Evidential equilibria: Heuristics and biases in static games of complete information
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al-Nowaihi, Ali; Dhami, Sanjit Article Evidential equilibria: Heuristics and biases in static games of complete information Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: al-Nowaihi, Ali; Dhami, Sanjit (2015) : Evidential equilibria: Heuristics and biases in static games of complete information, Games, ISSN 2073-4336, MDPI, Basel, Vol. 6, Iss. 4, pp. 637-676, https://doi.org/10.3390/g6040637 This Version is available at: https://hdl.handle.net/10419/167963 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Games 2015,6, 637-676; doi:10.3390/g6040637 OPEN ACCESS games ISSN 2073-4336 www.mdpi.com/journal/games Article Evidential Equilibria: Heuristics and Biases in Static Games of Complete Information Ali al-Nowaihi and Sanjit Dhami ∗ Department of Economics, University Road, University of Leicester, Leicester LE1 7RH, UK; E-Mail: [email protected] *Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +44-116-2522086; Fax: +44-116-2522908. Academic Editors: Andrew M. Colman and Briony D. Pulford Received: 6 August 2015 / Accepted: 5 November 2015 / Published: 16 November 2015 Abstract: Standard equilibrium concepts in game theory find it difficult to explain the empirical evidence from a large number of static games, including the prisoners’ dilemma game, the hawk-dove game, voting games, public goods games and oligopoly games. Under uncertainty about what others will do in one-shot games, evidence suggests that people often use evidential reasoning (ER), i.e., they assign diagnostic significance to their own actions in forming beliefs about the actions of other like-minded players. This is best viewed as a heuristic or bias relative to the standard approach. We provide a formal theoretical framework that incorporates ER into static games by proposing evidential games and the relevant solution concept: evidential equilibrium (EE). We derive the relation between a Nash equilibrium and an EE. We illustrate these concepts in the context of the prisoners’ dilemma game. Keywords: evidential reasoning; game theory; cognitive bias; prisoners’ dilemma game; oligopoly games; conservative heuristics; radical heuristics; decision making JEL classifications: D03 (behavioural microeconomics: underlying principles); C7 (game theory and bargaining theory)
Games 2015,6638 1. Introduction A considerable body of evidence shows that the predictions of the standard equilibrium concepts in game theory are not borne out by a significant fraction of experimental subjects. See Camerer [1] for a book-length treatment. For prisoners’ dilemma games, see Lewis [2], Howard [3], Rapoport [4], Shafir and Tversky [5], Cooper et al. [6], Croson [7], Li and Taplin [8], Acevedo and Krueger [9], Busemeyer [10], Zhong et al. [11], Histrova and Grinberg [12] and Khadjavi and Lange [13]. For oligopoly games, see Fouraker and Siegel [14], Huck et al. [15], Bosch-Domènech and Vriend [16] and Duersch et al. [17]. For public goods games, see Dawes and Thaler [18], Fehr and Gächter [19] and Gächter and Thöni [20]. For voting games, see Quattrone and Tversky [21], Grafstein [22], Forsythe et al. [23], Rassenti et al. [24], Krueger and Acevedo [25], Koudenburg [26], Requate and Waichman [27] and Delavande and Manski [28]. For p-beauty contests, see Moulin [29], Nagel [30] and de Sousa et al. [31]. For auctions, see Ivanov [32]. For the hawk-dove game, see Rubinstein and Salant [33]. For the give-some game, see Krueger [34]. For further results on these, and other games, that present a challenge for both mainstream and behavioural game theory, see Lucas et al. [35]. In this paper, we are interested in static games of complete information. Examples include the prisoners’ dilemma, the voting game, the public goods game, the hawk-dove game and oligopoly games. Let us briefly note the nature of the violations of the standard equilibrium concepts in some static games of interest. A more detailed treatment is given in the main body of the paper. More than half the subjects in the prisoners’ dilemma game play the dominated action “cooperate”. Voters vote in elections when it is clear to them that they will not be pivotal. Under traditional preferences, if there is a cost to voting, the act of voting is dominated by not-voting. The dominant action in public goods games is to free-ride. Yet, we can elicit near first-best levels of contributions with like-minded players.1 The evidence from the publications cited above suggests the following stylized facts that any reasonable theory of static games may aspire to explain. We consider the empirical evidence behind these claims in more detail in the main body of the paper. S1. A significant fraction of players behave in a manner that is consistent with the predictions of classical game theory. For instance, many players defect in a prisoners’ dilemma game; many people abstain from voting; and many people do not contribute at all in public goods games. S2. An even larger fraction of players violates the predictions of classical game theory, and they often seem to behave in an apparently non-strategic manner. For instance, the action “cooperate” in a prisoner’s dilemma game is a dominated action, but these actions by players, jointly, lead to higher payoffs. These findings are fairly robust even in the other static games that we do not consider in this paper. S3. In environments where players think that they are playing with other like-minded players, evidence shows that they impute diagnostic significance to their own actions when forming beliefs about the actions of others. For instance, those who cooperate (respectively defect) in prisoners’ dilemma games think that the vast majority of other players will also cooperate (respectively defect). 1In Bayesian games, like-minded players are often used to denote the case where players share the same priors. However, in this paper we use like-mindedness in its more general psychological sense.
Games 2015,6639 Similarly, despite publicly-available information on election polls, those who vote Democrat (respectively Republican) in the U.S. Presidential elections believe that a significant majority of other voters will also vote Democrat (respectively Republican). Each of the static games that we consider has an exceedingly simple structure. Hence, we believe that the anomalies relative to the predictions of classical game theory are less likely to arise from mistakenly playing the incorrect action. For this reason, our focus is not on behavioural alternatives, such as quantal response equilibrium (QRE) in which players play a noisy best response, but otherwise have consistent beliefs. The main focus of our paper is on S3, which also allows us to shed light on S1 and S2. There is always considerable uncertainty about what others will do in one-shot games. Think of being in an experiment where you are playing a prisoner’s dilemma game, or a voting game, or a public goods game. How do you infer what the other players are likely to play? Considerable evidence, which we shall review later, suggests that people often use evidential reasoning (ER)2,i.e., they assign diagnostic significance to their own actions in forming beliefs about the actions of other like-minded players. We stress that a player using ER does not believe that his or her actions influence the actions of other players3. ER merely influences a player’s own belief about which unobserved action other players are likely to take. ER is best viewed as a heuristic or bias. In the early 1970s, Kahneman and Tversky proposed the heuristics and biases approach. A substantial literature developed subsequently that identified a rich range of heuristics and generated evidence that they are used by human subjects. See, for instance, Kahneman et al. [36] and Kahneman [37]. ER, like these heuristics, is fast and frugal in the use/processing of information and in cognitive requirements. As with all heuristics, players who use ER may find ex post that their initial beliefs were incorrect. ER can lead to the violation of some of the principles of classical decision theory, for example Savage’s sure thing principle (Savage [38]). However, these violations have been well documented. For example, the sure thing principle is violated in the Ellsberg paradox (Ellsberg [39]; which, however, is a non-game theoretic situation). Evidence supports the interpretation of evidential reasoning as a heuristic. People who use evidential reasoning are not aware of using it despite their behaviour being obviously consistent with evidential reasoning. Evidential reasoning appears to arise as an automatic, rather than a deliberate effort or intention, e.g., it does not require awareness. Evidence supporting this view comes from experiments that show that evidential reasoning was not hampered by cognitive load or time required to complete an action; see Krueger [40]. Furthermore, other evidence, also reported in Krueger [40], suggests that considerable cognitive effort is required to suspend evidential reasoning. The evidence from Acevedo and Krueger [9] indicates that evidential reasoning applies to human-human interaction, but not to human-non-human interaction. Players using evidential reasoning do not believe that their actions cause the action of others; it merely informs their belief about the actions taken by others. Another feature of evidential reasoning is that individuals continue to behave in a self-interested manner. 2Also known as social projection (see Krueger [34]). 3According to Krueger ([34], p. 291) “... people use their own choices to predict the choices of others and then select the strategy that is best for them”. Note well that Krueger uses the word predict, not cause.
Games 2015,6640 It might be useful to distinguish between to types of heuristics. The first involves no violation of the standard assumptions of game theory. They propose extra conditions that are consistent with the standard assumptions of game theory, but whose aim is to reduce the multiplicity of Nash equilibria. Examples include payoff dominance and risk dominance (Harsanyi and Selten [41]). Note that in some cases, these heuristics are in conflict (in some games, payoff-dominance could select one equilibrium, but risk-dominance could select another). In fact, the whole program of refinements of Nash (van Damme [42]) may be viewed in this light. We may call these conservative heuristics. The second type, which we may call radical heuristics, involve relaxation of some of the standard assumptions of game theory. Examples include Stackelberg reasoning (Colman and Bacharach [43]; Colman et al. [44]). Evidential reasoning is firmly in the group of radical heuristics. Section 2gives a formal treatment of evidential reasoning and proposes several concepts that we will find useful in the rest of the paper. An evidential game is simply a game where players use evidential reasoning. An evidential equilibrium is one where each player chooses to optimize given his beliefs about the behaviour of the other players (inferred from his own behaviour in accordance with evidential reasoning). A consistent evidential equilibrium is an evidential equilibrium where beliefs turn out to be correct. Our formulation of evidential reasoning yields causal reasoning, the mode of reasoning assumed in the traditional framework in economics (and, indeed, generally) as a special case. If players use causal reasoning, then a consistent evidential equilibrium corresponds to a Nash equilibrium in the ordinary sense. We introduce the concept of a social projection function (SPF) and give formal definitions of like-mindedness, ingroup and outgroup. Section 3argues that evidential reasoning is a useful heuristic rather than a valid method of inference. Sections 4–6show that evidential reasoning can answer the following questions: Why do people voluntarily contribute to public goods? Why do people vote? Why is there so much cooperation in the prisoners’ dilemma game? Section 7examines the uniqueness of outcomes under evidential reasoning in the context of the Nash demand game. Oligopoly games are considered in Section 8. In Section 9, we argue that causal reasoning cannot adequately explain cooperation in the prisoners’ dilemma game. Section 10 concludes. 2. Evidential Equilibrium in Static Games of Complete Information 2.1. Elements of Standard Game Theory4 Consider the following standard description of a static game of complete information, {N, A,π}. N={1,2, ..., n}is the set of players. Ai⊆Ris the set of actions open to player i. We denote a typical member of Aiby ai.A=×n i=1Aigives all possible action profiles of the players. A−i⊆Rn−1is the set of vectors of actions open to the other players. Denote by Sithe set of probability distributions over the set of actions Ai. We denote a typical element of Siby siand call it a strategy. si(ai)is the probability with which player iplays ai∈Ai, so si(ai)≥0and Pai∈Aisi(ai) = 1. In particular, if si(ai) = 1 (hence, si(a0 i)=0for a0 i6=ai), then we call sa pure strategy, and we identify it with the action ai. 4See, for example, Fudenberg and Tirole [45], Part I.
Games 2015,6641 A profile of strategies of all players is denoted by s= (s1, s2, ..., sn)∈S, where S=×n i=1Siis the set of all possible profiles of strategies. A particular profile of strategies of other players is denoted by s−i= (s1, ..., si−1, si+1, ..., sn)∈S−i=×j∈N−{i}Sj. The payoff to player iis a mapping πi:S→R. Let πbe the vector of payoffs to all players. Given a strategy profile, s= (si,s−i)∈S, the payoff to player iis πi(si,s−i)∈R. The structure of the game, {N, A, π}, is common knowledge among the players. In an experimental setup, common knowledge can be achieved by a public announcement of {N, A, π}. This is the sense in which this is a game of complete information. However, when each player, i, chooses his strategy, si, he does not know the strategies, s−i, that have been, or will be, chosen by the other players. This is the sense in which this is a static game. Definition 1. :s∗ i∈Siis a dominant strategy for player i∈Nif πis∗ i,s−i≥πi(si,s−i)for each si∈Siand each s−i∈S−i. If πis∗ i,s−i> πi(si,s−i)for each si∈Si−s∗ iand each s−i∈S−i, then s∗ iis a strictly dominant strategy for player i. Definition 2. (Nash [46,47]): A strategy profile s∗=s∗ 1, s∗ 2, ..., s∗ n∈Sis a Nash equilibrium in the game Γ = {N, A, π}if s∗ imaximizes πisi,s∗ −iwith respect to si, given s∗ −i, for each i∈N, i.e., πis∗ i,s∗ −i≥πisi,s∗ −ifor all si∈Si Note that there is no role for beliefs about the strategies of others in the game {N, A, π}, nor in the definition of a Nash equilibrium (Definition 2). Hence, we augment the game {N, A, π}with a profile of “social projection functions”, P, that specify the beliefs of players; this is undertaken in Section 2.2, below. 2.2. Social Projection Functions We would like to define a function that captures the beliefs that a player has about the strategies of the other players, conditional on his own strategy. We will call such a function a social projection function.5 Definition 3. (Social projection functions (SPF)): A social projection function for player iis a mapping Pi:Si→S−ithat assigns to each strategy, si∈Si, for player i, an n−1vector of strategies for the other players. We write Pij (aj|si)for the subjective belief of player ithat player jplays aj∈Aj, conditional on player iplaying si∈Si. Hence, Pij (aj|si)≥0and Paj∈AjPij (aj|si)=1. We may write Pi(si) =se −i(si)to indicate that Pi(si)is the (n−1) vector of strategies that player i anticipates that the other players will follow if player iadopts the strategy si. We now define causal reasoning, the mode of reasoning assumed in classical game theory. Then, we define evidential reasoning. Definition 4. (Causal reasoning): We say that player iuses causal reasoning if Pij (aj|si)is independent of sifor each aj∈Ajand each j6=i, i.e., if Pi(si) = Pis0 ifor all si, s0 i∈Si. 5We use the term social projection function because, on the one hand, it is obviously connected to social projection and evidential reasoning and, on the other hand, to distinguish it from the term projection function as commonly used in mathematics.
Games 2015,6642 Definition 5. (Evidential reasoning) We say that player iuses evidential reasoning if it is not necessarily the case that Pi(si) = Pis0 ifor all si, s0 i∈Si. Remark 1: (a)In a static game of complete information, players are uncertain of the actions taken by others. Under evidential reasoning, player iresolves this uncertainty by assigning diagnostic significance to his or her own choice of strategy, si, in inferring the strategies of the other players, s−i, using his or her social projection function, Pi. For this reason, Definition 5allows for Pi(.|si)to change as si changes. However, it is crucially important to realize that there is no causal connection between siand s−i. The choice of siby player imerely influences that player’s belief about the strategies, s−i, of the other players. In particular, players who use evidential reasoning know that their own actions have no causal effects in altering the actions of others when they change their own actions. (b) An SPF (Definition 3) specifies the beliefs of a player for all possible actions of others, including out-of-equilibrium actions. The beliefs of a player need not turn out to be fulfilled in equilibrium. In this respect, ER is similar to other disequilibrium-in-belief models, such as the level-kmodel (Example 3, below). (c) If player iuses causal reasoning as in classical game theory (see Definition 4), then he or she assigns no diagnostic significance to his or her own strategy, si, in inferring the strategies, s−i, followed by the other players. Thus, under causal reasoning, Pi(si)remains fixed as sichanges. From Definitions 4 and 5, causal reasoning is a special case of evidential reasoning. In a dynamic game (under causal reasoning), if (say) Player 1moves first, choosing the strategy s1, followed by Player 2, who chooses strategy s2, having observed a realization of s1, then s2may very well depend on s1. When choosing s1, Player 1will take into account the influence of his choice on the future behaviour of Player 2. This should not be confused with evidential reasoning. Many different types of social projection functions (SPFs) are possible. There are possibly various degrees of like-mindedness. However, a particularly salient SPF is one where a player believes that a like-minded player will play a strategy that is identical to his own. We call this the identity social projection function. This SPF seems important when choices are low dimensional and players play symmetric games. Examples include cooperate or defect in a prisoners’ dilemma game, vote Democrats or Republicans in U.S. Presidential elections, coordinate or fail to coordinate in coordination games or play hawk or dove in the hawk-dove game. Definition 6. (Identity social projection function): Let M⊆Nbe a subset of players. Suppose that all players in Mhave the same action set, i.e., Ai=Aj=Afor all i, j ∈M. Let Pibe the social projection function for player i∈M. Recall that Pij (a|si)is the probability that player iassigns to player jplaying action awhen the strategy of player iis given by si. If Pij (a|si) = si(a)for all a∈Aand all j∈M− {i}, then we say that Piis an identity social projection function on M. If M=N, then we say that Piis an identity social projection function. Example 1 (self-similarity in the hawk-dove game): Consider the following game between two players. If both choose hawk (H), then each gets zero. If both choose dove (D), then each gets two. If one chooses H(the hawk) and the other chooses D(the dove), then the hawk gets three and the dove gets one. These payoffs are summarized by Table 1, where the row player plays Dwith probability
Games 2015,6643 p∈[0,1], while the column player plays Dwith probability q∈[0,1]. Table 1. A hawk-dove game. D(q)H(1 −q) D(p) 2,2 1,3 H(1 −p) 3,1 0,0 Under causal reasoning (Definition 4), the row player should use a constant social projection function. However, the experimental results reported by Rubinstein and Salant [33] show that the higher the probability, p, with which the row player plays D, the higher the probability, q, she or he thinks the column player will play D. This can be formalized by the row player adopting the identity social projection function (Definition 6): P12 (D|p) = p(1) 2.3. Ingroups, Outgroups and Evidential Reasoning Players need not impute diagnostic significance to their actions when others are perceived not to be like-minded; the next definition formalizes this idea. Definition 7. (Ingroups and outgroups): Suppose that players use evidential reasoning. (a)Player iregards player j(j6=i) as an outgroup member if Pij (aj|si)is independent of si, i.e., if Pij (aj|si) = Pij aj|s0 ifor all si, s0 i∈Siand all aj∈Aj. Otherwise, player iregards player j (j6=i) as an ingroup member. (b) Let M⊂Nbe a non-empty set of players. If every player in Mregards every other player in M as an ingroup member, then Mis an ingroup. (c) Let L⊂Nand M⊂Nbe disjoint non-empty sets of players. Suppose every player in Lregards every player in Mas an outgroup member. Then, we say that Mis an outgroup relative to L. Remark 2: Player iplays action ai∈Aiwith probability si(ai)and believes that player jwill play action aj∈Ajwith probability Pij (aj|si)(the latter is conditional on si). Hence, player ibelieves that the joint probability of aiand ajbeing played is Pij (ai, aj|si) = si(ai)Pij (aj|si). Suppose that player iregards player jas an outgroup member. Then (and only then), Pij (aj|si)is independent of si∈Si. Hence, in this case, Pij (ai, aj|si) = si(ai)Pij (aj|si). Thus, if player iregards player jas an outgroup member, then player ibelieves that the probability with which he or she (player i) plays ai∈Aiis independent of the probability that he or she believes jwill play aj∈Aj. In particular, if player iuses causal reasoning, then he or she regards all others as outgroup members, and hence, he or she believes that his or her actions are independent of the actions of all other players. Definition 8. (Perfect ingroups): Let M⊆Nbe a subset of players. Suppose that all players in M have the same action set, i.e., Ai=Aj=Afor all i, j ∈M. Let Pibe the social projection function for player i∈M. If Piis an identity social projection function on M, for each player i∈M, then Mis a perfect ingroup.
Games 2015,6644 Definition 9. (Evidential game): Consider the static game of complete information, {N, A, π}. Let P = (P1,P2, ..., Pn)be a profile of social projection functions, where Piis the social projection function of player i∈N(Definition 3). Then, we denote the game augmented with the vector of social projection functions, P, by Γ = {N, A, π, P}, and we call it an evidential game. We say that players in such a game use evidential reasoning. Definition 10. If each Pi(si)is independent of si, then we say that Γis a causal game. Remark 3: (a)From Definition 9, a causal game is a special case of an evidential game. (b) Suppose Pi(si)is independent of si, for each player, i, so that Γ = {N, A, π, P}is a causal game. Γis still richer than the static game of complete information, {N, A, π}, because Γincorporates players’ beliefs about other players’ actions, as given by P. Example 2 (Matching pennies): Consider the matching pennies game. H T H−1,1 1,−1 T1,−1−1,1 The set of players is N={1,2}. The action sets are A1=A2={H, T }. Player 1, the row player, plays Hand Twith respective probabilities p,1−p. Player 2, the column player, plays Hand Twith respective probabilities q,1−q. The sets of possible strategies are S1={p: 0 ≤p≤1} for Player 1and S2={q: 0 ≤q≤1}for Player 2. If Player 1plays H(say) and Player 2plays T, then the payoff is one to Player 1and −1to Player 2. For any profile of strategies (p, q),p, q ∈[0,1], the payoff functions of the players are π1(p, q) = −(1 −2p) (1 −2q) and π2(p, q) = (1 −2p) (1 −2q). The following are examples of social projection functions: (P12 (H|p) = p, for all p∈[0,1] P21 (H|q) = 0.5, for all q∈[0,1] (2) According to Equation (2), Player 1, who uses evidential reasoning, believes that if he or she (Player 1) plays Hwith probability p, then so will Player 2for any p∈[0,1]. Hence, Player 1has an identity social projection function. It is critical to note that these are the “beliefs” of Player 1. There is no presumption that these beliefs will turn out to be justified ex post. Player 2, who uses causal reasoning, believes that Player 1will play Hwith probability 0.5, whatever strategy, q, Player 2chooses. Hence, Player 1regards Player 2as an ingroup member, but Player 2regards Player 1as an outgroup member, so N={1,2}fails to be an ingroup. On the other hand, if both players had identity social projection functions, then N={1,2}would be an ingroup (in fact, a perfect ingroup). By contrast, if in Equation (2) we had, say, P12 (H|p) = 0.3for all p∈[0,1], then both players would exhibit causal reasoning; and this example would become a causal game. Example 3 (An application of the level-kmodel to the p-beauty contest6): A large number of contestants are asked to choose an integer between zero and 100, inclusive. It is announced that the 6Moulin [29], Nagel [30].
Games 2015,6651 Estimating the respective fractions of voters who follow evidential and causal reasoning is an interesting and open empirical question, but one that lies outside the scope of our paper. Similar comments also apply to the other experimental games that we consider in which some people cooperate while others do not. 6. Explaining the Prisoners’ Dilemma under Evidential Reasoning We have two players; hence, N={1,2}. Each player has two actions: cooperate (C) or defect (D). Hence, A1=A2={C, D},A = {C, D} × {C, D}={(C, C),(C, D),(D, C),(D, D)}. Therefore, if Player 1chooses, say, Cand Player 2chooses D, then Player 1gets zero and Player 2 gets 10. 6.1. The Prisoners’ Dilemma under Causal Reasoning Each player has a strictly dominant action, D(Definition 1); thus, the unique Nash equilibrium of this game is (D, D)(Definition 2). By contrast, the empirical evidence, reviewed in Section 6.3 below, shows that 50% or more of the outcomes involve the play (C, C). 6.2. The Prisoners’ Dilemma under Evidential Reasoning A strategy for player iis entirely determined by the probability piwith which he or she plays C. His or her expected payoff, πi(pi, pj), can be found from Table 4: πi(pi, pj) = 4 + 6pj−4pi+ 2pipj, i 6=j(7) Table 4. A prisoner’s dilemma game. C D C8,8 0,10 D10,0 4,4 We consider the following four cases. Case 1: Both players use evidential reasoning (Definition 5). Consider the social projection function (Definition 3) for player i: Pij (C|pi) = pi, i 6=j(8) Remark 4: The social projection Function (8) should not be interpreted as saying that by playing Cwith probability pi, player ican induce player j6=ito play Cwith probability pi; indeed, there is no such causal link. Player idoes not know what action the other player will take or has taken. Rather, Equation (8) is a heuristic device. Player imay reason as follows “I would like to cooperate with probability pi. Since player jis like-minded, I believe he or she will also choose cooperate with probability pi, just like me”. None of the players attempts to strategically exploit the SPF of the other
Games 2015,6652 players. Indeed, there is no requirement in an evidential game that there even be mutual knowledge of the social projection functions. From Equation (8), we see that both players use evidential reasoning (Definition 5), so Γ = {N, A, π, P}is an evidential game (Definition 9). In particular, each player uses his identity social projection function (Definition 6). Together, both players form an ingroup (Definition 7). In fact, Nforms a perfect ingroup (Definition 8). Proposition 2. : For the prisoners’ dilemma game, Table 4,(C, C)is a consistent evidential equilibrium under the identity social projection Function (8). Proof of Proposition 2: Substituting from Equation (8) into Equation (7), we get: πi(pi, Pij (C|pi)) = 4 + 2pi+ 2p2 i, pi∈[0,1] , i 6=j(9) From Equation (9), we see that πi(pi, Pij (C|pi)) is maximized when pi= 1. It follows that Cis the unique optimal choice for player i∈N(Definitions 11). Hence, (C, C)is the unique evidential equilibrium of this game (Definition 12). Each player expects the other to play C, which turns out to be correct, ex post. Therefore, (C, C)is a mutually consistent vector of strategies (Definition 13). Hence, (C, C)is a consistent evidential equilibrium (Definition 14). In contrast, (C, C)is not the Nash equilibrium of the game (Definition 2). Indeed, (C, C)requires each player to play a strictly dominated strategies. However, (C, C)is Pareto optimal. Note that under evidential reasoning, one does not need repeated game arguments to justify cooperation in the static prisoners’ dilemma game. Moreover, this is consistent with the play of the cooperative strategy by a majority of the players (see Section 6.3 below). This suggests that a majority of the players may be using evidential reasoning. Case 2: Player 1uses evidential reasoning (Definition 5), but Player 2uses causal reasoning (Definition 4). In this case, the SPF for each player is given by: P12 (C|p1) = p1, for all p1∈[0,1] (10) P21 (C|p2) = 1, for all p2∈[0,1] (11) From Equation (10), we see that Player 1uses evidential reasoning (Definition 5) and, in particular, his or her identity social projection function (Definition 6), as in Case 1 above. On the other hand, from Equation (11), we see that Player 2uses causal reasoning (Definition 4) and, in particular, mistakenly assumes that Player 1will always cooperate. This is an evidential game. The unique evidential equilibrium (Definition 12) is (C, D). It is an evidential equilibrium because each player’s chosen action is optimal, given his beliefs, which are captured by his social projection function. It is not a consistent evidential equilibrium (Definition 14) because the belief of Player 1turns out to be mistaken in equilibrium (P12 (C|C) = 1, but Player 2plays Dinstead). By contrast, the belief of Player 2that Player 1plays Cturns out to be correct in equilibrium. Case 3: Both players use causal reasoning (Definition 4), but beliefs turn out to be wrong ex post.
Games 2015,6653 In this case, the SPF for each player is given by: P12 (C|p1) = 1, for all p1∈[0,1] (12) P21 (C|p2) = 1, for all p2∈[0,1] (13) Both players use causal reasoning (Definition 4), so this is a causal game (Definition 10). Given these social projection functions, the unique payoff maximizing strategy for each player is to play D (Definition 11). Hence, (D, D)is the unique evidential equilibrium (Definition 12). It is also, of course, the unique Nash equilibrium of this game. However, (D, D)is not a mutually consistent vector of strategies (Definition 13) because each player expects his opponent to play Cin response to D, but the opponent’s response is D. Hence, (D, D)is not a consistent evidential equilibrium (Definition 14). Case 4: Both players use causal reasoning (Definition 4), and beliefs turn out to be correct ex post. In this case, the SPF for each player is given by: P12 (C|p1) = 0, for all p1∈[0,1] (14) P21 (C|p2) = 0, for all p2∈[0,1] (15) Both players use causal reasoning (Definition 4), so this is a causal game (Definition 10). Given his social projection function, playing Dis the unique optimal strategy for Player 1(Definition 11) and similarly for Player 2. Hence, (D, D)is the unique evidential equilibrium (Definition 12). Furthermore, (D, D)is a mutually consistent vector of strategies (Definition 13) because each player expects his or her rival to play D, and in fact, his or her rival does play D. Hence, (D, D)is a consistent evidential equilibrium (Definition 14). The unique Nash equilibrium of this game is, of course, (D, D). Hence, this case illustrates Proposition 1a, namely a Nash equilibrium of the game {N, A, π}is also a consistent evidential equilibrium (Definition 14) of the game {N, A, π, P}with a suitable choice of social projection functions, P. Remark 5: When players are randomly matched to play the one-shot prisoners’ dilemma game, the weight of the evidence, reviewed in the next Section, indicates a cooperation rate of at least 50%. To our minds, the only satisfactory explanation is provided by evidential reasoning with players using their identity social projection function (Definition 6). However, substantial numbers also defect, and this can be explained (as usual) by causal reasoning. Thus, the evidence can best be explained by a mixture of players, some of whom use causal reasoning, and the others use evidential reasoning. 6.3. Evidence of Cooperation in the Prisoners’ Dilemma Game In the static prisoners’ dilemma game, defection (D) is a strictly dominant strategy; recall Table 4. Hence, a player using causal reasoning should defect. However, experimental evidence indicates high cooperation rates. Rapoport [4] finds cooperation rates of 50% in the prisoners dilemma game. Zhong et al. [11] show that the cooperation rates in prisoners’ dilemma studies go up to 60% when positive labels are used (such as a “cooperative game”, rather than a “prisoners’ dilemma game”). When purely generic labels are used (such as C and D), then the cooperation rates are about 50%. Khadjavi and Lange [13] find that, while the cooperation rates among students playing the static prisoners’ dilemma game is 37%, the cooperation rate among prison inmates is 56%.
Games 2015,6654 Lewis [2] used evidential reasoning to explain the unexpected levels of cooperation in the one-shot prisoners’ dilemma game. Mutual cooperation is better than mutual defection. If players use evidential reasoning, they may take their own preference for mutual cooperation as diagnostic evidence that their rival also has a preference for mutual cooperation, in which case both players are more likely to cooperate. These views are borne out by the evidence. Cooperators believe that the probability of other players cooperating is between 0.6and 0.7. Similarly, players who defect believe that other players will defect with probabilities between 0.6to 0.7; see Krueger [40]. Like-mindedness is compatible with both outcomes, Cand D, and we do observe both outcomes in the PD game. Those who play C(respectively D) also believe that a disproportionately large share of the other players will play C(respectively D). However, why then do we observe so much cooperation in the static prisoners’ dilemma game? According to Gintis [54], page 145, humans have evolved the desire to cooperate with other humans. However, this cannot be the only cause, for it cannot explain why the rate of defection increases in the PD game when players know that their rivals have cooperated (see Section 9.5, below). Rapoport [55], pp. 139–141, argued that each player takes his own belief that rational players deserve the cooperative outcome as evidence that similarly rational players will also cooperate. This is similar to evidential reasoning. Howard [3] tests the assertion by Rapoport [55], pp. 139–141, by running a contest between two computer programs. One computer program is designed to play the dominant strategy, defect. Another computer program, called the MIRROR program, is able to recognize if it is playing another MIRROR program, in which case it also cooperates; otherwise, it plays defect. There are five copies each of the programs that plays a tournament, and not surprisingly, the MIRROR program achieves higher payoffs. In effect, what the MIRROR program is doing is replicating the notion that people would cooperate with other like-minded people. In the conclusion, Howard [3], p. 212, gives an argument that is identical in spirit to the evidential reasoning argument: “If all players use the self-recognition program listed in the Appendix, and play cooperatively only if they recognize their opponents as their twins, then every game will be played cooperatively.” In contrast to the standard explanations (Section 9, below), the explanation of cooperation based on evidential reasoning appears to be quite plausible. The fact that a sizeable fraction of the experimental subjects also defect suggests that the results are best accounted for by a mixture in the population of people who use evidential reasoning and causal reasoning. 7. The Nash Demand Game 7.1. Non-Uniqueness of Outcomes in the Nash Demand Game under Causal Reasoning and under Evidential Reasoning Consider the Nash demand game (Nash, [56,57]). Two players share a cake of size one. Player 1 demands x∈[0,1] and Player 2 simultaneously demands y∈[0,1]. If the demands are feasible, i.e., if x+y≤1, then each player receives what she or he demanded. However, if the demands are not feasible, i.e., if x+y > 1, then each player gets zero. Any pair, (x, y), such that x+y= 1 is a Nash non-cooperative equilibrium. Thus, the Nash non-cooperative equilibrium concept does not pin down a unique solution in the Nash demand game.
Games 2015,6655 The following theorem shows that a similar problem occurs with evidential reasoning.8 Proposition 3. : If we allow arbitrary social projection functions, then any outcome, (a, b), such that a+b= 1,a > 0and b > 0, is an outcome of a consistent evidential equilibrium for the Nash demand game for suitably-chosen social projection functions. Proof of Proposition 3: Let a+b= 1,a > 0,b > 0. We shall construct social projection functions under which (a, b)is a consistent evidential equilibrium. Consider the following social projection functions. Let λ12 >0and λ21 >0, where λ12λ21 = 1. The restriction λ12λ21 = 1 ensures that the shares demanded by both players sum up to one; see below. For Player 1, set P12 (y|x) = 1 ⇔y=λ12x,i.e., if Player 1 makes the demand x, then she expects Player 2 to make the demand y=λ12x. For Player 2, set P21 (x|y)=1⇔x=λ21y,i.e., if Player 2 makes the demand y, then she expects Player 1 to make the demand x=λ21y. Thus, Player 1 maximizes x subject to x+λ12x≤1. The unique solution to this maximization problem is x=1 1+λ12 . Similarly, the unique solution to Player 2’s maximization problem, maximize ysubject to λ21y+y≤1, is y=1 1+λ21 . It is straightforward to check that 1 1+λ12 ,1 1+λ21 is a consistent evidential equilibrium for the chosen social projection functions. Finally, choosing λ12 =1−a aand λ21 =1−b bgives the outcome (a, b). Corollary 4. :1 2,1 2is the unique consistent evidential equilibrium for the Nash demand game under the identity social projection functions: P12 (y|x) = 1 ⇔y=xand P21 (x|y) = 1 ⇔x=y. Proof of Corollary 4: Take λ12 =λ21 = 1 in the proof of Proposition 3. Corollary 4shows that a particularly salient social projection function, the identity function, leads to an equal division of the pie. This might be of empirical interest, particularly when studying norms of equal division. To overcome the non-uniqueness problem highlighted by Proposition 3, we need criteria to select social projection functions. One such criterion is to select the identity social projection function for symmetric evidential games. The identity social projection function appears plausible, maybe even compelling, for symmetric evidential games. However, what further criteria would help? One possibility is to appeal to the Nash bargaining axioms (Nash [56,57]; Osborne and Rubinstein [58]).9These axioms are introduced in Section 7.2, below, followed by application to the Nash demand game under evidential reasoning, in the subsequent Section 7.3. 7.2. Nash’s Axioms and Nash’s Theorem First, some definitions. Definition 15. : A bargaining problem is a pair hB,di, where B⊂R2is compact and convex and d∈B. We require that, for some b∈B,d1< b1and d2< b2. 8We are grateful to Ludovic Renou for drawing our attention to this problem. 9“One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely.” Nash [57].
Games 2015,6656 The bargaining problem hB,dimay be given the following interpretation. Bis the set of possible payoffs resulting from agreement. dis the pair of payoffs to the players if they fail to agree. Definition 16. : Let ßbe the set of all bargaining problems. A bargaining solution is a mapping, F :ß→R2, where FhB,di ∈ B. Definition 17. :b0∈Bis Pareto optimal if, for all b∈B,b1≤b0 1and b2≤b0 2. Definition 18. : A bargaining problem, hB,di, is symmetric if d1=d2and if (b1, b2)∈ B⇔(b2, b1)∈B. Definition 19. : We say that the bargaining solution, F, satisfies the independence of irrelevant alternatives if for all bargaining problems, hB,diand hB0,di, such that B⊂B0and FhB0,di ∈ B, we have FhB,di= F hB0,di. Definition 20. : Let hB,di,hB0,d0i ∈ß,α1, α2, β1, β2∈R, β1>0, β2>0. Let f : Bonto →B0, f1(b1) = α1+β1b1, f2(b2) = α2+β2b2. Then, we say that fis a positive affine transformation of hB,dionto hB0,d0i. 7.2.1. The Nash Axioms Let ßbe the set of all bargaining problems, Fa bargaining solution and hB,dia bargaining problem. We introduce the following axioms. Pareto: FhB,diis Pareto optimal. Symmetry: If hB,diis symmetric, then F1hB,di=F2hB,di. Independence: Fsatisfies the independence of irrelevant alternatives. Invariance: If f (b) = (α1+β1b1, α2+β2b2)is a positive affine transformation of hB,dionto hB0,d0i, then F1hB0,d0i=α1+β1F1hB,di, F2hB0,d0i=α2+β2F2hB,di. 7.2.2. The Nash Theorem Proposition 5. : There is a unique bargaining solution, F∗:ß→R2, satisfying the axioms: Pareto, symmetry, independence and invariance. It is given by: F∗hB,di= arg max (d1,d2)≤(b1,b2)∈B (b1−d1) (b2−d2) 7.3. Application to the Nash Demand Game under Evidential Reasoning We now illustrate, by an example, how the Nash axioms, Pareto, symmetry, independence and invariance, can reduce the choice among social projection functions sufficiently so as to generate a unique outcome for the Nash demand game. Example 5 (The Nash demand game under evidential reasoning) Assume that the utility of Player 1 is π1(x) = xαand that of Player 2 is π2(y) = yβ, where α > 0, β > 0. Consider the
Games 2015,6657 following social projection functions. Let λ12 >0and λ21 >0, where λ12λ21 = 1. For Player 1, set P12 (y|x) = 1 ⇔y=λ12x,i.e., if Player 1 makes the demand x, then he or she expects Player 2 to make the demand y=λ12x. For Player 2, set P21 (x|y)=1⇔x=λ21y,i.e., if Player 2 makes the demand y, then he or she expects Player 1 to make the demand x=λ21y. Thus, Player 1 maximizes xαsubject to x+λ12x≤1. The unique solution to this maximization problem is x=1 1+λ12 . Similarly, the unique solution to Player 2’s maximization problem, maximize yβsubject to λ21y+y≤1, is y=1 1+λ21 . It is straightforward to check that 1 1+λ12 ,1 1+λ21 is a consistent evidential equilibrium for the chosen social projection functions. To make progress, we need to select λ12, λ21. If α=β, then the game is symmetric, and λ12 =λ21 (and hence, λ12 =λ21 = 1) appear compelling. This give the plausible outcome x=y=1 2. However, if α6=β,then the game is not symmetric, and it is not clear, a priori, how to choose λ12, λ21. If we invoke, in addition to symmetry, the other Nash axioms, Pareto, independence and invariance, then, by Proposition 5, we must get the unique outcome that is determined by maximizing the Nash product π1(x)π2(y) = xαyβ(here, d1=π1(0) = 0 and d2=π2(0) = 0). We then show that this unique outcome can be supported by the appropriate choice of λ12, λ21. It is straightforward to show that the problem: choose xand y, so as to maximize xαyβsubject to x+y≤1, having the unique solution x=α α+β, y =β α+β. This, in turn, determines the unique values λ12 =β α, λ21 =α β. Thus, social projection functions compatible with the Nash axioms are P12 (y|x) = 1 ⇔y=β αx, and P21 (x|y) = 1 ⇔x=α βy. Note that these social projection functions are not unique (because they were chosen to be linear). However, the outcome x=α α+β, y = β α+βis unique, whatever social projection functions are chosen, provided they satisfy the Nash axioms. 8. Oligopoly Games Consider a market for a single homogeneous good. The total industrial output, Q, is produced by a fixed number of firms, n. Let qibe the output of firm i, then Q=Pn i=1 qi. All consumers are price takers. The unit price, P(Q), is given by: P(Q) = A−aQ,a > 0,A > 0(16) There are zero fixed costs, and the marginal cost of firm iis a constant, ci,i= 1,2, ..., n, where: 0≤c1≤c2≤... ≤cn< A (17) Hence, the profit of firm iis πi= (P−ci)qi,i= 1,2, ..., n, which, using Equation (16), can be written as: πi(qi,q−i) = A−ci−aXj6=iqjqi−aq2 i,i= 1,2, ..., n (18) where q−iis the vector of outputs of firms other than firm i. Maximizing πi(qi,q−i)with respect to qi, given q−i, leads to firm i’s reaction function: q∗ i(q−i) = A−ci 2a−1 2Xj6=iqj,i= 1,2, ..., n (19)
Games 2015,6658 8.1. Causal Reasoning The next proposition summarizes the results under causal reasoning on the part of all firms. Proposition 6. : (a) Under perfect competition, P=c1,Q=A−c1 a. (b) The monopoly outcome is given by P=A+c1 2,Q=A−c1 2a. (c) The Cournot output level of any firm is: qC i=A+Pj6=icj−nci (n+ 1) a, i = 1,2, ..., n (20) (d) In a Stackelberg leader-follower model where Firm 1 is the leader while Firm 2 is the follower, the equilibrium output levels of the leader and the follower, respectively, are: qL 1=A+c2−2c1 2a(leader) (21) qF 2=A+ 2c1−3c2 4a(follower) (22) Proof of Proposition 6: (a) Under perfect competition, each firm produces at the minimum cost, c1, and the price is set equal to c1; Equation (16) then gives Q=A−c1 a. (b) Suppose we have a single firm (the monopolist) that produces at minimum cost, c1. Setting Pj6=iqj= 0 and q∗ i(q−i) = Q(there are no other firms) in Equation (19) gives Q=A−c1 2a; Equation (16) then gives P=A+c1 2. (c) In a Cournot equilibrium, each firm chooses its out, qC i, so as to maximize its profit Equation (18), given the outputs, qC j, of the other firms. Set q∗ i(q−i) = qC iand qj=qC jin Equation (19) and solve the resulting system of simultaneous linear equations to get Equation (20). (d) Consider a duopoly with Firm 1 acting as leader and Firm 2 acting as follower. The follower chooses its output, qF 2, to maximize its profit given the output level, qL 1, of the leader; Equation (19) then gives qF 2qL 1=A−c2 2a−1 2qL 1. Substitute this into the profit function of the leader (from (Equation 18)) to get πL 1qL 1, qF 2qL 1=A−c1−aA−c2 2a−1 2qL 1qL 1−aqL 12. Maximize this with respect to qL 1to get qL 1=A+c2−2c1 2a, and hence, qF 2=A+2c1−3c2 4a. Remark 6: Note that the perfectly competitive, monopoly and Cournot games are all single-stage games, i.e., one-shot games. The firms choose their actions simultaneously (Parts a, b and c of Proposition 6). However, the Stackelberg leader-follower model is a two-stage game: The leader moves first choosing its output level correctly anticipating the reaction of the follower. The follower then moves having observed the output level of the leader. 8.2. Evidential Reasoning Consider the consequences of evidential reasoning for the producers. All consumers are causal reasoners, i.e., each consumer regards every other consumer and every firm as an outgroup member (recall Definition 7). We also assume that each firm regards each consumer as an outgroup member. Thus, if Cis the set of consumers and F is the set of firms, then each is an outgroup relative to the
Games 2015,6659 other (Definition 7c). This also allows us to continue to assume that the market demand curve is given by Equation (16). If we allowed consumers to use evidential reasoning, then a single consumer could reason as follows “If I cut my demand, then probably each like-minded consumer would also cut his or her demand. The aggregate result would be a reduction in price for all of us”. Consumers would then be able to collude. The consequence would be that we would no longer have an oligopoly model (as classically defined), but a bargaining model. While this is very interesting, it lies beyond the scope of this paper and, in fact, deserves a paper on its own. We now describe an evidential equilibrium, q∗, with the following properties. Suppose that firm i is considering a deviation, qi, from q∗ i. Firm ireasons as follows. “If I am tempted to deviate by an amount qi−q∗ iand if I believe that my rival, firm j,j6=i, is like-minded, then the rival is probably also tempted to deviate by an amount qj−q∗ j=λij qi−q∗ i”; the interpretation of λij is given below in more detail. We formalize such reasoning by the following social projection function (Definition 3): Pij (qj|qi) = 1 ⇔qj−q∗ j=λij qi−q∗ i,j6=i(23) The social projection specified in Equation (23) is quite general. It nests several subcases, as we show below. The generality of Equation (23) should not be taken to mean that the predictive content of the evidential reasoning model of oligopoly is empty. Rather, as in prisoners’ dilemma games, individuals display a wide variation in choices when they are asked to play the oligopoly game (see Section 8.4 below). Variations in the parameter λij in Equation (23) offer a parsimonious way of capturing this heterogeneity by varying the degree of like-mindedness. In particular, perfect like-mindedness, λij = 1, gives rise to the identity social projection function (Definition 6); each firm believes that the other will deviate from q∗by an identical distance. The other extreme arises when no like-mindedness is perceived by firms, as in models of causal reasoning. This corresponds to λij = 0. Intermediate cases of like-mindedness correspond to values 0< λij <1and to λij <0. The distribution of values of λij in any population is ultimately an empirical question that cannot be answered in a theoretical model. The next proposition gives the solution under evidential reasoning. Proposition 7. : (a) Given the social projection Functions (23), the unique evidential equilibrium (Definition 12), q∗, is characterized by the following set of simultaneous linear algebraic equations: 2 + Pj6=1 λ1j1... 1 1 2 + Pj6=2 λ2j... 1 ... ... ... ... 1 1 ... 2 + Pj6=nλnj q∗ 1 q∗ 2 ... q∗ n = A−c1 a A−c2 a ... A−cn a (24) (b) Furthermore, q∗is a mutually consistent vector of strategies (Definition 13) and, hence, a consistent evidential equilibrium. (c) Conversely, given any vector of outputs, q∗, satisfying q∗ i>0and Pn i=1 q∗ i≤A−c1 a, there exits a profile of social projection of the form Equation (23), such that q∗is a consistent evidential equilibrium. In particular, λij =λi,i, j = 1,2, ..., n,j6=i, where λi=A−ci (n−1) aq∗ i −2 n−1−1 (n−1) q∗ iX j6=i q∗ j,i= 1,2, ..., n (25)
Games 2015,6660 Proof of Proposition 7: (a) Substituting qjfrom Equation (23) into Equation (18) gives: πi(qi,Pi(.|qi)) = (A−ci−aX j6=ihq∗ j+λij qi−q∗ ii)qi−aq2 i,i= 1,2, ..., n (26) which, after simplification, gives: πi(qi,Pi(.|qi)) = A−ci+aq∗ iX j6=i λij −aX j6=i q∗ j!qi−a 1 + X j6=i λij!q2 i,i= 1,2, ..., n (27) Equation (27) shows how a player who uses the heuristic of evidential reasoning translates an essentially strategic problem into a decision theoretic problem. Maximizing Equation (27) with respect to qigives the optimal (pure) strategy for firm i(Definition 11), given his social projection Function (23): qi=A−ci+aq∗ iPj6=iλij −aPj6=iq∗ j 2a1 + Pj6=iλij,i= 1,2, ..., n (28) Setting qi=q∗ i,i= 1,2, ..., n and simplifying gives the following set of simultaneous linear algebraic equations, 2 + X j6=i λij!q∗ i+X j6=i q∗ j=A−ci a,i= 1,2, ..., n (29) which can be written in the matrix form Equation (24). (b) From Equation (23), we see that Pij qj|q∗ i= 1 ⇔qj=q∗ j. In effect, when firm iproduces the output q∗ i, it believes that firm jwill produce q∗ j.Ex post, firm ifinds that firm jindeed did produce an output level q∗ j, thus vindicating its ex ante belief. Hence, q∗is a mutually consistent vector of strategies and, hence, a consistent evidential equilibrium. (c) Rewrite Equation (29) in the form: X j6=i λij =A−ci aq∗ i −2−1 q∗ iX j6=i q∗ j,i= 1,2, ..., n (30) Equation (30) has many solutions, for example Equation (25). We now show how one may obtain the market outcomes under causal reasoning (Proposition 6) also under evidential reasoning by choosing suitable values for λij,j6=i, in Equation (25) of Proposition 7. Corollary 8. : (a) Setting c1=c2=... =cnand λij =−1 n−1, for all i, j and i6=j, gives the perfectly competitive output levels, Pn i=1 q∗ i=Q∗=A−c1 a(Proposition 6a). Here, each firm regards every other firm as an ingroup member and the set of firms forms an ingroup (Definition 7). We may call this a competitive ingroup and the resulting social projection functions competitive social projection functions. This is in line with the ideas considered in Section 2.3 and, in particular, is an illustration of the contrast effect. (b) Setting c1=c2=... =cnand λij = 1, for all i, j and i6=j, gives q∗ i=A−c 2na ,i= 1,2, ..., n.
Games 2015,6667 the game, so as to establish a reputation for cooperation. However, because defect is a dominant strategy, such a player will defect in the final period. This contradicts the observation of significant amounts of cooperation in the final period (Cooper et al. [6]). In particular, reputation (on its own) is unable to explain cooperation in the static (one-shot) prisoners’ dilemma game. Since “cooperate” is a dominated strategy, it is not rationalizable and cannot be supported in a correlated equilibrium either. In Level-kmodels, any player with level k,k≥1, will never play a dominated strategy. Cooperation in a prisoners’ dilemma game cannot be explained in evolutionary games either. The reason is that the set of evolutionary stable equilibria is a subset of the set of Nash equilibria of the game. Stackelberg reasoning cannot explain cooperation in the prisoners’ dilemma either. To see this, consider the payoff matrix in Table 4. According to Stackelberg reasoning, if Player 1, say, chooses C, then Dis the best response for Player 2, giving Player 1 the payoff zero; and if Player 1 chooses D, then, again, Dis the best response for Player 2, giving Player 1 the payoff 4>0. Hence, according to Stackelberg reasoning, Player 1 will always choose D; and similarly for Player 2. Therefore, cooperation will never be the outcome of the prisoners’ dilemma game under Stackelberg reasoning. 9.2. Team Reasoning Several elements are key to team reasoning (Bacharach, [63]): 1. The team agrees on a common objective. 2. The task for each member of the team is agreed upon by all members. 3. The way the surplus is divided among the members of the team is agreed upon by all members. The team is consolidated if all members carry out their tasks. However, serious failure of a significant number of members is liable to cause the team to breakup. As an illustration, consider the two-player game whose payoff matrix is given by Table 8, below. This, clearly, has the structure of a prisoners’ dilemma game. Table 8. A prisoner’s dilemma game. C D C5,5 2,6 D6,2 4,4 The interpretation of the payoffs in Table 8is as follows. If a player works on her or his own, she or he gets the payoff of four. If both work as a team, then the joint payoff is 10, which is shared equally to give each a payoff of five. However, if one player defects (shirks), she or he gets six. This is because she or he gets four for herself or himself from working on her or his own. The other player (who has not defected) generates a payoff of four for the team by working on her or his own. By the sharing rule, this payoff is shared equally, giving the defector a total payoff of six and the non-defector a payoff of two. Dis the strictly dominant strategy for each player. Hence, this game has the unique Nash equilibrium,
Games 2015,6668 (D, D), which gives each player the payoff four. However, if both players internalize the objective of the team, to maximise joint payoff, then each gets five. 9.3. Other-Regarding Preferences Consider, for example, other-regarding preferences in the model of Fehr and Schmidt [65,66]. Suppose that we have nplayers with incomes: y1≤y2≤... ≤yn. We concentrate on the linear version, which has had considerable empirical support. The Fehr–Schmidt utility function of an individual with income yj∈Yis given by: U(yj) = yj−β n−1 j−1 X i=1 (yj−yi)−α n−1 n X k=j+1 (yk−yj), α ≥0,0≤β < 1(36) Individual jcares for his own payoff, yj, as under selfish preferences. However, he or she also suffers disutility from being ahead of others (altruism) and from being behind others (envy). β≥0and α≥0 are sometimes known as the parameters of, respectively, advantageous and disadvantageous inequity. When α=β= 0, we have purely selfish preferences. Evidence indicates that disadvantageous inequity is more important than advantageous inequity (α > β), and one never benefits by throwing away one’s own income (β < 1). Let us apply FSpreferences, with β= 0.3, to the prisoners’ dilemma game (n= 2), whose payoff matrix is given by Table 4, above. It can then be easily seen that Cis a strict best reply to C, and Dis a strict best reply to D. Hence, other-regarding preferences can, potentially, explain cooperation in the prisoners’ dilemma game. We return to this in Section 9.5, below. 9.4. Altruism Cooper et al. [6] give an explanation of cooperation in the prisoners’ dilemma game based on altruism. They consider three types of players. Egoists who always defect; dominant strategy altruists who always cooperate; and best response altruists who respond to cooperate with cooperate and to defect with defect. Because there are three types of players and because when a player takes his or her action, he or she does not know the type of his or her opponent, this is a (static) game of incomplete information. The relevant solution concept (under causal reasoning) is the Bayesian Nash equilibrium. Each type of each player chooses his or her strategy so as to maximize his or her expected payoff, given the strategies of all of the types of all of the other players. The joint probability of all types is common knowledge. Each type of each player uses this, and his or her knowledge of his or her own type, to update his or her belief about the types of his or her rivals using Bayes’ Law. See, for example, Fudenberg and Tirole [45], Part III. To rationalize the behaviour of these types, Cooper et al. [6] assume that player ienjoys an extra amount of utility (warm glow), δi≥0, from playing cooperate (irrespective of the action played by his or her rival). If δiis sufficiently small, then player ibehaves as an egoist, i.e., defect is a dominant action for him or her. If δiis sufficiently large, then player ibehaves as a dominant strategy altruist, i.e., cooperate is a dominant strategy for him or her. Finally, for δiin the intermediate range, player i behaves as a best response altruist, i.e., for him or her, cooperate is the best response to cooperate and
Games 2015,6669 defect is the best response to defect. As an illustration, assume that, in addition to the payoffs in Table 4, player ienjoys warm glow, δi, from cooperation. The modified payoff matrix is now given by Table 9: Table 9. A prisoner’s dilemma game in the presence of warm glow. C D C8 + δ1,8 + δ2δ1,10 D10, δ24,4 The following is easy to check. 1. If δi<2, then defect is a strictly dominant action for player i. In this case, player ibehaves as an egoist. 2. If δi>4, then cooperate is a strictly dominant action for player i. In this case, player ibehaves as a dominant strategy altruist. 3. If 2< δi<4(e.g., δi= 3), then cooperate is a strict best response for player ito player j, j6=i, playing cooperate. Defect is a strict best response for player ito player j,j6=i, playing defect. Player ibehaves as a best response altruist. Reviewing the evidence, including the evidence from their own experiments, Cooper et al. [6] conclude that behaviour in the prisoners’ dilemma game can best be explained if players are either egoists (always defect) or best response altruists (respond to cooperate with cooperate and to defect with defect). To illustrate this, take δi= 0 if player iis an egoist and δi= 3 if player iis a best response altruist. Consider the game between two best response altruists. Putting δ1=δ2= 3, the payoff Table 9becomes: Table 10. A prisoner’s dilemma game with warm glow. C(q)D(1 −q) C(p) 11,11 3,10 D(1 −p) 10,3 4,4 In Table 10, Player 1plays Cwith probability pand Player 2plays Cwith probability q. Let a∈(0,1) be the probability of a player being the best response altruist. We have four cases to consider: 1. Suppose Player 1(a best response altruist) plays C. 1.1 Player 1meets an egoist with probability 1−a. An egoist always defects, giving Player 1 the payoff three. 1.2 Player 1meets a best response altruist with probability a. If Player 2(a best response altruist) plays C(probability q), then the payoff to Player 1is 11. If Player 2plays D(probability 1−q), then the payoff to Player 1is 3. 2. Suppose Player 1plays D.
Games 2015,6670 2.1 If Player 1meets an egoist (probability 1−a), who always defects, then the payoff to player 1will be four. 2.2 Suppose Player 1meets a best response altruist (probability a). If Player 2(a best response altruist) plays C(probability q), then the payoff to Player 1is 10. If Player 2plays D(probability 1−q), then the payoff of Player 1is four. Hence, the expected utility of Player 1is, U1(p, q) = p{3 (1 −a) + a[11q+ 3 (1 −q)]}+(1 −p){4 (1 −a) + a[10q+ 4 (1 −q)]} (37) which simplifies to: U1(p, q) = 4 + 6aq −p+ 2apq (38) from which we get: ∂U1(p, q) ∂p = 2aq −1(39) and hence: q > 1 2a⇒∂U1(p, q) ∂p >0 q < 1 2a⇒∂U1(p, q) ∂p <0 To simplify the discussion, let us concentrate on the pure strategy Bayesian Nash equilibria in a population with a fraction, a∈(0,1), of best response altruists and a fraction, 1−a, of egoists. If a < 1 2, then we have one pure strategy Bayesian Nash equilibrium, where all players defect. If a≥1 2, then we have two pure strategy Bayesian Nash equilibria. In one equilibrium (as before), all payers defect. In the second equilibrium, all best response altruists cooperate and all egoists defect. Hence, for the particular payoffs in Table 10, cooperation can be sustained as a Bayesian Nash equilibrium in the one-shot prisoners’ dilemma game if the percentage of best response altruists is ≥50%. See Cooper et al. [6] for a more extensive analysis, including mixed strategies and repeated prisoners’ dilemma games. 9.5. Can Team Reasoning, Altruism or Other-Regarding Preferences Explain Cooperation in the Prisoners’ Dilemma Game? Shafir and Tversky [5] presented experimental subjects with the usual one-shot prisoners’ dilemma game. They then considered the following two variants. 1. A player whose competitor had defected was informed of this and offered the chance to revise her or his decision. 2. A player whose competitor had cooperated was informed of this and offered the chance to revise her or his decision. Shafir and Tversky [5] found that when a player did not know what the opponent had chosen, then the player defected in 63% of games. If the player was informed that the opponent had defected (Case 1,
Games 2015,6671 above) then in 97% of the games, the player defected. Thus, defection increased. on the other hand, if the player was informed that the opponent cooperated (Case 2, above), then in 84% of the games, the player defected. Thus, defection, again, increased. These results were replicated and extended by Croson [7], Li and Taplin [8], Busemeyer et al. [10] and Histrova and Grinberg [12]. We shall argue that these results are consistent with evidential reasoning, but not with team reasoning, altruism or other regarding preferences. First, consider Case 1, above. Here, all four theories predict an increase in defection following a player being told that her or his rival has defected, in line with the evidence. However, the reasons are different. Once a player has been told the move of her or his rival uncertainty is resolved, she or he no longer needs recourse to evidential reasoning. Since she or he continues to play selfishly and since defect is a strictly dominant strategy, she or he defects. An observation of defection is liable to destroy team spirit (Section 9.2). Hence, team reasoning is consistent with the observation of an increase in defection. According to other-regarding preferences, as in Section 9.3 above, a player who originally chose Dwill continue to choose Dafter she or he has been told that her or his rival had defected. However, a player who had originally chosen Cwill change to Dafter being told that her or his rival had defected. Hence, the other-regarding preferences model will predict an increase in defection, in line with the evidence. Finally, consider altruism, as in Section 9.4 above. The egoists will not change their behaviour; they will continue to defect. Likewise, the dominant strategy altruists will not change their behaviour; they will continue to cooperate. The best response altruists who chose to defect will continue to choose defect after being informed that their rival has defected. However, each best response altruist who had chosen cooperation will now switch to defection. Hence, the amount of defection will increase in line with the evidence. Second, consider Case 2 above. Here, evidential reasoning predicts an increase in defection, in line with the evidence. The reason is that uncertainty is resolved once a player has been told the move of her or his rival. She or he no longer needs recourse to evidential reasoning. If she or he had defected, she or he will choose defect again. If she or he had cooperated, she or he will now defect. However, the other three theories predict a decline in defection, contrary to the evidence. For team-reasoning, the team spirit is strengthened once a player is told that her or his rival has cooperated. Hence, a player who had cooperated continues to cooperate. A team member who had initially chosen to defect may now change to cooperate. According other-regarding preferences, as in Section 9.3, a player who originally chose Cwill continue to choose Cafter she or he has been told that her or his rival had cooperated. However, a player who had originally chosen Dwill change to Cafter being told that her or his rival had cooperated. Hence, the other-regarding preferences model will predict a reduction in defection, contrary to the evidence. Finally, consider altruism, as in Section 9.4. The egoists will not change their behaviour; they will continue to defect. Likewise, the dominant strategy altruists will not change their behaviour; they will continue to cooperate. The best response altruists who chose to cooperate will continue to cooperate. However, each best response altruist who had chosen defect will now find it in their best interest to cooperate. Hence, the extent of defection will decrease, contrary to the evidence.
Games 2015,6672 10. Conclusions A large number of experimental subjects do not play a Nash equilibrium in well-known games, such as prisoners’ dilemma, the hawk-dove game, voting games, public goods games and oligopoly games. It would seem that this constitutes strong grounds for game theory to be open to alternative equilibrium concepts in static games. Aumann and Brandenburger [67] gave epistemic conditions under which the play of a game would result in a Nash equilibrium. A violation of Nash equilibrium is also a violation of the epistemic conditions that imply it. In static game players are uncertain about which actions the other players will take (or have taken). A great deal of evidence suggests that in resolving uncertainty about what other players will do (or have done), players assign diagnostic significance to their own actions. Such reasoning is described as evidential reasoning. Players using evidential reasoning do not believe that their actions cause the action of others; it merely informs their belief about the actions taken by others. However, players who use evidential reasoning can violate standard rationality assumptions, such as Savage’s sure thing principle. Thus, evidential reasoning is best viewed as a heuristic rather than a sound rational principle. The aim of our paper is to explore the significance of evidential reasoning for the class of static games of complete information. We define evidential games in which some players use evidential reasoning. We also propose the relevant solution concepts for such games: evidential equilibrium and consistent evidential equilibrium. The evidence shows that the cooperative outcome in the prisoners’ dilemma game occurs more than 50% of the time, despite cooperation being strictly dominated by defection. Other-regarding preferences and altruism under causal reasoning can both explain this. However, neither team-reasoning, other-regarding preferences nor altruism can explain why the amount of defection increases when players are told that their rivals have, in fact, cooperated. We find that the evidence can best be explained by a mixture of players, some who use evidential reasoning and others who use causal reasoning. Our proposal of an identity social projection function appears adequate for symmetric games. However, our general notion of a social projection functions appears too general (Proposition 3). We illustrated how the Nash bargaining axioms restrict social projections functions sufficiently to get unique outcomes in the Nash demand game (Example 5) and oligopoly games (Proposition 9). Our framework can naturally be extended to incomplete information games and dynamic games, but we lack a body of evidence that could underpin such extensions. Hence, we leave such developments for future research as more evidence accumulates. Acknowledgments We are very grateful to Subir Bose, Andrew Colman, Vincent Crawford, Herbert Gintis, Briony Pulford, Ludovic Renou, Chris Wallace, two anonymous referees and the editor for their comments on the paper. We remain responsible for all errors and omissions. Author Contributions Both authors contributed equally to this article.
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