تجزیه و تحلیل بیزی سلسله مراتبی از نتیجه و فرآیند مبتنی بر ترجیحات اجتماعی و باورها در بازی دیکتاتور و معماهای زندانی پی در پی

Abstract In this paper, using a within-subjects design, we estimate the utility weights that subjects attach to the outcome of their interaction partners in four decision situations: (1) binary Dictator Games (DG), second player’s role in the sequential Prisoner’s Dilemma (PD) after the first player (2) cooperated and (3) defected, and (4) first player’s role in the sequential Prisoner’s Dilemma game. We find that the average weights in these four decision situations have the following order: (1)>(2)>(4)>(3)(1)>(2)>(4)>(3). Moreover, the average weight is positive in (1) but negative in (2), (3), and (4). Our findings indicate the existence of strong negative and small positive reciprocity for the average subject, but there is also high interpersonal variation in the weights in these four nodes. We conclude that the PD frame makes subjects more competitive than the DG frame. Using hierarchical Bayesian modeling, we simultaneously analyze beliefs of subjects about others’ utility weights in the same four decision situations. We compare several alternative theoretical models on beliefs, e.g., rational beliefs (Bayesian-Nash equilibrium) and a consensus model. Our results on beliefs strongly support the consensus effect and refute rational beliefs: there is a strong relationship between own preferences and beliefs and this relationship is relatively stable across the four decision situations.

Introduction Social dilemmas are an important research area in sociology (e.g., Dawes, 1980 and Kollock, 1998). Standard rational choice models explain the emergence and persistence of cooperation in embedded settings with several factors such as network embeddedness, conditional cooperation, rewards, sanctions, termination of the relation, and renegotiation of profits (e.g., Axelrod, 1984, Schuessler, 1989, Heckathorn, 1990, Weesie and Raub, 1996, Fudenberg and Maskin, 1986 and Buskens and Raub, 2002). Yet, quite some social dilemma situations take place in non-embedded settings and among strangers where actors interact only once and will not see each other in the future. Such non-embedded settings lack the previously mentioned factors that could sustain cooperation. Thus, given classical models in the rational choice literature, one should not observe cooperation in non-embedded social dilemmas. However, we consistently observe otherwise (e.g., Sally, 1995, Camerer, 2003 and Aksoy and Weesie, 2013b). Explaining cooperation in non-embedded settings, thus, has been a puzzle. A rich body of literature, especially in social psychology and experimental economics but to a lesser extent in rational choice sociology, suggests that the emergence and persistence of cooperation in non-embedded settings are due to social preferences. That is, in non-embedded settings cooperation is observed because (some) people do not try to maximize only own outcomes but are interested also in others’ outcomes or hold other non-monetary motivations such as reciprocity. Many models of social preferences have been proposed to capture such non-selfish social preferences (for an overview see Fehr and Schmidt, 2006). One can distinguish roughly two types of social preferences: outcome-based and process-based McCabe et al., 2003. Outcome-based social preferences are about how actors evaluate a certain outcome distribution between self and others. Social value orientations, e.g., individualism, cooperativeness, altruism, competitiveness ( Schulz and May, 1989), and inequality aversion ( Fehr and Schmidt, 1999 and Bolton and Ockenfels, 2000) are examples of outcome-based preferences. In process-based social preferences, actors take the history of previous interactions into account. Responding kind intentions with more pro-social behavior (positive reciprocity) and unkind intentions with less pro-social behavior (negative reciprocity) are related to underlying process-based social preferences ( Gautschi, 2000, Falk and Fischbacher, 2006 and Vieth, 2009). In social dilemmas both outcome and process-based preferences could be at work. For example, in a sequential Trust Game when the trustor places trust, the trustee could be motivated by the objective outcomes that both actors would get in case she honors or abuses trust. But if trust is placed, the trustee may also want to reciprocate the kindness of the trustor irrespective of the monetary outcomes in the game. To understand cooperation in non-embedded settings, one should consider both outcome and process-based social preferences. Social preferences are only part of the explanation. Social dilemmas are interdependent situations. In interdependent situations, behavior depends not only own (social) preferences but also on beliefs about others’ choices. For example, one may not cooperate, however socially motivated, if one expects that others will free ride on one’s cooperation. Thus, to predict the cooperative choice of people we should also deal with their beliefs about the choices of others. Although the economics and rational choice literatures on social preferences are vastly developed, the literature on beliefs is relatively scarce (see also Blanco et al., 2009, Aksoy and Weesie, 2013a and Aksoy and Weesie, 2013b). In experimental economics and rational choice sociology, beliefs are often dealt with as an ingredient of the Bayesian-Nash equilibrium concept (Harsanyi, 1968). The Bayesian-Nash equilibrium is based on very strong assumptions about the beliefs that people have. For instance, people are assumed to know the distribution of social preferences in the population and that the interaction partner is a random draw from this distribution. Consequently, one’s beliefs about others’ social preferences are independent of one’s own social preferences. These strong assumptions ensure that in the Bayesian-Nash equilibrium beliefs and choices are consistent. Throughout the paper we will use the term “rational beliefs” to denote beliefs that satisfy the aforementioned assumptions of Bayesian-Nash equilibrium ( Bellemare et al., 2008). Although being mathematically elegant, in reality people’s beliefs deviate from rational beliefs. There is a strong empirical relationship between preferences and beliefs which refute Bayesian-Nash beliefs (e.g., Blanco et al., 2009, Aksoy and Weesie, 2013a, Aksoy and Weesie, 2013b and Aksoy and Weesie, 2012). Still, the behavioral consequences of ignoring this relationship between preferences and beliefs is yet to be studied. To be clear, if theoretical models which incorporate the rational beliefs assumption, thus ignore the relationship between preferences and beliefs, do not yield behavioral predictions that are far off from actual behavior, one might be content with the theoretical model despite the fact that the rational beliefs assumption is wrong. We should note that there are studies in the experimental economics literature that elicit beliefs experimentally rather than relying on Bayesian-Nash equilibrium (e.g., Bellemare et al., 2008 and Blanco et al., 2009). These studies restrict the focus exclusively on the beliefs about the choices of others (see for a brief overview Aksoy and Weesie, 2013a). In our view, as one explains choices through a micro-model of social preferences, one should also explain beliefs about others’ choices through the same micro-model of social preferences. That is, beliefs about the choices of others should be explained by beliefs about social preferences of others given by the model of social preferences. Extending the use of a model of social preferences to explain beliefs about the choices of others will, firstly, facilitate the empirical test of the social preference model ( Aksoy and Weesie, 2013b). Secondly, explaining choices and beliefs about others’ choices using the same social preference model provides a more parsimonious account than taking beliefs about others’ choices as exogenous variables measured empirically. In this paper, we employ a within-subjects experimental design with a set of binary Dictator Games and a set of non-embedded sequential Prisoner’s Dilemma (PD) games, see Fig. 1. Using a simple model with a single social value orientation parameter, our analysis focuses on the following three questions. First, how does the social value orientation parameter differ between situations with and without relationship history (process)? For example, is there a change in the social value orientation parameter of Ego after Alter’s cooperative or defective behavior in line with positive or negative reciprocity? Second, how does the belief about the social value orientation parameters of others vary with own preferences, and does the relationship between own preferences and beliefs vary across histories. Third, if there is a relationship between one’s own social value orientations and one’s beliefs about others’ social value orientations, and hence the Bayesian-Nash equilibrium does not hold, how much harm does assuming rational beliefs do in predicting choices in non-embedded social dilemmas? Answering these questions, we take advantage of hierarchical Bayesian statistical modeling. Games used in the experiment. DG, PD, C, D are symbols that denote the decision ... Fig. 1. Games used in the experiment. DG, PD, C, D are symbols that denote the decision nodes.

Discussion and conclusions In this paper, using a within-subjects design, we estimate the utility weights (“social orientations”) that the subjects attach to the outcome of their interaction partners in four decision situations: binary Dictator Games (DG), the second player’s role in the sequential Prisoner’s Dilemma after the first player cooperated (C) and defected (D), and the first player’s role in the sequential Prisoner’s Dilemma game (PD). In addition, we analyze the relationship between subjects’ social orientations and their beliefs about the social orientations of others in the first three of the decision situations. We first discuss the findings on social orientations, then discuss the findings on subjects’ beliefs about others’ social orientations. In line with many studies, (e.g., Schulz and May, 1989, Simpson, 2004 and Fehr and Schmidt, 2006) we find significant variation between subjects with respect to social orientations. In addition, social orientations differ significantly between the four decision nodes: the decision context and the relationship “history” between Ego and Alter influence social preferences (Gautschi, 2000, Falk and Fischbacher, 2006 and Vieth, 2009). Yet, the social orientations of a subject across the four decision nodes are highly correlated. This shows that social orientations are partially dispositional traits and partially states influenced by the decision context ( Steyer et al., 1999). Furthermore, comparing several specifications of the effects of context and history on social orientations, we find that the effects of context and history on social orientations differ between subjects. Thus, social orientations vary not only between subjects and contexts, but also how much social orientations vary between contexts varies between subjects. In other words, there is an interaction between subject and context variables (see also DeCremer and VanVugt, 1999 and VanLange, 2000). We now discuss the nature of the influence of history and decision context on social orientations. First consider the social orientations in the three decision situations in a sequential Prisoner’s Dilemma (C, D, and PD). We find that the mean of social orientations in these three decision situations have the following order: μC>μPD>μDμC>μPD>μD. This shows that there is both positive and negative reciprocity. The average social orientation is higher after Alter cooperated (positive history) than in the situation where there is no positive or negative history, i.e., when Ego decides as the first player, PD. The average social orientation is by far the lowest after Alter defected (negative history). These findings are in line with past research on history effects (Gautschi, 2000, Falk and Fischbacher, 2006 and Vieth, 2009). We also find that negative reciprocity is stronger than positive reciprocity, that is, μPD-μD>μC-μPDμPD-μD>μC-μPD. This is in line with a recent study which also reports that in one-shot situations negative reciprocity is stronger than positive reciprocity (Al-Ubaydli et al., 2010). Additionally, we find that the Prisoner’s Dilemma context makes subjects more competitive than the Dictator Game context. In fact, the average social orientation in the Dictator Game is larger than the average social orientation in all three decision situations in the sequential Prisoner’s Dilemma. Moreover, the average social orientation is negative in the sequential Prisoner’s Dilemma, even after Alter cooperated. It has been shown that subtle features of the game, or how the game has been presented to the subjects, may influence social preferences—“explicit” framing effects. (e.g., Lindenberg, 2008, Liberman et al., 2004 and Burnham et al., 2000). The asymmetric investment game framework that we used in the experiment may have made subjects less pro-social, i.e, decreased their social orientations, compared to the more naturally presented Prisoner’s Dilemma. Aksoy and Weesie (2013b) use the same asymmetric investment game framework in a simultaneous play Prisoner’s Dilemma and report that the weight for Alter’s “payoff” is also negative in that case. Another possible explanation for our subjects being more cooperative in a Dictator Game than in a Prisoner’s Dilemma concerns the notion of responsibility as discussed by Camerer, 2003 and Blanco et al., 2011 or the notion of power (Handgraaf et al., 2008), i.e., “implicit” framing effects. In a Dictator Game, the decision maker is fully responsible for the outcomes for Ego and Alter. Consequently, the decision maker in DG may try to make a fair decision by placing a high weight to the outcome of Alter. In the Prisoner’s Dilemma, however, the outcomes for Ego and Alter are determined by the decisions of both Ego and Alter, thus the feeling of responsibility is likely lower. This may, in turn, have reduced the weight attached to the outcomes of Alter in the Prisoner’s Dilemma. This alternative responsibility explanation can be studied with the following simple “partial Dictator” game. In a partial Dictator game, the player in the Dictator role makes a decision, say in a binary Dictator Game as the ones included in our design. Then, the experimenter tosses a coin. If it is heads, the decision of the Dictator is implemented and if it is tails, one of the binary decisions in the Dictator Game is randomly implemented. In this partial Dictator game, the Dictator is not fully responsible for Ego’s and Alter’s outcomes. If the responsibility explanation is correct, then social orientations would be lower in the partial Dictator game than in the conventional Dictator Game. We find it interesting to study further how social preferences vary across different types of games, not only Dictator Game and the Prisoner’s Dilemma, but also other types of games and which features of games influence social preferences the most. With our current design, we cannot differentiate explicit or implicit framing effects. In addition to social orientations, we simultaneously analyze beliefs of subjects about others’ social orientations in three decision situations, DG, C, and D. We compare several alternative theoretical models on beliefs, e.g., some variants of a consensus model and rational beliefs (Bayesian-Nash equilibrium). Our results on beliefs support the model that incorporates a form of consensus effect and reject rational beliefs: there is a strong relationship between own preferences and the mean of beliefs. In addition, the relationship between a subject’s own social orientations and her beliefs about others’ social orientations is relatively stable across the decision situations. This means that differences in beliefs about others’ social orientations between the decision situations are mainly due to differences in own social orientations between these decision situations. However, although own social orientations explain the variance of beliefs to a large extent—in fact explains more than 50% of the interpersonal variance in most cases, there is still significant unexplained variance. We do not analyze other factors than own social orientations that could potentially explain the variation of beliefs. No clear factors come to mind that can explain the variance of beliefs other than own social orientations. Consequently, we interpret this unexplained variance in beliefs as noise. Irrespective of its source, accounting for this noise proves to be important for model fit. Ignoring this noise in beliefs by, for instance, imposing a strict version of the false consensus effect which omits unexplained interpersonal variation in beliefs, deteriorates fit substantially. Although the rational beliefs assumption employed in the Bayesian-Nash equilibrium concept is clearly refuted, the evidence for negative consequences of using rational beliefs for the quality of predictions of behavior in our case is not crystal clear. We find that using obviously wrong rational beliefs in predicting the first players’ PD choices does not seem to deteriorate much the fit—relative to complexity—of the part of the statistical model for social orientations. (Yet, it does deteriorate a lot the fit of the part of the model for beliefs). This satisfactory fit—relative to complexity—may be due to the fact that the rational belief assumption yields substantial parsimony. The rational beliefs assumption implies that beliefs about the distribution of others’ social orientations correspond to the true distribution of social orientations. Consequently, parameters describing beliefs and parameters describing the distribution of social orientations are collapsed, yielding a huge reduction in model complexity. Yet, when subjects’ beliefs are considered, the rational beliefs assumption is flatly rejected. Moreover, imposing directly the consensus effect yields a model as parsimonious as imposing rational beliefs, since under the consensus effect a subject’s belief about others’ social orientations is a projection of her own social orientation. Additionally, the model that directly imposes the consensus effect yields a marginally better fit than the model that imposes rational beliefs, both in predicting beliefs and behaviors of subjects. Consequently, we recommend analyzing behavior taking potential ego-centered biases in beliefs into account rather than automatically assuming rational beliefs. We acknowledge that allowing for deviations from rational beliefs opens the door for adjusting beliefs ex-post to fit any theory to data. That is, one may explain many phenomena by changing the assumptions about subjects’ beliefs ex-post. For an extensive discussion on the consequences of dropping the rational beliefs assumption see Morris (1995). However, we do not suggest abolishing the rational beliefs assumption and letting the researcher arbitrarily adjust the assumptions on subjective beliefs of subjects. We recommend replacing the rational beliefs assumption with a well defined model that reflects ego-centered biases in beliefs. Another issue that comes about when one deviates from rational beliefs is higher order beliefs. Higher order beliefs are crucial in modeling behavior in multistage and simultaneous action games. Under the rational beliefs assumption, first and higher order beliefs are all given by a common prior. When one abolishes the rational beliefs assumption, modeling higher order beliefs becomes difficult. Precisely because of this reason, we did not analyze beliefs about the social orientations of others in the first player’s role in the sequential Prisoner’s Dilemma (node PD). Recall that in order to assess the social orientation θPDθPD in node PD we had to deal with the first order beliefs of subjects about the social orientations of the second players, θCθC and θDθD. If we want to deal with beliefs of subjects about the θPDθPD of others, we need to deal with second order beliefs: what subjects believe about other first players beliefs about θCθC and θDθD. We leave a theoretical and empirical treatment of higher order beliefs to future research, as the paper is already dense. Yet, the following “egocentric” model will be a good starting point to model higher order beliefs taking ego-centered biases into account ( McKelvey and Palfrey, 1992). Recall that we model first order beliefs about others’ social orientations as a normally distributed random variable, centered around a subject’s own social orientation. In principle, the same ego-centered normal distribution can be used to derive all higher order beliefs. Using a subject’s ego-centered normally distributed beliefs to derive all higher order beliefs, an equilibrium analogous to the Bayesian-Nash equilibrium can be calculated. In turn, using the distribution of social orientations and the theoretical model that specifies beliefs about others’ social orientations, a distribution of Bayesian-Nash equilibria can be obtained, which then can be contrasted with experimental data. In addition to higher order beliefs, there are other open issues. For instance, in our analyses we focused on a single parameter social orientation model, but had to ignore other types of social preferences, such as inequality aversion (Schulz and May, 1989, Fehr and Schmidt, 1999, Bolton and Ockenfels, 2000 and Aksoy and Weesie, 2012). We expect that ignoring inequality aversion does not influence our results on the social orientations and beliefs about others’ social orientations as Aksoy and Weesie (2012) explicitly show. Yet, it would be interesting to study how history might influence inequality aversion as it is theoretically not clear what reciprocity implies for inequality aversion. For example, it is unclear if a negative history between Ego and Alter makes Ego less inequality averse or a positive history makes Ego more inequality averse. Perhaps, in this case, it makes more sense to distinguish between advantageous and disadvantageous inequality aversion (Fehr and Schmidt, 1999): probably a positive (negative) history will decrease (increase) disadvantageous inequality aversion and increase (decrease) advantageous inequality aversion. Another open issue, not only for this particular paper but for the social preference literature in general, is explaining the variance in social preferences, both as traits and states. We show that social preferences vary between subjects and contexts. Moreover, the effect of context on social preferences varies between subjects. Yet, we leave providing an explanation for these inter-personal and inter-contextual variances in preferences for future research. A third open issue is the following. Although we showed that the rational beliefs assumption gives a poor description the beliefs of subjects, more work is needed to clearly document consequences of assuming rational beliefs for the predictions of behavior. Among the four decision situations we investigated in this study, DG, C, D, and PD, only in the last one beliefs of subjects directly influence their behavior. Extending our work to other decision situations where beliefs about others’ preferences have potentially strong effects on behavior will help study further behavioral consequences of making wrong assumptions about beliefs. Examples of such situations include the proposer behavior in the Ultimatum Game (Güth et al., 1982), the trustor’s decision in the Trust Game (e.g., Snijders, 1996), and contribution decisions in Public Goods Games with non-linear production functions (Erev and Rapoport, 1990). Another further research direction will be analyzing repeated games or sequential games with several decision nodes. In such games, at least theoretically, actors would update their beliefs about others’ social preferences based on others’ previous choices and take into account that actors’ own behaviors also affect others’ beliefs. Analyzing such cases, one can also combine outcome-based social preferences and process-based social preferences in a single model. For example, consider the following approach. Assume that people have “effective” social preferences. These are preferences that actors use when they make a decision. Effective preferences are a function of a person’s “true” social preferences and her beliefs about others’ social preferences. For example, a “true” cooperative actor, i.e., an actor who assigns a large weight to the outcome of the other, may lower her effective weight on the other’s outcome if she believes that the other is selfish, i.e., has a zero “true” weight for others’ outcomes. Note that this model can also be applied to non-dynamic settings, such as Dictator Games or the PD studied in this paper. However, in dynamic games, depending on others’ observed behavior, people’s beliefs about others’ social preferences change. Hence, their “effective” social preferences also change. This paper has, what we believe, a methodological strength. Here we present several examples of model testing using hierarchical Bayesian statistical analysis of social preferences and beliefs. Hierarchical Bayesian methods are quite flexible. The simultaneous analysis of own social orientations and beliefs about others’ social orientations in several different decision situations is almost impossible within the frequentist paradigm (Aksoy and Weesie, 2013a). Yet, as we show in this paper, such complex analyses can be carried out within the Bayesian statistical framework. Because of their flexibility, in the future we expect to see more applications of Bayesian methods in social science research. As we showed in this paper and others elsewhere (e.g., Falk and Fischbacher, 2006 and Vieth, 2009), in social dilemmas both outcome and process-based social preferences are at work. Moreover, social preferences are not enough to explain behavior in social dilemmas as theoretically behavior in interdependent situations depends also on beliefs about others’ decisions. In addition to outcome and process-based social preferences, to account fully for behavior in social dilemmas one needs to tackle beliefs of actors about others’ decisions and preferences. We believe that rather than talking about a social preferences model, it makes much more sense to talk about a social preferences-belief model, where preferences and beliefs are explicitly modeled. As we demonstrate here, accounting for various forms of social preferences and at the same time explicitly modeling beliefs is complex but possible.