making. 2. b. Much of what happens to me is probably a matter of luck. b. I control my own fate. 3. a. The world is so complicated that I just cannot figure things out. b. The world is really complicated all right, but I can usually work things out by effort and persistence.
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of Economic Psychology I5 (19941 351-374
373
4. a. It is silly to think one can really change another’s basic attitudes.
b. When I am right I can convince others. 5. a. The points I have earned with the games are completely determined by chance. b. The points I have earned with the games are entirely my own responsibility.
References Antonides, G., 1991. Psychology in Economics and Business. Dordrecht: Kluwer Academic Publishers. Antonides, G. and A.J.M. Hagenaars, 1989. The distribution of welfare in the household. Papers on Economic Psychology 82. Rotterdam: Erasmus University. Becker, G.S.. 1981. A Treatise on the Family. London: Harvard University Press. Belle, F. and P. Qckenfels, 1990. Prisoner’s Dilemma as a game with incomplete information. Journal of Economic Psychology 11, 69-S4. Colman, A.M., 1982. Game Theory and Experimental Games. Oxford: Pergamon Press. Dawes, R.M., 1980. Social dilemmas. Annual Review of Psychology 31, 169-193. Dawes, R.M., J. McTavish and H. Shacklee, 1977. Behavior, communication and assumptions about other people’s behavior in a commons dilemma situation. Journal of Personality and Social Psychology 35, 1- 11. Kahneman, D. and A. Tversky, 1979. Prospect theory: An analysis of decision under risk. Econometrica J-I, 263-291. Kahneman, D. and A. Tversky, 1984. Choices, values and frames. American Psychologist 39. 341-350. Liebrand, W.B.G., D.M. Messick and H.A.M. Wilke, 1992. Social Dilemmas; Theoretical and Research Findings. Oxford: Pergamon Press. Maddala, G.S., 1983. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge: Cambridge University Press. Marwell, G. and R.E. Ames, 1981. Economists free ride, does anyone else? Journal of Public Economics 15, 295-310. McClelland, DC., 1975. Power: The Inner Experience. New York: Irvington. McClintock, CC., R.M. Kramer and L.J. Kiel, 1984. Equity and social exchange in human relationships. Advances in Experimental and Social Psychology 17, 183-228. NCBS (Netherlands Central Bureau of Statistics), 1992. Statistical Yearbook. Phares, E.J., 1978. ‘Locus of control.’ In: H. London and J.E. Exner Jr. (Eds.), Dimensions of Personality (pp. 263-304). New York: Wiley. Pruitt, D.G., 1967. Reward structure and cooperation: The decomposed Prisoner’s Dilemma game, Journal of Personality and Social Psychology 7, 21-27. Pruitt, D.G., 1970. Motivational processes in the decomposed prisoner’s dilemma game. Journal of Personality and Social Psychology 14, 227-238. Rotter, J.B., 1966. Generalized expectancies for internal versus external control of reinforcements, PsychoIogicai Monographs 80 (If. Sawyer, J., 1965. The altruism scale: A measure of cooperative, individualistic, and competitive interpersonal orientation. The American Journal of Sociology 71, 407-416.
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Schneider, F. and W.W. Pommerehne, 1981. Free riding and collective action: An experiment in public microeconomics. Quarterly Journal of Economics 689-704. Schoemaker, P.J.H., 1982. The expected utility model: Its variants, purposes, evidence and limitations. Journal of Economic Literature 20, 529-563. Stroebe, W. and B.S. Frey, 1982. Self-interest and collective action: The economics and psychology of public goods. British Journal of Social Psychology 21, 121-137. Thaler, R., 1985. Mental accounting and consumer choice. Marketing Science 4, 199-214. Van Lange, P.A.M., W.B.G. Liebrand and D.M. Kuhlman, 1990. Causal attribution in three n-person prisoner’s dilemmas. Journal of Experimental Social Psychology 26, 34-48. Van Praag, B.M.S., 1968. Individual Welfare Functions and Consumer Behavior. Amsterdam: NorthHolland.
Journal
of Economic
Psychology
15 (1994) 351-374
Mental accounting in a sequential Prisoner’s Dilemma game * Gerrit
Antonides
**
Dept. of Economic Sociology and Psychology, Erasmus Utzinillersity, P. 0. Box 1738, 3000 DR Rotter-dam, The Netherlands Received
November
19,1992;
accepted
February
28, 1994
Abstract The two-person Prisoner’s Dilemma (PD) game is decomposed in three different ways. The games differ with respect to positive and negative outcomes, whether they are played simultaneously or sequentially and whether one or the other person is allowed to play the sequential game first. The observed strategies of 5? participants playing these games in a computer experiment are explained by the values of the payoffs, the expectations regarding the other person’s moves and by personal information. It appears that a subjective expected utility model can be used to test different psychological models of choice. A model assuming segregated evaluation of the payoffs in the sequential games gives a better explanation of the choices than a model assuming integration of payoffs. This implies some rules for giving rewards for behavior in the social dilemma situations concerned.
1. Introduction Choices in social dilemmas are predicted by game theory from the normative rational model. In the Prisoner’s Dilemma (PD) game, this results in collectively inefficient solutions. In psychology, many factors have
* Bouke den Hoed is gratefully acknowledged for creating the dataset. I appreciate the computational assistance by Edwin Jansen van Rosendaal. I thank Henk Elffers, Werner Giith, Charles Noussair and two anonymous referees for their helpful comments. ** Fax: +31 10 452 2646, E-mail: [email protected] 0167-4870/94/$07.00 0 1994 Elsevier SSDI 0167-4870(94)00006-V
Science
B.V. All rights reserved
352
G. Antonides /Joumal of Economic kychology 15 (1993) 351-374
been investigated to explain deviations from the game-theoretic solution. These include, inter a&, altruism, conscience and norms, communication, group size, noticeability and expectations (see, for example, Dawes, 1980; Liebrand et al., 1992). These factors may prevent society from disasters such as, for example, overpopulation or depletion of natural resources. Also, they influence the provision of public goods. Here, we shall not deal with large scale consequences of choices in social dilemmas but concentrate on the mental processes involved in making a choice. In particular, we shall consider situations in which the payoff is not obtained instantaneousty but depends on the choice made by another person at a later point in time. These situations are common in dynamic work environments (e.g. streetworkers who help each other at different points in time; see Pruitt, 1967) and in other situations where reactions of other people to a person’s behavior are delayed (e.g. in competitive markets, agency problems, venture capital markets, etc.). A number of experiments have been run to estimate the effects of the psychological factors mentioned above (cf. Dawes, 1980; Colman, 1982; Liebrand et at., 1992). Typically, in these experiments, the factors are manipulated and the behavior is observed. Although this procedure obtains reliable results, the mental choice process is barely considered. In the present paper, choice models are derived based on different mental processes taking place in decision making. In experimental studies of the Prisoner’s Dilemma, assumptions about the utility of the choice alternatives are made but utilities usually are not estimated. However, knowledge of the utility function of choice alternatives is important in understanding the choice process. Expectations regarding the opponent’s choice are considered to be among the determinants of choice (Dawes, 1980). Here, expectations will be investigated systematically in relation to the choices made. The explanation of these expectations will also be considered. Utilities of alternatives and expectations concerning the outcomes are combined in subjective expected utility models to explain the choices of the participants in an experiment including Prisoner’s Dilemma games. Rather than asking participants about the reasons for their choices (e.g. Pruitt, 19701 subjective information has been obtained during the process in order to test hypotheses regarding the choices made. Our concern will be choices in one-shot games rather than choices in a dynamic context. Hence, the results may be applicable in situations where the players do not build relationships.
G. Antonides /Journal
2. The sequential
Prisoner’s
353
of Economic Psychology 15 (1994) 351-374
Dilemma game
All the games considered here are symmetric, that is, the distinction between the players is arbitrary - both face the same payoff matrix. Although there is no compelling need to use symmetric games, we decided to base our experiments on Pruitt’s games (1967) for reasons of comparability. The symmetric simultaneous PD game is shown in Fig. 1 in its strategic form representation. In the PD game, the condition a > b > c > d applies by definition. Individuals i and j both choose simultaneously from the strategies C (cooperative) and D (defecting). The individual payoffs are known directly after choices have been made. The SPD game in Fig. 1 is run in two stages. First, Individual 1 chooses C or D, getting the corresponding i (self) payoff for him/herself and giving the j (other) payoff to Individual 2. Second, after being informed about l’s choice, Individual 2 makes a choice, getting the corresponding i payoff for him/herself and giving the j payoff to Individual 1. The summed payoffs resulting from the two choices equal the payoffs in the simultaneous PD. It follows that the symmetric SPD is subject to the conditions p + q = b, r+s=c,p+s=dandr+q=a,andhencethata+d=b+cisanecessary condition for a PD to be decomposed symmetrically. However, an infinite number of different values of p, q, r and s will generate the same PD. Since different decompositions constitute the same PD, game theory would predict the same strategy (D) in each decomposition. If the sign of the payoffs in Fig. 1 is reversed, the dominant strategy of the players is changed. After transposing the rows and the columns of the payoff matrix, the cooperative and defecting strategies appear as usual. However, Pruitt
individual other
j’s choice
Simultaneous
self
PD
Fig. 1. Payoff matrices of simultaneous and sequential j = 1, 2; i # j.) (C = cooperate, D = defect.)
Sequential symmetric
Prisoner’s
PD
Dilemma
games.
(i = 1, 2;
354
G. Antonides /Journal
of Economic Psychology 15 (1994) 351-374
(1970) obtained different proportions of defecting strategies on presenting three different SPD’s to his participants. From the reports given by the participants in retrospect, their behavior was inspired by different motives, i.e. tactical altruism, fair division and greed. Since the assignment of participants to the SPD’s was random, the framing of the SPD’s should be responsible for eliciting the different motives. The next sections deal with the question of how different strategies can be explained by the different frames. 2. I. Utilities of payoffs Usually, social dilemma games are investigated empirically by providing the participants with payoffs resulting from their choices. Since the payoffs are of only moderate financial value, one could argue that the choices would not matter to the participants very much. However, as Dawes et al. (1977) report, participants may show hostile behavior to non-cooperative persons even if moderate payoffs are at stake. This suggests that the payoffs should not be considered as marginal increments of their wealth but as isolated outcomes in their own right. Kahneman and Tversky (1979) also notice the isolation effect in two-stage games. This implies that utilities should be defined on the payoffs of the game, rather than on end states of wealth including the payoffs. As an alternative to the economic utility function, Kahneman and Tversky (1979) propose a value function which is concave for gains and convex for losses. The inflection point is located at the point of reference, i.e. the present state of affairs. In essence, they propose their value function as an alternative to the economic utility function which is defined over total assets. Alternatively, one could say that utility theory assumes only one utility function for the evaluation of payoffs, whereas prospect theory assumes two different functions depending on whether the prospect is framed positively or negatively. This suggests that the utilities associated with positive and negative payoffs in PD games may not be described by the same utility function. 2.2. Subjective
expected utilities of payoffs
Choice in a PD game is associated with uncertainty regarding the opponent’s strategy. It is hypothesized here that the probability of making a cooperative choice depends both on the utilities of the payoffs and the
G. Antonides/Journal
of Economic Psychology 15 (1994) 351-374
35.5
subjective expectation that the opponent chooses a particular strategy. It appears that the explanation of behavior in a PD calls for a subjective expected utility model (cf. Schoemaker, 1982). More specifically, the subjective expected utility of a C choice may be compared with the subjective expected utility of a D choice. The outcome of the comparison may be associated with the decision to make a C choice. Let the value of a C-C outcome for an individual be V(b) and the value of a C-D outcome V(d). Analogously, the value of a D-C outcome is V(a) and the value of a D-D outcome is V(c). If an individual is 100% sure that the opponent will choose C, the subjective expected utility (SEU) of a C choice is V(b) and the SEU of a D choice is V(a). We assume that the higher V(b) relative to V(a), the more likely a C choice by the individual will be. Furthermore, we assume that a vector of other variables (X), such as psychological factors, will affect the probability of choosing C. Usually, individuals are poor at predicting the choices made by the other player in the PD game (Dawes, 1980). For this reason, perceived probabilities of outcomes, i.e. expectations about the choices to be made by the other player, become important in explaining the choice behavior. If 7~ is the perceived probability that the other player will choose C, the SEU of a C choice is TV(~) + (1 - ~)l/(d) and the SEU of a D choice is VI’(~) + (1 - r).)I/(c). We assume that the difference between the two SEU’s, together with personal factors X, is related to the probability of a C choice, P(C). Applying the probit model to the discrete choice between C and D (Maddala, 1983) we have:
P(C) =N((dqb) -(7TV(a)
+ (1 - %-)I+)) + (1 - 7T)V(c)) +x’p),
(1)
where N is the cumulative standard normal distribution function and p is a vector of parameters. Eq. (1) assumes that the probability of a C choice depends on the difference between the two SEU’s and on the value of X’p (one of the X’s equals one to allow for a constant term). We observe that a high expectation of the other player’s C choice does not necessarily induce a high probability of a C choice by the individual. Eq. (1) implies that cooperative choice depends on the interactions between the subjective probabilities that the other player chooses cooperatively and the payoff values. Without knowledge of the value function, one cannot predict whether expectations positively or negatively affect cooperative choice.
ml
G. Antonides /Journal
of Economic Psychology 15 (1994) 351-374
Despite conclusions from specific studies regarding the effect of expectations (e.g. Dawes et al., 1977; Marwell and Ames, 1981) it appears that no general rules can be given on this matter. Eq. (1) shows the basic model for evaluating choice probabilities. The same type of modeling is applied to the sequential PD games in Appendix A.
3. Mental accounting
of payoffs
In the PD, integrated payoffs are evaluated. However, since the payoffs in the sequential PD are given separately, they may be accounted separately. As Thaler (1985) has argued, the mental accounting of separate payoffs may proceed differently from neo-classical computation of utilities. Mental accounting (or mental arithmetic) is a psychological mechanism for evaluating the balances of different accounts. Money values are frequently not cleared within the same mental account (cf. Antonides, 1991). In particular, Thaler (1985) assumes that V(x) + V(y) differs from V(x + y) generally. This implies that the summed evaluations of the two different stages of the SPD may be different from the evaluation of the integrated payoffs. For example, in the SPD, the equation p + q = b must hold. If payoffs are integrated, V(p + q) = V(b) should also hold. However, if the payoffs in the two stages of the game are evaluated separately and the function V is nonlinear, then in general V(p) + V(q) # V(b). If decision makers in an SPD integrate the payoffs, the choice model is different than if they evaluate the payoffs separately. This is shown more fully in Appendix A. We need to consider two variants of the SPD, depending on whether we are focusing on the first or the second player. The second player makes his/her choice after the other player has moved. In this case, the payoffs resulting from the other person’s behavior are known and expectations do not come into play. This game is referred to here as SPD2. The choice models that apply to it are also derived in Appendix A. We characterize the games where the different models apply: - The choice model for the simultaneous PD involves the same expectations about the other player’s behavior, whether one chooses C or D. - The choice models for the SPD include different expectations about the other player’s behavior after a C choice than after a D choice, because the other player moves second.
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- The choice models for integrated or segregated - The choice models for first, differ according to evaluated by integration
4. Personal
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357
the SPD game differ according to whether evaluation of payoffs is assumed. the SPD game, where the other player moves whether payoffs in the first round are ignored, or evaluated by segregation.
factors in the Prisoner’s
Dilemma game
Since the game-theoretic solution of the PD is frequently violated, other factors should be added to explain cooperative choice. Dawes (1980) describes a number of factors influencing choices in social dilemmas, including personal factors, X above. A personal characteristic influencing choices is strategy preference (this is similar to the concept of social value orientation in Liebrand et al., 1992, p. 17). A number of basic preferences can be distinguished, four of which are of practical relevance: altruism, cooperation, competition and individualism (cf. McClintock et al., 1984). Altruism is characterized by maximizing the payoffs of the other player, regardless of one’s own payoff. Cooperation is characterized by maximizing the players’ joint payoffs, competition by maximizing the difference between the players’ payoffs, and individualism by maximizing the individual payoff regardless of the other player’s payoff. Alternatively, concern for the other player may be modeled by including the second player’s outcomes in the utility function of the first player (cf. Becker, 1981; Antonides and Hagenaars, 1989; Sawyer, 1965). If the payoffs of the second player are valued within the utility function of the first player, the expected utility of the second player (as considered by the first player) can be added to the models considered before, weighted by a coefficient, 4, indicating the extent of concern for the second player (see Appendix A). Locus of Control refers to the disposition to attribute the results of one’s behavior to one’s own actions (internal control) or to the environment (external control) (cf. Rotter, 1966; Phares, 1978). Internal control may increase cooperative choice because it assumes that one is able to influence the other player’s strategy. External control may increase defecting choice because one believes that one cannot influence the other player. Frequently, in experimental games men have been found to be more cooperative than women (Colman, 1982, p. 124). Income may have a positive effect on cooperation (cf. Schneider and Pommerehne, 1981),
358
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possibly because at higher incomes the risk of a loss counts less. Education might have a positive effect on cooperation because it may help to overcome the inefficient equilibrium of the PD by insight. Finally, we assume that married people show more cooperation than singles because of selection effects (cooperative people marry more frequently) and learning effects (married people may learn to cooperate).
5. Experimental
procedures
In March 1992, an experiment was run with individuals from a neighborhood near Erasmus University. In total, 300 people were invited to participate by putting a letter in their mailbox. The letter said that on average Dfl.lO.- could be earned by playing a computer game with other people. A few days later, recipients to the letter were asked in person to make an appointment for the experiment. Failures to make appointments occurred because participants were not at home on two different occasions (100 cases), unable to make an appointment on the two scheduled dates or refused (133). Thus 67 people agreed to participate. Groups of four participants were scheduled on one of two consecutive Saturdays each for about one hour. Each participant was seated in a separate room in front of a personal computer. All the instructions, games and questionnaires were administered by computer. Participants were instructed that they were matched with another individual, who acted as the other player. Because we wanted to standardize the behavior of the other player, they actually played with the computer which made cooperative and defecting choices at mndom, each with 50% probability. A 50% probability is comparable with the average cooperation rates in first trials reported by Pruitt (1967, 1970). Participants received instructions for playing the PD and the SPD games and performed at least three practice trials in each type of game until they understood the game. Each point in the payoff matrices earned lOc, so the payoffs in each experimental trial ranged from -Dfl.1.80 to +Dfl.l.SO. They were told that a minimum of Dfl.5.- plus the total payoffs exceeding this amount could be taken home. The experiment lasted for about 45 minutes on average after which the participants were paid. Since the participants were likely to communicate with each other because they live in the same neighborhood, we could not reveal that they actually played
G. Antonides /Journal
of Economic Psychology 15 (1994) 351-374
359
with a computer. We did not keep records of the identities of the participants, so it was impossible to contact them after the experiment for debriefing or to ask further questions. Before each trial, participants were asked to state their expectations regarding the other person’s choices and the subjective probabilities that these expectations would come true. In the SPD, separate expectations were recorded for the cases where one would oneself choose a cooperative or defecting strategy, respectively. After each trial participants were informed about the total amount won or lost. In the SPD’s in which the participant made the second move, no expectations were elicited since the other player’s choice was already known. The participants played a number of different games; they were told that each game consisted of three trials. At the start of each game, a change of the other player was announced. The participants were distributed almost evenly over 12 conditions of the experimental design shown in Appendix C. PD I and SPD’s III, IV and V in Appendix B were used by Pruitt (19671. PD II, SPD’s VI, VII and VIII are similar, except for the sign of the payoffs. SPD’s IX-XIV are similar to SPD’s III-VIII, except that the player is allowed to move after the other player. All games involved payoffs that were multiples of 6, which facilitates the analysis of the value function. After the games, the participants completed several questionnaires regarding their socio-demographic situation, their strategy preference and their Locus of Control by means of the personal computer. To measure the strategy preferences, Van Lange et al. (1990) used a scale consisting of nine items, each with three different answers. One answer typically implies high joint payoffs, another answer implies a high difference of payoffs and the remaining item implies high individual payoff. Sub-scale scores associated with the strategy preferences range from zero to nine and the sum of scores equals 27, so two scales are sufficient to identify the strategy preferences. The motivation for cooperation is measured by counting the number of answers maximizing joint payoffs (MOTCOOP), whereas the motivation for competition is measured by counting the number of answers maximizing the difference between the players’ payoffs (MOTCOMP). The LOC scale was adapted from an abbreviated version by Phares (1978) consisting of five questions (see Appendix D). The fifth question of this scale deals with student grades and was not relevant for our sample. A question concerning the points earned with the games was included instead.
360
Pruitt’s outcomes Since our elicitation and all of
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/Journal
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Psycho&y
1.5 (1994) 351-374
questionnaires between trials (1967, 1970) did not result in different from those in a control group without questionnaires. games are similar to Pruitt’s experiment, it is assumed that the of expectations does not interfere with the participants’ behavior the participants were treated as explained above.
6. Results The experiment unintentionally took place when the weather was fine and as a result the number of participants was further reduced to 52. The sample composition was as follows: male: 54%; married: 31%; age 40 years or over: 38%; higher education: 54%; reported after-tax monthly household income over Dfl.2000: 56%. The sample included 16% less married people than the Dutch population (cf. NCBS, 19921, possibly due to family obligations on Saturday. Also, the sample included about 20% less people with an after-tax household income over Dfl.2000, possibly due to underreporting of income to some extent. 6.1. Effect of game order Since the dynamics of experimental games is not the object of study, we first tested for the presence of dynamic effects. If dynamic effects are present, we expect the position of a game within the session to affect the probability of a cooperative move. Table 1 shows the results of a probit analysis on the observed strategies (0 = defecting choice, 1 = cooperative choice). The analysis included 1248 choices (52 individuals times 8 different games times 3 repetitions). Dummy variables associated with the type of the game were included and the logarithm of the sequential position of the game (ranging from 1 to 24) should capture dynamic effects. Several dummies had significant effects on the probability of cooperation, notably those associated with the negative simultaneous game and the third SPD. The effect of the positive simultaneous PD game was included in the constant term. Apparently, the order of the game in the experiment did not significantly affect cooperation. A quadratic order function obtained similar results. It appears that on average the participants acted as if each game was played with a different person, unlike experiments where players were allowed to play a number of games with each other (e.g.
G. Antonides /Journal Table 1 Probit analysis
of dynamic
effects
Constant PD(l -) SPDfl + ) SPD(2 +) SPD(3 + ) SPDfl -) SPD(2 - ) SPD(3 - 1 SPD2f 1 + ) SPD2(2 + ) SPD2f3 +) SPD2( 1 - ) SPD2(2 - ) SPD2(3 - ) Inforder of the game)
of Economic Psycholow 15 (1994) 351-374
in cooperative
choice (t-values
361
in parentheses)
- 1.19 (-7.62) 0.38 (2.46) 0.11 (0.65) - 0.08 ( - 0.44) 0.50 (2.90) 0.20 (1.16) 0.13 (0.71) 0.57 (3.38) 0.16 (0.72) 0.27 (1.19) 0.32 (1.46) 0.04 (0.16) - 0.23 (- 0.90) 0.27 (1.19) -0.02(-0.41)
PD( .) refers to simultaneous PD’s. SPD( .) refers to sequential PD’s where the individual moves first. SPD2( .) refers to sequential PD’s where the individual moves second.
Pruitt, 1967, 1970). In what follows, it is assumed that the order of the games did not affect the choice of strategy. 6.2. Cooper-ation rate Since the number of trials in each game is known and finite, game theory would predict non-cooperative choice. However, the average cooperation rate was 26.2% overall. The average cooperation rates in the first, second and third trials of the games were 26.0%, 26.2% and 26.4%, respectively. These cooperation rates are not significantly different from each other. The cooperation rate was lower than cooperation rates on the first trial obtained by Pruitt (1967, 1970) with undergraduate students (ranging 45%-80% in different games). Pruitt’s participants (1970) were not told the exact number of trials, which may account for the higher cooperation rate at least in part. Cooperation rates different from zero have also been found by Dawes (19801, Dawes et al. (19771, Stroebe and Frey (1982) and Pruitt (1967), among others. The cooperation rates differed across the games to some extent (see Table 2). The average cooperation rate of the SPD (27.6%) was not significantly different, statistically, from the SPD2 where the player moves
362
G. Antonides /Journal
Table 2 Cooperation parentheses)
PD(+) PDt-) Average
rates
PD
and
expectations
of Economic Psychology 15 (1994) 351-374
of the
other
player’s
cooperative
choice
(standard
Cooperation rate
Expectation of the other player’s cooperative choice
Total number of trials
19.9% (3.2) 32.1% (3.7) 26.0% (2.5)
39.6% (2.2) 47.1% (2.5) 43.4% (1.7)
156 156
SPD(l+) SPD(2 + ) SPD(3 +) SPDtl -) SPD(2 - ) SPD(3 -) Average SPD
23.2% 17.7% 36.3% 25.9% 23.5% 39.2% 27.6%
(4.1) (3.8) (4.8) (4.2) (4.2) (4.8) (1.8)
SPDU 1 + ) SPD2(2 + 1 SPD2(3 + 1 SPD2t 1 - ) SPD2(2 - ) SPD2(3 - ) Average SPD2
24.1% 20.4% 27.5% 13.7% 29.4% 27.5% 24.4%
(5.8) (5.6) (6.3) (4.7) (6.4) (6.3) (2.4)
Expectation in case of choosing C
Expectation in case of choosing D
40.2% 41.1% 48.4% 40.2% 45.8% 44.0% 43.2%
34.5% 39.1% 41.9% 36.6% 38.7% 37.7% 38.0%
(3.0) (3.1) (3.4) (3.9) (3.3) (3.3) (1.3)
(3.3) (3.3) (3.3) (3.4) (3.3) (3.6) (1.4)
errors
in
108 102 102 108 102 102
54 51 51 54 51 51
PD( . ) refers to simultaneous PD’s. SPD( .) refers to sequential PD’s where the individual moves first. SPD2( .) refers to sequential PD’s where the individual moves second.
second (24.4%). On average, it does not seem to matter if the players move first or second. This result is not expected from game theory, although it is consistent with the results obtained by Bolle and Ockenfels (1990). 6.3. Expectations Table 2 shows the average expectations of the players regarding the other player’s move for each type of game. The expectation is the perceived probability r, that the other player will choose cooperatively. The average expectations were lower than the actual percentage of cooperative moves made by the computer (51.0%), i.e. the true percentage was underestimated. On the other hand, the observed cooperation rates were lower than the expectations regarding the other player’s cooperative
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363
strategy, indicating exploitative motives to some extent. In the SPD, the average expectation of cooperative choice was 5% higher when participants had themselves made a C choice than when they had made a D choice (t = 2.72). The average difference of 5% applies to each individual’s choices, since at each move the individual was asked to state an expectation conditional on a cooperative choice and an expectation conditional on a defecting choice. The fact that expectations differ according to the choice made indicates a belief that the other player will react to one’s own move (i.e. that the other player does not follow the game-theoretic solution). 6.4. Estimation of the choice models The models in Appendix A were estimated by means of probit analysis. In this analysis, the probabilities of the choices were related to the observed expectations of the other players’ choices and to personal factors. The evaluations of payoffs were estimated as sample constants. In each model, each of the choices made by an individual in 24 games were analyzed for the sample of 52 individuals (1068 observations in total). Note that the effects of personal factors were assumed to be constant over all games and individuals. Table 3 shows the results of three model estimations. Model 1 assumes segregation of payoffs in the sequential games, according to Eqs. (A.31 and (A.41 in Appendix A. Model 2 assumes integration of payoffs in all of the games. Model 3 adds the values of the other player’s payoffs to Model 1 (including Eqs. (A.la), (A.3a) and (A.4a) in Appendix A). In Games III and V, a negative payoff ( - 6) occurs in combination with positive payoffs, and it might be that this context would give it a different value from the one it took in Games VI-VIII, where the other payoffs were also negative. Similarly Games VI and VIII involve a positive payoff ( + 6) in a context of negative payoffs. To test for context effects, the value differences v(O) - V( - 6) and V(6) - V(0) were estimated separately for positive and negative contexts. If we arbitrarily set T/(O) equal to zero, these differences give the values of 1/(6) and V( - 6) directly. The x2 values reported in the table equal - 2 times the log likelihood ratio resulting from the specified model relative to a model including a constant term only, and can be used to assess the overall significance of the explanatory variables included in the model (see Maddala, 1983, p. 40). Model 1 yielded a very significant x2 and a pseudo-R2 of 0.126.
364 Table 3 Results of probit
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analyses
of cooperative
of Economic Psychology 15 (1994) 351-374
choice (absolute
Model 1 Segregation
t-values in parentheses)
Model 2 Integration
Model 3 Segregation + weight of other’s
2.05
(2.28)
0.43
(2.34)
-0.05 -0.03 0.01 - 0.00 0.30 0.61 -0.04 - 0.46
(1.21) (2.10) (0.69) (0.01) (2.68) (5.11) (2.12) (4.33)
2.25 0.61 0.35 0.33 0.66 0.39 0.36 0.11 0.10
Constant V(- 12)- V(- 18) V(-6)I’-12) I’(O)- I’-6) V ” (6)-V ’ (0) V”(O)-I’“(-6) V(6) - V(O) V(12)- V(6) W18)- V(12) 4 (weight of other’s payoff) LOC MOTCOMP MOTCOOP Male/Female Age Married/Unmarried Education Income
2.40 0.49 0.43 0.37 0.09 0.13 0.64 0.43 0.36
(2.63) (2.59) (3.24) (2.72) (0.65) (0.92) (4.71) (3.1 I) (1.62)
- 0.05 - 0.03 0.01 0.04 0.29 0.63 -0.04 - 0.46
(1.14) (2.14) (1.02) (0.46) (2.62) (5.25) (2.11) (4.30)
log(likelihood)
- 666.1
- 676.4
- 666.0
103.4 15 0.126
82.8 9 0.102
103.5 16 0.116
payoff
(2.34) (2.14) (1.60) (1.74) (4.70) (2.46) (1.61) (0.70) (0.74)
0.18 (0.48) - 0.04 (1.06) - 0.03 (2.08) 0.01 (1.09) 0.04 (0.46) 0.29 (2.53) 0.63 (5.23) - 0.04 (2.12) - 0.46 (4.26)
2
$f Pseudo-R’ a The sign of the payoff
differs
from the other
payoffs
in the games
concerned.
Model 2 assumes integrated payoffs in all the games, so that, for example, in SPD(l + > V(6) + V(6) = 1/(12), V(6) + V( - 6) = V(O), V(12) + V(6) = T/(18) and V(12) + V( -6) = V(6). That is, a linear value function is used. It was therefore only necessary to estimate one segment of the linear value function, V(6) - V(0). The x2 of this model was also very significant. Since Model 2 is a restricted version of Model 1, a x2 concerning the difference between Model 1 and Model 2 can be computed. The x2 equals 20.6 (df= 6) and is significant at the 1% level in favor of Model 1. The result indicates segregation of payoffs, i.e. mental accounting in the SPD games, rather than integration. Model 3 is a non-linear probit model because the weight given to the other player’s outcome, 4 is multiplied by the expected values of the other person’s payoffs (see Appendix A). The coefficient 4 was not significantly
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365
different from zero, so on average concern for the other player’s payoffs seems to be absent, once the motivation for cooperation (MOTCOOP) and motivation for competition (MOTCOMP) were taken into account. Above, a linear value function was rejected statistically. This seems to be due mainly to different evaluations of payoffs of + 6 and -6. If these payoffs occurred in games otherwise including payoffs with a different sign (indicated by a superscript ‘a’ in Table 31, the value was different from that found in games where all the payoffs had the same sign. It appears that the context of the payoff influences its value. In Model 1, the estimate of V(6) - V(O) is almost twice as large as V(O) - V( - 61, so the same difference of payoffs was evaluated differently in the positive area than in the negative area. According to prospect theory, the value function for losses is steeper than for gains (Kahneman and Tversky, 1979). Our result seems to contradict this assumption. Although the result was not reproduced in Model 3, in that case the two slopes are equal. According to all models, value differences act as coefficients of expectations. Therefore, significant t-values in the models indicate that expectations significantly affect the probability of cooperation. Although this is consistent with the theory of cooperative choice, our result shows that the effect of expectations is modified by the values of the payoffs. The Locus of Control did not significantly influence cooperative choice. The motivation to compete (MOTCOMP) with the other player significantly decreased cooperation, whereas the motivation to cooperate (MOTCOOP) and gender did not have significant effects. A higher age and a married status both increased cooperation and both education and income diminished cooperation significantly. Although Model 1 is favored over Model 2, one might object that it actually consists of three different models describing behavior in three different types of games. To deal with this objection, we considered the three game types separately (PD, SPD and SPD2, respectively). In each sequential game type, the restricted model fitted significantly worse than the unrestricted model, confirming that segregation was more likely than integration of payoffs in the SPD games. 6.5. Expectations and previous choices Finally, we consider the subjective probabilities regarding the other player’s cooperative move, excluding games where the individual moves
G. Antonides /Joumul
366 Table 4 Regressions
of expected
Constant Previous computer move minus previous expectation Next-to-previous computer move minus next-to-previous expectation R’ (adjusted) N
cooperation
of Economic Psychology I5 (1994) 351-374
(t-values
in parentheses)
PD
on previous PD
behavior
SPD own C choice
of the other
SPD own C choice
SPD own D choice
SPD own D choice
0.09 (2.65) 0.44 (21.55)
0.43 (14.95) 0.11 (2.35)
0.45 (28.17) 0.12 (4.89)
0.45 (19.66) 0.07 (1.70)
0.41 (25.25) 0.19 (7.41)
0.42 (18.23) 0.13 (3.29)
0.08 (1.57)
0.03 208
0.06 104
player
0.13 (3.46)
0.05 416
0.08 208
0.14 (3.69)
0.11 416
0.13 208
second. We tested the hypothesis that expectations are revised if the actual moves of the computer deviate from expectations in former trials. Table 4 shows the results of regressions of the expectations in trial t (E,) on the differences between actual strategies played by the computer and expectations in the former (two) move(s) (S,pl - E,_1 and S,_, - El_,, respectively). In case of the SPD, the participants stated two expectations, one conditional on a C choice and one conditional on a D choice. The ‘previous expectation’ in the regression is the one associated with the strategy one has actually played. For example, if S, _, is cooperative (or defecting) E, _ , is the expectation associated with one’s own cooperative (or defecting) choice. Since the participants were told that in each game they were matched with another person, only the second and third moves in each game could be explained by the preceding events, whereas only the third move in a game could be explained by the two former events. The table shows that expectations were significantly affected by the deviations of the previous move(s) of the other player from previous expectations, though the amount of variance accounted for in this way was small.
7. Discussion The choices in the Prisoner’s Dilemma games have been modeled assuming different hypotheses concerning the choice process. The hypothe-
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of Economic Psychology 15 (I 994) 351-374
367
sis of evaluating integrated payoffs has been rejected in favor of segregated evaluation. In general, this implies that the evaluation of two payoffs of the same magnitude is different from the evaluation of their sum. In the example of the two streetworkers, this means that it is better if both obtain half of a positive payoff on each of two occasions than if they obtain the total payoff on one occasion and zero payoff on another occasion (besides discounting the advancement or postponement of a reward). It appears that the choice models discriminate between the different mental choice processes concerned. Also, by estimating them, estimates of the value function were obtained. In our experiment, we did not find a convex value function for losses, as postulated by Kahneman and Tversky (1979). Since we only experimented with small payoffs, the hypothesis of a convex value function for losses may still hold for large negative payoffs. This would imply an inflection of the value function in this area. An inflection of the value function would be consistent with the lognormal utility function of income proposed by Van Praag (1968). For very low incomes (comparable with a very low payoff) the lognormal utility function is convex and for high incomes it is concave. However, this function applies to incomes rather than (small) wealth changes as used in our experiment. The value of a particular payoff seems to depend partly on the other payoffs included in the game. A positive payoff in a game otherwise including negative payoffs is evaluated as smaller than the same payoff in a game with otherwise positive payoffs and vice versa. This indicates a context or framing effect, resulting in underestimation of payoffs that are inconsistent with the frame. This result seems interesting enough to be studied in future research. We assumed a constant value function for all participants, with only additional effects of personal characteristics. Since much variation is captured by the personal information in the models, we think that this assumption is justified to some extent. In any case, it does not seem worse than utility functions in microeconomics where personal information is almost excluded. Since the participants each played eight different games, a different value function could in principle be estimated for each individual. However, since this would require many more parameters to be estimated, this is left for future research. The expectations regarding the other person’s strategy explain a significant part of the choice behavior (otherwise the estimated utility intervals would not have been statistically significant). However, expectations of a C
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choice by the other player do not directly increase the probability of a C choice by the individual; their effects are mediated by the value function in the SEU model. Only in the simultaneous PD does it seem that the higher cooperation rate in games with negative payoffs is due to a higher expectation that the other player will play cooperatively. The expectations in trial t can be explained to some extent by the differences between the actual outcomes and expectations in former trials (t - 1 or t - 2). This suggests a learning process despite the finite number of trials (three in each game). The non-zero cooperation rates in the experiment show that the strong form of the game-theoretic hypothesis of playing a defecting strategy in the PD has to be rejected. Even in the SPD where the individual moves second, the cooperation rate is not significantly different from SPD’s where the individual moves first. On average, however, exploitation occurs, because the cooperation rate is about half of the expectation that the other player chooses cooperatively. Although the computer chose C in 51% of the cases, the average expectation that the other player chooses C was 43% in the PD and the SPD. The objective probability of 51% was underestimated by the participants. This is in agreement with the subjective weighting function of probabilities in Kahneman and Tversky (1979), who also find that objective probabilities around 50% are underestimated. Several personal factors influenced cooperative choice in the experiment. The motivation to compete with the other player has a negative effect, whereas the motivation to cooperate does not have a significant effect. The weight associated with the other player’s payoffs is not significantly different from zero. The latter result suggests a lack of altruistic motivation on the average. Locus of Control did not have a significant effect on choice. Possibly, the need to exert power (McClelland, 1975) should be distinguished from the attribution of control. Age and married status both affected cooperative choice positively and income and education both had a negative effect.
8. Appendix A: Choice models in simultaneous Dilemma games
and sequential
Prisoner’s
Let the subjective probability that the opponent chooses C in the PD be r,(P,(C)>, with pj(C> being the probability that individual j chooses C and
G. Antonides /Journal
T~( *> the subjective estimate of this probability 1). For the ease of presentation rj(Pj(C)) Furthermore, the individual values of payoffs refers to a vector of personal factors. Simultaneous PD The probability specified as:
that an individual
P(C) = F{7TV(b) -TV(a) with F some probability
369
of Economic Psychology 15 (1994) 351-374
chooses
by individual i (0 I V& - ) I will be abbreviated to r. will be denoted by I’( *>. X
C in the PD, P(C) has been
+ (1 - $7)V(d) - (1 - ~)l/(c) function
+X’/3},
and p a vector
(Al) of coefficients.
Sequential PD with integration of payoffs Eq. (A.l) deals with choice in a simultaneous PD in which the payoffs resulting from the choices of both players are integrated. According to expected utility theory, integration of payoffs should occur in the SPD, too. However, in the SPD the expectations regarding the other person’s strategy may differ according to the player’s own choice of strategy. In particular, it may be assumed that the expectation that the other person chooses C is higher if one chooses C oneself than if one chooses D. The probability that an individual chooses C in an SPD, P(C) has been specified as: P(C) = F{rcV(b) -(I
+ (1 - n,)V(d)
- %)J+)
- r,V(a)
+m),
(A4
with rc the subjective expectation that the other player chooses C (0 I rc( - ) I 1) if one chooses C oneself and 7r,, if one chooses D oneself. Eq. (A.21 is similar to Eq. (A.11 except that r differs according to whether one chooses C or D oneself. Sequential PD with segregation of payoffs Given the assumption of segregated valuation of the payoffs mental accounting hypothesis, the probability that an individual in the SPD, P(C) has been specified as follows: P(C)
=FMW) -Q(W)
Inspection restrictions
+ %))
+ (1 - %)(I+4
+ V(s))
+ I’(q))
- (1 -Q)(W)
+ I+))
under the chooses C
+X’P}.
(A.3)
shows that Eq. (A.21 is a restricted version of Eq. (A.31, the concerning the integration of payoffs. The probabilities of
G. Antonides /Journal
370
of Economic Psychology 15 (1994) 351-374
cooperative choice, P(C) in Eqs. (A.2) and (A.31 may be different for a particular individual because of the different decompositions of the game. Sequential PD, the individual moves second
Assuming that the payoffs in the first round are neglected, the probability that an individual chooses C in these games, P(C) is derived from Equation (A.3) with the expectations replaced by the other player’s actual behavior: P(C) =F{V(p)
- V(r) +x’p}.
(A4
In this case, the individual does not include the payoffs resulting from the other player’s behavior in making a choice. Sequential PD, the individual moves second and values payoffs from the first move
Assuming integration of payoffs, Eq. (A.2) is modified, resulting in Eq. (AS). Eq. (AS) is similar to Eq. (A.21, except that now the expectations rr are substituted by facts, reflected by D. D equals 1 if the other player has moved cooperatively and 0 otherwise. P(C) = F{DV(b) + (1 - D)V(d) - DV(a) -(l - D)V(c) +X’p}. (A-5) Evaluation of the other players payoffs
In Eqs. (A.l)-(AS), no weight is given to the payoffs of the other player. However, if cooperation is assumed, the payoffs of the other player should be evaluated besides the effects of motivation for cooperation and motivation for competition. The equations corresponding to Eqs. (A.11, (A.31 and (A.41, including the value of the other player’s payoffs are, respectively: P(C) = F{~l/(b) + (1 - r)V(d) - TV(a) - (1 - z-)V(c) +4(rV(6) + (1 - r)V(a) - TV(d) (A.la) -(l - 7r)V(c)) +x’p}. P(C) = WQMP) -TOYr) +Vq))
+ V(q))
+ (I-
%)(VP)
+ V(s))
+ V(q))
- (I-
TAG
+ V(s)) + +4%)
+ (I-
- (1 - r&v(r)
P(C)
=F(~(P)
-v(r)
Tz)(V(q)
+ V(r))
+ v(s))) +4(v(s)
- T#Ys)
+ VP))
(A.3a)
+X’PL -
v(s)
+ V(P)
-
v(r)>
+X’P}.
(A.4a)
371
9. Appendix B: Games used in the experiments
Game (I)
Your Gains
Other’s Gains
Your Gains
c
c
D
D Game (III)
Your Gains
Other’s Gains
Game (II)
Your Gains
Other’s Gains
Other’s Gains
~~~ Game (IV)
Your Gains
Game (V)
Your Gains
Other’s Gains
Other’s Gains
C ~~1
D Game (VI)
Game (VII)
Games IX-XIV are similar to Games III-VIII moves aster the other player.
Game (VIII)
except that the player
372
G. Antonides /Journal
10. Appendix
C: Experimental
(Rows refer to conditions,
11. Appendix
(Adapted
of Ecorzomic I?yw/dogy 15 (1994) 351-374
design
cells refer to the games in Appendix B.)
D: Short scale of internal-external
Locus of Control
from Phares, 1978.)
I more strongly believe that: 1. a. Many people can be described as victims of circumstances. b. What happens to other people is pretty much of their own making. 2. b. Much of what happens to me is probably a matter of luck. b. I control my own fate. 3. a. The world is so complicated that I just cannot figure things out. b. The world is really complicated all right, but I can usually work things out by effort and persistence.
G. Antonides /Journal
of Economic Psychology I5 (19941 351-374
373
4. a. It is silly to think one can really change another’s basic attitudes.
b. When I am right I can convince others. 5. a. The points I have earned with the games are completely determined by chance. b. The points I have earned with the games are entirely my own responsibility.
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