Interdependent preferences and endogenous reciprocity

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Highlights

  • We use an indirect approach to examine the evolutionary stability of interdependent preferences.

  • We also explore the stability of reciprocity and reciprocal preferences.

  • We study how individuals equipped with intrinsic preferences adjust their behavior depending on who they interact with.

  • The key aspect of our method is that behavioral preferences are choice variables that optimally evolve, accounting for strategic interaction.

  • There is a continuum of evolutionary stable interdependent preference profiles: At least one player behaves spitefully, and at most one acts selfishly.

Abstract

This paper employs an indirect approach to formally examine the evolutionary stability of interdependent preferences when players randomly engage in pairwise interactions. Following the model specification for altruism and spitefulness in experiments proposed by Levine (1998), we also explore the stability of reciprocity and reciprocal preferences. In particular, we study how individuals equipped with intrinsic preferences such as altruism, selfishness or spitefulness adjust their behavior depending on who they interact with. The key aspect of our method is that behavioral preferences are choice variables that optimally evolve, accounting for strategic interaction. Our model predicts that in a specific economic framework characterized by negative externalities and strategic substitutes, there is a continuum of evolutionary stable interdependent preference profiles: At least one player behaves spitefully, and at most one acts selfishly. The emergence of altruism as an evolutionarily stable preference crucially depends on how large the support for preferences is. When players have reciprocal preferences, altruism might arise even in meetings where one player is intrinsically spiteful, but not necessarily from the intrinsically altruistic player.

Introduction

Economic theory usually dictates that agents are self-interested and rational. However, the experimental literature (Güth, Schmittberger, Schwarze, 1982, Isaac, Walker, 1988, Fehr, Schmidt, 1999, Charness, Rabin, 2002 among others) suggests that agents are better characterized as having interdependent preferences and being concerned about the payoffs of others.1 Additional supporting evidence for this interpretation is also provided by Brandts and Solà (2001), Güth et al. (1998), Sadrieh and Schröder (2016) and Thunström et al. (2016). Furthermore, the seminal experimental work of Levine (1998) suggests that agents behave as if they have reciprocal preferences, a more specific type of interdependent preference. Agents with these preferences adjust the concern they express for others based on their perceptions of how they are being treated by their opponents. Of course, these perceptions do not have to remain unchanged. To wit, the behavior of players evolves, as they may perceive intentions differently based on who the opponent is. This experimental evidence inspired our work to analytically solve Levine’s model to more accurately predict how preferences evolve.

In this paper, we employ an indirect approach to analytically explore the evolutionary stability of reciprocal preferences using the specification that Levine used to address experimental evidence. In our model, a large population of individuals are continuously and randomly matched in pairs. Players interact in a strategic environment that shows negative externalities and strategic substitutes. Matched players preferences are common knowledge; they determine players’ choices, which in turn determine outcomes and payoffs. Behavior is guided by adjusted utility maximization, whereas the stability of preferences is driven exclusively by material payoff maximization. We follow (Levine, 1998) and assume that the adjusted utility functions are linear on both individual material payoffs. We use a quadratic monetary payoff specification, as this may represent many social dilemmas in which the individual choice that maximizes individual payoffs differs from the one that maximizes group payoffs. In addition, its tractability allows us to derive a closed form solution for optimal strategies, which is an appealing property.

So equipped, we first endogenize interdependent preferences instead of considering them as exogenously given. We find that when players choices are perfect substitutes, there is a continuum of evolutionarily stable interdependent preferences (Proposition 1). Altruism or selfishness might arise as optimally evolved preferences but only by one of the matched players. Instead, spiteful preferences might arise as evolved preferences by both players. That is, whenever one player behaves altruistically, his opponent always behaves spitefully. Otherwise, at least one player behaves spitefully and at most, one acts selfishly. More generally, stable preferences act as substitutes: When players strategically choose how to shape their preferences, they are downward sloping functions of their opponent’s preferences. As one player behaves more altruistically (less spitefully), the other behaves more spitefully (less altruistically). Aside from these predictions, our model quantitatively predicts how altruism and spitefulness impact players payoffs.

We then turn to reciprocity to model reciprocal preferences. We use the linear approach proposed by Levine (1998) and distinguish the intrinsic preferences of each player and their behavioral preferences. We consider the former as genes that are acquired through genetic inheritance (Güth, 1995) and the latter as weighted averages of intrinsic values. Our key observation is that some notion of fairness or reciprocity shapes players’ preferences. To wit, while players’ preferences depend to some extent on their genetics, they will be able to adjust their behavior depending on who they interact with. Therefore, we aim to address contextual behavior without assuming that players’ intrinsic preferences change. Instead, we let reciprocity evolve, which indirectly determines preferences and behavior. This is more consistent with recent empirical works, which suggest that culture (reciprocity), rather than genes (intrinsic preference), provides a greater scope for large-scale human evolution (Bell et al., 2009). Similarly, Boyd, Richerson, 1988, Boyd, Richerson, 2006 argue that cultural adaptation — the ability to create non-genetic evolution — is what makes the human species different from others. This cultural adaptation process is much faster than genetic evolution and leads to highly adaptive behavior.

In this more specific version of the model, pairwise meetings occur between players from two large populations, each characterized by its own intrinsic preference parameters (players types). We propose that each population intrinsic preference parameter is common knowledge, (as in Güth, Napel, 2006, Menicucci, Sacco, 2009, Sethi, Somanathan, 2001). While this can be seen as a rather strong assumption, it can be noted that there is psychological evidence that several observable physical symptoms, such as posture, respiration, voice, and facial muscle tone and expression, can provide some indication of a person’s disposition towards others (Frank, 1987, Frank, 1988). These physical symptoms act as signals that can condition people’s behavior towards others. Alternatively, this perfect information assumption may be replaced with an assumption of sufficiently accurate pre-play signals or the intrinsic preferences information is available for both players at sufficiently small costs.2

For our tractable quadratic material payoff function, we compute the set of evolutionarily stable reciprocal preferences that would arise in the setting proposed by Levine (1998). Players adjust their behavior depending on who they interact with and sometimes, their behavioral preferences coincide with their intrinsic ones. More specifically, we find that in each meeting, at least one player acts reciprocally and that his concern about his opponent’s material payoff depends on the intrinsic preferences of his opponent. Furthermore, we find that genes might restrict the induced behavioral preferences. When they do, strong reciprocity (when a player’s behavioral preference equals his opponent’s intrinsic value) together with no reciprocity (when a player’s behavioral preference coincides with his own intrinsic value) both arise as an evolutionarily stable strategy profile. This occurs when both players are either too altruistic or too spiteful (Proposition 2).

The emergence of altruism as an evolved preference crucially depends on how intrinsic preferences are restricted. Our specification includes an extended support [,1] that includes models of altruistic and egoistic behavior, such as those in Bester and Güth (1998), in which intrinsic values are restricted to be in [0,1], as well as models of spitefulness, as suggested by Bolle (2000) and Possajennikov (2000). We find that if intrinsic values are only allowed to be in [1,1], as in Levine (1998), then altruism only arises in meetings between altruistic players (Proposition 3 and Proposition 4). However, with an extended support for spite, altruism may also arise in meetings between an altruistic player and a highly spiteful player. More surprisingly, in these meetings, altruism may arise as an evolved behavioral preference by the intrinsically spiteful player, and when this happens, the intrinsically altruistic player may have spiteful preferences. In fact, in these meetings, multiple more natural combinations of preferences might also arise as evolutionarily stable. These include both players behaving spitefully or each player behaving according to their intrinsic value. That is, the spiteful player behaving spitefully and the altruistic behaving altruistically.

Literature Review: There are as many ways to model interdependent preferences as there are players concerns other than their own payoffs (Sobel, 2005). Fehr and Schmidt (1999) propose a model of inequality aversion, when players avoid inequitable outcomes, whereas (Güth and Napel, 2006) further explore the evolution of inequality aversion. Charness and Rabin (2002) and Alger and Weibull (2013) study interdependent preferences when players have a concern for efficiency. Kokesen et al. (2000) set up a model for negatively interdependent preferences. They focus on players behaving either selfishly or spitefully, and they consider conditions under which those who behave spitefully earn higher material payoffs than those behaving selfishly. In an alternative approach, Dekel et al. (2007) focus on the stability of outcomes and show that when preferences are observable, only efficient outcomes are stable. That is, the efficiency of outcomes is necessary for the stability of preferences. Herold and Kuzmics (2009) extends this result accounting for preferences that may depend on the opponent’s preferences. Unlike us, they consider exogenously specified preferences that do not account for players’ optimizing behavior or strategic interaction.3

Our paper contributes to the extensive game theoretic literature on reciprocal preferences (Sethi, Somanathan, 2001, Sethi, Somanathan, 2003, Levine, 1998). In the reciprocity games literature, it is widely accepted that reciprocity is driven by perceived kindness. This raises the question of whether agents consider intentions, consequences or both when they evaluate kindness and at the moment of reciprocation. Rabin (1993) develops a theory of fairness equilibria, where a player’s reciprocity is exclusively driven by the underlying belief about his opponent’s intention. In Falk and Fischbacher (2006), reciprocity is not only based on intentions but also driven by the observed consequences of actions. Our focus on reciprocity and altruism considers exclusively intentions — summarized in players types — rather than consequences. As in Levine’s static setting, observed choices change as reciprocity and preferences evolve (and not the other way around); our reciprocity coefficients cannot depend on observed consequences of actions. This approach greatly improves (Rabin, 1993) tractability by replacing beliefs about intentions with beliefs about the players’ intrinsic altruism (types). Similar to our study, (Sethi and Somanathan, 2001) conduct an evolutionary analysis of reciprocity using a slight variation of Levine’s specification to model preferences. Unlike this paper, they consider only two types of players, materialists and reciprocators, and provide sufficient conditions for stable preferences. Furthermore, all reciprocators are altruists and have the same intrinsic preferences. That is, meetings between reciprocators translates into a symmetric game. We model reciprocal preferences more generally and do not require that reciprocators have the same intrinsic preferences; therefore, interactions always occur between heterogeneous players. More importantly, they do not endogenize reciprocity or preferences, which we see as a key feature of our model.

We present the model in Section 2 and offer theoretical predictions for evolutionarily stable interdependent preferences in Section 3. We then explore the reciprocal preferences case in Section 4. Finally, we present our conclusions in Section 5. Brief and instructive proofs are in the text, and lengthier ones are provided in the appendix.

Section snippets

The model

A large population of individuals are continuously and randomly matched in pairs. We label matched players as player i and player j, with i, j ∈ {1, 2}. In a meeting, they independently choose quantities xi,xjR+. Conditional on his opponent’s choice xj, player i receives a material payoff πi(xi,xj)=xi(1xixj). That is, the strategic environment allows for negative externalities and strategic substitutes.

Preferences are interdependent and players care about the material payoff of their

Evolutionarily stable interdependent preferences

We first explore interdependent preferences that are evolutionarily stable. Ultimately, we wish to predict the behavior of the players and understand how much concern they express for others. First, let preferences be given by βi for player i. In a meeting, the players simultaneously maximize their adjusted utility by choosing quantities; player i chooses xi ≥ 0.5

Reciprocity and reciprocal preferences

We now use the indirect evolutionary approach to study reciprocity and explore evolutionarily stable reciprocal preferences.

Our key underlying observation is that individuals are better characterized by an intrinsic preference coefficient — like genes, which are acquired through genetic inheritance — yet they are able to adjust their behavior depending on who they interact with. That is, preferences evolve as reciprocity adjusts in each meeting, but intrinsic values do not vary at all. This

Conclusions

Levine’s experimental work suggests that agents are better characterized as having reciprocal preferences. We used an indirect evolutionary approach to formally examine the evolutionary stability of interdependent preferences as well as the stability of reciprocity to explore how it shapes individual preferences and behavior. Unlike previous papers, we let reciprocity and preferences optimally evolve, and we do not take them as given. This strategic aspect of the preferences is the source of

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    • Strategic reciprocity and preference formation

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      However, as we account for intrinsic types we are able to show that spite and altruism emerge at its lowest intensity in games of complements and substitutes, respectively. This insight is in stark contrast to previous theoretical work on endogenous preferences, which mainly predict that positive concern for others (altruism) arises only when the strategic context is one of the strategic complements; otherwise, only negative concern (spitefulness) will arise (Bester and Güth, 1998; Bolle, 2000; Possajennikov, 2000; Carrasco et al., 2018). Furthermore, when the intrinsic types of players differ significantly among themselves (i.e., only one is an altruist or only one is spiteful), then altruism can only emerge in games of strategic complements; otherwise, in games of strategic substitutes only spiteful preferences can emerge.

    • A game-theoretic model of reciprocity and trust that incorporates personality traits

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      In game theory, positive and negative reciprocity are typically modeled as “reciprocal preferences:” parameters of heterogeneous utility functions that take into account the material welfare of others; positively if the have been kind, negatively if they have been hostile. ( Cox et al., 2007; Carrasco, et al., 2018; Dufwenberg and Kirchsteiger, 2004; Falk and Fischbacher, 2006; Rabin, 1993). Trust, on the other hand, is modeled as a subjective probabilistic belief about the trustworthiness of others (Ashraf et al., 2006; Buchan et al., 2008; Eckel and Wilson, 2004; Fetchenhauer and Dunning, 2009).

    We thank the editor and two anonymous referees for many helpful comments and suggestions. The authors declare that they have no relevant or material financial interests that relate to the research described in this paper.

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