Mental equilibrium and strategic emotions

Mental Equilibrium and Strategic Emotions∗ Eyal Winter, Luciano M´endez-Naya, Ignacio Garc´ıa-Jurado Abstract We model mental states as part of an equilibrium notion. In a mental equilibrium each player “selects” an emotional state that determines the player’s preferences over the outcomes of the game. These preferences typically differ from the players’ material preferences. The emotional states interact to play a Nash equilibrium and in addition each player’s mental state must be a best response to the mental states of the others (in the sense of maximazing material payoffs). We discuss the concept behind the definition of mental equilibrium and examine it in the context of some of the most popular games discussed in the experimental economics literature. In particular our approach allows us to identify the mental states (the psychology) that lead players to play various prominent experimental outcomes. We provide necessary and sufficient conditions for mental equilibria to be sustained by material preferences. Finally, we discuss the concept of collective emotions, which is based on the idea that players can coordinate their mental states. Keywords: Games, Equilibrium, Behavioral Economics, Emotions 1 Introduction Over the past three decades several interesting and important models have been developed to reconcile the discrepancy between experimental economic results and game-theoretic predictions, without neglecting the idea that ∗The authors wish to thank Itai Arieli, Ken Binmore, Werner Gueth, Sergiu Hart, Eric Maskin, Assaf Romm, Reinhard Selten, and Jean Tirole for their comments and suggestions on an earlier draft of this paper. We also thank audiences at Bocconi, Copenhagen, Harvard, Johns Hopkins, Max Planck in Jenna, Northwestern, Michigan, Minnesota, Paris School of Economics, Tel Aviv, UBC, UCLA, Wisconsin, The Behavioral Game Theory Workshop at Stony Brook, and the Fifth International Meeting on Experimental and Behavioral Economics in Granada for numerous suggestions and comments. Ignacio Garc´ıaJurado acknowledges the financial support of Ministerio de Econom´ıa y Competitividad through projects MTM2011-27731-C03 and MTM2014-53395-C3-1-P. 1 players behave strategically. The common objective of these models is to re-evaluate the outcomes of the game for each player, while taking into account non-pecuniary factors such as inequality aversion, spitefulness, and envy, so that in the new set of utility functions the equilibrium behavior is closer to the experimental observations (see Fehr and Schmidt 1999, and Bolton and Ochenfels 2000). The main challenge of this strand of literature is to identify the set of parameters that best explains the experimental results and to use these parameters to understand players’ motives in the underlying games. A somewhat different approach was proposed by Rabin (1993) with the concept of fairness equilibrium. Here the material payoffs are also altered to incorporate fairness into the utility function. However, the measure of fairness depends on the players’ actions and beliefs, which are determined in equilibrium. In this paper we attempt to take an endogenous approach to non-monetary preferences by allowing these preferences to arise endogenously from equilibrium conditions. Immaterial preferences often tend to be contextual and even tailored to the underlying strategic environment. Rustichini and Villeval (2014) demonstrate that individuals’ fairness judgments can be strongly affected by their bargaining positions. Meshulam, Winter, and Ben Shachar (2012) show how anger can be “synthesized” when agents have monetary incentives to become angry. Gneezy and Imas (2013) further show that sometimes subjects are aware of the effects of emotions, such as anger, on strategic interactions and try to gain a strategic advantage by manipulating these emotions. In this paper we attempt to investigate theoretically the role of such strategic emotions. Our main focus here is on the role of commitment in strategic emotions. We shall use the term mental state to represent these non-monetary preferences and embed them in an equilibrium concept called mental equilibrium. Much of the focus of our analysis will be on deriving players’ behavioral preferences through the equilibrium conditions. The linkage we have here between mental preferences and emotions requires some discussion. Mental preferences can arise from a variety of social and psychological generators. We use “emotions” in this paper as a general term for the mechanisms that generate mental preferences, including attitudes like fairness and reciprocity1. The term emotion is also meant to capture two additional im1Substantial evidence in neuroscience (starting with Damasio 1994) indicates that at2 portant aspects of mental preferences in our model: their role in securing commitment and the fact that they need to be transparent (at least to some degree). As we shall argue later, an emotional reaction, say, anger, allows players who experience it to credibly commit to retaliate against the cause of anger even when such retaliation is costly and therefore materially irrational. Furthermore, emotional reactions transmit social cues and signals to others and they are therefore, to a certain extent, transparent. Hence, they can be used as a commitment device. The concept of mental equilibrium can be described as follows. Each player, who we assume seeks to maximize only his material/monetary payoffs, is assigned a mental state. A mental state is simply a utility function over the outcomes of the game (i.e., the set of strategy profiles) that is typically different from the material utility function. A strategy profile sof the game is said to be a mental equilibrium if two conditions hold: firstly, shas to be a Nash equilibrium with respect to players’ mental states. Secondly, each player’s mental state is a best response to the mental states of the other players, given his material and selfish preferences. We offer two valid interpretations of our equilibrium concept. The first involves the idea of the evolution of norms and emotions. Essential to our model is the fact that the benchmark preferences of a player are selfish and material. It is not unreasonable to assume that human emotions like anger, envy, and revenge, which play a role in many interactive situations, have been developed through an evolutionary process to increase individuals’ fitness to the social environment in which they live. Our equilibrium concept can be viewed as a theoretical foundation for this feature. We are not proposing any specific evolutionary model to this effect, but, conceptually, mental equilibrium can be viewed as a stability concept arising from an evolutionary process.2Evolutionary selection reinforces different mental states in different strategic environments, and material payoffs in the game can be viewed as a measure of fitness. This interpretation is in line with the indirect evolutionary approach proposed by Gueth and Yaari (1992). The second interpretation of our equilibrium concept is that of rational emotions. In strategic environments individuals may be endowed with a titudes such as fairness and reciprocity can be related to emotional reactions that are processed in the pre-frontal cortex. 2Our equilibrium conditions are necessary but not sufficient conditions for evolutionary stability. 3 certain emotional state that serves their interest. Emotional states are often induced through cognitive reasoning whether in full or in partial awareness and they can serve as a commitment device. In order for the commitment to be credible, the emotional state has to be genuine and not feigned.3To illustrate this point we suggest a thought experiment that demonstrates how emotions are triggered by incentives. Imagine that you are informed at the airport that your flight has been canceled and that you should report to the airline desk the next day. Consider the following two scenarios: in scenario A you observe most of the passengers leaving the terminal quietly. In scenario B you run across an acquaintance who tells you that he was rerouted to a different flight after explaining to the airline employees, in a very assertive and determined manner, that he had to arrive at his destination that day. In scenario B you are most likely to find yourself in a very different emotional state from the one in which you would have been in scenario A. You are likely to exhibit signs of anger quite quickly in scenario B; in fact, these won’t be mere signs, you will actually be angry. You have been offered incentives to be angry and as a consequence you “choose” to be angry. The example above suggests that in certain environments mental states can be thought of as outcomes of a cognitive choice. We refer the reader to an experimental testing of rational emotions by Winter et al. (2009), which shows that the objective emotional reactions of receivers in a dictator game strongly depend on the presence of incentives. Under the interpretation of rational emotions one can think of mental equilibrium as an equilibrium in an amended game of credible commitments. The material payoffs here are standard payoffs in a game and not a measure of evolutionary fitness. The two interpretations we propose are very distinct. The evolutionary interpretation fits emotions that are global and robust, while under the rational emotions interpretation they can be specific and fragile. However, we shall be subscribing to both interpretations and shall not argue in favor of one or the other as we believe that the appeal of each of these interpretations is context-dependent. In particular, in explaining the foundation of conventions and norms in games vaguely defined and robust to whether players can see each other or not, the evolutionary approach seems more appropri3A considerable body of recent work in the psychology literature discusses the conscious control and regulation of emotions (see Demasio et al. 2000, Ochsner and Gross 2005) Tice and Bratslavsky (2000) suggest specific types of emotion control tasks (such as “getting into” and “getting out of” emotions) and discuss their regulation strategies. 4 ate (most “blind” experiments fall under this category). On the other hand, the rational emotions interpretation might be more relevant to situations that rely on mutual eye contact and are strongly responsive to incentives. We point out that the distinction between the two interpretations is akin to the distinction made by Aumann (2009) between rule rationality and act rationality. In both interpretations, however, we view emotions as a mechanism to promote self-interests. Our concept of mental equilibrium can also be viewed as a model of endogenous preferences. Players in our model select their preferences in view of their beliefs about the preferences of those with whom they interact. The remarkable feature of this concept is that while the choice of preferences is made from a self-centered point of view, the equilibrium choice of preferences may give rise to non-trivial social preferences in which the players’ behavior is very far from that of a self-centered player. Because we wish to fully endogenize non-monetary preferences we adopt a framework that puts no constraints on the set of potential preferences. An important part of our analysis is to identify “material” games, i.e., games in which all mental equilibria can be sustained by material preferences only. We show that all zero-sum games are material and we characterize the entire class of material games using the concept of “Stackelberg strategies” that appear in the literature on repeated games and reputation (e.g., Mailath and Samuelson 2006). An equilibrium outcome of a game in our model involves a combination of a vector of mental preferences and a strategy profile. Most of our attention in this paper will be given to the endogenous preferences rather than to the strategy profiles. While the set of strategy profiles supported by a mental equilibrium is very large, the set of mental preferences supporting a given profile are much more structured and informative. Hence our approach is particularly useful as a contribution to understanding the underlying motives of players’ strategies. Our approach allows us to take a strategy profile that is played frequently in laboratory or fields experiments and identify the mental preferences that support it. Put differently, it helps us reveal the psychology behind various prominent outcomes that emerge from the empirical data. This, we believe, is the most important advantage of our concept that other behavioral concepts lack as they treat preferences exogenously or assume specific functional forms. We demonstrate this point in Section 3 by 5 showing that for the class of all conflict games (which includes the Prisoner’s Dilemma) only “reciprocal” mental preferences can support cooperation as a mental equilibrium. This result rules out the possibility that cooperation in the Prisoner’s Dilemma is driven by altruism or inequality aversion. An implicit assumption that is built into the definition of mental equilibrium is that players must have correct beliefs regarding other players’ mental states when playing a game.4This is a critical issue when trying to answer the question of how a mental equilibrium emerges. It is of lesser importance if we treat the concept of mental equilibrium as a static stability concept (just like the Nash equilibrium). Nevertheless, there are two grounds on which this assumption can be justified. Firstly, a player’s choice of mental states involves some sort of pre-play communication game that we intentionally leave unspecified. A player signals his mental state in this game through body language, facial expression, intonation, and other actions. One cannot exclude deception, but it makes sense to assume that while our ability to identify the mental state of the other is imperfect, our ability to deceive is imperfect as well. In Section 8 we bring some empirical evidence to this effect and analyze a model of noisy detection of mental states. But even without direct eye contact players may still form consistent beliefs about the mental states of their counterparts. Just as with the learning literature that explains how consistent beliefs leading to Nash equilibrium emerge, it is conceivable that various dynamic models exist that converges to consistent beliefs about mental states. While interesting and important in themselves, these learning models are beyond the scope of this paper. In addition to its relation to the literature on social preferences discussed above, our work is related to two other strands of literature. The first is the literature on delegation pioneered by Fershtman, Judd, and Kalai (1990). This paper discusses strategic environments in which players can choose delegates to play a game on their behalf. By setting up the incentives to delegates properly, players can support strategic outcomes that are not standard Nash equilibria (see also Fershtman and Kalai, 1997, and Bester and 4If mental preferences and actions are chosen simultaneously, so that players have strategic uncertainty about the mental states, the set of mental equilibria will boil down to be the set of Nash equilibria of the game. This is because mental preferences would lose their role as a commitment devise if the other players cannot monitor them. However, as we show in Section 8 it is enough to have informative but noisy signals about the mental preferences to facilitate the role of mental states as a commitment device. 6 Sakovic, 2001). The second strand of literature concerns papers that discuss the evolutionary foundation of preferences. Gueth and Yaari (1992) introduced a game of cooperation between two players and showed how preferences for cooperation (which in their model boils down to being the value of a parameter in the utility function) can emerge through evolution (see also Gueth and Kliemt, 1999). This approach, known as the indirect evolutionary approach, was also present in Dekel et al. (2007), who develop a more general model than that in Gueth and Yaari (1992). Dekel et al. (2007) study the evolution of preferences using a notion of evolutionary stability that is much more stringent than Nash equilibrium. Both our model and Dekel et al.’s deal with endogenous preferences and both make the distinction between objective/material preferences and subjective/mental preferences. Moreover, both models adopt the approach by which the stability conditions imposed on subjective preferences intend to maximize objective outcomes. Thus, the two models are very similar. However, the two papers differ in many respects, both conceptually and technically. First, Dekel et al.’s results deal only with two-person symmetric games because their model relies on pairwise random matching within a single population. Since our main interest lies in the study of strategic preferences (and not in evolution), we impose simple Nash equilibrium conditions on preferences. This allow us to treat the entire class of normal form games for any finite number of players. However, the main difference between Dekel et al. (2007) and our work lies in the results and their implications. While their main interest is to characterize the equilibrium outcomes that survive their evolutionary process, our focus is on the endogenous preferences. Most our results are directed at identifying the mental preferences that support specific equilibrium outcomes, a direction on which Dekel et al. (2007) are almost silent. Several other papers use the indirect evolutionary approach in specific economic environments, such as Bergman and Bergman (2000) in the context of bargaining, Gueth and Ockenfels (2001) in the context of legal institutions, and Fershtman and Heifetz (2006) in the context of elections and political competition. Our paper departs from the two strands of literature discussed above in terms of motivation, interpretation, and formal modeling. Our objective is to study the role of mental states in strategic decision-making. Accordingly, much of our attention will be given to identifying the mental states that support specific strategic outcomes, mainly those which arise in 7 laboratory experiments. We shall show that our results are consistent with some prominent experimental results that cannot be explained by standard game-theoretic concepts. In terms of formal modeling our model differs from those used in the literature discussed above. It is formulated as a general equilibrium concept of normal form games with an arbitrary number of players. In contrast to other papers using the indirect evolutionary approach we do not impose evolutionary conditions for stability. Instead, our model involves two levels of equilibrium conditions. One level involves the mental game in which the payoffs are derived from players’ mental states, and the other level involves the selection of players’ mental states to maximize material preferences. At each of these levels agents are assumed to play Nash equilibria. As a consequence of the fact that the Nash equilibrium conditions for the selection of mental states are less stringent than Dekel et al.’s (2007) evolutionary conditions, our set of mental equilibria is typically larger than the set of stable outcomes `a la Dekel et al. (2007) and other related papers, and our model admits a mental equilibrium for any game. Finally, we expand the scope of applications by defining mental equilibrium variants to other solution concepts (beyond Nash equilibrium), including subgame perfect equilibrium and strong Nash equilibrium. In Section 2 we continue with the formal definition of mental equilibrium. In Section 3 we provide a useful characterization of mental equilibria in two-person games, which we later use to study mental equilibria in some prominent games for which experimental results have been accumulated. We then reflect on the mental states that support cooperation in a class of cooperation games that include the Prisoner’s Dilemma. We show that reciprocity-seeking preferences are both necessary and sufficient to sustain cooperation in a mental equilibrium for these games. Section 4 deals with “material” games. In this section we provide a characterization of those twoperson games in which all mental equilibria can be supported by material preferences. Section 5 is devoted to discussing the role of mental equilibrium in two games that have been prominently discussed in the experimental economics literature: the Trust game and the Ultimatum game. In Section 6 we deal with an alternative definition of mental equilibrium for n-person games. This amendment is motivated by showing that for games with four or more players the standard concept of mental equilibria loses its predictive power, since any strategy profile in such games is a mental 8 equilibrium. This follows from the fact that for some choices of mental states by the players the corresponding mental game may possess no pure Nash equilibria. We study properties of this amended concept and apply it to the game of voluntary contributions (the n-person Prisoner’s Dilemma). We show that in a mental equilibrium (according to the new definition) either no one contributes or the set of contributors is sufficiently large. These equilibria are supported by very intuitive mental states in which players experience substantial disutility when they contribute alone or together with a small group of contributors. In Section 7 we discuss collective emotions. These emotions emerge when a group of players coordinate their mental state to enhance the rational role of emotions as a commitment device. Our definition and analysis here builds on Aumann’s (1959) notion of strong equilibrium. Strong mental equilibrium, which is our main concept here, uniquely selects cooperation in the Prisoner’s Dilemma, quite unlike anything else in the plethora of gametheoretic solution concepts. Section 8 studies a variant of mental equilibrium that builds on the idea that the detection of others’ mental states is imperfect. We discuss this concept in the context of the famous TV game “Split or Steal,” which is a variant of the Prisoner’s Dilemma. We refer to the empirical observation of “mind-reading” according to which pre-play communication results in correlated actions in the Prisoner’s Dilemma, and show that our concept of mental equilibrium can explain this correlation. Section 9 concludes. 2 Basic Definitions Let G= (N, S, U) be a normal form game where Nis the set of players, S=S1×S2×. . . ×Snis the set of strategy profiles for the players, and U= (U1, ..., Un) are the players’ utility functions over strategy profiles. We refer to Uias the benchmark (selfish/material) utility function of the players and use uito represent the mental states’ utility functions. A profile of mental states is denoted by u= (u1, ...., un). For a given game Gwe denote by NE(G) the set of Nash equilibria of the game G. Definition 1 Amental equilibrium of the game G= (N, S, U)is a strategy profile s∈Ssuch that for some profile of mental states uthe following two conditions are satisfied: 9 Observation 4 has the important implication that players’ mental states must have the reciprocity-seeking property to sustain cooperation (generically) in any Prisoner’s Dilemma game. This is an important insight that cannot be derived from standard game-theoretic solution concepts. To elaborate on this point, we shall consider here two alternative types of mental preferences – the first involving altruism and the second based on inequality aversion – to demonstrate that none of these can explain cooperation at least for some Prisoner’s Dilemma games. Starting with altruism, consider the Prisoner’s Dilemma given by: D C D 1,1 5,0 C 0,5 4,4 We argue that mental preferences that sustain the cooperative outcome cannot be of the form ui=αiUi+βiUj. Based on the payoff function in our example above, these mental preferences result in the following mental game: D C Dα1+β1,α2+β25α1,5β2 C 5β1,5α24(α1+β1),4(α2+β2) For (C, C) to be an equilibrium in this mental game we need to have 4(α2+β2)≥5α2.This inequality implies 5β2≥α2+β2.But this means that player 1 is better off in terms of material payoffs if he adopts the mental state u1=U1.With such a mental state he will be able to sustain (D, C) as an equilibrium, which is the best possible outcome in terms of material payoffs. Note the difference between the preference given by ui=αiUi+βiUj and the one we used in Observation 4. The former represents a mental state with some degree of altruism (if βi>0) or spitefulness (if βi<0). By contrast, the mental preferences that we used to sustain (C, C) represent mental states for reciprocity-seeking behavior. These mental preferences sustain (C, C) regardless of the cardinal representation of the Prisoner’s Dilemma game. We next discuss inequality aversion (`a la Fehr and Schmidt 1999) and consider the following Prisoner’s Dilemma game: 16 D C D 35,50 45,45 C 30,65 40,60 We point out that an inequality-averse mental state of player 1 must satisfy u1(D, C)> u1(C, C). This is because (D, C) generates a greater (material) payoff to player 1, and involves a greater equality than the outcome (C, C). Hence, in light of our discussion above, there exists no profile of (inequality-averse) mental preferences that supports (C, C) as a mental equilibrium in this Prisoner’s Dilemma game. We conclude that reciprocity-seeking preferences can explain cooperation in every Prisoner’s Dilemma game, but altruism, spitefulness, or inequality aversion cannot. Example 2 Consider the finitely repeated Prisoner’s Dilemma of the following one-shot game: D C D1,1 10,0 C0,10 4,4 In this example we consider the one-shot Prisoner’s Dilemma above as well as the finitely repeated game. In the case of repetition we shall be referring to the game in its normal form and to the Nash equilibria and the mental equilibria of this normal form game. Observation 5 It is well known that (D,D) is the only profile played in a Nash equilibrium in both the one-shot game and the finitely repeated game. Furthermore, as established earlier, only (C,C) and (D,D) are the mental equilibria in the one-shot game. However, mental equilibria of the finitely repeated game (in its normal form) admit plays in which both (C,D) and (D,C) are played. One such mental equilibrium is for players to alternate between these two outcomes. Such an equilibrium exists in any repeated Prisoner’s Dilemma game where the total material payoff to the two players exceeds that of (D,D). Proof. The mental preferences supporting this equilibrium in our game above can be described as follows: (1) for the outcomes in which the two 17 players alternate between (C,D) and (D,C) the mental utility is identical to the material utility (and is 10k/2, with k being the even number of periods); (2) if only (C,C) or (D,D) are played along the path the mental and material preferences are again identical; (3) for any other profile (i.e., in which the two players choose different actions in the same period) the mental utility function yields a payoff of −1 to both players. 2 Interestingly, the mental preferences described above allow players in the repeated game to trade reciprocity within periods with reciprocity between periods. Substantial experimental evidence shows that players reciprocate by taking turns on their preferred outcomes in a variety of repeated interactions, including the repeated Prisoner’s Dilemma. Kaplan and Ruffle (2011) study a class of two-person entry games and show that when payoffs are sufficiently symmetric players tend to cooperate by means of turn taking and alternate between (C,D) and (D,C). Sibly, Tisdel, and Evans (2014) have also documented substantial turn-taking behavior in laboratory experiments of the repeated Prisoner’s Dilemma with and without cheap talks. Finally, Cason, Lau, and Mui (2013) also investigate the dynamics behind turn-taking behavior and show how it spreads through learning. 4 Material Games As we have seen, cooperation in the Prisoner’s Dilemma is sustained through mental states that represent reciprocity. Players are therefore required to depart from their material preferences in order to sustain cooperation as a mental equilibrium. In this sense the Prisoner’s Dilemma induces mental preferences that affect players’ behavior. Do all two-person games induce non-trivial mental preferences? Clearly games in which all mental equilibria can be supported by material preferences do not induce mental preferences that affect players’ behavior. We refer to such games as material games. In a material game all players can play according to their selfish and material payoffs in every mental equilibrium. It implies in particular that commitment plays no role in such games. In this section we shall characterize this class of games. Let G= (N, S, U) be a two-person game for which the following two vectors m, M ∈R2are well defined: 18 •mis the maxmin vector; i.e., mi= maxsi∈Siminsj∈SjUi(si, sj), where i, j ∈ {1,2},i6=j. •Mis the vector of Stackelberg values for the two players; i.e., it pays each player the maximal payoff under the assumption that the other player will best respond to his action. Formally, for i, j ∈ {1,2},i6=j, and all si∈Si, define Bj(si) = {sj∈Sj|Uj(si, sj)≥Uj(si,˜sj),∀˜sj∈Sj} and Mi= maxsi∈Simaxsj∈Bj(si)Ui(si, sj), where i, j ∈ {1,2},i6=j. Notice that mand Mare well defined in games with finite sets of strategies, and M≥m. Furthermore, M=mwhenever the game is zero-sum. We can now establish the following result. Proposition 3 Let G= (N, S, U)be a two-person game for which the two vectors m, M ∈R2are well defined. Then Gis a material game if and only if the following condition holds for every s∈S: U(s)≥m⇒U(s)≥Mand sis a Nash equilibrium of G.(1) Note that the condition specified in Proposition 3 applies to zero-sum games. Hence all zero-sum games are material. This is a rather intuitive observation. In zero-sum games commitments play no role whatsoever. If by committing himself to a certain mental state player 1 can get more than his maxmin value, this means that player 2 cannot guarantee his maxmin value, which is a contradiction. Let us see now the proof of Proposition 3. Proof. Let sbe a mental equilibrium. By Proposition 1 ssatisfies U(s)≥ m, and thus by (1) sis a Nash equilibrium with respect to the mental preferences ugiven by u=U. To show that u=Usupports sas a mental equilibrium consider a deviation by one player to an alternative mental state, say u0 i. Let s0be an equilibrium of the resulting mental game. By assumption Ui(s)≥Mi≥Ui(s0). So u=Usatisfies the second condition of the definition of mental equilibrium. Hence, all mental equilibria of Gare supported by material preferences. Conversely, assume that (1) does not 19 hold. Then, there exists s∈Swith U(s)≥mand such that U(s)6≥ Mor s is not a Nash equilibrium of G. Since U(s)≥m, it follows from Proposition 1 that sis a mental equilibrium. If it is not a Nash equilibrium then it obviously cannot be supported by material preferences. Assume by way of contradiction that sis a Nash equilibrium; then U(s)6≥ M, which means that Ui(s)< Mifor an i∈ {1,2}. Thus, scannot be supported by material preferences; to prove it, notice that ican choose mental state u0 igiven by •u0 i(ˆsi, s0 j) = 1, for all s0 j∈Sj, if maxsj∈Bj(ˆsi)Ui(ˆsi, sj) = Mi, •u0 i(s0 i, s0 j) = 0, for all s0 j∈Sj, if maxsj∈Bj(s0 i)Ui(s0 i, sj)< Mi. Clearly (N, S, (u0 i, Uj)) has a Nash equilibrium offering ia payoff of Mi> Ui(s). 2 5 Mental Equilibrium in Trust and Ultimatum Games We shall now discuss the role of mental equilibrium in two games that are prominently discussed in the experimental economics literature: the Trust game and the Ultimatum game. Example 3 The Trust game. A large body of experimental data has helped us understand the Trust game since the publication of Berg, Dickhaut, and McCabe (1995). In its most standard form the game can be described as follows. Player 1 has an endowment of x. He can transfer 0≤y≤xto player 2. If player 1 transfers y, player 2 receives 3y. Player 2 can now reward player 1 with a transfer of z≤3y. Finally, the payoff to player 1 is x−y+zand the payoff to player 2 is 3y−z. Observation 6 An outcome (a1, a2)is a mental equilibrium outcome if and only if a1≥xand a2≥0. Proof. Consider such an outcome (a1, a2). Since a1≥xplayer 2 can guarantee that player 1 gets no more than a1. This can be done by transferring no money back to player 1 if player 2 received any money from player 1. Furthermore, it is clear that player 1 can guarantee that player 2 receives no more than zero by simply making a zero transfer to player 2. In view of 20 Proposition 1, (a1, a2) is a mental equilibrium outcome. Consider a mental equilibrium outcome (a1, a2) such that either a1< x or a2<0. Then either player 1 or player 2 gets less than the maxmin value, which contradicts Proposition 1. 2 We note that the Trust game has a unique Nash equilibrium in which player 1 makes a zero transfer to player 2. Observation 6 suggests that any level of trust displayed by player 1 coupled with a level of trustworthiness that compensates player 1 for at least the level of his initial endowment can be supported by mental equilibria. We point out that experimental results support a considerable level of trust by player 1 and a considerable reciprocity by player 2 (see, e.g., Berg, Dickhaut, and McCabe 1995). We now discuss our concept of mental equilibrium in the context of another prominent game, the Ultimatum game. Example 4 The Ultimatum game. The game involves two players. Player 1 has an endowment of 1from which he has to make an offer to player 2. An offer is a number 0≤y≤1.Player 2 can either accept the offer or reject it. If player 2 accepts the offer player 1 receives 1−yand player 2 receives y. If player 2 rejects the offer both players receive a payoff of zero. The subgame perfect equilibrium of the game predicts a zero offer by player 1, which is accepted by player 2. A massive amount of experimental evidence starting with Gueth et al. (1982) has shown, however, that player 1 makes substantial offers, with the mode of the distribution being (0.5,0.5). To discuss the concept of mental equilibrium for this game we first introduce the subgame perfect version of mental equilibrium. Consider an extensive form game G= (T, U), where Tis the game form defined by a tree, and U= (U1, . . . , Un) are the payoff functions for players in N={1, . . . , n} assigning a payoff vector to each terminal node of the game. We denote by SPE(G) the set of subgame perfect equilibria of the game G. Definition 3 Amental subgame perfect equilibrium of the game Gis a strategy profile sof Gsuch that for some profile of mental states uthe following two conditions are satisfied: 1. s∈SPE(T, u). 21 2. There exist no player i, mental state u0 i, and strategy profile s0∈ SPE(T, u0 i, u−i)with Ui(s0)> Ui(s). Observation 7 Let Gbe an extensive form game. Every Nash equilibrium outcome of Gis a mental subgame perfect equilibrium outcome of G. Proof. Let sbe a Nash equilibrium of G. We construct the following mental (extensive form) game. For player i∈Nchoose a mental state uiin the following manner: for each terminal node dof the game, ui(d) = 1 if and only if the path leading to dincludes a decision node where player iplays according to the strategy si; in any other case, ui(d) = 0. It is clear that sconstitutes a subgame perfect equilibrium in the mental game based on the profile of mental states u. It is left to show that no player can unilaterally change his mental state in such a way that the new mental game will possess a subgame perfect equilibrium with a higher material payoff for this player. Suppose by way of contradiction that such a mental state u0 iexists, and take s0∈SPE(T, u0 i, u−i) with Ui(s0)> Ui(s).From the definition of uit is clear that s0 j=sjin all the information sets in the path of s0(for all j6=i). Hence, Ui(s0) = Ui(s0 i, s−i). But this cannot happen since sis a Nash equilibrium of G.2 Subgame perfect equilibria are often described as Nash equilibria with credible threats. Mental equilibria are, in some sense, based on commitment and, thus, can turn non-credible threats into credible ones. This is the basic insight of Observation 7. Returning to the Ultimatum game, it is easy to check that every distribution (y, 1−y) can be supported by a Nash equilibrium. Hence, in view of Observation 7, it holds that all allocations of an Ultimatum game are supported by a mental subgame perfect equilibrium. Let us characterize the corresponding mental states. Observation 8 Consider the outcome in which player 1 gets xand player 2 gets (1 −x). Then the following conditions are both necessary and sufficient for a pair of mental states u= (u1, u2)to sustain this outcome as a mental subgame perfect equilibrium. 1. If u1(y, 1−y)≥u1(0,0) then y≥x, and if u2(y, 1−y)≥u2(0,0) then x≥y. 22 2. u1(x, 1−x)≥u1(0,0) and u2(x, 1−x)≥u2(0,0). Proof. First we claim that under any pair of mental states satisfying the above conditions (x, 1−x) is supported by a subgame perfect equilibrium of the mental game. Note that, by condition 1, any offer ywith y > x will satisfy that u2(0,0) > u2(y, 1−y) and will be rejected by player 2. Adding condition 2 we get that (x, 1−x) is the most preferred offer for player 1 among those which are acceptable for player 2. Otherwise, if for some (z, 1−z) with z < x,u1(z, 1−z)≥u1(x, 1−x), then we have u1(0,0) > u1(z, 1−z)≥u1(x, 1−x)≥u1(0,0), which is a contradiction. Since both players prefer the outcome (x, 1−x) to the outcome (0,0) this must be a unique subgame perfect equilibrium outcome under (u1, u2). Note now that no player can deviate to a different mental state and increase his material payoff in subgame perfect equilibrium since no such an outcome is acceptable for the other player on the basis of the mental states (u1, u2). Hence, the two conditions are sufficient. We now show that they are also necessary. Suppose that condition 1 fails to hold with respect to player 1; then there exists an outcome (y, 1−y) with y < x such that u1(y, 1−y)≥u1(0,0). Consider the mental state u0 2of player 2 such that u0 2(y, 1−y)> u0 2(0,0) and u0 2(z, 1−z)< u0 2(0,0) for z6=y(i.e., player 2 is willing to accept only (y, 1−y)). Clearly (y, 1−y) is a subgame perfect equilibrium outcome of the mental game based on (u1, u0 2) and player 2’s material payoff has increased, which is a contradiction. Suppose now that condition 1 is violated with respect to player 2, i.e. that for some (y, 1−y) with y > x it holds that u2(y, 1−y)≥u2(0,0). Consider now a deviation of player 1 to the mental state u0 1such that u0 1(y, 1−y)≥u0 1(z, 1−z) for any distribution (z, 1−z). In that case (y, 1−y) is a subgame perfect equilibrium outcome of the mental game based on (u0 1, u2). Since this outcome increases player 1’s material payoff we again reach a contradiction. Finally, note that the necessity of condition 2 is immediate; if this condition is violated (x, 1−x) cannot be a is a subgame perfect equilibrium outcome under the mental states (u1, u2). 2 The insight provided by the observation above is that the mental preferences that sustain a given mental subgame perfect equilibrium outcome are characterized by the allocations that player doom as unacceptable; the way they compare two acceptable outcomes is irrelevant. 23 6n-Person Games Our model and results in this paper are based on pure strategies. 7The restriction to pure strategies implies that when a player deviates to a different mental state the resulting mental game may not possess (pure) Nash equilibria. This would rule out such deviations and may sustain a large set of artificial mental equilibria. We start this section by showing that the set of mental equilibria expands to the entire set of strategy profiles when the number of players is at least four. In order to restore the predictive power of the mental equilibrium concept we screen out these artificial mental equilibria. We shall thus introduce a minor amendment to the definition that allows for more deviations and hence for fewer equilibria. This amended definition coincides with our original definition for two-person games. We use the amended definition for our subsequent analysis in this section. We start by demonstrating the drawback of the original definition. Proposition 4 For every normal form game Gwith n≥4, every strategy profile is a mental equilibrium. Proof. For each player iwe select one strategy and denote it by 0. We denote the set of the remaining strategies by Tisuch that Si=Ti∪ {0}. We shall show that the profile (0,0, ..., 0) is a mental equilibrium. Since the strategy was selected arbitrarily it will show that every profile is a mental equilibrium. For a strategy profile s∈Swe denote d(s) = #{j∈N s.t. sj∈Tj}, i.e., the number of players choosing a strategy different from 0. For each integer kwe denote the parity of k(i.e., whether kis odd or even) by p(k). Consider now the following vector of mental states (u1, ..., un), where ui:S→ {0,1}: ui(0, ..., 0) = 1 for all i. For any strategy profile sdifferent from (0, ...., 0) we set ui(s) = 0 if and only if p(d(s)) = p(i). Otherwise ui(s)=1.We show that for any profile s6= (0, ...., 0), half of the players can profit by deviating.8 Indeed, each player who receives 0 can increase his payoff by changing his 7It turns out that mixed strategies pose a serious technical problem to our analysis, mainly for two reasons: (1) the space of mental states is a continuum and (2) f, the bestresponse correspondence at the level of the mental states, is highly non-convex, which makes existence hard to deal with. Olschewski and Swiatczak (2009), who build on our paper, address this issue for the case of 2 ×2 bimatrix games. 8This holds when nis even; if the number of players is odd, then at least n−1 2players will choose to deviate. 24 strategy from playing 0 to playing something else in Tior, if he is already playing a strategy in Ti, he should switch to playing 0. In so doing the deviator will trigger a new profile s0for which p(d(s0)) 6=p(i) and he will raise his own payoff from 0 to 1. To show that (0, ..., 0) is a mental equilibrium, first note that it is a Nash equilibrium with respect to the chosen mental states (u1, ..., un) as it globally maximizes the payoff to all players. Furthermore, if player ideviates and sends a different mental state u0 ihe will not be able to sustain a better equilibrium because the corresponding mental game will have no equilibrium different from (0, ..., 0). Regardless of what mental state player iplays, there will be at least one other mental state j6=ithat deviates. 2 As pointed out earlier, the main reason why a mental equilibrium supports all strategy profiles in games of four or more players is that it does not permit deviations that induce mental games without pure Nash equilibria. This requirement seems rather demanding. If a player can deviate to a mental state that guarantees him a higher material payoff regardless of what other players are doing, then the lack of a pure Nash equilibrium is not essential. Indeed, while the deviating player cannot have consistent beliefs about what the other players will do, he can realize that no matter what the others do he will be better off. Building on this insight we propose a minor amendment to the original definition, which will resolve the excessive multiplicity described in Proposition 4. Definition 4 Amental equilibrium of the game G= (N, S, U)is a strategy profile s∈Ssuch that for some profile of mental states uthe following two conditions are satisfied: 1. s∈NE(N, S, u). 2. There do not exist a player iand a mental state u0 isuch that (a) for a strategy profile s0∈NE(N, S, u0 i, u−i)it holds that Ui(s0)> Ui(s), or (b) ihas a dominant strategy s0 iin the game (N, S, u0 i, u−i)with Ui(s0 i, s0 −i)> Ui(s)for all s0 −i. In the sequel we shall use the amended definition presented here. In the Appendix we show that (1) a mental equilibrium (as defined here) always 25 mental equilibrium. Bachi, Ghosh, and Neeman (2013) introduce a related model that is based on a direct commitment on actions and not on mental states. We restrict the set of mental states in this section to include only two elements: {Ra, Re}.Ra (Rational) refers to a mental state that represents self-interest. Under Ra the player chooses D regardless of the signal he receives. A player endowed with Re (Reciprocal) chooses C if and only if he received a signal of Re from the other player. The interaction involves the following three stages: •In stage 1 players choose a mental state out of two possible mental states. These choices are potentially mixed. •In stage 2 a signal that reveals the mental state of player ito player jis generated. The signal involves a probability 1 −pof error; i.e., if player ichooses Re player jreceives the signal Re with probability p and the signal Ra with probability 1 −p. Likewise, if he chooses Ra the signal is Ra only with probability p. •In stage 3 the players choose an action C or D in the Prisoner’s Dilemma game. Equilibrium is now defined as follows via the following two conditions. 1. Actions taken in the mental game (after the choice of the mental states) form a Bayesian equilibrium. 2. Given the equilibrium expected in the mental game the choice of mental states forms a Nash equilibrium with respect to players’ material preferences. To avoid occurrence of multiple equilibria that arises from higher-order beliefs, we shall assume that an Re type who believes he is facing an Re type chooses C. Under these conditions, the normal form game played by the players is the following one: Ra Re Ra 1,1p+ 5(1 −p), p −4(1 −p) Re p−4(1 −p), p+ 5(1 −p) 4p2+ 5p(1 −p)−4p(1 −p) + (1 −p)2, 4p2+ 5p(1 −p)−4p(1 −p) + (1 −p)2 32 Simplifying the expressions in the payoff matrix we get:. Ra Re Ra 1,1 5 −4p, 5−4p Re 5 −4p, 5−4p4p2−p+ 1,4p2−p+ 1 It is easy to check that this game has a symmetric and totally mixed equilibrium given by (4p2+ 3p−4 4p2−2p+ 1,5(1 −p) 4p2−2p+ 1) provided that p∈(−3+√73 8,1). This equilibrium induces a probability distribution on the strategy profiles of the Prisoner’s Dilemma. This distribution generically involves correlation between the players’ actions. If, for example, p= 4/5,then the equilibrium is (24 49,25 49) and the corresponding distribution is D C D 0.167 0.0916 C 0.0916 0.65 and note that similarly to the empirical results in Kalay et al. (2003) this distribution over strategy profiles displays a significant correlation in players’ actions. 9 Discussion In his treatise Politics Aristotle makes the following observation about the emotion of anger: “Anyone can become angry—that is easy. But to be angry with the right person, to the right degree, at the right time, for the right purpose, and in the right way; this is not easy.” Anger, just like many other emotions, is an important component of strategic decision-making. In this paper we attempted to introduce a formal framework in which discuss the role of emotions in strategic interactions using the concept of mental equilibrium. Two promising directions seem to suggest themselves at this stage. 1. The role of mental states in sequential interactions. Our concept was mainly applied to normal form games although we have also proposed 33 the concept of mental subgame perfect equilibrium. However, it would be interesting to investigate the role of a new concept in which players can change their mental state during the course of the game. To capture the idea that mental states have a certain degree of persistence one can, for example, require that players commit to a mental state for a duration of k periods or, alternatively, that players have the same mental states in every two subgames that are isomorphic. It would indeed be interesting to examine such a model in the context of sequential bargaining. 2. Emotions often trigger values and norms. One can often think of social norms as mental states that apply to a class of games. Put differently, norms may arise from having players commit themselves to the same mental state over multiple, possibly similar, games. This brings us back to Aumann’s (2008) insight about the difference between “rule rationality” and “act rationality.” Our concept of mental equilibrium can lend itself to a formal model of rule rationality. Roughly, in a rule-rational equilibrium players are restricted to a small number of mental states, but they allocate these mental states to different games in a way that is “globally” optimal relative to some distribution of the occurrence of these games over the course of a life. The fact that players cannot freely change their mental state from one game to another facilitates the commitment device that can work in favor of their own material interests. 10 Appendix In order to prevent ambiguities, in this appendix we call r-mental equilibrium to the version of mental equilibrium in Definition 4. Proposition 5 Let G= (N, S, U)be an n-person game and take s∈Sto be an r-mental equilibrium of G. Then Ui(s)≥mifor all i∈N. Proof. Take s∈Swith Ui(s)< mifor some i∈Nand assume that there exists a profile uof mental states supporting sas an r-mental equilibrium. Take now s0 i∈Sito be a maxmin strategy of iand u0 ito be a mental state according to which playing s0 iis a dominant strategy for i. Then, since Ui(s0 i, s0 −i)≥mi> Ui(s) 34 for all s0 −i,udoes not support sas an r-mental equilibrium, which is a contradiction. 2 Proposition 6 Let G= (N, S, U)be an n-person game and take s∈Sto be a mental equilibrium of Gsuch that Ui(s)≥mifor all i∈N. Then sis an r-mental equilibrium of G. Proof. Take a profile uof mental states supporting sas a mental equilibrium. If udoes not support sas an r-mental equilibrium then there must exist i∈Nand s0 i∈Siwith Ui(s0 i, s0 −i)> Ui(s) for all s0 −i, but this clearly implies that Ui(s)< mi, which is impossible. 2 Corollary 2 Let G= (N, S, U)be an n-person game with n= 2. Then, s∈Sis an r-mental equilibrium if and only if Ui(s)≥mifor all i∈N. Proof. Proposition 5 shows the only if part. To prove the if statement, take into account that in two-person games every profile whose associated payoff vector is greater than or equal to the maxmin vector is a mental equilibrium; using this and Proposition 6, the if statement follows. 2 Corollary 2 and Proposition 1 imply that in two-person games mental equilibrium and r-mental equilibrium are equivalent concepts. Corollary 3 Let G= (N, S, U)be an n-person game with n≥4. Then, s∈Sis an r-mental equilibrium if and only if Ui(s)≥mifor all i∈N. Proof. Proposition 5 shows the only if part. To prove the if statement take into account that in n-person games with n≥4 every profile is a mental equilibrium; using this and Proposition 6, the if statement follows. 2 The case n= 3 is a singular case. The following example shows a threeperson games with a profile that is not an r-mental equilibrium in spite of the fact that its associated payoff vector is greater than or equal to the maxmin vector. 35 Example 6 Consider the following three-person game. a a b a 0,0,0 1,1,1 b 1,1,1 1,1,1 b a b a 1,1,1 1,0,1 b 0,1,1 1,1,0 Clearly m= (0,0,0) and thus U(a, a, a)≥m. However, (a, a, a)is not a mental equilibrium (and hence not r-mental). Suppose that it is. Then, there must exist a mental game satisfying that (i) for the strategy profiles (a, b, a), (b, a, a),(b, b, a),(a, a, b)there exist at least two players willing to deviate, and (ii) in (b, a, b)either player 1wants to deviate or players 2and 3want to deviate, and in (a, b, b)either player 2wants to deviate or players 1and 3want to deviate, and in (b, b, b)either player 3wants to deviate or players 1and 2want to deviate. It is an easy exercise to check that conditions (i) and (ii) cannot be simultaneously satisfied. In spite of this negative result, the following proposition shows that in threeperson games there always exists at least an r-mental equilibrium. Proposition 7 Every three-person game G= (N, S, U)has an r-mental equilibrium. Proof. The proof is constructive. First we define a strategy profile s∗and then we prove that it is an r-mental equilibrium. For every s3∈S3define S2(s3) as the set of maxmin strategies of player 2 in G(s3), the two-person game resulting from Gwhen we fix s3. Denote by ms3 2the maxmin payoff to player 2 in G(s3). Now, choose a strategy of player 2 in S2(s3) and denote it by s2(s3). Next, choose a best reply of player 1 to (s2(s3), s3) in Gand denote it by s1(s3). Finally, choose s∗ 3∈S3satisfying that U3(s1(s∗ 3), s2(s∗ 3), s∗ 3)≥U3(s1(s3), s2(s3), s3),for all s3∈S3. 36 Let us check that s∗:= (s1(s∗ 3), s2(s∗ 3), s∗ 3) is a mental equilibrium of G. Consider the mental game udefined below for every s∈S: u1(s) =      1 if s= (s1(s3), s2(s3), s3), 1 if s= (ˆs1,ˆs2, s3) with ˆs26=s2(s3), and U2(ˆs1,ˆs2, s3)≤ms3 2, 0 otherwise. u2(s) = (1 if s2=s2(s3), 0 otherwise. u3(s) = (1 if s3=s∗ 3, 0 otherwise. Obviously, s∗∈NE(N, S, u). •Player 1 cannot gain by choosing a different u0 1because players 2 and 3 are going to play (s2(s∗ 3), s∗ 3) in (N, S, u0 1, u2, u3) and then player 1 will also play s1(s∗ 3). •Player 2 cannot gain by choosing a different u0 2because in (N, S, u1, u0 2, u3) player 3 will play s∗ 3and player 1 will play ˆs1with U2(ˆs1,ˆs2, s∗ 3)≤ms∗ 3 2 for any ˆs26=s2(s∗ 3). Notice that ms∗ 3 2≤U2(s∗). •Player 3 cannot gain by choosing a different u0 3because in (N, S, u1, u2, u0 3), for any election s3of player 3, players 1 and 2 will play (s1(s3), s2(s3)) and, by definition of s∗ 3, player 3 will not gain more than U3(s∗). Hence, s∗is a mental equilibrium of G. 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