BBiased-Belief Equilibrium ∗ Yuval Heller † Eyal Winter ‡ June 30, 2020
Final pre-print of a paper published in
American Economic Journal: Microeconomics
Abstract
We investigate how distorted, yet structured, beliefs can persist in strategic situations. Specif-ically, we study two-player games in which each player is endowed with a biased-belief functionthat represents the discrepancy between a player’s beliefs about the opponent’s strategy and theactual strategy. Our equilibrium condition requires that: (1) each player choose a best-responsestrategy to his distorted belief about the opponent’s strategy, and (2) the distortion functionsform best responses to one another. We obtain sharp predictions and novel insights into the setof stable outcomes and their supporting stable biases in various classes of games.JEL classification: C73, D83.Keywords: commitment, indirect evolutionary approach, distortions, wishful thinking, strate-gic complements, strategic substitutes.
Standard models of equilibrium behavior attribute rationality to players at two different levels:beliefs and actions (see, e.g., Aumann and Brandenburger, 1995). Players are assumed to behave asif they form correct beliefs about the opponents’ behavior, and they choose actions that maximizetheir utility given the beliefs that they hold. Much of the literature in behavioral and experimentaleconomics that documents violations of the assumption that players have correct beliefs ascribesthese violations to cognitive limitations. However, in interactive environments where one person’sbeliefs affect other persons’ actions, belief distortions are not arbitrary, and they may arise to servesome strategic purpose.In this paper we investigate how distorted, yet structured, beliefs can persist in strategic situa-tions. Our basic assumption here is that distorted beliefs can persist because they offer a strategicadvantage to those who hold them even when these beliefs are wrong. More specifically, players ∗ The authors are very grateful to the anonymous referees for very helpful comments and suggestions. † Department of Economics, Bar Ilan University, Israel. [email protected]. URL:https://sites.google.com/site/yuval26/. The author is grateful to the
European Research Council for its finan-cial support (ERC starting grant ‡ ∼ mseyal/. The author is grateful to the German-IsraeliFoundation for Scientific Research and Google for their financial support. a r X i v : . [ ec on . T H ] J un often hold distorted beliefs as a form of commitment device that affects the behavior of their coun-terparts. The precise cognitive process that is responsible for the formation of beliefs is complex, andit is beyond the scope of this paper to outline it. We believe, however, that in addition to analyticassessment of evidence, preferences in the form of desires, fears, and other emotions contribute tothe process and, to an extent, facilitate belief biases. If the evidence is unambiguous and decisive,or if the consequence of belief distortion is detrimental to the player’s welfare, preferences may playless of a role and learning may work to calibrate beliefs to reality. But when beliefs are biased inways that favor their holders by affecting the behavior of their counterparts, learning can actuallyreinforce biases rather than diminish them. Biased Beliefs
Standard equilibrium notions in game theory draw a clear line between preferencesand beliefs. The former are exogenous and fixed; the latter can be amended through Bayesianupdating but are not allowed to be affected by preferences. However, phenomena such as wishfulthinking (see, e.g., Babad and Katz, 1991) and overconfidence (see, e.g., Forbes, 2005; Barber andOdean, 2001; Malmendier and Tate, 2005; Heller, 2014), where beliefs are tilted toward what theirholder desires reality to be, suggest that in real life, beliefs and preferences can intermingle, and thatbiased beliefs may be persistent. Similarly, belief rigidity and belief polarization (see, e.g., Lord,Ross, and Lepper, 1979; Ross and Anderson, 1982) refer to situations in which two people withconflicting prior beliefs each strengthen their beliefs in response to observing the same data. Theparties’ aversion to depart from their original beliefs can also be regarded as a form of interactionbetween preferences and beliefs.It is easy to see how the belief biases described above can have strategic benefits in interactivesituations. Wishful thinking and optimism can facilitate cooperation in interactions that requiremutual trust. Overconfidence can deter competitors, and belief rigidity can allow an agent to supporta credible threat. An important objective of our analysis is to identify the strategic environments thatsupport biases such as wishful thinking as part of equilibrium behavior. It is worthwhile to note thatindividuals are not the only ones susceptible to strategically motivated belief biases. Governmentsare prone to be affected by such biases as well. The Bush administration’s unsubstantiated confidencein Saddam Hussein’s possession of “weapons of mass destruction” prior to the Second Gulf War andthe vast discrepancy between Israeli and US intelligence assessments of Iran’s nuclear intentionsprior to the signing of the Iran nuclear deal can be interpreted as strategically motivated beliefdistortion. Belief biases in strategic environments are also connected to self-interest biases regarding moraland ethical standards. Babcock and Loewenstein (1997) had participants in a lab experiment ne-gotiate a deal between a plaintiff and a defendant in a court case. When they asked participants There are other possible interpretations of these controversial real-life examples. In a dynamic real-life setup it ishard to have access to agents’ private information, and therefore it is very difficult to achieve direct empirical evidencefor persistent biased beliefs. There are a few lab experiments that elicit subjects’ beliefs (using monetary incentives andproper scoring rules) about the expected behavior of the opponent. Nyarko and Schotter (2002) demonstrate that theelicited forecasts of subjects about about the opponents’ future behavior substantially differ from the empirical playof opponents in the past. Palfrey and Wang (2009) present evidence that forecasts by players (about the opponent’sbehavior in a simple two-player game) are significantly different from the forecasts of external observers. Moreover, theplayers’ forecasts are systematically biased, and significantly less accurate than the forecasts of the external observers. to make predictions about the outcome of the real court case the authors found a significant beliefdivergence depending on the role participants were assigned to in the negotiations. A similar moralhypocrisy was revealed by Rustichini and Villeval (2014) who showed that subjects’ subjective judg-ments regarding fairness in bargaining depended on the bargaining power they were assigned in theexperiment.A different body of empirical evidence consistent with strategic beliefs is offered by the psychi-atric literature on “depressive realism” (e.g., Dobson and Franche, 1989). This literature comparesprobabilistic assessments conveyed by psychiatrically healthy people with those suffering from clini-cal depression. Participants in both categories were requested to assess the likelihood of experiencingnegative or positive events in both public and private setups. Comparing subjects’ answers with theobjective probabilities of these events revealed that in a public setup clinically depressed individualswere more realistic than their healthy counterparts for both types of events. The apparent belief biasamong healthy individuals can be reasonably attributed to the strategic component of beliefs. Mooddisorders negatively affect strategic reasoning (Inoue, Tonooka, Yamada, and Kanba, 2004), which,to a certain extent, may diminish strategic belief distortion among clinically depressed individualsrelative to their healthy counterparts.For biased beliefs to yield a strategic advantage to the agents holding them, it is essential that(1) agents be committed to follow their biased beliefs, and (2) agents best-reply to the perceivedbehavior induced by their counterparts’ biases (both on and off the equilibrium path). For the sakeof tractability, we shall avoid formalizing a concrete dynamic model that describes how biased beliefsare formed, and how agents credibly commit to these biased beliefs. Instead, we shall adopt a staticapproach by imposing equilibrium conditions on the agents’ beliefs and the opponents’ interpretationof their beliefs. (We discuss our modeling approach and its evolutionary interpretation in Section 3.6,and we present a formal evolutionary foundation in Appendix B.) This static approach is consistentwith a large part of the literature on endogenous preferences (see, e.g., the literature cited below).Nevertheless, we mention a few mechanisms that can facilitate these processes and turn biased beliefsinto a credible commitment device.1. Refraining from accessing or using biased sources of information, e.g., subscribing to a news-paper with a specific political orientation, consulting biased experts, and reading Facebook’spersonalized news feeds, which are typically biased due to friends who hold similar beliefs.2. Passionately following a religion, a moral principle, or an ideology that has belief implicationson human behavior.3. Possessing personality traits that have implications on beliefs (e.g., narcissism or naivety).The mechanisms described above are likely not only to induce belief biases, but also to generatesignals sent to the player’s counterparts about these biases with a certain degree of verifiability.These mechanisms, the signals they induce, and their interpretation are the main forces that facilitatebiased-belief equilibrium.
Solution Concept
Our notion of biased-belief equilibrium (henceforth, BBE) uses a two-stageparadigm. In the first stage each player is endowed with a biased-belief function. This function represents the discrepancy between a player’s beliefs about the strategy profile of other playersand the actual profile. In the second stage the players play the biased game induced by theirdistortion functions, in which each player chooses a best-reply strategy to his biased belief about theopponent’s strategy (the chosen strategy profile is referred to as the equilibrium outcome). Finally,our equilibrium condition requires that the distortion functions not be arbitrary, but form bestreplies to one another.If one of the players deviates to being endowed with a different biased-belief function, then theremight be multiple Nash equilibria in the new biased game induced by this deviation. Our weaknotion ( weak BBE ) requires the deviator to be outperformed in at least one equilibrium of the newbiased game. Our strong notion ( strong BBE ) requires (1) each agent to have a monotone biasedbelief, according to which he assigns a higher probability to his opponent playing a certain strategythan this probability actually is, and (2) a deviator to be outperformed in all
Nash equilibria of thenew biased game. Our main notion,
BBE , lies in between these two notions, and it requires (1) eachagent to have a monotone biased belief, and (2) the deviator to be outperformed in at least oneplausible Nash equilibrium of the new biased game, where we rule out implausible Nash equilibriain which the non-deviator behaves differently even though he does not observe any change in thedeviator’s perceived strategy.In Section 2.5 we present our main evolutionary interpretation of our solution concept, accordingto which the endowed biased beliefs are the result of an evolutionary process of social learning(the interpretation is formalized in Appendix B). In addition, we present an alternative, delegationinterpretation of the model (which is formalized in Appendix C).
Nash Equilibrium and BBE
We begin our analysis by studying the relations between BBEoutcomes and Nash equilibria. We show that any Nash equilibrium can be implemented as theoutcome of a BBE, though in some cases this requires that the players have biased beliefs that areaccurate on the equilibrium path, but that they be blind to some deviations of the opponent offthe equilibrium path. This, in particular, implies that every game admits a BBE. Next, we showthat introducing biased beliefs does not change the set of equilibrium outcomes in games in whichat least one of the players has a dominant action. By contrast, BBE admits non-Nash behavior inmost other games.
Main Results
Our main results show that the notion of BBE induces substantial predictivepower in various classes of interval games. In these classes of games the strategy of each playeris a number in a bounded interval, where a higher strategy (interpreted as a higher investment)induces a higher payoff for the opponent. We begin by characterizing the set of BBE in games withstrategic complements (Bulow, Geanakoplos, and Klemperer, 1985), such as price competition withdifferentiated goods (Example 2), input games (Example 9 in Appendix A.4), and stag hunt games(Example 10 in Appendix A.4). We show three key properties of any BBE: (1) overinvestment :the strategy of each agent is (weakly) higher than the best reply to the opponent’s (real) strategy,(2) ruling out bad outcomes : both players invest more than their investments in the worst Nashequilibrium of the underlying game, and (3) wishful thinking : each agent perceives his opponent asinvesting (weakly) more than the opponent’s real investment.Next, we characterize the set of BBE in games with strategic substitutes, such as Cournotcompetitions (Example 3) and hawk-dove games (Example 5 in Appendix A.5). We show three keyproperties of any BBE: (1) underinvestment : the strategy of each agent is (weakly) higher thanthe best reply to the opponent’s (real) strategy, (2) ruling out excellent outcomes : at least one ofthe players invests less than his investments in one of the Nash equilibria of the underlying game,and (3) wishful thinking : each agent perceives his opponent as investing (weakly) more than theopponent’s real investment.Finally, we characterize the set of BBE in a class of games (which are less common in economicinteractions), in which the strategy of player 1 is a complement of player 2’s strategy, while thestrategy of player 2 is a substitute of player 1’s strategy (e.g., duopolistic competition in which onefirm chooses its quantity while the opposing firm chooses its price (Singh and Vives, 1984), andvarious classes of asymmetric contests (Dixit, 1987)). We show that in this class of games agentspresent pessimism in any BBE : each agent perceives his opponent as investing (weakly) less thanthe opponent’s real investment. Additional Results
Our next result shows an interesting class of BBE that exist in all games.We say that a strategy is undominated Stackelberg if it maximizes a player’s payoff in a setupin which the player can commit to an undominated strategy, and his opponent reacts by best-replying to this strategy. We show that every game admits a BBE in which one of the players is“strategically stubborn” in the sense of having a constant belief about the opponent’s strategy, andalways playing his undominated Stackelberg strategy, while the opponent is “rational” in the senseof having undistorted beliefs and best-replying to the player’s true strategy.Section 7.2 shows that unless one imposes both requirements on the definition of a BBE, namely,monotonicity and ruling out implausible equilibria, then the set of BBE outcomes is very large invarious classes of games. Specifically, Proposition 8 shows that for a large class of finite games, astrategy profile is a monotone weak BBE iff (1) no player uses a strictly dominated strategy, and (2)the payoff of each player is above the minmax payoff of the player in a setup in which both playersare restricted to choose only undominated strategies (i.e., strategies that are not strictly dominated).Proposition 9 shows a similar folk theorem result for non-monotone strong BBE in a large class ofinterval games.
Empirical Predictions
Our main results imply two empirical predictions. First, they suggestthat efficient (non-Nash equilibrium) outcomes are easier to support in games with strategic comple-ments, relative to games with strategic substitutes. This prediction is consistent with the experimen-tal findings of Potters and Suetens (2009), which show that there is significantly more cooperationin games with strategic complements than in the case of strategic substitutes.Our second empirical prediction is that wishful thinking is strategically stable in many com-mon environments, though some (less common) strategic interactions may induce pessimism. Thisempirical prediction is consistent with the experimental evidence that people tend to present wish-ful thinking, while the presented level of wishful thinking may substantially differ between various environments; see, e.g., Babad and Katz (1991); Budescu and Bruderman (1995); Bar-Hillel andBudescu (1995) and Mayraz (2013).
Structure
The structure of this paper is as follows. We discuss the related literature in Section2. Section 3 describes the model. In Section 4 we analyze the relations between BBE and Nashequilibria. Section 5 defines games with strategic complements/substitutes and wishful thinking. Weanalyze these games and present our main results in Section 6. In Section 7 we present additionalresults: (1) the relation between BBE and strategies played by a Stackelberg leader, and (2) folktheorem results when relaxing the definition of BBE. We conclude in Section 8. All the appendicesof the paper appear in the online supplementary material. Appendix A presents various interestingexamples. We formally present the evolutionary interpretation of our solution concept in AppendixB, and the delegation interpretation in Appendix C. Appendix D relaxes the assumption that biasedbeliefs have to be continuous. Appendix E shows how to extend our results to a setup with partialobservability. Appendix F presents our formal proofs.
Our paper aims at making a contribution to the behavioral game theory literature. Much of thisliterature concerns behavioral equilibrium concepts that depart from the framework of Nash equi-librium by introducing weaker rationality conditions. This has been done primarily at the levelof preferences (e.g., Güth and Yaari, 1992; Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000;Acemoglu and Yildiz, 2001; Heifetz, Segev, et al., 2004; Dekel, Ely, and Yilankaya, 2007; Heifetz,Shannon, and Spiegel, 2007a; Friedman and Singh, 2009; Herold and Kuzmics, 2009; Heller andWinter, 2016; Winter, Garcia-Jurado, and Mendez-Naya, 2017). But it has also been done at thelevel of beliefs (e.g., Geanakoplos, Pearce, and Stacchetti, 1989; Rabin, 1993; Battigalli and Dufwen-berg, 2007; Attanasi and Nagel, 2008; Battigalli and Dufwenberg, 2009; Battigalli, Dufwenberg, andSmith, 2015; Gannon and Zhang, 2017). This latter literature deals with belief-dependent prefer-ences, and focuses primarily on the way players’ beliefs about the intentions of others affect theirpreferences and behavior.Our equilibrium concept also operates on beliefs rather than preferences but is based on aninherently different approach. Preferences in our model are not affected by beliefs but beliefs arebiased in a way that serves players’ strategic purposes. Our analysis of biased belief goes beyondcharacterizing equilibrium outcomes. An additional important objective is to identify the beliefbiases that support these equilibrium outcomes in different strategic environments. Central to ouranalysis are belief-distortion properties, such as wishful thinking and pessimism, that sustain BBEin different strategic environments.The existing literature has presented various prominent solution concepts that assume that play-ers have distorted beliefs. Some examples include models of level- k and cognitive hierarchy (see, e.g.,Stahl and Wilson, 1994; Nagel, 1995; Costa-Gomes, Crawford, and Broseta, 2001; Camerer, Ho, andChong, 2004), analogy-based expectation equilibrium (Jehiel, 2005), cursed equilibrium (Eyster andRabin, 2005), and Berk-Nash equilibrium (Esponda and Pouzo, 2016). These equilibrium notionshave been helpful in understanding strategic behavior in various setups, and yet these notions posea conceptual challenge to our understanding of the persistence of distorted beliefs, even in view ofthe empirical evidence for such persistence. If players can infer the truth ex post why don’t theycalibrate their beliefs toward reality? Much of the literature presenting such models points to cogni-tive limitations as the source of this rigidity. Our model and analysis offer an additional perspectiveto this issue by suggesting that belief biases that yield a strategic advantage in the long run arelikely to emerge in equilibrium. In this sense our approach can be viewed as providing a tool toexplain why some cognitive limitations persist while others do not (see Example 11 in Appendix A,in which we show how level-1 behavior can be supported as part of a BBE outcome in the traveler’sdilemma).Our notion of BBE is related to the notion of conjectural equilibrium (Battigalli and Guaitoli,1997, originally written in 1988) insofar as both solution concepts relax the Nash equilibrium’s re-quirement that beliefs need to be consistent with actual play (while still requiring that an agent’saction has to be optimal given the agent’s belief). A conjectural equilibrium is defined in an en-vironment in which players do not observe each other’s actions but rather observe signals of eachother’s actions, according to an exogenous feedback correspondence . In a conjectural equilibriumeach player best replies to his belief about the opponent’s action, and this belief is required to beconsistent with the signal observed by the player. There are two key structural differences betweena BBE and a conjectural equilibrium. First, a BBE is defined in an environment in which there isno exogenous feedback correspondence; rather, the feedback correspondence is implicitly defined aspart of the solution concept by the agents’ biased-belief functions. These biased-belief functions arenot restricted by a consistency requirement with respect to an exogenous feedback mechanism, butrather they are are restricted by the requirement that each biased-belief function has to be a bestreply against the opponent’s biased belief. The second structural difference is that while a BBEdescribes what would be the agent’s belief for any feasible action of the opponent, a conjecturalequilibrium describes only the agent’s belief about the equilibrium action of the opponent.Despite these structural differences, it is interesting to discuss relations between the equilib-rium behavior induced by each solution concept, i.e., the relations between a BBE outcome and aconjectural equilibrium outcome. Without restricting the feedback correspondence, the notion ofconjectural equilibrium is rather broad (it rules out only strictly dominated strategies), and, ac-cordingly, any BBE outcome is a conjectural equilibrium outcome. Fudenberg and Levine’s (1993)notion of self-confirming equilibrium deals with extensive-form games, and refines conjectural equi-librium by requiring that the feedback correspondence is the one in which each player observesthe opponent’s realized actions (but does not observe the opponent’s behavior off the equilibriumpath). In the setup of two-player one-shot games, which is the focus of the present paper, the setof self-confirming equilibria coincides with the set of Nash equilibria (whereas the set of BBE out-comes is broader and includes non-Nash outcomes). Another refinement of conjectural equilibriumis the rationalizable conjectural equilibrium (Rubinstein and Wolinsky, 1994; the notion has beengeneralized to games with structural uncertainty in Esponda, 2013). This concept requires that theagents’ beliefs be consistent with the common knowledge that all players maximize utility given theirsignals. There is no inclusion relation between the set of BBE outcomes and the set of rationalizable conjectural equilibrium outcomes. Specifically, in games with a unique rationalizable action profile,such as price competitions with differentiated goods and Cournot competitions, the unique rational-izable conjectural equilibrium outcome is the Nash equilibrium (for any feedback correspondence),while the set of BBE outcomes is substantially larger (see Examples 2 and 3). By contrast, in gamessuch as stag hunt and hawk–dove, when the feedback correspondence is non-informative any actionprofile is a conjectural equilibrium outcome, while the set of BBE outcomes is much more restricted(see Examples 10 and 12 in Appendix A).
Let i ∈ { , } be an index used to refer to one of the players in a two-player game, and let j bean index referring to the opponent. Let G = ( S, π ) be a normal-form two-player game (henceforth, game ), where S = ( S , S ) and each S i is a convex compact set of strategies. Specifically, we focuson two cases:1. Finite games : Each S i is a simplex over a finite set of pure actions, where each strategycorresponds to a mixed action (i.e., A i is a finite set of pure actions, and S i = ∆ ( A i )), and thevon Neumann–Morgenstern payoff function is linear with respect to the mixing probability.2. Interval games : Each S i is a bounded interval in R (e.g., each player chooses a real numberrepresenting quantity, price, or effort).We denote by π = ( π , π ) players’ payoff functions; i.e., π i : S → R is a function assigning eachplayer a payoff for each strategy profile. We use s i to refer to a typical strategy of player i . Weassume each payoff function π i ( s i , s j ) to be continuously twice differentiable in both parameters andweakly concave in the first parameter ( s i ).Let BR (resp., BR − ) denote the (inverse) best-reply correspondence; i.e., BR ( s i ) = argmax s j ∈ S j ( π j ( s i , s j ))is the set of best replies against strategy s i ∈ S i , and BR − ( s i ) = { s j ∈ S j | s i ∈ BR ( s j ) } is the set of strategies for which s i is a best reply against them.In a finite game, we use a i ∈ A i to denote also the degenerate mixed action that assigns mass oneto a i . When the set of actions of a player is given as an ordered set A i = (cid:0) a i , a i , ..., a ni (cid:1) , we identifya mixed action with a vector s i = ( α , α , ..., α n ), where 0 ≤ α k = s i (cid:16) a ki (cid:17) for each 1 ≤ k ≤ n , and P k α k = 1. Given two strategies s i , s i ∈ S j and α ∈ [0 , α · s i + (1 − α ) · s i be the mixture ofthe two strategies: ( α · s i + (1 − α ) · s i ) ( a i ) = α · s i ( a i ) + (1 − α ) · s i ( a i ).When there are two (ordered) actions for each player (say, A i = { c i , d i } ), we identify a mixedaction s i with the probability it assigns to the first pure action s i ( c i ), and we identify the set of .2 Biased-Belief Function S i with the interval [0 , We start here with the definition of biased-belief functions that describe how players’ beliefs aredistorted. A biased belief ψ i : S j → S j is a continuous function that assigns to each strategy ofthe opponent, a (possibly distorted) belief about the opponent’s play. That is, if the opponentplays s j , then player i believes that the opponent plays ψ i ( s j ). We call s j the opponent’s realstrategy, and we call ψ i ( s j ) the opponent’s perceived (or biased) strategy. Formally, the continuityrequirement is that if ( s j,n ) n → n →∞ s j , then ( ψ i ( s j,n )) n → n →∞ ψ i ( s j ) (where in a finite game, we saythat ( s j,n ) n → n →∞ s j iff ( s j,n ( a )) n → n →∞ s j ( a ) for each action a ). Remark . Two reasons motivate us to require that a biased belief be continuous: (1) continuityimplies that each biased game (defined below) admits a Nash equilibrium, which allows us to simplifythe definition of BBE, and (2) continuity reflects a plausible restriction that a small change inthe opponent’s strategy should induce a small change in the perceived strategy. In Appendix Dwe present an alternative (and somewhat more complicated) definition of a BBE that relaxes theassumption that biased beliefs must be continuous, and we show that all the BBE characterized inthe results of the paper remain BBE when we allow deviators to use discontinuous biased beliefs.We say that a biased belief ψ i : S j → S j is monotone if:1. In interval games: s j ≥ s j implies ψ i ( s j ) ≥ ψ i (cid:16) s j (cid:17) for each strategy s j ∈ S j .2. In finite games: If the opponent plays a j more often, while keeping the same proportion ofplaying the remaining actions, then the perceived probability that the opponent plays anyother action weakly decreases (which implies, in particular, that the perceived probabilitythat the opponent plays a j weakly increases); that is,( ψ i ((1 − α ) · s j + α · a j )) (cid:16) a j (cid:17) ≤ ( ψ i ( s j )) (cid:16) a j (cid:17) for each α ∈ [0 , a j ∈ A j , each action a j = a j , and each strategy s j ∈ ∆ ( A j ).In particular, when the game has two actions for each player, a biased belief ψ i is monotoneiff ψ i is weakly increasing in α j ; i.e., α j ≥ α j implies that ψ i ( α j ) ≥ ψ i (cid:16) α j (cid:17) .Monotone biased beliefs reflect a plausible restriction on the distortion of agents, namely, that if theopponent changes his real strategy in some direction, the agent captures the direction of the changecorrectly, but may have the wrong perception about the magnitude of the change.Let I d be the undistorted (identity) function, i.e., I d ( s ) = s for each strategy s . A biased belief ψ is blind if the perceived opponent’s strategy is independent of the opponent’s real strategy, i.e., if ψ ( s j ) = ψ (cid:16) s j (cid:17) for each s j , s j ∈ S j . With a slight abuse of notation we use s i to denote also theblind biased belief ψ j that is always equal to s i .0 An underlying game and a profile of biased beliefs jointly induce a biased game in which the (biased)payoff of each player is determined by the perceived strategy of the opponent. Formally:
Definition 1.
Given an underlying game G = ( S, π ) and a profile of biased beliefs ( ψ i , ψ j ), let the biased game G ψ = ( S, ψ ◦ π ) be defined as the game with the following payoff function ( ψ ◦ π ) i : S i × S j → R for each player i : ( ψ ◦ π ) i ( s i , s j ) = π i ( s i , ψ i ( s j )) . A Nash equilibrium of a biased game is defined in the standard way. Formally, a pair of strategies s ∗ = ( s ∗ , s ∗ ) is a Nash equilibrium of a biased game G ψ = ( S, ψ ◦ π ), if each s ∗ i is a best reply againstthe perceived strategy of the opponent, i.e., s ∗ i = argmax s i ∈ S i (cid:16) π i (cid:16) s i , ψ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) . Let
N E ( G ψ ) ⊆ S × S denote the set of all Nash equilibria of the biased game G ψ . Observe that the set of strategies of a biased game is convex and compact, and the payoff function( ψ ◦ π ) i : S i × S j → R is weakly concave in the first parameter and continuous in both parameters.This implies (due to a standard application of Kakutani’s fixed-point theorem) that each biasedgame G ψ admits a Nash equilibrium (i.e., N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) = ∅ .) We are now ready to define our equilibrium concept. A weak biased-belief equilibrium (abbr. weakBBE) is a pair consisting of a profile of biased beliefs and a profile of strategies, such that: (1)each strategy is a best reply to the perceived strategy of the opponent, and (2) each biased beliefis a best reply to the opponent’s biased belief, in the sense that any agent who chooses a differentbiased-belief function is outperformed in at least one equilibrium in the new biased game (relativeto the agent’s payoff in the original equilibrium). Formally:
Definition 2. A weak BBE is a pair ( ψ ∗ , s ∗ ), where ψ ∗ = ( ψ ∗ , ψ ∗ ) is a profile of biased beliefs and s ∗ = ( s ∗ , s ∗ ) is a profile of strategies satisfying: (1) (cid:16) s ∗ i , s ∗ j (cid:17) ∈ N E ( G ψ ∗ ), and (2) for each player i and each biased belief ψ i , there exists a strategy profile (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , such that thefollowing inequality holds: π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) .The notion of weak BBE is arguably too permissive because it allows incumbents: (1) to haveimplausible non-monotone beliefs, and (2) to outperform the deviators in a single Nash equilibriumof the biased game (while, possibly, the incumbents are outperformed by the deviators in manyother equilibria). Proposition 8 (in Section 7.2.2) demonstrates that this single Nash equilibrium,in which the deviators are outperformed, may be implausible due to allowing the incumbents to“discriminate” against the deviators, even though the deviators exhibit exactly the same perceivedbehavior as the rest of the population. .5 BBE all equilibriaof the induced biased game. Formally: Definition 3. A weak BBE ( ψ ∗ , s ∗ ) is a strong BBE if (1) each biased function ψ ∗ i is monotone,and (2) the inequality π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) holds for every player i , every biased belief ψ i , andevery strategy profile (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) . Finite games typically induce multiple Nash equilibria. This is often the case also with respect tobiased games. This suggests that the refinement of strong BBE may be too restrictive, as there areare potentially many Nash equilibria of many biased games, and the requirement of the deviatorsbeing outperformed in all these equilibria might be too demanding. Our main solution concept,BBE, lies in between weak BBE and strong BBE.In a BBE, the deviator is required to be outperformed in at least one plausible equilibrium of thenew biased game. Roughly speaking, in a plausible equilibrium of the new biased game induced bya deviation of player i to a different biased belief, player j is allowed to choose a new strategy onlyif he distinguishes between i ’s original strategy and i ’s new strategy. More precisely, implausibleequilibria are defined as follows. We say that a Nash equilibrium of a biased game induced by adeviation of player i is implausible if (1) player i ’s strategy is perceived by the non-deviating player j as coinciding with player i ’s original strategy, (2) player j plays differently relative to his originalstrategy, and (3) player j playing his original strategy induces an equilibrium of the biased game.That is, implausible equilibria are those in which the non-deviating player j plays differently againsta deviator even though player j has no reason to do so: player j does not observe any change inplayer i ’s behavior, and player j ’s original behavior remains an equilibrium of the biased game.Formally: Definition 4.
Given weak BBE ( ψ ∗ , s ∗ ), deviating player i , and biased belief ψ i , we say that a Nashequilibrium of the biased game (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) is implausible if: (1) ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ),(2) s ∗ j = s j , and (3) (cid:16) s i , s ∗ j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) . An equilibrium is plausible if it is not implausible.Let P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) be the set of all plausible equilibria of the biased game G ( ψ i ,ψ ∗ j ).Note that it is immediate from Definition 4 and the nonemptiness of N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) that P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) is nonempty. Definition 5.
Weak BBE ( ψ ∗ , s ∗ ) is a BBE if (1) each biased function ψ ∗ i is monotone, and (2)for each player i and each biased belief ψ i , there exists a plausible Nash equilibrium (cid:16) s i , s j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , such that π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) .2 A strategy profile s ∗ = ( s ∗ , s ∗ ) is a (resp., strong, weak) BBE outcome if there exists a profile ofbiased beliefs ψ ∗ = ( ψ ∗ , ψ ∗ ) such that ( ψ ∗ , s ∗ ) is a (resp., strong, weak) BBE. In this case we saythat the biased belief ψ ∗ supports (or implements) the outcome s ∗ . Evolutionary/Learning Interpretation
Biases can emerge in a learning process that reinforcesbiases that yield a strategic advantage to their holders. Specifically, we interpret a BBE to be areduced-form solution concept capturing the essential features of an evolutionary process of culturalor social learning. Our methodology follows the extensive literature that studies the stability ofendogenous preferences using the “indirect evolutionary approach” (see, e.g., Güth and Yaari, 1992;Güth, 1995; Fershtman and Weiss, 1998; Dufwenberg and Güth, 1999; Koçkesen, Ok, and Sethi,2000; Guttman, 2003; Güth and Napel, 2006; Heifetz, Shannon, and Spiegel, 2007b; Friedman andSingh, 2009; Herold and Kuzmics, 2009; Alger and Weibull, 2013; Heller and Mohlin, 2017). Weapply this modeling approach to the study of endogenous biased beliefs in a setup in which biasedbeliefs induce behavior, behavior determines “success,” and success regulates the evolution of biasedbeliefs.In Appendix B we formally adapt the definition of a stable population state from Dekel, Ely,and Yilankaya (2007) to our setup, and show that the adapted definition is equivalent to a strongBBE. In what follows we briefly and informally present our evolutionary interpretation. Considertwo large populations of agents: agents who play the role of player 1, and agents who play the roleof player 2. In each round agents from each population are randomly matched to play a two-persongame against opponents from the other population. Each agent in each population is endowed witha biased-belief function. For simplicity, we focus on “homogeneous” populations, in which all agentsin the population have the same monotone biased-belief function. Agents distort their perceptionabout the behavior of the agents in the other population according to their endowed biased-belieffunctions, and they play a Nash equilibrium of the biased game.With small probability a few agents (“mutants”) in one of the populations (say, population 1)may be endowed with a different biased-belief function due to a random error or experimentation.We assume that agents of population 2 observe whether their opponents are mutants or not, and thatthe agents of population 2 and the mutants of population 1 gradually adapt their play against eachother into an equilibrium of the new biased game. Note that a dynamic adaptation into playing aNash equilibrium of the biased game requires agents of population 2 to know the perceived strategycurrently being played by the mutants of population 1, but the agents do not need to know thebiased beliefs of the mutants of population 1.Finally, we assume that the total “success” (fitness) of agents is monotonically influenced by their(unbiased) payoff in the underlying game, and that there is a slow process in which the compositionof the population evolves. This slow process might be the result of a slow flow of new agents whojoin the population. Each new agent randomly chooses one of the incumbents in his own populationas a “mentor” (and mimics the mentor’s biased belief), where the probabilities are such that agentswith higher fitness are more likely to be chosen as mentors. If the original population state is nota BBE, it implies that there are mutants who outperform the remaining incumbents in their own .6 Discussion of the Model
Variants of the Solution Concept
The main solution concept we use in the paper is BBE .In Section 7.2 we demonstrate that unless one applies both requirements of Definition 5, namely,monotonicity and ruling out implausible equilibria, then the set of BBE is very large (folk theoremresults), and some of the biased beliefs that support some of these equilibria seem implausible. Theintuition for the monotonicity requirement is quite straightforward (ruling out peculiar biased beliefsin which an opponent who deviates to play a higher strategy is perceived as deviating to play a lowerstrategy). The second requirement rules out implausible equilibria in which a player responds to hisopponent’s deviation in spite of not being able to perceive itIn what follows we sketch a dynamic justification for the second requirement of ruling out im-plausible equilibria (following the evolutionary interpretation described above). Consider a BBE(( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )). Assume that both ( s , s ) and ( s , s ∗ ) are Nash equilibria of the biased game G ( ψ ,ψ ∗ ). In what follows, we briefly, and informally, explain why ( s , s ) is not a plausible equi-librium of the new biased game (and, thus, why it is ruled out in the definition of BBE). Considera deviation of some agents in the population playing in the role of player 1 to having the biasedbelief ψ . Following this deviation, strategy s ∗ might not be a best reply to the perceived strategy ofplayer 2 (i.e., s ∗ BR ( ψ ( s ∗ ))) and, as a result, the deviating agents might change their strategy to s , which is a best reply to the perceived strategy of player 2 (i.e., s ∈ BR ( ψ ( s ∗ ))). The currentstrategy profile ( s , s ∗ ) is a Nash equilibrium of the biased game (i.e., ( s , s ∗ ) ∈ N E (cid:16) G ( ψ ,ψ ∗ ) (cid:17) ). Inorder to move from this equilibrium to ( s , s ), agents of population 2, who are matched against thedeviators, have to change their behavior from s ∗ to s , but there is no reason for them to do so, astheir current behavior (namely, s ∗ ) is already a best reply to the perceived strategy of the deviators(i.e., s ∗ ∈ BR ( ψ ∗ ( s ))), as well as being how they are used to playing against non-deviators. Delegation Interpretation
A different interpretation of our solution concept relies on strategicdelegation. The literature on strategic delegation (see, e.g., Fershtman, Judd, and Kalai, 1991;Dufwenberg and Güth, 1999; Fershtman and Gneezy, 2001) deals with players who strategically useother agents to play on their behalf, where the agents so used may have different preferences thanthe players using them. We adapt this approach to our setup in which agents differ in their biasedbeliefs (rather than in their preferences). Specifically, in Appendix C we show that the notion ofweak BBE is equivalent to a subgame-perfect equilibrium of a two-stage game in which in stage oneeach unbiased player strategically chooses the biased belief of his agent, and in the second stage thebiased agents play on behalf of the players (and each agent can observe the opposing agent’s biasedbeliefs).4
Partial Observability
The requirement that an agent be able to observe that his opponentbelongs to a group of “mutant” agents who have different biased beliefs than the rest of the populationcan be explained by pre-play social cues and messages that facilitate this observation. In Appendix Ewe show that this observability need not be perfect. We generalize the model to partial observabilityby studying a setup in which, when an agent is matched with a mutant opponent, the agent privatelyobserves the opponent to be a mutant with probability 0 < p ≤
1. We show that all our results holdin this extended setup for p sufficiently close to one (and some of the results hold also for low levelsof p ). In this section we study the relations between Nash equilibria and BBE outcomes.
We begin with a simple observation that shows that in any weak BBE in which the outcome is nota Nash equilibrium, at least one of the players must distort the opponent’s perceived strategy. Thereason for this observation is that if both players have undistorted beliefs, then it must be that eachagent best-replies to the opponent’s strategy, which implies that the outcome is a Nash equilibriumof the underlying game.The following example demonstrates that even Nash equilibria may require distorted beliefsto be supported as BBE outcomes. Specifically, Example 1 shows that this is the case for Nashequilibrium in a Cournot competition. The intuition behind Example 1 is straightforward. TheCournot equilibrium cannot be supported by undistorted beliefs because such pairs of beliefs willinduce one of the players to adopt a distorted belief by which he expects his opponent not to produceat all, and to best-reply to this distorted belief by producing the monopoly quantity. This in turnwill force the opponent to reduce his production substantially below the Cournot level, making thedeviator better off.
Example 1 ( Cournot equilibrium cannot be supported by undistorted beliefs, yet it can be supportedby blind beliefs ) . Consider the following symmetric Cournot game G = ( S, π ): S i = [0 ,
1] and π i ( s i , s j ) = s i · (1 − s i − s j ) for each player i . The interpretation of the game is as follows. Each s i is interpreted as the quantity chosen by firm i , the price of both goods is determined by thelinear inverse demand function p = 1 − s i − s j , and the marginal cost of each firm is normalized tobe zero. The unique Nash equilibrium of the game is s ∗ i = s ∗ j = , which yields a payoff of toboth players. Assume to the contrary that this outcome can be supported as a weak BBE by theundistorted beliefs ψ ∗ i = ψ ∗ j = I d . Consider a deviation of player 1 to the blind belief ψ ≡
0. Theunique equilibrium of the biased game G (0 ,I d ) is s = , s = , which yields a payoff of > tothe deviator. The unique Nash equilibrium s ∗ i = s ∗ j = can be supported as the outcome of thestrong BBE (cid:16)(cid:16) , (cid:17) , (cid:16) , (cid:17)(cid:17) with blind beliefs, in which each agent believes the opponent is playing regardless of the opponent’s actual play, and the agent plays the unique best reply to this belief,which is the strategy . .2 Any Nash Equilibrium is a BBE Outcome Remark . We interpret an undistortedbelief as describing an agent who has an accurate belief about the opponent’s behavior on theequilibrium path, and, in addition, the agent keeps looking for cues that his opponent might havea different type, and if the agent observes such a cue, the agent evaluates the opponent’s likelybehavior, and best-replies to this assessment. Example 1 shows that the Cournot equilibrium cannotbe supported by a population in which each agent keeps looking for cues for his opponent’s type.In such a population, deviators would strictly earn by having a blind biased belief that induces thedeviator to play the Stackelberg strategy. The incumbents will identify the mutants’ type, and theywill respond by playing the Stackelberg follower action, which will benefit the deviators.By contrast, the second part of Example 1 (and its generalization in Proposition 1 below) showsthat any Nash equilibrium can be supported by a blind belief, which is accurate on the equilibriumpath. We interpret such a belief as describing an agent who understands correctly the equilibriumbehavior of the opposing player, and ignores signals that suggest that his opponent is about to dosomething else. Our observation that it is rather equilibrium that supports belief rigidity, a prevalentbehavioral phenomenon, and not disequilibrium is, we believe, quite interesting.
The following result generalizes the second part of Example 1, and shows that any (strict) Nashequilibrium is an outcome of a (strong) BBE in which both players have blind beliefs that areaccurate on the equilibrium path.
Proposition 1.
Let ( s ∗ , s ∗ ) be a (strict) Nash equilibrium of the game G = ( S, π ) . Let ψ ∗ ≡ s ∗ and ψ ∗ ≡ s ∗ . Then (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a (strong) BBE.Proof. The fact that ( s ∗ , s ∗ ) is a Nash equilibrium of the underlying game implies that ( s ∗ , s ∗ ) isan equilibrium of the biased game G ( ψ ∗ ,ψ ∗ ). The fact that the beliefs are blind implies that forany biased belief ψ i , there is an equilibrium in the biased game G ( ψ i ,ψ ∗ j ) in which player j plays s ∗ j and player i gains at most π i (cid:16) s ∗ i , s ∗ j (cid:17) , which implies that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE. Moreover, if( s ∗ , s ∗ ) is a strict equilibrium, then in any equilibrium of any biased game G ( ψ i ,ψ ∗ j ), player j plays s ∗ j and player i gains at most π i (cid:16) s ∗ i , s ∗ j (cid:17) , which implies that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a strong BBE.An immediate corollary of Proposition 1 is that every game admits a BBE. Corollary 1.
Every game admits a BBE.
Recall that a game is zero sum if there exists c ∈ R + such that π i ( s i , s j ) + π j ( s i , s j ) = c for eachstrategy profile ( s i , s j ) ∈ S .The following simple result shows that the unique Nash equilibrium payoff of a zero-sum gameis also the unique payoff in any weak BBE. Claim . The unique Nash equilibrium payoff of a zero-sum game is also the unique payoff in anyweak BBE.6
Proof.
Let v i be the unique Nash equilibrium payoff of player i in the underlying zero-sum game.Assume to the contrary that there exists a weak BBE ( ψ ∗ , s ∗ ) in which the payoff of player i isstrictly lower than v i . Consider a deviation of player i into the undistorted bias function ψ i = I d .The assumption that ( ψ ∗ , s ∗ ) is a weak BBE implies that the deviator gets strictly less than v i ina Nash equilibrium (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , but this is impossible as the definition of v i impliesthat there exists ˆ s i satisfying π i (cid:16) ˆ s i , s j (cid:17) ≥ v i > π i (cid:16) s i , s j (cid:17) .Example 6 in Appendix A.2 shows that even though the weak BBE payoff must be the Nashequilibrium payoff in a zero-sum game, the strategy profile sustaining it need not be a Nash equi-librium. Next we show that if at least one of the players has a dominant strategy, then any weak BBEoutcome must be a Nash equilibrium. Formally:
Proposition 2.
If a game admits a strictly dominant strategy s ∗ i for player i , then any weak BBEoutcome is a Nash equilibrium of the underlying game.Proof. Observe that s ∗ i is the unique best reply of player i to any perceived strategy of player j ,and, as a result, player i plays the dominant action s ∗ i in any weak BBE. Assume to the contrarythat there is a weak BBE in which player j does not best-reply against s ∗ i . Consider a deviation ofplayer j to choosing the undistorted belief I d . Observe that player i still plays his dominant action s ∗ i , and that player j best-replies to s ∗ i in any Nash equilibrium of the induced biased game, and, asa result, player j achieves a strictly higher payoff, and we get a contradiction.Proposition 2 implies, in particular, that defection is the unique weak BBE outcome in theprisoner’s dilemma game. Example 7 in Appendix A.1 demonstrates that a relatively small changeto the prisoner’s dilemma game, namely, adding a third weakly dominated “withdrawal” strategythat transforms “cooperation” into a weakly dominated strategy, allows us to sustain cooperationas a strong BBE outcome. In this section we present a large class of games with monotone externalities and monotone differ-ences, and define the notions of wishful thinking and pessimism, which will be analyzed in Section6.
We say that an interval game is monotone if it satisfies two conditions:1.
Monotone externalities : the payoff function of each player is strictly monotone in the oppo-nent’s strategy. Without loss of generality, we assume that the externalities are positive , i.e., .2 Wishful Thinking ∂π i ( s i ,s j ) ∂s j > i and each pair of strategies s i , s j . The assumption of positive externalities (givenmonotone externalities) is indeed without loss of generality because if originally the external-ities with respect to player j are negative, then we can redefine player j ’s strategy to be itsinverse, and obtain positive externalities; for example, defining the difference between maximalcapacity and quantity to be the strategy of each player in a Cournot competition yields a gamewith positive externalities.In a game with positive externalities we refer to a player’s strategy as his investment , andwhen s i increases we refer to this increase a larger investment by as player i .2. Monotone differences : For each player i , the derivative of the player’s payoff with respect tohis own strategy (i.e., ∂π i ( s i ,s j ) ∂s i ) is strictly monotone in the opponent’s strategy. Specifically,we divide the set of monotone games into three disjoint and exhaustive subsets:(a) Strategic complements (increasing differences, supermodular games) : ∂π i ( s i ,s j ) ∂s i is strictlyincreasing in s j for each player i and each strategy s i (or, equivalently, ∂ π i ( s i ,s j ) ∂s i · ∂s j > s i , s j ). Games with strategic complements are common in the economics literature,and include, in particular, price competitions with differentiated goods (Example 2), inputgames (Example 9 in Appendix A.4), and stag-hunt games (Example 10 in AppendixA.4). Finite games with a payoff structure that resembles a discrete variant of strategiccomplements include the traveler’s dilemma (Example 11 in Appendix A.4).(b) Strategic substitutes (decreasing differences, submodular games ): ∂π i ( s i ,s j ) ∂s i is strictly de-creasing in s j for each player i and each strategy s i (or, equivalently, ∂ π i ( s i ,s j ) ∂s i · ∂s j < s i , s j ). Games with strategic substitutes are common in the economics literature, andinclude, in particular, Cournot (quantity) competitions (Example 3 below) and hawk-dovegames (see Example 12 in Appendix A.5).(c) Opposing differences : ∂π i ( s i ,s j ) ∂s i is decreasing in s j (for each strategy s i ), while ∂π j ( s i ,s j ) ∂s j isincreasing in s i (for each strategy s j ). Games with opposing differences are less commonin the economics literature. Examples of these games include (1) duopolies in which onefirm chooses its quantity, while the other firm chooses its price (see, e.g., Singh and Vives,1984), and (2) asymmetric contests, in which it is often the case that a commitment ofthe favorite (underdog) player to exert more (less) effort induces the opponent to exertless effort (see, e.g., Dixit, 1987). We say that player i exhibits wishful thinking if the perceived opponent’s strategy yields a higherpayoff to the player relative to the real strategy the opponent plays. Formally: Definition 6.
Player i exhibits wishful thinking in weak BBE (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) if π i ( s i , ψ ∗ i ( s j )) ≥ π i (cid:16) s i , s ∗ j (cid:17) for each s i ∈ S i . Remark . Note that in a game with positive externalities player i exhibits wishful thinking in weakBBE (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) iff ψ ∗ ( s ∗ ) ≥ s ∗ and ψ ∗ ( s ∗ ) ≥ s ∗ .8 Similarly, we define the opposite notion, that of exhibiting pessimism. We say that a BBEexhibits pessimism if the perceived opponent’s strategy yields a lower payoff to the player relativeto the real opponent’s strategy for all strategy profiles. It exhibits pessimism in equilibrium if itsatisfies this property with respect to the strategy the opponent plays on the equilibrium path.Formally:
Definition 7.
A weak BBE (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) exhibits pessimism if π i ( s i , ψ ∗ i ( s j )) ≤ π i (cid:16) s i , s ∗ j (cid:17) forall s i ∈ S i . In what follows we present two definitions that will be used in the analysis in the following sections:undominated Pareto optimality, and biased-belief minmax payoff.We say that a strategy profile is undominated Pareto optimal if it is (1) undominated, and (2)Pareto optimal among all undominated strategy profiles. Formally:
Definition 8.
Strategy profile ( s ∗ , s ∗ ) is undominated Pareto optimal if (1) s ∗ i ∈ S Ui for each player i , and (2) there does not exist ( s , s ) ∈ S U × S U with a payoff that Pareto dominates ( s ∗ , s ∗ ) , i.e., π ( s ∗ , s ∗ ) ≤ π ( s , s ) and π ( s ∗ , s ∗ ) ≤ π ( s , s ) where at least one of these inequalities is strict. A biased-belief minmax payoff for player i (denoted by ˜ M Ui ) is the maximal payoff player i canguarantee to himself in the following process: (1) player j chooses an arbitrary perceived strategy ofplayer i , and (2) player i chooses a strategy profile, under the constraint that player j ’s strategy is abest reply to the perceived strategy chosen above. That is, ˜ M Ui is the payoff player i can guaranteehimself no matter how his opponent (player j , she) perceives player i’s action, assuming that player j best-replies to what he believes player i is doing (and if there are multiple best replies, then weassume that player j chooses the best reply that is optimal for player i ). Formally: Definition 9.
Given game G = ( A, u ), let ˜ M Ui , the biased-belief minmax payoff of player i , bedefined as follows: ˜ M Ui = min s i ∈ S Ui max ( s i ,s j ) ∈ S i × BR ( s i ) π i ( s i , s j ) ! . Observe that the biased-belief minmax is weakly larger than the undominated maxmin (Defini-tion 10), i.e., ˜ M Ui ≥ M Ui with an equality if the strategy of player j that guarantees that player i ’spayoff is at most M Ui is a unique best reply against some strategy of player i (which is the case, inparticular, if the payoff function is strictly concave). Our main results characterize the set of BBE and BBE outcomes in three classes of games: gameswith strategic complements, games with strategic substitutes, and games with strategic opposites. .1 Preliminary Result: Necessary Conditions for a Weak BBE Outcome We begin by defining undominated strategies and the undominated minmax payoff, which will beused to characterize necessary conditions for a strategy profile to be a weak BBE outcome.Strategy s i of player i is undominated if it is a best reply of some strategy of the opponent, i.e., ifthere exists strategy s j ∈ S j , such that s i ∈ BR ( s j ). We say that a strategy profile is undominated if both strategies in the profile are undominated. Recall that in a finite game, due to the minmaxtheorem, a strategy is undominated iff it is not strictly dominated by another strategy.Let S Ui ∈ S i denote the set of undominated strategies of player i . Observe that S Ui is notnecessarily a convex set.An undominated minmax payoff for player i is the maximal payoff player i can guarantee tohimself in the following process: (1) player j chooses an arbitrary undominated strategy, and (2)player i chooses a strategy (after observing player j ’s strategy). Formally: Definition 10.
Given game G = ( S, u ), let M Ui , the undominated minmax payoff of player i , bedefined as follows: M Ui = min s j ∈ S Uj (cid:18) max s i ∈ S i π i ( s i , s j ) (cid:19) . Observe that the undominated minmax is weakly larger than the standard maxmin, i.e., M Ui ≥ min s j ∈ S j (max s i ∈ S i π i ( s i , s j )) with an equality if player j does not have any strictly dominatedstrategy (i.e., if S Uj = S j ).The following simple result (which will be helpful in deriving the main results in the followingsubsections) shows that any weak BBE outcome is an undominated strategy profile that yields apayoff above the player’s undominated minmax payoff to each player. Proposition 3.
If a strategy profile s ∗ = ( s ∗ , s ∗ ) is a weak BBE outcome, then (1) the profile s ∗ isundominated and (2) π i ( s ∗ ) ≥ M Ui .Proof. Assume that s ∗ = ( s ∗ , s ∗ ) is a biased-belief equilibrium outcome. This implies that each s ∗ i is a best reply to the player’s distorted belief, which implies that each s ∗ i is undominated. Assumeto the contrary, that , π i ( s ∗ ) < M Ui . Then, by deviating to the undistorted function I d , player i canguarantee a fitness of at least M Ui in any distorted equilibrium. Our first main result characterizes the set of BBE outcomes in games with strategic complements.It shows that a strategy profile is a BBE outcome essentially iff (I) it is undominated, (II) it yields apayoff above the undominated/biased-belief minmax payoff to both players, and (III) both playersoverinvest (i.e., use a weakly higher strategy than the best reply to the opponent). Formally:
Proposition 4.
Let G be a game with strategic complements and positive externalities . The undominated minmax payoff might be strictly higher than the undominated maxmin payoff due to the non-convexity of S jU ; i.e., player i might be able to guarantee only a lower payoff in a setup in which player j is allowed tochoose his undominated strategy after observing player i ’s chosen strategy.
1. Let ( s ∗ , s ∗ ) be a BBE outcome. Then ( s ∗ , s ∗ ) has the following properties: (I) it is undominated,and it satisfies for each player i : (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) ≥ M Ui , and (III) overinvestment: s ∗ i ≥ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) .2. Let ( s ∗ , s ∗ ) be an undominated profile that satisfies, for each player i : (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui ,and (III) s ∗ i ≥ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Then, ( s ∗ , s ∗ ) is a BBE outcome.Moreover, if π i ( s i , s j ) is strictly concave in s i (i.e., ∂π i ( s i ,s j ) ∂s i > ) then ( s ∗ , s ∗ ) is a strongBBE outcome.Sketch of Proof (formal proof in Appendix F.1). Part 1:
Proposition 3 implies (I) and (II). To prove (III, overinvestment), assume to the contrarythat s ∗ i < min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Consider a deviation of player i that induces him to invest slightlymore than s ∗ i . The fact that s ∗ i < min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) implies that player i strictly earns from his owndeviation. The assumption that the biased belief of the opponent is monotone implies that theagent’s deviation induces the opponent to invest more and, thereby to further improve the agent’spayoff. Thus, the agent gains from the deviation, and ( s ∗ , s ∗ ) cannot be a BBE outcome. Part 2:
The strategy profile ( s ∗ , s ∗ ) is supported as a BBE outcome by a profile of biasedbeliefs ( ψ ∗ , ψ ∗ ) in which each biased belief ψ ∗ j satisfies: (1) blindness to good news: ψ ∗ j distorts any s i ≥ s ∗ i into BR − (cid:16) s ∗ j (cid:17) , and (2) overreaction to bad news: ψ ∗ j distorts any s i < s ∗ i to a sufficientlylow strategy ψ j ( s i ), such that player i loses in any strategy profile (cid:16) s i , s j (cid:17) in which player j best-replies to the perceived strategy of player i (i.e., s j ∈ BR ( ψ j ( s i ))). These properties imply that(( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE (and a strong BBE if the payoff function is strictly concave).Recall that a game with strategic complements admits a lowest Nash equilibrium ( s , s ) in whichboth players invest less than in any other Nash equilibrium, i.e., s i ≥ s i for each player i and eachstrategy s i that is played in a Nash equilibrium (see, e.g., Milgrom and Roberts, 1990).An immediate corollary of Prop. 4 is that in each BBE outcome, both players invest more thanin any Nash equilibrium. Formally: Corollary 2.
Let G be a game with strategic complements and positive externalities with a lowestNash equilibrium ( s , s ) that satisfies s < max ( S i ) for each player i . Let ( s ∗ , s ∗ ) be a BBE outcome.Then s i ≤ s ∗ i for each player i .Proof. The result is immediate from part (1.III) of Proposition 4 (namely, that both agents weaklyoverinvest in any BBE outcome), and the observation (which is formally proved in Lemma 1 inAppendix F.2) that s ∗ i < s i implies that at least one of the players strictly underinvests.Corollary 2 shows that the notion of BBE rules out socially bad outcomes in which one (orboth) of the players invests less effort than the lowest Nash equilibrium. In particular, in a pricecompetition with differentiated goods (see Example 2 below), the corollary implies that the pricechosen by any player in any BBE is at least the player’s price in the unique Nash equilibrium of thegame. .2 Games with Strategic Complements Corollary 3.
Let G be a game with positive externalities and strategic complements. Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a BBE. If s ∗ i / ∈ { min ( S i ) , max ( S i ) } , then player i exhibits wishful thinking(i.e., ψ ∗ i (cid:16) s ∗ j (cid:17) ≥ s ∗ j ). Proof.
Assume to the contrary that ψ ∗ i (cid:16) s ∗ j (cid:17) < s ∗ j . The strategic complementarity implies thatmax (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) ≤ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) with an equality only ifmax (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) ∈ { min ( S i ) , max ( S i ) } (see Lemma 2 in Appendix F.3 for a formal proof of this claim). Part 1 of Proposition 4 and thedefinition of a BBE imply thatmax (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) ≥ s ∗ i ≥ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . The previous inequalities jointly imply thatmax (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) = s ∗ i = min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) ∈ { min ( S i ) , max ( S i ) } , which contradicts the assumption that s ∗ i / ∈ { min ( S i ) , max ( S i ) } .Next, we apply our analysis of games with strategic complements to price competition with dif-ferentiated goods (the linear city model `a la Hotelling). Specifically, we show that (1) players chooseprices above the unique Nash equilibrium price in all BBE, and (2) any undominated symmetricprice profile above the Nash equilibrium price can be supported as a strong BBE. In Appendix A.4we present three additional examples: input games, stag hunt games, and the traveler’s dilemma. Example 2 ( Price competition with differentiated goods; see a textbook analysis in Mas-Colell,Whinston, and Green, 1995, Section 12.C ) . Consider a mass one of consumers equally distributedin the interval [0 , i chooses price s i ∈ [0 , M ] for its widgets. The totalcost of buying a widget from firm i is equal to its price s i plus t times the consumer’s distance from Corollary 3 allows for pessimism of player i in a BBE only if player i plays an extreme strategy (either, the minimalfeasible strategy or the maximal feasible strategy) and his pessimism does not affect his play; i.e., the best reply againstthe real opponent’s strategy and the best reply against the perceived opponent’s strategy coincide in being the sameextreme strategy. For example, this is the case in the biased beliefs that support the action profile ( s i , s j ) in the staghunt game analyzed below. the firm, where t ∈ [0 , M ]). Each buyer buys a widget from the firm with the lower total buyingcost. This implies that the total demand for good i is given by function q i ( s i , s j ): q i ( s i , s j ) = s j − s i + t · t < s j − s i + t · t < s j − s i + t · t < s j − s i + t · t > , The payoff (profit) of firm i is given by π i ( s i , s j ) = s i · q i ( s i , s j ). Observe that the payoff functionis strictly concave in s i for any non-extreme s j (and it is weakly concave for the extreme values of s j ). One can show that the game has strategic complements, and that the best-reply function ofeach player is: s i ( s j ) = s j + t s j < · ts j − t s j ≥ · t. It is well known that the unique Nash equilibrium of this example is given by s i = s j = t, whichyields a payoff of t to each firm.Observe that the set of undominated strategies of each player is the interval h t , M + t i (where t is the best reply against 0 and M + t is the best reply against M ). This implies that the undominatedminmax of each player is equal to π i (cid:16) · t, t (cid:17) = · t · = · t . Proposition 4 implies that a strategyprofile ( s i , s j ) is a BBE outcome if for each player i : (1) s i ∈ h t , M + t i (undominated strategy),(2) π i ( s i , s j ) > · t (payoff above the undominated minmax payoff), and (3) overinvestment: s i ≥ s j + t .Figure 1 shows the set of BBE outcomes (which coincides with the set of strong BBE outcomes,due to the strict concavity of the payoff function), for t = 1 and M = 3.Observe that the sum of the payoffs to the two firms, s i · q i ( s i , s j ) + s j · q j ( s i , s j ), is a mixedaverage of s i and s j . The fact that the Nash equilibrium is in the bottom left corner of the set ofBBE outcomes implies that all BBE outcomes (except the Nash equilibrium itself) strictly improvesocial welfare relative to the Nash equilibrium (as measured by the sum of payoffs of the two firms).Next, we make two observations regarding the implications of the extent of wishful thinking onthe players’ payoffs (both observations hold also for the input games in Example 9 in AppendixA.4):1. Increasing the wishful thinking of both players improves the players’ payoffs. Specifically,with respect to symmetric BBE outcomes, a higher level of wishful thinking induces a higherequilibrium price and a higher payoff to the players: a wishful thinking level of x ∗ ≡ ψ ∗ ( s ∗ ) − s ∗ ∈ [0 ,
1] induces the symmetric BBE price x ∗ + t = x ∗ + 1 (which is implied by the perceivedbast-reply equation s ∗ = ψ ∗ ( s ∗ )+ t = s ∗ + x ∗ + t ), which yields a payoff of x ∗ +12 to each player. One can show that the constraint on s i implied by π i ( s i , s j ) > · t is nonbinding. The constraint is s i ∈ s j + t − p ( s j + t ) − . · t , s j + t − p ( s j + t ) − . · t ! . .2 Games with Strategic Complements t = 1, M = 3)2. When the wishful thinking levels of the two players differ, the player with the higher wishfulthinking level has a lower payoff. This is because the difference between the payoffs of a firmwith price s i and an opponent with price s j < s i is equal to: π i − π j = s i · (cid:18) s j − s i + 12 (cid:19) − s j · (cid:18) s i − s j + 12 (cid:19) = 0 . s j ( s j − − s i ( s i − < . Intuitively, wishful thinking is like a public good in this setup: (1) a higher level of wishful thinkingis beneficial to social welfare, and (2) if the two players have different levels of wishful thinking, theplayer with the higher level obtains a lower payoff.We conclude the example by presenting a symmetric biased belief ψ ∗ = ψ ∗ that supports theoutcome (2 ,
2) as the BBE (( ψ ∗ , ψ ∗ ) , (2 , M = 3 and t = 1: ψ ∗ i ( s j ) = s j > · s j − s j ∈ [0 . , s j < . . Observe that: (1) this BBE yields a payoff of 1 to each player and (2) the biased belief presentswishful thinking, i.e., ψ ∗ i (2) = 3 >
2. Further observe that a player with biased belief ψ ∗ i plays thesame strategy as the opponent (regardless of the opponent’s biased belief) in any equilibrium of thebiased game in which the opponent plays any intermediate value of s j (i.e., s j ∈ [0 . , s i ( ψ ∗ i ( s j )) = s i (3) = 2 s j > s i (2 · s j −
1) = 0 . · (2 · s j − s j s j ∈ [0 . , s i (0) = 0 . s j < . . This implies that the equilibrium payoff of a deviating player j who plays strategy s j is equal to: π j ( s j , s i ( ψ ∗ i ( s j ))) = s j · . · (2 − s j + 1) = s j · . · (3 − s j ) < s j > s j · . · q ( s j , s j ) = 0 . · s j s j ∈ [0 . , s j · . · (0 . − s j + 1) = s j · . · (1 . − s j ) < . s j < . , and it is at most 1, which implies that a deviator cannot gain from his deviation.Finally, note that Figure 1 shows that the two Stackelberg-leader equilibria (the unique subgame-perfect equilibrium of the sequential games in which one of the players plays first, and the opponentreplies after observing the leader’s strategy) are included in the set of BBE, as is proven in generalin Proposition 7. Our next result characterizes the set of BBE outcomes in games with strategic substitutes (andpositive externalities). It shows that a strategy profile is a BBE outcome essentially iff (I) it isundominated, (II) it yields a payoff above the undominated/biased-belief minmax payoff to bothplayers, and (III) both players underinvest (i.e., use a weakly lower strategy than the best reply tothe opponent). Formally:
Proposition 5.
Let G be a game with strategic substitutes and positive externalities .1. Let ( s ∗ , s ∗ ) be a BBE outcome. Then ( s ∗ , s ∗ ) has the following properties: (I) it is undomi-nated, and if satisfies for each player i : (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) ≥ M Ui and (III) s ∗ i ≤ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) (underinvestment).2. Let ( s ∗ , s ∗ ) be an undominated profile that satisfies, for each player i : (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui ,and (III) s ∗ i ≤ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Then, ( s ∗ , s ∗ ) is a BBE outcome.Moreover, if π i ( s i , s j ) is strictly concave then ( s ∗ , s ∗ ) is a strong BBE outcome. The proof, which is analogous to the proof of Proposition , is presented in Appendix F.4.An immediate corollary of Proposition 4 is that in each BBE outcome, at least one of the playersinvests less relative to his maximal Nash equilibrium investment. Formally: Corollary 4.
Let G be a game with strategic substitutes and positive externalities. Let ( s ∗ , s ∗ ) be aBBE outcome. Then, there exists a Nash equilibrium of the underlying game ( s e , s e ) , and a player i such that s ei ≥ s ∗ i .Proof. The result is immediate from part (1-III) of Proposition 4 (namely, that both agents weaklyunderinvest in any BBE outcome), and the observation (which is formally proved in Lemma 3 inAppendix F.5) that if the effort of each player s ∗ i is strictly below all of his Nash equilibrium efforts,then at least one of the players strictly underinvests.Corollary 4 shows that the notion of BBE rules out socially good outcomes in which both playersinvest more effort relative to their maximal Nash equilibrium effort. In particular, in a Cournot .3 Games with Strategic Substitutes empirical prediction of our modeland the notion of BBE: efficient (non-Nash equilibrium) outcomes are easier to support in gameswith strategic complements, relative to games with strategic substitutes. This prediction is consistentwith the experimental findings of Potters and Suetens (2009), which show that there is significantlymore cooperation in games with strategic complements than in games with strategic substitutes.The following corollary shows that in games with strategic substitutes, as in games with strategiccomplements, there is the a close relation between BBE and wishful thinking. Specifically, it showsthat any biased belief in any BBE (with a non-extreme outcome) of a game with strategic substitutesexhibits wishful thinking. The intuition is that wishful thinking causes an agent to believe that theopponent is playing a higher action, which induces the agent to respond with a lower action, which,in turn, causes the opponent to respond by playing a higher action, which benefits the agent.
Corollary 5.
Let G be a game with positive externalities and strategic substitutes. Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a BBE. If s ∗ i / ∈ { min ( S i ) , max ( S i ) } , then player i exhibits wishful thinking (i.e., ψ ∗ i (cid:16) s ∗ j (cid:17) ≥ s ∗ j ).The proof, which is analogous to the proof of Corollary 3, is presented in Appendix F.6.The following example characterizes the set of BBE outcomes in a Cournot competition. Ap-pendix A.5 presents an analysis of another game of strategic substitutes: the hawk-dove game. Example 3 ( Cournot competition with linear demand ) . Consider a symmetric Cournot competition,where we relabel the set of strategies to describe unused capacity, rather than quantity, in orderto follow the normalization of positive externalities. Formally, let G = ( S, π ): S i = [0 ,
1] and π i ( s i , s j ) = (1 − s i ) · ( s i + s j −
1) for each player i . Each s i is interpreted as the unused capacity(= one minus the quantity, i.e., s i = 1 − q i ) chosen by firm i , the price of both goods is determinedby the linear inverse demand function p = 1 − q i − q j = s i + s j −
1, and the marginal cost of eachfirm is normalized to be zero.Observe that:1. BR ( s i ) = 1 − s i , and the unique Nash equilibrium of the game is s ∗ = s ∗ = , which yields apayoff of to both players.2. The set of undominated strategies of each player is the interval [0 . ,
1] (where 1 is the bestreply against 0, and 0.5 is the best reply against 1).3. The symmetric Pareto optimal profile (which is also undominated) is s i = s j = , yielding apayoff of to each player.4. The undominated minmax payoff M Ui = , which is achieved by the opponent playing hislowest undominated strategy s i = 0 . s , s ) is π ( s , s ) + π ( s , s ) =(2 − ( s i + s j )) · (( s i + s j ) − s i + s j in the domain ofundominated strategies s i , s j ≥ . Applying the analysis of the previous subsection to a Cournot competition shows that strategy profile( s , s ) is a BBE outcome iff it satisfies for each player i : (1) the strategy is undominated: s i ≥ . − s i ) · ( s i + s j − ≥ = M Ui ,and (3) underinvestment relative to the best reply against the opponent: s i ≤ BR ( s j ) = 1 − s j .Due to having a strictly concave payoff function, the set of BBE outcomes coincides with the set ofstrong BBE outcomes. Figure 1 shows this set of BBE outcomes (the strategy profiles that satisfythe above three conditions).Figure 2: The Set of (Strong) BBE Outcomes in a Cournot CompetitionObserve that the unique Nash equilibrium (cid:16) , (cid:17) is the profile that maximizes the sum s i + s j within the set of BBE. This implies that all other BBE outcomes yield lower social welfare (asmeasured by the sum of payoffs) relative to the Nash equilibrium.Next, we make two observations regarding the implications of the level of wishful thinking onthe players’ payoffs.1. Increasing the wishful thinking of both players decreases the players’ payoffs. Specifically,when focusing on symmetric BBE outcomes, a higher level of wishful thinking induces a lowerlevel of unused capacity and a lower payoff to both players; the higher level of productionis induced by the false assessment of each firm that the other firm is producing less than itactually does.
2. When the wishful thinking levels of the two players differ, the player with the higher wishfulthinking has a higher payoff. This is because the difference between the payoffs of a firm with A wishful thinking level of x ∗ ≡ ψ ∗ ( s ∗ ) − s ∗ ∈ [0 , .
28] induces a symmetric BBE unused capacity of s ∗ = − x ∗ (which is implied by the perceived best-reply equation s ∗ = 1 − ψ ∗ ( s ∗ ) = 1 − s ∗ + x ∗ ). .4 Pessimism in Games with Opposing Differences s i and an opponent with price s j < s i is equal to π i − π j = s i · (cid:18) s j − s i + 12 (cid:19) − s j · (cid:18) s i − s j + 12 (cid:19) = 0 . s j ( s j − − s i ( s i − < . Thus, a higher level of wishful thinking is beneficial to social welfare, but harms the player with thehigher level (relative to the opponent’s payoff).Finally, note that Figure 2 shows that the two Stackelberg-leader equilibria (the unique subgame-perfect equilibria of the sequential games in which one of the players plays first, and the opponentreplies after observing the leader’s strategy) are included in the set of BBE, as is proven in generalin Proposition 7.
The results of the previous two subsections present a strong tendency of BBE to exhibit wishfulthinking both in games with strategic complements and in games with strategic substitutes. Thisraises the question of which class of games induces pessimism. In this section we show that theanswer to this question is games with strategic opposites. Recall that these are games in whichthe strategy of player 1 is a complement of player 2’s strategy, while the strategy of player 2 is asubstitute of player 1’s strategy, e.g., duopolistic competitions in which one firms chooses its quantitywhile the opposing firm chooses its price (Singh and Vives, 1984) and various classes of asymmetriccontests (Dixit, 1987).Proposition 6 characterizes the set of BBE outcomes in games with strategic opposites (andpositive externalities).It shows that a strategy profile is a BBE outcome essentially iff (I) it is undominated, (II) ityields a payoff above the undominated/biased-belief minmax payoff to both players, and (III) player1 (for whom player 2’s strategy is a complement) underinvests, while player 2 (for whom player 1’sstrategy is a substitute) overinvests. Formally:
Proposition 6.
Let G be a game with positive externalities and strategic opposites : ∂ π ( s ,s ) ∂s ∂s > ∂ π ( s ,s ) ∂s ∂s < s , s .1. Let ( s ∗ , s ∗ ) be a BBE outcome. Then ( s ∗ , s ∗ ) is (I) undominated: (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) ≥ M Ui foreach player i , and (III) s ∗ ≤ max ( BR ( s ∗ )) and s ∗ ≥ min ( BR ( s ∗ )) (i.e., player 1 underinvestsand player 2 overinvests relative to the best reply to the opponent).2. Let ( s ∗ , s ∗ ) be a profile satisfying the following conditions: (I) ( s ∗ , s ∗ ) is undominated, (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui for each player i , and (III) s ∗ ≤ max ( BR ( s ∗ )) and s ∗ ≥ min ( BR ( s ∗ )) .Then, ( s ∗ , s ∗ ) is a BBE outcome. The proof, which is analogous to the proof of Proposition , is presented in Appendix F.7.The following corollary shows that in games with strategic opposites, there is a close relationbetween BBE and pessimism. Specifically, it shows that any biased belief in any BBE (with anon-extreme outcome) of a game with strategic opposites exhibits pessimism. The intuition is that8 pessimism causes player 1 to believe that player 2 is playing a lower action, which induces player 1to respond with a lower action, which, in turn, causes player 2 to respond by playing a higher action,which benefits player 1. Similarly, pessimism causes player 2 to believe that player 1 is playing alower action, which induces player 2 to respond with a higher action, which, in turn, causes player1 to respond by playing a higher action, which benefits player 2. Corollary 6.
Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a BBE of a game with positive externalities and strate-gic opposites (i.e., ∂ π ( s ,s ) ∂s ∂s > and ∂ π ( s ,s ) ∂s ∂s < for each pair of strategies s , s ). If s ∗ i / ∈{ min ( S i ) , max ( S i ) } , then player i exhibits pessimism (i.e., ψ ∗ i (cid:16) s ∗ j (cid:17) ≤ s ∗ j ). The proof, which is analogous to the proof of Corollary 3, is presented in Appendix F.8.Next, we present an example of a game with strategic opposites, and we characterize the set ofBBE in this game.
Example 4 ( Matching pennies with positive externalities ) . The game presented in Table 1, a variantof the matching pennies game, is played as follows:1. Player 1 (player 2) gains 1 utility point from matching (mismatching) his opponent.2. Each player i induces a gain of 3 utility points to his opponent by choosing heads (action h i ).The game admits a unique Nash equilibrium (0 . , .
5) with a payoff of 1.5 to each player. The(undominated) minmax payoff of each player is 1 (obtained when the opponent plays t j ). Observethat the game has positive externalities, that the strategy of player 2 is a strategic complement forplayer 1, while the strategy of player 1 is a strategic substitute for player 2.Table 1: Matching Pennies with Positive Externalities h t h , − , t , , − ψ ∗ , ψ ∗ ) , ( β , β )) satisfies for each player i : (I) β i ∈ [0 . ,
1] (i.e., both players play headsmore frequently than in the unique Nash equilibrium), (II) pessimism: ψ ∗ i ( β j ) = 0 . < β j ,and (III) one-sided blindness: ψ ∗ ( α ) = 0 . α ≥ β ; ψ ∗ ( α ) < . α < β ; ψ ∗ ( α ) = 0 . α ≤ β ; and ψ ∗ ( α ) < . α > β .2. A class in which player 1 mixes while giving more weight to tails, while player 2 plays heads.Both players exhibit pessimism. Specifically, each BBE in this class (( ψ ∗ , ψ ∗ ) , ( β , β )) satisfiesfor each player i : (I) β ∈ [0 , .
5] and β = 1 (i.e., player 1 plays tails more frequently than inthe unique Nash equilibrium, while player 2 always plays heads), (II) pessimism for player 1: .5 Empirical Prediction Regarding Wishful Thinking ψ ∗ ( β = 1) = 0 . <
1, and ψ ∗ ( β ) = 0 . ψ ∗ ( α ) > . α > β .Observe that any profile ( β , β ), where β < . β < . β < β < . β = 0, then player 2’s payoff is negative, and less than his undominatedminmax payoff of 1.2. If β < . β >
0, then player 1 can gain by deviating to ψ ≡
0, as the only possibleequilibria of the new biased game are (0 , ) and (0 , β ), both of which induce a higher payoffto player 1 relative to ( β , β ).3. If β < . β <
1, then player 2 can gain by deviating to ψ ≡
0, as the only possibleequilibria of the new biased game are (0 , ) and ( β , ), both of which induce a higher payoffto player 2 relative to ( β , β ). Arguably, the class of games with strategic opposites (which induces pessimism) is less commonin strategic interactions than the classes of games with strategic complements/substitutes (both ofwhich induce wishful thinking). This observation suggests the following empirical predictions of ourmodel: (1) wishful thinking is more common than pessimism, and (2) there are some (less common)strategic interactions that induce pessimism. This empirical prediction is consistent with the ex-perimental evidence that people tend to present wishful thinking, although, the extent of wishfulthinking may substantially differ across different environments and may disappear in some environ-ments (see, e.g., Babad and Katz, 1991; Budescu and Bruderman, 1995; Bar-Hillel and Budescu,1995; Mayraz, 2013).
In this subsection we present an interesting class of BBE that exist in all games. In this class, oneof the players is “strategically stubborn” in the sense that he plays his undominated Stackelbergstrategy (defined below) and has blind beliefs, while his opponent is “flexible” in the sense of havingunbiased beliefs.A strategy is undominated Stackelberg if it maximizes a player’s payoff in a setup in which theplayer can commit to an undominated strategy, and his opponent reacts by choosing the best replythat maximizes player i ’s payoff. Formally: Definition 11.
The strategy s i is an undominated Stackelberg strategy if it satisfies s i = argmax s i ∈ S Ui (cid:16) max s j ∈ BR ( s i ) ( π i ( s i , s j )) (cid:17) . Let π Stac i = max s i ∈ S Ui (cid:16) max s j ∈ BR ( s i ) ( π i ( s i , s j )) (cid:17) be the undominated Stackelberg payoff. Observethat π Stac i ≥ π i ( s ∗ , s ∗ ) for any Nash equilibrium ( s ∗ , s ∗ ) ∈ N E ( G ).Our next result shows that every game admits a BBE in which one of the players: (1) has a blindbelief, (2) plays his undominated Stackelberg strategy, and (3) obtains his undominated Stackelbergpayoff. The opponent has undistorted beliefs. Moreover, this BBE is strong if the undominatedStackelberg strategy is a unique best reply to some undominated strategy of the opponent.The intuition behind Proposition 7 is as follows. The “strategically stubborn” player i cannot gainfrom a deviation, because player i already obtains the highest possible payoff under the constraintthat player j best-replies to player i ’s strategy. The “flexible” player j cannot gain from a deviation,because the “blindness” of player i implies that player i ’s behavior remains the same regardless ofplayer i ’s deviation, and, thus, player i cannot do better than best-replying to player i ’s strategy. Proposition 7.
Game G = ( S, π ) admits a BBE (cid:16) ( ψ ∗ i , Id ) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) for each player i with thefollowing properties: (1) ψ ∗ i is blind, (2) s ∗ i is an undominated Stackelberg strategy, and (3) s ∗ j = max s j ∈ BR ( s ∗ i ) ( π i ( s ∗ i , s j )) .Moreover, (cid:16) ( ψ ∗ i , Id ) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) is a strong BBE if { s ∗ i } = BR − (cid:16) s ∗ j (cid:17) .Proof. Let s ∗ i be an undominated Stackelberg strategy of player i . Let s ∗ j = argmax s j ∈ BR ( s ∗ i ) ( π i ( s ∗ i , s j )) . Let ˆ s j ∈ BR − ( s ∗ i ) ( { ˆ s j } = BR − ( s ∗ i ) with the additional assumption of the “moreover” part).We now show that (cid:16) ( ψ ∗ i ≡ ˆ s j , Id ) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) is a (strong) BBE. It is immediate that (cid:16) s ∗ i , s ∗ j (cid:17) ∈ N E (cid:16) G (ˆ s j ,Id ) (cid:17) , and that both biased beliefs are monotone.Next, observe that for any biased belief ψ j there is a plausible equilibrium (in any equilibrium)of the biased game G ( ˆ s j ,ψ j ) in which player i plays s ∗ i , and player j gains at most π j (cid:16) s ∗ i , s ∗ j (cid:17) , whichimplies that the deviation to ψ j is not profitable to player j in this plausible equilibrium (in anyequilibrium) of the new biased game.If player i deviates to a biased belief ψ i , then in any equilibrium of the biased game G ( ψ i ,Id )player i plays some strategy s i and gains a payoff of at most max s j ∈ BR ( s i ) (cid:16) π i (cid:16) s i , s j (cid:17)(cid:17) , and thisimplies that player i ’s payoff is at most π Stac i , and that he cannot gain by deviating. This showsthat ((ˆ s j , Id ) , ( s ∗ , s ∗ )) is a (strong) BBE.We demonstrate this class of equilibria in a Cournot competition. Example 5 ( Well-behaved BBE that yields the Stackelberg outcome in a Cournot competition ) . Consider the symmetric Cournot game with linear demand in Example 1: G = ( S, π ): S i = R + and π i ( s i , s j ) = s i · (1 − s i − s j ) for each player i . Then (cid:16) (0 , I d ) , (cid:16) , (cid:17)(cid:17) is a strong well-behaved BBEthat induces the Stackelberg outcome (cid:16) , (cid:17) , and yields the Stackelberg-leader payoff of to player1 and yields the follower payoff of to player 2. This is because: (1) (cid:16) , (cid:17) ∈ N E (cid:16) G (0 ,I d ) (cid:17) , (2)for any biased belief ψ , player 1 keeps playing and as a result player 2’s payoff is at most , and(3) for any biased belief ψ , player 2 will best-reply to player’s 1 strategy, and thus player 1’s payoffwill be at most his Stackelberg payoff of . .2 Folk Theorem Results In this subsection we present various folk theorem results (i.e., general feasibility results) that showthat relaxing either of the two requirements in the definition of a BBE (namely, monotonicityand ruling out implausible equilibria) yields little predictive power in various classes of games.Specifically, we show that in those games a strategy profile is a monotone weak BBE outcome (resp.,non-monotone strong BBE outcome) essentially iff it is (1) undominated, and (2) induces a payoffabove the undominated minmax payoff.
We begin by defining the notions of monotone weak BBE, and of non-monotone strong BBE.
Definition 12.
A weak BBE ( ψ ∗ , s ∗ ) is a monotone weak BBE if each biased belief ψ ∗ i is monotonefor each player i .A weak BBE ( ψ ∗ , s ∗ ) is a non-monotone strong BBE if the inequality π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) holdsfor every player i , every biased belief ψ i , and every strategy profile (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) .Note that (1) a monotone weak BBE is a weakening of the notion of a BBE, which relaxes therequirement of ruling out implausible equilibria, and (2) a non-monotone strong BBE is a weakeningof the notion of strong BBE, which relaxes the requirement of monotonicity. We say that a finite game G admits best replies with full undominated support , if, for each player i ,there exists an undominated strategy s i ∈ S Ui with a support that includes all undominated actions,i.e., supp ( s i ) = A i ∩ S Ui . Two classes of games that admit best replies with full undominatedsupport are:1. All two-action games . The reason for this is as follows. If player i has a dominant action, then,trivially, the dominant action a i is an undominated strategy with a support that includes allundominated actions. If player i does not have a dominant action, then there must be astrategy of the opponent for which the player is indifferent between his two actions, whichimplies that there exists an undominated strategy with full support.2. Any game with a totally mixed equilibrium (e.g., a rock-paper-scissors game).Our next result focuses on finite games that admit best replies with full undominated support,and shows that in such games a strategy profile s ∗ is a monotone weak BBE outcome iff (I) s ∗ isundominated, and (II) the payoff of is s ∗ above the undominated minmax payoff.The sketch of the proof is as follows. Each player has a blind belief that his opponent plays herpart of the Nash equilibrium with full undominated support. This implies that each player is alwaysindifferent between all undominated actions and, as such, can (1) play s ∗ i on the equilibrium path,and (2) play a punishing strategy that guarantees the opponent a payoff of at most her undominatedminmax payoff following any deviation of the opponent.2 Proposition 8 ( Folk Theorem result for monotone weak BBE outcomes ) . Let G be a finite game thatadmits best replies with full undominated support. Then the following two statements are equivalent:1. Strategy profile ( s ∗ , s ∗ ) is a monotone weak BBE outcome.2. Strategy profile ( s ∗ , s ∗ ) is (I) undominated and (II) π i ( s ∗ , s ∗ ) ≥ M Ui .Proof. Proposition 3 implies that “1. ⇒ ⇒ s ∗ , s ∗ ) is undom-inated, and π i ( s ∗ , s ∗ ) ≥ M Ui . For each player j , let s pj be an undominated strategy that guaranteesthat player i obtains, at most, his minmax payoff M Ui , i.e., s pj = argmin s j ∈ S Uj (max s i ∈ S i π i ( s i , s j )) . For each player j , let s ej ∈ S Uj be a best-reply strategy with full undominated support, i.e., supp (cid:16) s ej (cid:17) = A i ∩ S Ui . For each player i , let s di ∈ BR − (cid:16) s ej (cid:17) . The fact that s ej ∈ BR (cid:16) s di (cid:17) im-plies that s ∗ j , s pj ∈ ∆ (cid:16) S Uj (cid:17) = ∆ (cid:16) supp (cid:16) s ej (cid:17)(cid:17) ⊆ BR (cid:16) s di (cid:17) .We conclude by showing that (cid:16)(cid:16) s d , s d (cid:17) , ( s ∗ , s ∗ ) (cid:17) is a monotone weak BBE (in which both playershave blind beliefs). It is immediate that ( s ∗ , s ∗ ) ∈ N E (cid:16) s d , s d (cid:17) . Next, observe that for any deviationof player i to a different biased belief ψ i , there is a Nash equilibrium of the biased game G ( ψ i ,s ej )in which player j plays s pj , and, as a result, player i obtains a payoff of at most M Ui , which impliesthat the deviation is not profitable. Thus, ( s ∗ , s ∗ ) is a BBE outcome.Proposition 8 suggests that the notion of monotone weak BBE is too weak. The folk theoremresult relies on the incumbents “discriminating” against deviators who have exactly the same per-ceived behavior as the rest of the population: the incumbents of population j “punish” deviators byplaying s pj against them, while continuing to play s ∗ j against the incumbents, even though both thedeviators and the incumbents are perceived to behave the same (i.e., ψ ∗ j ( s ei ) = ψ ∗ j ( s ∗ i )).Example 8 in Appendix A.3 demonstrates that the folk theorem result does not necessarily holdfor games that do not admit best replies with full undominated support. In this section we show a folk theorem result for strong BBE in a broad family of interval gamesin which each payoff function π i ( s i , s j ) is (1) strictly concave in s i and (2) weakly convex in s j .Examples of such games include Cournot competitions, price competitions with differentiated goods,public good games, and Tullock contests.The following result shows that in this class of interval games, any undominated strategy profile( s ∗ , s ∗ ) that induces each player a payoff strictly above the player’s undominated minmax payoff canbe implemented as an outcome of a strong BBE. Formally: Proposition 9.
Let G = ( S, π ) be an interval game. Assume that for each player i , π i ( s i , s j ) isstrictly concave in s i and weakly convex in s j . If ( s ∗ , s ∗ ) is undominated and π i ( s ∗ , s ∗ ) > M Ui foreach player i , then ( s ∗ , s ∗ ) is a non-monotone strong BBE outcome. The sketch of the proof is as follows (the formal proof is presented in Appendix F.9).Each player j has a biased belief ψ ∗ j that (I) distorts s ∗ i into BR − (cid:16) s ∗ j (cid:17) , and (II) distorts any s i that is not in a small neighborhood of s ∗ i , to BR − (cid:16) s pj (cid:17) , where s pj is a “punishing” strategy that3guarantees that player i obtains at most his undominated minmax payoff. Part (I) implies that( s ∗ , s ∗ ) is an equilibrium of the biased game. Part (II) implies that following any deviation of player i to a different biased belief, if player i plays a strategy that is not in a small neighborhood of s ∗ i ,then player i loses from the deviation. Finally, the assumption that the payoff function π i ( s i , s j ) isconvex in s j implies that we can “complete” a continuous description of ψ ∗ j for s i that are in a smallneighborhood around s ∗ i , such that a player cannot gain from deviating to playing strategies in thissmall neighborhood. The results of this section show that the notion of weak BBE has little predictive power in the sensethat, essentially, any undominated strategy profile with a payoff above the undominated minmaxpayoff is a weak BBE outcome. Moreover, we show that this multiplicity of BBE outcomes holds inlarge classes of games also when applying a refinement of monotonicity (Prop. 8), or when applyinga refinement of strongness (Prop. 9). By contrast, in Section 6 we show that the combination oftwo plausible requirements, namely, monotonicity and ruling out implausible equilibria, allows us toachieve sharp predictions for the set of BBE outcomes in various interesting classes of games andfor the set of biased beliefs that support these outcomes.Our folk theorem results have similar properties to the famous folk theorem results for repeatedgames and sufficiently discounted players (see, e.g., Fudenberg and Maskin, 1986). This is so be-cause it allows for implicit punishments similar to those used in repeated games in order to sustainequilibria. This is because our model assumes that when a player deviates to a different biased beliefhis opponent can react to the deviation and deter against it.Observe that our result has somewhat stronger predictive power than the folk theorem resultfor repeated games, in the sense that the set of monotone weak BBE in one-shot finite games andthe set of non-monotone strong BBE in one-shot interval games are each smaller than the set ofsubgame-perfect equilibria of repeated games between patient players. In particular, the followingstrategy profiles can be supported as the subgame-perfect equilibrium outcomes of a repeated gamebetween patient players, but they cannot be the outcome of a weak BBE outcome of a one-shotgame: (1) strategy profiles in which one of the players plays a strategy that is strictly undominatedin the underlying (one-shot) game, and (2) strategy profiles in which some of the players obtain apayoff between the standard minmax payoff and the (higher) undominated minmax payoff.In Appendix D we show that if one relaxes the assumption that the biased beliefs must becontinuous, then one can obtain a folk theorem result in broader classes of games, namely, (1) in allfinite games, and (2) in all interval games with strictly concave payoffs.
Decision makers’ preferences and beliefs may intermingle. In strategic environments distorted beliefscan take the form of a self-serving commitment device. Our paper introduces a formal model for thepersistence of such beliefs and proposes an equilibrium concept that supports them. Our analysischaracterizes BBE in a variety of strategic environments, such as games with strategic complements4
REFERENCES and games with strategic substitutes. In particular, we show that agents present wishful thinkingin all BBE in both of these common environments.Our analysis here deals with simultaneous games of complete information, but the idea of strate-gically distorted beliefs may play an important role also in sequential games and in Bayesian games.In these frameworks, belief distortion may violate Bayesian updating, and our concept here canpotentially offer a theoretical foundation for some of the cognitive biases relating to belief updating.It can also potentially identify the strategic environments in which these biases are likely to occur.We view this as an important research agenda that we intend to undertake in the future.A different research track that might shed more light on strategic belief distortion is the experi-mental one. Laboratory experiments often conduct belief elicitation with the support of incentivesfor truthful revelation. Strong evidence for strategic belief bias in experimental games can be ob-tained by showing that players assign different beliefs to the behavior of their own counterpart inthe game and to a person playing the same role with someone else. In general, our model predictsthat beliefs about a third party’s behavior are more aligned with reality than those involving one’scounterpart in the game. Laboratory experiments can also test whether specific types of beliefdistortions (such as wishful thinking) arise in the strategic environments that are predicted by ourmodel.Finally, we point out that strategic beliefs may play an important role in the design of mechanismsand contracts. Belief distortions may destroy the desirable equilibrium outcomes that a standardmechanism aims to achieve. Mechanisms that either induce unbiased beliefs or adjust the rules ofthe game to account for possible belief biases are expected to perform better.
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Online AppendicesA Additional Examples
A.1 A Non-Nash Strong BBE Outcome in a Zero-Sum Game
The following example shows that although the weak BBE payoff must be the Nash equilibriumpayoff in a zero-sum game, the strategy profile sustaining it need not be a Nash equilibrium.
Example 6.
Consider the symmetric rock–paper–scissors zero-sum game described in Table 2.We show that (cid:16)(cid:16) I d , (cid:16) , , (cid:17)(cid:17) , (cid:16) R, (cid:16) , , (cid:17)(cid:17)(cid:17) is a strong BBE, in which the player 1 (he) hasTable 2: Symmetric Rock-Paper-Scissors Zero-Sum Game PayoffsR P SR 0 , , , R , while player 2 (she) has a blind belief that the opponent alwaysmixes equally, and she mixes equally. It is immediate that (cid:16) R, (cid:16) , , (cid:17)(cid:17) ∈ N E (cid:16) G ( I d , ( , , )) (cid:17) ,and the equilibrium payoff to each player is zero. Next, observe that after any deviation of player1 to a biased belief ψ , there is an equilibrium of the game G ( ψ , ( , , )) in which player 2 mixesequally and player 1 obtains a payoff of zero. Finally, observe that after any deviation of player 2 toa biased belief ψ , player 1 obtains a payoff of at least zero (her minmax payoff in G ( I d ,ψ )) in anyNash equilibrium in G ( I d ,ψ ), which implies that player 2 obtains a payoff of at most zero, and, asa result, she does not gain from the deviation. A.2 Prisoner’s Dilemma with a Weakly Dominated Withdrawal Strategy
Proposition 2 implies, in particular, that defection is the unique weak BBE outcome in the prisoner’sdilemma game. The following example demonstrates that a relatively small change to the prisoner’sdilemma game, namely, adding a third weakly dominated “withdrawal” strategy that transforms“cooperation” into a weakly dominated strategy, can allow us to sustain cooperation as a strongBBE outcome. This is done by means of biases under which a player believes that his opponent isplanning to withdraw from the game whenever he intends to cooperate, which makes cooperation arational move.
Example 7.
Consider the variant of the prisoner’s dilemma game with a third “withdrawal” actionas described in Table 3. In this symmetric game both players get a high payoff of 10 if they bothplay action c (interpreted as cooperation). If one player plays d ( defection ) and his opponent plays c , then the defector gets 11 and the cooperator gets 0. If both players defect, then each of themgets a payoff of 1. Finally, if either player plays action w (interpreted as withdrawal ), then bothplayers get 0. Observe that defection is a weakly dominant action, and that the game admits twoNash equilibria: ( w, w ) and ( d, d ), inducing respective symmetric payoffs of zero and one. A ADDITIONAL EXAMPLES
Table 3: Prisoner’s Dilemma Game with a Withdrawal Action c d wc d w α c , α d , α w ), where α c ≥ α d ≥ , α w ≥ c (resp., d , w ). For each player i , let ψ i be the followingbiased-belief function: ψ ∗ i ( α c , α d , α w ) = (0 , α d , α c + α w ) . We now show that (( ψ ∗ , ψ ∗ ) , ( c, c )) is a non-monotone strong BBE in which both players obtaina high payoff of 10 (which is strictly better than the best Nash equilibrium payoff, and strictlybetter than the Stackelberg payoff of each player). Observe first that c ∈ BR ( ψ ∗ i ( c )) = BR ( w ),which implies that ( c, c ) ∈ N E (cid:16) G ( ψ ∗ ,ψ ∗ ) (cid:17) . Next, consider a deviation of player i to biased belief ψ i . Observe that player i can gain a payoff higher than 10 only if he plays action d with positiveprobability, but this implies that the unique best reply of player j to his biased belief about player i ’s strategy is defection, which implies that player i obtains a payoff of at most one. A.3 The Folk Theorem Result Does not Hold for All Finite Games
The following example demonstrates that the folk theorem result (Proposition 4) does not necessarilyhold for games that do not admit best replies with full undominated support.
Example 8.
Consider the three-action symmetric game described in Table 4. Observe that allTable 4: A Game in which ( a, a ) is not a Monotone Weak BBE Outcome a b ca
2, 2 2, 3 1.1, 3 b
3, 2 3, 3 1, 0 c a ( c ) is a best reply only to his opponent’s strategiesthat assign a probability of at least 90% to action c ( a ), which implies that actions a and c cannotbe best replies simultaneously. Observe that the undominated minmax payoff of each player is equalto 1 (because the opponent can play the undominated action c , and by playing this the opponentguarantees that the player gets a payoff of at most 1).Consider the undominated action profile ( a, a ) (which induces a payoff strictly above the undom-inated minmax payoff to each player). We will show that ( a, a ) is not a monotone weak BBE (whichdemonstrates that the folk theorem result of Proposition 8 does not hold in this game). Assume tothe contrary that ( a, a ) is a monotone weak BBE. Let (( ψ ∗ , ψ ∗ ) , ( a, a )) be a monotone weak BBE. .4 Examples of Games with Strategic Complements a, a ) ∈ N E (cid:16) G ( ψ ∗ ,ψ ∗ ) (cid:17) implies that ψ ∗ ( a ) ( c ) > ψ = b . Observe that player 2 plays action b in any equilibrium of G ( ψ ∗ ,ψ ). The monotonicity of ψ ∗ implies that ψ ∗ ( b ) ( a ) ≤ ψ ∗ ( a ) ( a ) ≤ − ψ ∗ ( a ) ( c ) ≤ ψ ∗ ( b )) does not haveaction c in its support. This implies that player 1 gains a payoff of at least 3 in any Nash equilibriumof the new biased game G ( ψ ∗ ,ψ ), which contradicts (( ψ ∗ , ψ ∗ ) , ( a, a )) being a monotone weak BBE. A.4 Examples of Games with Strategic Complements
In this subsection we analyze three examples of games with strategic complements: input games,stag hunt games, and the traveler’s dilemma.Our first example demonstrates how to implement the undominated Pareto optimal profile as astrong BBE in an input (or partnership game).
Example 9 ( Input games ) . Consider the following input game (closely related games are analyzedin, among others, Holmstrom, 1982 and Heller and Sturrock, 2017). Let S i = S j = [0 , π i ( s i , s j , ρ ) = s i · s j − s i ρ , where the parameter ρ is interpreted as the cost ofeffort. One can show that (1) the best-reply function of each agent is to exert an effort that is ρ <1times smaller than the opponent’s (i.e., BR ( s j ) = ρ · s j ), (2) in the unique Nash equilibrium eachplayer exerts no effort s i = s j = 0, (3) the highest undominated strategy of each player i is s i = ρ ,and (4) the undominated strategy profile ( ρ, ρ ) is Nash improving and yields the best payoff to bothplayers out of all the undominated symmetric strategy profiles. Let ψ ∗ i be the following biased-belieffunction: ψ ∗ i ( s j ) = s j ρ s j < ρ s j ≥ ρ. Observe that ψ ∗ i is monotone and exhibits wishful thinking. We now show that (( ψ ∗ , ψ ∗ ) , ( ρ, ρ )) isa strong BBE. Observe that BR ( ψ ∗ i ( s j )) = BR (cid:16) s j ρ (cid:17) = s j for any s j ≤ ρ , and that BR ( ψ ∗ i ( s j )) = BR (1) = ρ for any s j ≥ ρ . This implies that ( ρ, ρ ) ∈ N E (cid:16) G ( ψ ∗ ,ψ ∗ ) (cid:17) , and that for any player i , anybiased belief ψ i , and any Nash equilibrium ( s , s ) of the biased game G ( ψ i ,ψ j ), s j = min ( s i , ρ ). Thisimplies that π i ( s , s ) ≤ π i ( ρ, ρ ), which shows that (( ψ ∗ , ψ ∗ ) , ( ρ, ρ )) is a strong BBE. Observe thatthis BBE induces only a small distortion in the belief of each player, assuming that ρ is sufficientlyclose to one: | ψ ∗ i ( s j ) − s j | < (cid:12)(cid:12)(cid:12)(cid:12) s j ρ − s j (cid:12)(cid:12)(cid:12)(cid:12) < − ρρ . Our second example characterizes the set of BBE outcomes (and their supporting beliefs) in staghunt games.
Example 10 ( Stag hunt games ) . Stag hunt is a two-action game describing a conflict between safetyand social cooperation. Specifically, each player i has two actions: s i (“stag”) and h i (“hare”), andhis ordinal preferences are ( s i , s j ) (cid:31) i ( h i , s j ) (cid:23) i ( h i , h j ) (cid:31) i ( s i , h j ). Table 5 presents the payoff of ageneral stag hunt game, where we have normalized, without loss of generality, the payoff of eachplayer when playing action profile ( s i , s j ) (( h i , h j )) to be one (zero), and where each g i is positiveand each l i is in the interval (0 , A ADDITIONAL EXAMPLES
Table 5: Stag Hunt Game ( g , g ∈ (0 ,
1] and l , l > s h s , − l , g h g , − l , s i , s j ), ( h i , h j ), and ( α ∗ , α ∗ ), with α ∗ i = l j l j + (1 − g j ) ∈ (0 , , where each α i represents the probability that player i plays s i .Applying the analysis of the previous section shows that the game admits 3 classes of BBE: • Hunting the hare:(( ψ ∗ , ψ ∗ ) , (0 , ψ ∗ i is an arbitrary monotone biased belief thatsatisfies ψ ∗ i (1) ≥ α ∗ i . • Hunting the stag. (( ψ ∗ , ψ ∗ ) , (1 , ψ ∗ i is an arbitrary monotone biased belief thatsatisfies ψ ∗ i (1) ≤ α ∗ i . • Mixing with less weight to hunting the stag, wishful thinking, and responsiveness to badnews: (( ψ ∗ , ψ ∗ ) , ( β , β )), where for each player i : (1) the payoff is above the minmax payoff: π i ( β i , β j ) ≥
0, (2) the players hunt the stag less often in the unique Nash equilibrium: β i ∈ (0 , α ∗ i ), (3) wishful thinking: ψ ∗ i ( β j ) = α ∗ j > β j , (4) responsiveness to bad news: ψ ∗ i ( α ) = α ∗ j for each α ≥ β j , and ψ ∗ i ( α ) < α ∗ j for each α < β j .Observe that any profile ( β , β ), where β i ∈ ( α ∗ i , β j = 1, thenplayer i can gain by deviating to ψ i ≡
1, as the unique equilibrium of the new biased game is (1 , i relative to ( β i , β j ). If β j <
1, then player j can gain bydeviating to ψ j ≡
1, as the only possible equilibria of the new biased game are (1 ,
1) and ( β i , j relative to ( β i , β j ).Our third example deals with the traveler’s dilemma game, in which each agent has 100 pureordered actions that have a discrete payoff structure that resembles strategic complementarity ininterval games. We demonstrate how to implement the undominated Pareto optimal profile in thisgame as a strong BBE outcome that presents wishful thinking. Example 11 ( Implementing the undominated Pareto optimal profile as a strong BBE in the trav-eler’s dilemma ) . .4 Examples of Games with Strategic Complements A i = { , ..., } ), and the payoff function of each player is π i ( a i , a j ) = a i + 2 a i < a j a i a i = a j a j − a i > a j . The interpretation of the game is as follows. Two identical suitcases have been lost, each ownedby one of the players. Each player has to evaluate the value of his own suitcase. Both players geta payoff equal to the minimal evaluation (as the suitcases are known to have identical values), and,in addition, if the evaluations differ, then the player who gave the lower (higher) evaluation gets abonus (malus) of 2 to his payoff.It is well known that the unique Nash equilibrium is (1 , ψ ∗ i as follows: ψ ∗ i ( α , α ..., α , α ) = (cid:18) α , α , ..., α , α α (cid:19) . In what follows we show that (( ψ ∗ , ψ ∗ ) , (99 , ψ ∗ i (99) = (cid:16) , ..., , , (cid:17) , which implies that 99 ∈ BR ( ψ ∗ i (99)), and, thus, (99 , ∈ N E (cid:16) G ( ψ ∗ ,ψ ∗ ) (cid:17) . Let ψ be an arbitrary perception bias of player i . Observe that player i never plays action 100 in a anyNash equilibrium of any biased game, because action 100 is not a best reply against any strategy ofplayer j . Next observe that player i can obtain a payoff higher than 99 only if (1) player j choosesaction 99 with a positive probability, and (2) player i chooses action 98 with a probability strictlyhigher than his probability of playing action 100. However, the biased belief ψ ∗ j of player j impliesthat if player i chooses action 98 with a probability strictly higher than his probability of playing 100,then player j never chooses action 99 in any Nash equilibrium of the induced biased game becauseaction 99 yields a strictly lower payoff to player j than action 98 against the perceived strategy ofplayer i (because according to this perceived strategy, player i plays action 100 with a probabilitystrictly less than player i ’s probability of playing either action 98 or action 99).Note that the BBE equilibrium outcome (99 ,
99) is consistent with level-1 behavior in the level- k and cognitive hierarchy literature (see, e.g., Stahl and Wilson, 1994; Nagel, 1995; Costa-Gomes,Crawford, and Broseta, 2001; Camerer, Ho, and Chong, 2004), according to which each agent believes A ADDITIONAL EXAMPLES that his opponent is following a focal non-strategic action (the action 100 in the traveler’s dilemma),and best-replies to this belief. The notion of BBE can help explain why such level-k behavior inducesa strategic advantage in the long run, and why, therefore, it is likely to emerge in an equilibrium.
A.5 Hawk-Dove Game
The following example characterizes the set of BBE (and their supporting beliefs) in a hawk-dovegame (which is a game of strategic substitutes).
Example 12 ( The Hawk-dove game ) . The hawk-dove (or “chicken”) game is a two-action game inwhich each player i has two actions: d i (interpreted as a “dove”-like action of willingness to share aresource with the opponent) and h i (interpreted as a “hawk”-like action of insistence on getting thewhole resource, even if this requires fighting against the opponent), and where the ordinal preferencesof each player i are ( h i , d j ) (getting the resource) (cid:31) ( d i , d j ) (sharing the resource) (cid:31) ( d i , h j ) (notgetting the the resource) (cid:31) ( h i , h j ) (being involved in a serious fight). Table 6 presents the payoffof a general two-action hawk-dove game, where we have normalized, without loss of generality, thepayoff of each player when playing action profile ( d i , d j ) (( h i , h j )) to be one (zero), and where each g i positive and each l i is in the interval (0 , g , g > l , l ∈ (0 , d h d , − l , g h g , − l , d , h )and ( h , d ), and one mixed equilibrium ( α ∗ , α ∗ ), where the probability that player i plays action α ∗ i is α ∗ i = 1 − l j g j + (1 − l j ) ∈ (0 , , and π (cid:16) α ∗ i , α ∗ j (cid:17) = α ∗ j · (1 + g i ) = 1 − g i g i + (1 − l i ) · l i . The undominated minmax payoff of each player coincides with the minmax payoff of each player(as there are no dominated actions), and it is equal to M Ui = 1 − l i , which is obtained when theopponent plays h j .Applying the analysis of the previous section shows that the game admits 3 classes of BBE: • Pure equilibrium hawk-dove: (cid:16)(cid:16) ψ ∗ i , ψ ∗ j (cid:17) , (0 , (cid:17) , where (1) ψ ∗ i is an arbitrary monotone biasedbelief that satisfies ψ ∗ i (0) ≥ α ∗ i , and (2) ψ ∗ j is an arbitrary monotone biased belief that satisfies ψ ∗ i (0) ≤ α ∗ i . • Mixing (with less weight to playing dove), wishful thinking, and one-directional blindness:(( ψ ∗ , ψ ∗ ) , ( β , β )), where for each player i : (1) the payoff is above the minmax payoff: π i ( β i , β j ) ≥ − l i , (2) β i ∈ (0 , α ∗ i ) (i.e., agents play dove less often in the unique Nashequilibrium), (3) wishful thinking: ψ ∗ i ( β j ) = α ∗ j > β j , and (4) responsiveness only to goodnews: ψ ∗ i ( α ) = α ∗ j for each α ≤ β j , and ψ ∗ i ( α ) > α ∗ j for each α > β j .Observe that any profile ( β , β ) where β i ∈ ( α ∗ i ,
1) cannot be a BBE outcome. If β j = 1, then player i can gain by deviating to ψ i ≡
1, as the unique equilibrium of the new biased game is (0 i , j ), whichinduces a higher payoff to player i relative to ( β i , β j ). If β j <
1, then player j can gain by deviatinginto ψ j ≡
1, as the only possible equilibria of the new biased game are (1 i , j ) and ( β i , j ), both ofwhich induce a higher payoff to player j relative to ( β i , β j ). B Evolutionary Interpretation of BBE
In this section we present a formal definition of strong BBE that is exactly analogous to the definitionof a stable configuration `a la Dekel, Ely, and Yilankaya (2007). This shows that our static solutionconcept of strong BBE captures evolutionary stability in the same way as the solution concepts usedin the literature on “indirect evolution of preferences.” Finally, we illustrate a detailed example of apossible learning dynamic that may result in convergence to strong BBE.
B.1 Evolutionary Definition of Strong BBE ` a la Dekel, Ely, and Yilankaya (2007) In this subsection we present a definition of a strong BBE that is completely analogous to thedefinition of a stable configuration a la Dekel, Ely, and Yilankaya (2007) (henceforth DEY) for thecase of perfect observability of the opponent’s type (i.e., p = 1 in DEY).In the adaptation of the notion of stable configuration `a la Dekel, Ely, and Yilankaya (2007) toour setup we change two aspects (and only these aspects):1. We deal with general two-player games played between two different populations, rather thanDEY’s setup that deals with symmetric two-player games played within a single population.2. Each agent in DEY’s model is endowed with a type that determines the agent’s subjectivepreferences. By contrast, in our setup each agent is endowed with a type that determines theagent’s monotone biased belief.3. We focus on homogeneous configurations. DEY’s general definitions allow one to deal withheterogeneous configurations (in which different incumbents may have different types). How-ever, their results mainly deal with homogeneous configurations (in which all incumbents havethe same type). Therefore, to ease notation, we focus on homogeneous configurations in ouradaptation of DEY’s definitions.After adapting DEY’s definition of a homogeneous configuration (page 689 in DEY) to the threeaspects mentioned above, their definition is as follows: Definition 13.
A (homogeneous) configuration is a pair (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )), where, for each player i , function ψ ∗ i is a monotone biased belief of player i and s ∗ i is a strategy of player i satisfying s ∗ i ∈ BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17) .It is immediate that any monotone weak BBE is a configuration. B EVOLUTIONARY INTERPRETATION OF BBE
Next, DEY present a notion of a balanced configuration (page 689 in DEY) that is triviallysatisfied by any homogeneous configuration.Consider two continuum populations of mass one that follow a configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )).Assume that one of these populations (say, population i ) is invaded by a small group of 0 < (cid:15) << ψ i = ψ ∗ i . DEY assume that (1) such a mutation can destabilizea configuration by resulting in the mutants achieving a higher fitness than the incumbents of thesame population i , and (2) the incumbents continue to play the same behavior among themselves(what DEY calls “focal equilibria”).Let Ψ i be the set of all biased beliefs of player i . Following DEY (page 690 in DEY) we define N i,(cid:15) ( ψ ∗ i , ψ i ) ∈ ∆ (Ψ i ) to be the set of distributions over biased beliefs in population i resulting fromentry by no more than (cid:15) mutants. Formally, N i,(cid:15) (cid:0) ψ ∗ i , ψ i (cid:1) = (cid:8) µ i ∈ ∆ (Ψ i ) | µ i = (cid:0) − (cid:15) (cid:1) · ψ ∗ i + (cid:15) · ψ i , (cid:15) < (cid:15) (cid:9) . Given a configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) and a post-entry distribution of biased beliefs in pop-ulation i ˜ µ i ∈ N i,(cid:15) ( ψ ∗ i , ψ i ), a post-entry focal configuration is a pair (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) , where (1) s i ∈ BR (cid:16) ψ i (cid:16) s j (cid:17)(cid:17) is interpreted as the mutant’s strategy, and (2) s j ∈ BR (cid:16) ψ ∗ j ( s i ) (cid:17) is interpretedas population j’s strategy against the mutants. The incumbents are assumed to play the same pre-entry strategies (cid:16) s ∗ i , s ∗ j (cid:17) when being matched among themselves. Let B (˜ µ i ) denote the set of allpost-entry focal configurations.Following DEY (Definition 3 on page 691 in DEY), we define DEY-stability of a configurationas follows. Definition 14.
Configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is DEY-stable if there exists (cid:15) > i , every biased belief ψ i , every post-entry distribution of biased beliefs ˜ µ i ∈ N i,(cid:15) ( ψ ∗ i , ψ i ),and every post-entry focal configuration (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) , the mutants are weakly outperformedrelative to the incumbents’ payoff (in their own population), i.e., π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) . B.2 Equivalence between the Definitions
The following result shows that the definition of a stable configuration coincides with our definitionof strong BBE.
Proposition 10.
A configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is DEY-stable iff it is a strong BBE.Proof. “If” part: Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a strong BBE. Let (cid:15) > i ∈ { , } , and ψ i ∈ Ψ i .Let ˜ µ i ∈ N i,(cid:15) (cid:16) ψ ∗ j , ψ i (cid:17) be a post-entry distribution of biased beliefs. Let (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) be apost-entry focal configuration. The fact that (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) is a post-entry focal configurationimplies that s i ∈ BR (cid:16) ψ i (cid:16) s j (cid:17)(cid:17) and s j ∈ BR (cid:16) ψ ∗ j ( s i ) (cid:17) . The fact that it is a strong BBE impliesthat π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) , which shows that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is DEY-stable. Under imperfect observability, a mutant can destabilize a configuration by unraveling the original equilibriumbehavior, thereby causing the incumbents’ strategies to substantially diverge following the mutant’s entry into thepopulation. This cannot happen under perfect observability, as the incumbents can always exhibit the same equilibriumbehavior when being matched against other incumbents (see, page 690 in DEY for a discussion of focal equilibria). .2 Equivalence between the Definitions ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be DEY-stable configuration. Let i ∈ { , } and ψ i ∈ Ψ i .Let (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) be an equilibrium of the new biased game. Let (cid:15) >
0. Let ˜ µ i ∈ N i,(cid:15) ( ψ ∗ i , ψ i ) be a post-entry distribution of biased beliefs. For each (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , let (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) be a post-entry focal configuration. The assumption that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) isDEY-stable implies that π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) . This implies that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a strongBBE. Remark (Allowing multiple simultaneous invasions of mutants) . The definition of DEY-stabilitypresented above is unaffected when various groups of mutants simultaneously invade one of thepopulations. By contrast, if one were to require a stable configuration to resist simultaneous invasionsof two groups of mutants, one invasion of each population, it would require a refinement of the conceptof strong BBE, in the spirit of Maynard-Smith and Price’s (1973) notion of evolutionary stability,such that if both ψ and ψ are best replies against configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )), then (1) ψ ∗ should be a strictly better reply against ψ (relative to ψ ), and (2) ψ ∗ should be a strictly betterreply against ψ (relative to ψ ).Similarly, one can formulate a definition of stability equivalent to that of monotone BBE byrequiring the mutants to be weakly outperformed in at least one post-entry focal configuration. Definition 15.
Configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is weakly stable if there exists (cid:15) > i , every biased belief ψ i , and every post-entry distribution of biased beliefs ˜ µ i ∈ N i,(cid:15) ( ψ ∗ i , ψ i ), there exists a post-entry focal configuration (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) in which the mutantsare weakly outperformed relative to the incumbents’ payoff, i.e., π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) . The following result shows that the definition of a weakly stable configuration coincides withour definition of weak BBE. The simple proof, which is analogous to the proof of 10, is omitted forbrevity.
Proposition 11.
A configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is weakly stable iff it is a monotone weak BBE. Finally, one can formulate a definition of stability equivalent to that of a BBE by requiring themutants to be weakly outperformed in at least one plausible post-entry focal configuration.
Definition 16.
Given configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )), (cid:15) > i ∈ { , } , biased belief ψ i , and apost-entry distribution of biased beliefs ˜ µ i ∈ N i,(cid:15) ( ψ ∗ i , ψ i ), we say that a post-entry focal configuration (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) is implausible if: (1) ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ), (2) s j = s ∗ j , and (3) (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s ∗ j (cid:17)(cid:17) isa post-entry focal configuration. A post-entry focal configuration is plausible if it is not implausible. Definition 17.
Configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is plausibly stable if there exists (cid:15) > i , every biased belief ψ i , and every post-entry distribution of biased beliefs ˜ µ i ∈ N i,(cid:15) ( ψ ∗ i , ψ i ), there exists a plausible post-entry focal configuration (cid:16)(cid:16) ˜ µ i , ψ ∗ j (cid:17) , (cid:16) s i , s j (cid:17)(cid:17) in which themutants are weakly outperformed relative to the incumbents’ payoff, i.e., π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) . The following result shows that the definition of a plausibly stable configuration coincides withour definition of BBE. The simple proof, which is analogous to the proof of 10, is omitted for brevity.
Proposition 12.
A configuration (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is plausibly stable iff it is a BBE. B EVOLUTIONARY INTERPRETATION OF BBE
B.3 Illustration of the Evolutionary Interpretation
Similar to DEY, we have presented a reduced-form static notion of evolutionary stability, withoutformally modeling a detailed dynamics according to which the biased beliefs and the strategiesco-evolve. In Section 3.6 we present the essential features of this evolutionary process, which areanalogous to DEY’s essential features (see first paragraph in Section 2.2 in DEY): agents are endowedby biased beliefs, these biased-beliefs induce equilibrium behavior in the biased game (presumablyby a relatively quick adjustment of the biased players that leads to equilibrium behavior in the biasedgame), behavior determines “success,” and success (the material payoffs) regulates the evolution ofbiased beliefs (presumably by a slow process in which agents occasionally die and are replaced bynew agents who are more likely to mimic the biased beliefs of more successful incumbents).In what follows, we illustrate this evolutionary process and its underlying dynamics in an exam-ple. Specifically, we present a strong BBE in an “input” game and we illustrate how this strong BBEcan persist, given plausible evolutionary dynamics through which the composition of the populationevolves.
Example 13 ( Example 9 revisited ) . Consider the following “input” game. Let S i = S j = [0 , π i ( s i , s j , ρ ) = s i · s j − s i ρ , where the parameter ρ is interpreted as the costof effort, and we assume that ρ ∈ (0 . , ρ times smaller than the opponent’s (i.e., BR ( s j ) = ρ · s j ), (2)in the unique Nash equilibrium of the unbiased game each player exerts no effort s i = s j = 0, and(3) the strategy profile ( ρ, ρ ) yields a payoff of ρ − ρ >
0, which is the highest symmetric payoffamong all strategy profiles in which agents do not use strictly dominated strategies. Let ψ ∗ i be thefollowing biased-belief function: ψ ∗ i ( s j ) = s j ρ s j < ρ s j ≥ ρ. In Example 9 we have shown that (( ψ ∗ , ψ ∗ ) , ( ρ, ρ )) is a strong BBE. In what follows we illustrate howthis strong BBE can persist. Consider a small group of mutants of population i who have undistortedbeliefs. Assume that, initially, the incumbents of population j use the same strategy against themutants as they use against the incumbents of population i (i.e., strategy ρ ), and the mutantsgradually learn to best reply to the incumbents’ behavior by playing ρ . Recall that we assume thatthe agents of population j identify the mutants as a separate group of agents who behave differentlythan the rest of population j (without assuming that the incumbents of population j know anythingabout the biased beliefs of the mutants). These incumbents perceive the mutants’ play as ρ (dueto the incumbents’ biased beliefs), and gradually learn to best reply to this perceived strategy byplaying ρ . This, in turn, induces the mutants to adapt their play to playing ρ , and, in response,the incumbents of population j adapt their play against the mutants and play ρ (the best replyto the mutants’ perceived strategy ρ ). This mutual gradual adaptation process continues until theplay in the matches between incumbents of population j and mutants of population i converges to(0 , , i ( ρ − ρ >
0) in the underlying game, their fitness is expected to belower, and they are much less likely to be chosen as mentors. As a result the share of mutants inthe population slowly shrinks until they disappear from the population.
C Principal-Agent (Subgame-Perfect) Definition of BBE
In this appendix we present an equivalent definition of BBE as a subgame-perfect equilibrium of atwo-stage game in which in the first round each player chooses the biased belief of the agent whowill play on his behalf in the second round.
C.1 The Two-Stage Game Γ G Given an underlying two-player normal-form game G = ( S, π ) define Γ G as the following four-playertwo-stage extensive-form game. The four players in the game Γ are: principal 1 and principal 2(who choose representative agents for the second stage), agent 1 (who plays on behalf of principal 1in round 2), and agent 2 (who plays on behalf of principal 2 in round 2).The game Γ G has 2 stages. In the first stage, the principals simultaneously choose biased beliefsfor their agents. That is, each principal i chooses a biased belief ψ i : S j → S j for agent i . In thesecond stage the agents simultaneously choose their strategies. That is, each agent i chooses strategy s i ∈ S i . The payoff of each principal i is π i ( s i , s j ). The payoff of each agent i is π i ( ψ i ( s i ) , s j ) . LetΨ i be the set of all feasible (monotone) biased beliefs of agent i .A pure strategy profile of Γ G (henceforth Γ G -strategy profile) is a tuple ( ψ , ψ , σ , σ ), whereeach ψ i is a biased belief, and each σ i : Ψ × Ψ → S i is a function assigning a strategy to each pairof (monotone) biased beliefs. Let SP E (Γ G ) denote the set of all subgame-perfect equilibria of Γ. C.2 Subgame-Perfect Definition of Weak BBE
The following result shows that a weak BBE is equivalent to a subgame-perfect equilibrium of Γ.Formally:
Proposition 13.
Let G be a game. Strategy profile (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a weak BBE of G iff thereexists a subgame-perfect equilibrium (( ψ ∗ , ψ ∗ ) , ( σ ∗ , σ ∗ )) of Γ G satisfying σ ∗ i ( ψ ∗ i ) = s ∗ i for each player i. Proof. “If side”: Let (( ψ ∗ , ψ ∗ ) , ( σ ∗ , σ ∗ )) ∈ SP E (Γ G ) be a subgame-perfect equilibrium of Γ sat-isfying σ ∗ i ( ψ ∗ i ) = s ∗ i for each player i. Let ψ i be a biased belief of player i . Let s = σ ∗ (cid:16) ψ i , ψ ∗ j (cid:17) and s = σ ∗ (cid:16) ψ i , ψ ∗ j (cid:17) . The fact that (( ψ ∗ , ψ ∗ ) , ( σ ∗ , σ ∗ )) ∈ SP E (Γ G ) implies that (1) ( s , s ) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) and (2) π i ( s , s ) ≤ π i ( s ∗ , s ∗ ). This implies that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a weak BBEof G .2 C PRINCIPAL-AGENT (SUBGAME-PERFECT) DEFINITION OF BBE “Only if side”: Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a weak BBE of G . We define ( σ ∗ , σ ∗ ) as follows: (1) σ ∗ i ( ψ ∗ , ψ ∗ ) = s ∗ i , (2) for each biased belief ψ i = ψ ∗ i , define σ ∗ i (cid:16) ψ i , ψ ∗ j (cid:17) = s i and σ ∗ j (cid:16) ψ i , ψ ∗ j (cid:17) = s j such that (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) and π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) (such a pair (cid:16) s i , s j (cid:17) exists due to(( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) being a weak BBE of G), and (3) for each pair of biased beliefs ψ i = ψ ∗ i and ψ j = ψ ∗ j , define σ ∗ i (cid:16) ψ i , ψ j (cid:17) = s i and σ ∗ j (cid:16) ψ i , ψ j (cid:17) = s j such that (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ j ) (cid:19) . Thedefinition of ( σ ∗ , σ ∗ ) immediately implies that (( ψ ∗ , ψ ∗ ) , ( σ ∗ , σ ∗ )) ∈ SP E (Γ G ). C.3 Subgame-Perfect Definition of BBE
Next, we present an equivalent definition of a BBE as a refinement of a subgame-perfect equilibriumof Γ G . Specifically, a subgame-perfect equilibrium ( ψ ∗ , ψ ∗ , σ ∗ , σ ∗ ) is required to remain a subgame-perfect equilibrium even after changing the off-the equilibrium path behavior to a different Nashequilibrium of the induced subgame in which (I) a single player (say, player j ) has deviated to adifferent biased-belief, (II) the non-deviator perceives the deviator’s strategy in the same way as theoriginal on-the-equilibrium path opponent’s strategy, and (III) the non-deviator changes his behaviorsuch that after the change it coincides with his on-the-equilibrium path behavior. Formally, Definition 18.
A subgame-perfect equilibrium ( ψ ∗ , ψ ∗ , σ ∗ , σ ∗ ) ∈ SP E (Γ G ) is a plausible subgame-perfect equilibrium if (I) the biased beliefs ψ ∗ and ψ ∗ are monotone, and (II) ( ψ ∗ , ψ ∗ , σ , σ ) ∈ SP E (Γ) for each pair of second-stage strategies σ , σ satisfying: (1) (cid:16) ψ , ψ , σ , σ (cid:17) ∈ SP E (Γ G ) forsome pair of first-stage strategies (cid:16) ψ , ψ (cid:17) (i.e., second-stage behavior is consistent with equilibriumbehavior in all subgames) and (2) if σ i (cid:16) ψ , ψ (cid:17) = σ ∗ i (cid:16) ψ , ψ (cid:17) , then: (I) ψ i = ψ ∗ i and ψ j = ψ ∗ j , (II) ψ ∗ i (cid:16) σ j (cid:16) ψ , ψ (cid:17)(cid:17) = ψ ∗ i (cid:16) σ ∗ j (cid:16) ψ , ψ (cid:17)(cid:17) , and (III) σ j (cid:16) ψ , ψ (cid:17) = σ ∗ j (cid:16) ψ , ψ (cid:17) . Proposition 14.
Let G be a game. Strategy profile (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE of G iff there existsa plausible subgame-perfect equilibrium (( ψ ∗ , ψ ∗ ) , ( σ ∗ , σ ∗ )) of Γ G satisfying σ ∗ i ( ψ ∗ i ) = s ∗ i for eachplayer i. The simple proof, which is analogous to the proof of Proposition 13, is omitted for brevity.
C.4 Subgame-Perfect Definition of Strong BBE
Finally, we present an equivalent definition of a strong BBE as a refinement of a subgame-perfectequilibrium of Γ, which remains an equilibrium even after changing off the equilibrium path insubgames to other Nash equilibria of the induced subgames. Formally,
Definition 19.
A subgame-perfect equilibrium ( ψ ∗ , ψ ∗ , σ ∗ , σ ∗ ) ∈ SP E (Γ G ) is a strong subgame-perfect equilibrium if (I) the biased beliefs ψ ∗ and ψ ∗ are monotone, and (II) ( ψ ∗ , ψ ∗ , σ , σ ) ∈ SP E (Γ G ) for each pair of second-stage strategies σ , σ satisfying: (1) (cid:16) ψ , ψ , σ , σ (cid:17) ∈ SP E (Γ G )for some pair of first-stage strategies ψ , ψ (i.e., second-stage behavior is consistent with equilib-rium behavior in all subgames) and (2) σ i ( ψ ∗ , ψ ∗ ) = σ ∗ i ( ψ ∗ , ψ ∗ ) (i.e., behavior after ( ψ ∗ , ψ ∗ ) isunchanged). The definition of ( σ ∗ , σ ∗ ) relies on the axiom of choice. Proposition 15.
Let G be a game. Strategy profile (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a strong BBE of G iffthere exists a strong subgame-perfect equilibrium (( ψ ∗ , ψ ∗ ) , ( σ ∗ , σ ∗ )) of Γ satisfying σ ∗ i ( ψ ∗ i ) = s ∗ i foreach player i. The simple proof, which is analogous to the proof of Proposition 13, is omitted for brevity.
D Discontinuous Biased Beliefs
In this appendix we present an alternative definition of BBE that relaxes the assumption that biasedbeliefs have to be continuous. We show that all BBE characterized in the main text remain BBEwhen deviators are allowed to use discontinuous biased beliefs.
D.1 Adapted Definitions: Quasi-equilibria
We redefine a biased belief ψ i : S j → S j to be an arbitrary (rather than continuous) function thatassigns to each strategy of the opponent a (possibly distorted) belief about the opponent’s play. Thedefinition of a configuration ( ψ ∗ , s ∗ ) is left unchanged (i.e., we require that (cid:16) s ∗ i , s ∗ j (cid:17) ∈ N E ( G ψ ∗ )).Recall that a configuration is a BBE if each biased belief is a best reply to the opponent’s biasedbelief, in the sense that an agent who chooses a different biased belief is weakly outperformed inthe induced equilibrium of the new biased game. Allowing discontinuous beliefs implies that somebiased games G ( ψ ,ψ ) in which one (or both) of the biases are discontinuous may not admit Nashequilibria. This requires us to adapt the definition of a BBE to deal with behavior in biased gamesthat do not admit Nash equilibria. We do so by assuming that the resulting behavior in a biasedgame that does not admit a Nash equilibrium is a “ j -quasi-equilibrium,” in which the non-deviator(player j ) best replies to the perceived behavior of the deviator (player i ), while the deviator isallowed to play arbitrarily. Formally: Definition 20.
Let ( ψ i , ψ j ) be a profile of biased beliefs, and let j be one of the players (interpretedas the non-deviator); then we define QE j (cid:16) G ( ψ i ,ψ j ) (cid:17) as the set of j -quasi-equilibria of the biasedgame G ( ψ i ,ψ j ) as follows: QE j (cid:16) G ( ψ i ,ψ j ) (cid:17) = N E (cid:16) G ( ψ i ,ψ j ) (cid:17) N E (cid:16) G ( ψ i ,ψ j ) (cid:17) = ∅{ ( s i , s j ) | s j ∈ BR ( ψ j ( s i )) } N E (cid:16) G ( ψ i ,ψ j ) (cid:17) = ∅ . Note that any biased game admits a j -quasi-equilibrium. D.2 Adapted Definitions: BBE We redefine our notions of BBE as follows, and write them as BBE . In a strong BBE , the deviator(player i ) is required to be outperformed in all j -quasi-equilibria, and biased beliefs are required4 D DISCONTINUOUS BIASED BELIEFS to be monotone. In a weak BBE , the deviator is required to be outperformed in at least one j -quasi-equilibrium. The notion of a BBE is in between these two notions. Specifically, in a BBE ,the biased beliefs are required to be monotone, and, in addition, the deviator (player i ) is requiredto be outperformed in at least one plausible j -quasi-equilibrium of the new biased game, whereimplausible j -quasi-equilibria are defined as follows. We say that a j -quasi-equilibrium of a biasedgame induced by a deviation of player i is implausible if (1) player i ’s strategy is perceived by thenon-deviating player j as coinciding with player i ’s original strategy, (2) player j plays differentlyrelative to his original strategy, and (3) if player j were playing his original strategy, this wouldinduce a j -quasi-equilibrium of the biased game. That is, implausible j -quasi-equilibria are those inwhich the non-deviating player j plays differently against a deviator even though player j has noreason to do so: player j does not observe any change in player i ’s behavior, and player j ’s originalbehavior remains an equilibrium of the biased game. Formally: Definition 21.
Given configuration ( ψ ∗ , s ∗ ), deviating player i , and biased belief ψ i , we say thata j -quasi-equilibrium of the biased game (cid:16) s i , s j (cid:17) ∈ QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) is implausible if: (1) ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ), (2) s ∗ j = s j , and (3) (cid:16) s i , s ∗ j (cid:17) ∈ QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) . A j -quasi-equilibrium is plausible if it isnot implausible. Let P QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) be the set of all plausible j -quasi-equilibria of the biasedgame G ( ψ i ,ψ ∗ j ).Note that it is immediate from Definition 21 and the nonemptiness of QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) that P QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) is nonempty. Definition 22.
Configuration ( ψ ∗ , s ∗ ) is:1. a strong BBE if (I) each biased belief ψ ∗ i is monotone, and (II) π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) forevery player i , every biased belief ψ i , and every j -quasi-equilibrium (cid:16) s i , s j (cid:17) ∈ QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) ;2. a weak BBE if for every player i and every biased belief ψ i , there exists a j -quasi-equilibrium (cid:16) s i , s j (cid:17) ∈ QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , such that π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) ;3. a BBE if (I) each biased belief ψ ∗ i is monotone, and (II) for every player i and every biasedbelief ψ i , there exists a plausible j -quasi-equilibrium (cid:16) s i , s j (cid:17) ∈ P QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , such that π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) .It is immediate that any strong BBE is a BBE , and that any BBE is a weak BBE .(resp., strong, weak) BBE ( ψ ∗ , s ∗ ) is continuous if each biased function ψ ∗ i is continuous. Note,that deviators are allowed to choose discontinuous biased beliefs. D.3 Robustness of BBE to Discontinuous Biased Beliefs
In what follows we observe that all the BBE that we characterize in all the results of the paper arealso BBE . That is, all of our BBE are robust to allowing deviators to use discontinuous biased5beliefs. Specifically, any BBE (resp., weak BBE, strong BBE) that is characterized in any result(or example) in the paper, is a continuous BBE (resp., weak continuous BBE , strong continuousBBE ).The reason why this observation is true is that in all the arguments in the proofs of the paper’sresults for why a configuration (cid:16)(cid:16) ψ ∗ i , ψ ∗ j (cid:17) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) is a BBE, when we show that a deviator (player i ) is outperformed after deviating to biased belief ψ i and after the players play strategy profile (cid:16) s i , s j (cid:17) , we rely only on the assumption that the non-deviator (player j ) best replies to the deviator(i.e., that s j ∈ BR (cid:16) ψ ∗ j ( s i ) (cid:17) , which is implied by assuming (cid:16) s i , s j (cid:17) ∈ QE j (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) ), and we donot use in any of the arguments the assumption that the deviator plays a best reply (i.e., we do notrely on s i ∈ BR (cid:16) ψ i (cid:16) s j (cid:17)(cid:17) in any of the proofs). E Partial Observability
Throughout the paper we assume that if an agent deviates to a different biased belief, then theopponent always observes this deviation. In this appendix, we relax this assumption, and show thatour results hold also in a setup with partial observability (some results hold for any level of partialobservability, while others hold for a sufficiently high level of observability).
E.1 Restricted Biased Games
Let p ∈ [0 ,
1] denote the probability that an agent who is matched with an opponent who deviatesto a different biased belief privately observes the opponent’s deviation (henceforth, observationprobability ). If an agent does not observe the deviation, then he continues playing his originalconfiguration’s strategy.Our definitions of configuration and biased game remain unchanged. We now define a restrictedbiased game G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) as a game with a payoff function according to which (1) each player’s payoffis determined by the opponent’s perceived strategy, and (2) the non-deviator is restricted to playing s ∗ j with probability p (i.e., when not observing the opponent’s deviation). Formally: Definition 23.
Given an underlying game G = ( S, π ), a profile of biased beliefs (cid:16) ψ i , ψ ∗ j (cid:17) , and astrategy s ∗ j of player j (interpreted as the non-deviator), let the restricted biased game G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) = (cid:16) S, ˜ π (cid:16) ψ i , ψ ∗ j , s ∗ j , p (cid:17)(cid:17) be defined as follows:˜ π i (cid:16) ψ i , ψ ∗ j , s ∗ j , p (cid:17) ( s i , s j ) = p · π i (cid:0) s i , ψ i ( s j ) (cid:1) + (1 − p ) · π i (cid:16) s i , ψ i (cid:16) s ∗ j (cid:17)(cid:17) , and˜ π j (cid:16) ψ i , ψ ∗ j , s ∗ j , p (cid:17) ( s i , s j ) = p · π j (cid:16) s j , ψ ∗ j ( s i ) (cid:17) + (1 − p ) π i (cid:16) s ∗ j , ψ ∗ j ( s i ) (cid:17) . A Nash equilibrium of a p -restricted biased game is defined in the standard way. Formally, apair of strategies s ∗ = ( s , s ) is a Nash equilibrium of a restricted biased game G ( ψ i ,ψ ∗ j ,s ∗ j ,p ), if each s i is a best reply against the perceived strategy of the opponent, i.e., s i = argmax s i ∈ S i (cid:16) ˜ π i (cid:16) ψ i , ψ ∗ j , s ∗ j , p (cid:17) (cid:16) s i , s j (cid:17)(cid:17) . E PARTIAL OBSERVABILITY
Let
N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ) (cid:19) ⊆ S × S denote the set of all Nash equilibria of the restricted biasedgame G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) . Observe that the set of strategies of a biased game is convex and compact, and the payoff function˜ π i (cid:16) ψ i , ψ ∗ j , s ∗ j , p (cid:17) : S i × S j → R is weakly concave in the first parameter and continuous in bothparameters. This implies (due to a standard application of Kakutani’s fixed-point theorem) thateach restricted biased game G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) admits a Nash equilibrium (i.e., N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) = ∅ ). E.2 p -BBE We are now ready to define our equilibrium concept. Configuration ( ψ ∗ , s ∗ ) is a p -BBE if each biasedbelief is a best reply to the opponent’s biased belief, in the sense that an agent who chooses a differentbiased belief is weakly outperformed in the induced equilibrium of the new restricted biased game.We present three versions of p -BBE that differ with respect to the equilibrium selection when thenew biased game admits multiple equilibria. In a strong p -BBE (I) each biased-belief is monotone,and (II) the deviator is required to be outperformed in all Nash equilibria of the new restricted biasedgame. In a weak BBE, the deviator is required to be outperformed in at least one equilibrium ofthe new restricted biased game.The notion of a p -BBE is in between these two notions. Specifically, in at p -BBE (I) each biased-belief is monotone, and (II) the deviator is required to be outperformed in at least one plausibleequilibrium of the new restricted biased game, where implausible equilibria are defined as follows.We say that a Nash equilibrium of a restricted biased game induced by a deviation of player i isimplausible if (1) player i ’s strategy is perceived by the non-deviating player j as coinciding withplayer i ’s original strategy, (2) player j plays differently relative to his original strategy, and (3) ifplayer j were playing his original strategy, this would induce an equilibrium of the biased game.That is, implausible equilibria are those in which the non-deviating player j plays differently againsta deviator even though player j has no reason to do so: player j does not observe any change inplayer i ’s behavior, and player j ’s original behavior remains an equilibrium of the biased game.Formally: Definition 24.
Given configuration ( ψ ∗ , s ∗ ), deviating player i , and biased belief ψ i , we say that aNash equilibrium of the restricted biased game (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) is implausible if: (1) ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ), (2) s ∗ j = s j , and (3) (cid:16) s i , s ∗ j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) . An equilibrium is plausible if it is not implausible. Let P N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) be the set of all plausible equilibria of the biasedgame G ( ψ i ,ψ ∗ j ,s ∗ j ,p ).Note that it is immediate from Definition 24 and the nonemptiness of N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) that P N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) is nonempty. Definition 25.
Configuration ( ψ ∗ , s ∗ ) is: .3 Extension of Results a strong p -BBE if (I) each biased belief ψ ∗ i is monotone, and (II) π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) forevery player i , every biased belief ψ i , and every Nash equilibrium (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) ;2. a weak p -BBE if for every player i and every biased belief ψ i , there exists a Nash equilibrium (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) , such that π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) ;3. a p -BBE if (I) each biased belief ψ ∗ i is monotone, and (II) for every player i and every biasedbelief ψ i , there exists a plausible Nash equilibrium (cid:16) s i , s j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) (cid:19) , such that π i (cid:16) s i , s j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) .It is immediate that: (1) any strong p -BBE is a p -BBE, and that any p -BBE is a weak p -BBE, and(2) the definition of 1-BBE (resp., weak 1-BBE, strong 1-BBE) coincides with the original definitionof BBE (resp., weak BBE, strong BBE). E.3 Extension of Results
In what follows we sketch how to extend our results to the setup of partial observability. Theadaptations of the proofs are relatively simple, and, for brevity, we only sketch the differences withrespect to the original proofs.
E.3.1 Adaptation of Section 4 (Nash Equilibria and BBE Outcomes)
The example that some Nash equilibria cannot be supported as the outcomes of weak P -BBE withundistorted beliefs can be extended for any p > Example 14 ( Example 1 revisited. Cournot equilibrium cannot be supported by undistorted beliefs ) . Consider the following symmetric Cournot game with linear demand G = ( S, π ): S i = [0 ,
1] and π i ( s i , s j ) = s i · (1 − s i − s j ) for each player i . The unique Nash equilibrium of the game is s ∗ i = s ∗ j = , which yields both players a payoff of . Fix observation probability p >
0. Assume to the contrarythat this outcome can be supported as a weak p -BBE by the undistorted beliefs ψ ∗ i = ψ ∗ j = I d . Fixa sufficiently small 0 < (cid:15) <<
1. Consider a deviation of player 1 to the blind belief ψ i ≡ − · (cid:15) .Note that this blind belief has a unique best reply: s i = + (cid:15) . The unique equilibrium of therestricted biased game G ( ψ i ,ψ ∗ j ,s ∗ j ,p ) is s j = − (cid:15) , s i = + (cid:15) , which yields the deviator a payoff of + (cid:15) − (cid:15) with probability p (when his deviation is observed by player 2) and a payoff of − (cid:15) withprobability 1 − p (when his deviation is not observed by player 2). For a sufficiently small (cid:15) > .All the results of Section 4 hold for any observation probability p ∈ [0 ,
1] with minor adaptationsto the proofs.
Proposition 16 (Proposition 1 extended) . Let ( s ∗ , s ∗ ) be a (strict) Nash equilibrium of the game G = ( S, π ) . Let ψ ∗ ≡ s ∗ and ψ ∗ ≡ s ∗ . Then (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a (strong) p -BBE for any p ∈ [0 , .Claim . The unique Nash equilibrium payoff of a zero-sum game is also theunique payoff in any weak p -BBE for any p ∈ [0 , E PARTIAL OBSERVABILITY
Proposition 17 (Proposition 2 extended) . If a game admits a strictly dominant strategy s ∗ i forplayer i , then any weak p -BBE outcome is a Nash equilibrium of the underlying game. E.3.2 Adaptation of Section 6 (Main Results)Adaptation of Subsection 6.1 (Preliminary Result)
Minor adaptations of the proof of Propo-sition 3 show that it holds for any p ∈ [0 , Proposition 18.
Let p ∈ [0 , . If a strategy profile s ∗ = ( s ∗ , s ∗ ) is a weak p -BBE outcome, then(1) the profile s ∗ is undominated and (2) π i ( s ∗ ) ≥ M Ui . Adaptation of Subsection 6.2 (Games with Strategic Complements)
Minor adaptationsto the proofs of the results of Subsection 6.2 show that most of these results (namely, part (1) ofProposition 4 and Corollaries 2 and 3) hold for any p ∈ [0 , p -s sufficiently close to one. Formally: Proposition 19 ( Proposition . Let G be a game with strategic substitutes and positiveexternalities .1. Fix p ∈ [0 , . Let ( s ∗ , s ∗ ) be a p -BBE outcome. Then ( s ∗ , s ∗ ) is (I) undominated, and for eachplayer i : (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) ≥ M Ui , and (III) s ∗ i ≤ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) (underinvestment).2. Let ( s ∗ , s ∗ ) be an undominated profile satisfying for each player i : (II’) π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui , and(III) s ∗ i ≤ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Then there exists ¯ p < such that ( s ∗ , s ∗ ) is a p -BBE outcome forany p ∈ [¯ p, .Moreover, if π i ( s i , s j ) is strictly concave then ( s ∗ , s ∗ ) is a strong p -BBE outcome for any p ∈ [¯ p, . Corollary 7.
Fix p ∈ [0 , . Let G be a game with strategic complements and positive externalitieswith a lowest Nash equilibrium ( s , s ) satisfying s < max ( S i ) for each player i . Let ( s ∗ , s ∗ ) be a p -BBE outcome. Then s i ≤ s ∗ i for each player i . Corollary 8.
Fix p ∈ [0 , . Let G be a game with positive externalities and strategic complements.Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a p -BBE. If s ∗ i / ∈ { min ( S i ) , max ( S i ) } , then player i exhibits wishful thinking(i.e., ψ ∗ i (cid:16) s ∗ j (cid:17) ≥ s ∗ j ).One can also adapt the examples of Section 6.2 (and, similarly, the examples of Sections 6.3 and6.4) to sufficiently high p s. Adaptation of Section 6.3 (Games With Strategic Substitutes)
Minor adaptations tothe proofs of the results of Subsection 6.3 show that most of these results (namely, part (1) ofProposition 5 and Corollaries 4 and 5) hold for any p ∈ [0 , p -s sufficiently close to one. Formally: .3 Extension of Results Proposition 20 ( Proposition . Let G be a game with strategic substitutes and positiveexternalities.1. Fix p ∈ [0 , . Let ( s ∗ , s ∗ ) be a p -BBE outcome. Then ( s ∗ , s ∗ ) is (I) undominated, and for eachplayer i : (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) ≥ M Ui , and (III) s ∗ i ≥ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) (overinvestment).2. Let ( s ∗ , s ∗ ) be an undominated profile satisfying for each player i : (II’) π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui , and(III) s ∗ i ≥ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Then there exists ¯ p < such that ( s ∗ , s ∗ ) is a p -BBE outcome forany p ∈ [¯ p, .Moreover, if π i ( s i , s j ) is strictly concave then ( s ∗ , s ∗ ) is a strong p -BBE outcome for any p ∈ [¯ p, . Corollary 9.
Fix p ∈ [0 , . Let G be a game with strategic substitutes and positive externalities. Let ( s ∗ , s ∗ ) be a BBE outcome. Then, there exists a Nash equilibrium of the underlying game ( s e , s e ) ,and a player i such that s ei ≥ s ∗ i . Corollary 10.
Fix p ∈ [0 , . Let G be a game with strategic substitutes and positive externalities.Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a p -BBE. If s ∗ i / ∈ { min ( S i ) , max ( S i ) } , then player i exhibits wishfulthinking (i.e., ψ ∗ i (cid:16) s ∗ j (cid:17) ≥ s ∗ j ). Adaptation of Section 6.3 (Games With Strategic Opposites)
Minor adaptations to theproofs of the results of Subsection 6.3 show that most of these results (namely, part (1) of Proposition6, and Corollary 6) hold for any p ∈ [0 , p -s sufficientlyclose to one. Formally: Proposition 21.
Let G be a game with positive externalities and strategic opposites : ∂π ( s ,s ) ∂s > ∂π ( s ,s ) ∂s < s , s .1. Fix p ∈ [0 , . Let ( s ∗ , s ∗ ) be a p -BBE outcome. Then ( s ∗ , s ∗ ) is (I) undominated: (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) ≥ M Ui for each player i , and (III) s ∗ ≤ max ( BR ( s ∗ )) and s ∗ ≥ min ( BR ( s ∗ )) (underinvestment of player 1 and overinvestment of player 2).2. Let ( s ∗ , s ∗ ) be a profile satisfying: (I) undominated, (II) π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui for each player i ,and (III) s ∗ ≤ max ( BR ( s ∗ )) and s ∗ ≥ min ( BR ( s ∗ )) . Then there exists ¯ p < such that ( s ∗ , s ∗ ) is a p -BBE outcome for any p ∈ [¯ p, . Corollary 11.
Fix p ∈ [0 , . Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) be a p -BBE of a game with positive externalities and strategic opposites (i.e., ∂π ( s ,s ) ∂s > ∂π ( s ,s ) ∂s < s , s ). If s ∗ i / ∈ { min ( S i ) , max ( S i ) } , then player i exhibits pessimism (i.e., ψ ∗ i (cid:16) s ∗ j (cid:17) ≤ s ∗ j ). E.3.3 Adaptation of Section 7 (Additional Results)Adaptation of Subsection 7.1 (BBE with Strategic Stubbornness)
In what follows weshow how to extend Example 5 to the setup of partial observability (while we leave the extension ofthe general result, Proposition 7, to future research). The example focuses on Cournot competition.0
E PARTIAL OBSERVABILITY
We show that for each level of partial observability p ∈ [0 , p ), while the opponent has undistorted beliefs. The first player’s (resp., opponent’s)payoff is strictly increasing (resp., decreasing) in p : it converges to the Nash equilibrium payoff when p →
0, and it converges to the Stackelberg leader’s (resp., follower’s) payoff when p → Example 15 ( Example 5 revisited ) . Consider the symmetric Cournot game with linear demand: G = ( S, π ): S i = R + and π i ( s i , s j ) = s i · (1 − s i − s j ) for each player i . Let p ∈ [0 ,
1] be theobservation probability. Then (cid:18)(cid:18) − p − p , I d (cid:19) , (cid:18) − p , − p · (3 − p ) (cid:19)(cid:19) is a strong BBE that yields a payoff of − p · (3 − p ) to player 1, and yields a payoff of (cid:16) − p · (3 − p ) (cid:17) to player2. Observe that player 1’s payoff is increasing in p , and it converges to the Nash equilibrium (resp.,Stackelberg leader’s) payoff of ( ) when p → p → p , and it converges to the Nash equilibrium (resp., Stackelberg follower’s) payoffof ( ) when p → p → (cid:16)(cid:16) − p − p , I d (cid:17) , (cid:16) − p , − p · (3 − p ) (cid:17)(cid:17) is a strong BBE issketched as follows: (1) n(cid:16) − p , − p · (3 − p ) (cid:17)o = N E (cid:18) G (cid:0) − p − p ,I d (cid:1)(cid:19) (because − p is the unique best replyagainst − p − p and − p · (3 − p ) is the unique best reply against − p ) ; (2) for any biased belief ψ , player 1keeps playing − p due to having a blind belief, and as a result player 2’s payoff is at most (cid:16) − p · (3 − p ) (cid:17) ;and (3) for any biased belief ψ inducing a deviating player 1 to play strategy x , player 2 plays − x (the unique best reply against x ) with probability p (when observing the deviation), and player 2plays − p · (3 − p ) (the original configuration strategy) with the remaining probability of 1 − p . Thus, thepayoff of a deviating player 1 who deviates into playing strategy x is π ( x ) := p · x · (cid:18) − x (cid:19) + (1 − p ) · x · (cid:18) − x − − p · (3 − p ) (cid:19) = (cid:18) − p (cid:19) · x · (1 − x ) − (2 − p ) · (1 − p )2 · (3 − p ) · x, where this payoff function π ( x ) is strictly concave in x with a unique maximum at x = − p (theunique solution to the FOC 0 = ∂π∂x = (cid:0) − p (cid:1) · (1 − · x ) − (2 − p ) · (1 − p )2 · (3 − p ) ). Extending the Folk Theorem Results for Sufficiently High p -s The main results of Subsec-tion 7.2, show folk theorem results for: (1) monotone BBE in games that admit best replies with fullundominated support, and (2) strong BBE in interval games with a payoff function that is strictlyconcave in the agent’s strategy, and weakly convex in the opponent’s strategy. Minor adaptationsof each proof can show that each result can be extended to p -s that are sufficiently close to one.Formally: Proposition 22 ( Proposition . Let G be a finite game that admits best replies with fullundominated support. Let ( s ∗ , s ∗ ) be an undominated strategy profile that induces for each player apayoff above his minmax payoff (i.e., π i ( s ∗ , s ∗ ) > M Ui ∀ i ∈ { , } ). Then there exists ¯ p < , suchthat ( s ∗ , s ∗ ) is a monotone weak p -BBE outcome for each p ∈ [¯ p, . Proposition 23 ( Proposition . Let G = ( S, π ) be an interval game. Assume that foreach player i , π i ( s i , s j ) is strictly concave in s i and weakly convex in s j . If ( s ∗ , s ∗ ) is undominatedand π i ( s ∗ , s ∗ ) > M Ui for each player i , then there exists ¯ p < , such that ( s ∗ , s ∗ ) is a strong p -BBEoutcome for each p ∈ [¯ p, .Sketch of adapting the proofs of Propositions
22 and 23 to the setup of partial observability.
Observethat the gain of an agent who deviates to a different biased belief, when his deviation is unobservedby the opponent, is bounded (due to the payoff of the underlying game being bounded). When thedeviation is observed by the opponent, the agent is strictly outperformed, given the BBE constructedin the proofs of Propositions 8 and 9 . This implies that there exists ¯ p < p ∈ [¯ p, F Proofs
F.1 Proof of Proposition 4
Part 1:
Proposition 3 implies (I) and (II). It remains to show (III, overinvestment). Let (cid:16)(cid:16) ψ ∗ i , ψ ∗ j (cid:17) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) be a BBE. Assume to the contrary that s ∗ i < min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Consider a deviation ofplayer i to a blind belief that the opponent always plays strategy s ∗ j (i.e., ψ i ≡ s ∗ j ). Let (cid:16) s i , s j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) be a plausible equilibrium of the new biased game. Observe first that s i ∈ BR (cid:16) ψ i (cid:16) s j (cid:17)(cid:17) = BR (cid:16) s ∗ j (cid:17) . This implies that s i > s ∗ i , and, thus, due to the monotonicity of ψ ∗ j we have: ψ ∗ j ( s i ) ≥ ψ ∗ j ( s ∗ i ). We consider two cases:1. If ψ ∗ j ( s i ) > ψ ∗ j ( s ∗ i ), then the strategic complementarity implies that s j ≥ min (cid:16) BR (cid:16) ψ ∗ j ( s i ) (cid:17)(cid:17) ≥ max (cid:16) BR (cid:16) ψ ∗ j ( s ∗ i ) (cid:17)(cid:17) ≥ s ∗ j , and this, in turn, implies that player i strictly gains from his devi-ation: π i (cid:16) s i , s j (cid:17) ≥ π i (cid:16) s i , s ∗ j (cid:17) > π i (cid:16) s ∗ i , s ∗ j (cid:17) , a contradiction.2. If ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ), then (cid:16) s i , s ∗ j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) and π i (cid:16) s i , s ∗ j (cid:17) > π i (cid:16) s ∗ i , s ∗ j (cid:17) , whichcontradicts that (cid:16)(cid:16) ψ ∗ i , ψ ∗ j (cid:17) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) is a BBE. Part 2:
Assume that strategy profile ( s ∗ , s ∗ ) satisfies I, II, and III. For each player i let s ei =min (cid:0) BR − ( s ∗ i ) (cid:1) . For every player i and every strategy s i < s ∗ i define X ( s i ) as the set of strategies s i for which player i is worse off (relative to π i ( s ∗ , s ∗ )) if he plays strategy s i , while player j playsa best-reply to s i . Formally: X s ∗ ( s i ) = n s i ∈ S i | π i ( s i , s j ) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) ∀ s j ∈ BR (cid:0) s i (cid:1)o . The assumption that π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui implies that X s ∗ ( s i ) is nonempty for each s i . The as-sumption of strategic complements implies that X s ∗ ( s i ) is an interval starting at min ( S i ). Let φ s ∗ ( s i ) = sup ( X s ∗ ( s i )). The assumption that the payoff function is continuously twice differen-tiable implies that φ s ∗ ( s i ) is continuous. The assumption that s ej = min (cid:0) BR − ( s ∗ i ) (cid:1) implies that2 F PROOFS lim s i % s ∗ i ( φ s ∗ ( s i )) = s ei . These observations imply that for each player j there exists a monotonebiased belief ψ ∗ j satisfying (1) ψ ∗ j ( s i ) = s ei for each s i ≥ s ∗ i and (2) ψ ∗ j ( s i ) ≤ φ s ∗ ( s i ) for each s i < s ∗ i with an equality only if φ s ∗ ( s i ) = min ( S i ).We now show that these properties of ( ψ ∗ , ψ ∗ ) imply that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE (a strongBBE if π i ( s i , s j ) is strictly concave in s i ). Consider a deviation of player i into an arbitrary biasedbelief ψ i . For each s i ≥ s ∗ i , and each (cid:16) s i , s j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) ( (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) ), thefact that ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ) implies that s j = s ∗ j , and due to assumption (III) of overinvestment andthe concavity of the payoff function: π i (cid:16) s i , s j (cid:17) = π i (cid:16) s i , s ∗ j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) . For each s i < s ∗ i , and each (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , the fact that ψ ∗ j ( s i ) ≤ φ s ∗ ( s i ) with an equality only if φ s ∗ ( s i ) = min ( S i )(and, thus, ψ ∗ j ( s i ) ∈ X s ∗ ( s i )) implies that π i (cid:16) s i , s j (cid:17) ≤ π i ( s ∗ , s ∗ ). This shows that player i cannotgain from his deviation, which implies that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a (strong) BBE. F.2 Proof of a Lemma Required for Corollary 2
Lemma 1.
Let G be a game with strategic complements and positive externalities with a lowestNash equilibrium ( s , s ) satisfying s i < max ( S i ) i for each player i . Let s ∗ < s . Then for each s ∗ ∈ S either (1) s ∗ < min ( BR ( s ∗ )) or (2) s ∗ < min ( BR ( s ∗ )) .Proof. Assume first that s ∗ > s . The fact that s ∈ BR ( s ) and s ∗ > s , together with the strategiccomplements, imply that s ∗ < s < min ( BR ( s ∗ )). We are left with the case where s ∗ ≤ s . Considera restricted game in which the set of strategies of each player i is restricted to being strategies thatare at most s ∗ i . The game is a game of strategic complements, and, thus, it admits a pure Nashequilibrium (cid:16) s i , s j (cid:17) . The minimality of ( s , s ) implies that (cid:16) s i , s j (cid:17) cannot be a Nash equilibriumof the unrestricted game. The strategic complements and the concavity of the payoff jointly implythat if (cid:16) s i , s j (cid:17) is not a Nash equilibrium of the unrestricted game, then there is player i for which s ∗ i = s i < min (cid:16) BR (cid:16) s j (cid:17)(cid:17) ≤ min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . F.3 Proof of a Lemma Required for Corollary 3
Lemma 2.
Let G be a game with positive externalities and strategic complementarity of the payoffof player i (i.e., ∂ π i ( s i ,s j ) ∂s i · ∂s j > for each s i , s j ). Then s j < s j implies that max (cid:16) BR (cid:16) s j (cid:17)(cid:17) ≤ min ( BR ( s j )) with an equality only if max (cid:16) BR (cid:16) s j (cid:17)(cid:17) = min ( BR ( s j )) ∈ { min ( S i ) , max ( S i ) } .Proof. The inequality s j < s j and the strategic complementarity of the payoff of player i im-plies that ∂π i ( s i ,s j ) ∂s i < ∂π i ( s i ,s j ) ∂s i for each s i ∈ S i , which implies that whenever max (cid:16) BR (cid:16) s j (cid:17)(cid:17) / ∈{ min ( S i ) , max ( S i ) } , thenmax (cid:16) BR (cid:16) s j (cid:17)(cid:17) = max s ∗ i ∈ S i | ∂π i (cid:16) s i , s j (cid:17) ∂s i = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i < min ( s ∗ i ∈ S i | ∂π i ( s i , s j ) ∂s i = 0 (cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i ) ≤ min ( BR ( s j )) . .4 Proof of Proposition 5 (cid:16) BR (cid:16) s j (cid:17)(cid:17) / ∈ { min ( S i ) , max ( S i ) } . Itremains to show that the weak inequality (namely, max (cid:16) BR (cid:16) s j (cid:17)(cid:17) ≤ min ( BR ( s j ))) holds whenmax (cid:16) BR (cid:16) s j (cid:17)(cid:17) ∈ { min ( S i ) , max ( S i ) } . If max (cid:16) BR (cid:16) s j (cid:17)(cid:17) = min ( S i ) then this is immediate. As-sume that max (cid:16) BR (cid:16) s j (cid:17)(cid:17) = max ( S i ). Then:max ( S i ) = max s ∗ i ∈ S i | ∂π i (cid:16) s i , s j (cid:17) ∂s i ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i . ≤ min ( s ∗ i ∈ S i | ∂π i ( s i , s j ) ∂s i ≥ (cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i ) ≤ min ( BR ( s j )) . F.4 Proof of Proposition 5
The proof is analogous to the proof of Proposition 4, and is presented for completeness.
Part 1:
Proposition 3 implies (I) and (II). It remains to show (III) (underinvestment). Let (cid:16)(cid:16) ψ ∗ i , ψ ∗ j (cid:17) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) be a BBE. Assume to the contrary that s ∗ i > max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Consider adeviation of player i to a blind belief that the opponent always plays strategy s ∗ j (i.e., ψ i ≡ s ∗ j ). Let (cid:16) s i , s j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) be a plausible equilibrium of the new biased game. Observe first that s i ∈ BR (cid:16) ψ i (cid:16) s j (cid:17)(cid:17) = BR (cid:16) s ∗ j (cid:17) . This implies that s i < s ∗ i , and, thus, due to the monotonicity of ψ ∗ j we have: ψ ∗ j ( s i ) ≤ ψ ∗ j ( s ∗ i ). We consider two cases:1. If ψ ∗ j ( s i ) < ψ ∗ j ( s ∗ i ), then the strategic substitutability implies that s j ≥ min (cid:16) BR (cid:16) ψ ∗ j ( s i ) (cid:17)(cid:17) ≥ max (cid:16) BR (cid:16) ψ ∗ j ( s ∗ i ) (cid:17)(cid:17) ≥ s ∗ j , and this, in turn, implies that player i strictly gains from hisdeviation: π i (cid:16) s i , s j (cid:17) ≥ π i (cid:16) s i , s ∗ j (cid:17) > π i (cid:16) s ∗ i , s ∗ j (cid:17) , a contradiction.2. If ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ), then (cid:16) s i , s ∗ j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) and π i (cid:16) s i , s ∗ j (cid:17) > π i (cid:16) s ∗ i , s ∗ j (cid:17) , whichcontradicts that (cid:16)(cid:16) ψ ∗ i , ψ ∗ j (cid:17) , (cid:16) s ∗ i , s ∗ j (cid:17)(cid:17) is a BBE. Part 2:
Assume that strategy profile ( s ∗ , s ∗ ) satisfies I, II, and III. For each player i let s ei =max (cid:0) BR − ( s ∗ i ) (cid:1) . For each player i and each strategy s i > s ∗ i define X ( s i ) as the set of strategies s i for which player i is worse off (relative to π i ( s ∗ , s ∗ )) if he plays strategy s i , while player j playsa best-reply to s i . Formally: X s ∗ ( s i ) = n s i ∈ S i | π i ( s i , s j ) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) ∀ s j ∈ BR (cid:0) s i (cid:1)o . The assumption that π i (cid:16) s ∗ i , s ∗ j (cid:17) > ˜ M Ui implies that X s ∗ ( s i ) is nonempty for each s i . The as-sumption of strategic substitutes implies that X s ∗ ( s i ) is an interval ending at max ( S i ). Let φ s ∗ ( s i ) = inf ( X s ∗ ( s i )). The assumption that the payoff function is continuously twice differen-tiable implies that φ s ∗ ( s i ) is continuous. The assumption that s ej = max (cid:0) BR − ( s ∗ i ) (cid:1) implies thatlim s i & s ∗ i ( φ s ∗ ( s i )) = s ei . These observations imply that for each player j there exists a monotone4 F PROOFS biased belief ψ ∗ j satisfying (1) ψ ∗ j ( s i ) = s ei for each s i ≤ s ∗ i and (2) ψ ∗ j ( s i ) ≥ φ s ∗ ( s i ) for each s i > s ∗ i with an equality only if φ s ∗ ( s i ) = max ( S i ).We now show that these properties of ( ψ ∗ , ψ ∗ ) imply that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE (a strongBBE if π i ( s i , s j ) is strictly concave in s i ). Consider a deviation of player i to an arbitrary biasedbelief ψ i . For each s i ≤ s ∗ i , and each (cid:16) s i , s j (cid:17) ∈ P N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) ( (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) ), thefact that ψ ∗ j ( s i ) = ψ ∗ j ( s ∗ i ) implies that s j = s ∗ j and, due to assumption (III) of underinvestment andthe concavity of the payoff function, it follows that π i (cid:16) s i , s j (cid:17) = π i (cid:16) s i , s ∗ j (cid:17) ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) . For each s i > s ∗ i , and each (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ ∗ j ) (cid:19) , the fact that ψ ∗ j ( s i ) ≥ φ s ∗ ( s i ) with an equality only if φ s ∗ ( s i ) = max ( S i ) (and, thus, ψ ∗ j ( s i ) ∈ X s ∗ ( s i )) implies that π i (cid:16) s i , s j (cid:17) ≤ π i ( s ∗ , s ∗ ). This showsthat player i cannot gain from his deviation, which implies that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a (strong)BBE. F.5 Proof of a Lemma Required for Corollary 4
Lemma 3.
Let G be a game with strategic substitutes and positive externalities. Let ( s ∗ , s ∗ ) be a strategy profile satisfying s ∗ i > s ei for each player i and each Nash equilibrium ( s e , s e ) ∈ N E ( G ) .Then, either (1) s ∗ > max ( BR ( s ∗ )) or (2) s ∗ > max ( BR ( s ∗ )) .Proof. Consider a restricted game in which the set of strategies of each player i is restricted to beingstrategies that are at least s ∗ i . The restricted game is a game with strategic substitutes, and, thus,it admits a pure Nash equilibrium ( s , s ) (recall, that after relabeling the set of strategies of oneof the players, the game becomes supermodular, and because of this the game admits a pure Nashequilibrium due to Milgrom and Roberts, 1990). The assumption that s ∗ i > s ei for each player i and each Nash equilibrium ( s e , s e ) ∈ N E ( G ) implies that ( s , s ) cannot be a Nash equilibrium ofthe unrestricted game. The concavity of the payoff and the strategic substitutes jointly imply thatif (cid:16) s i , s j (cid:17) is not a Nash equilibrium of the unrestricted game, then there is a player i for which s ∗ i = s i > max (cid:16) BR (cid:16) s j (cid:17)(cid:17) ≥ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . F.6 Proof of Corollary 5
The proof is analogous to Corollary 3, and is presented for completeness. Assume to the contrarythat ψ ∗ i (cid:16) s ∗ j (cid:17) < s ∗ j . Lemma 4 (below) implies that min (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) ≥ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) with anequality only if min (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) ∈ { min ( S i ) , max ( S i ) } . Part 1 of Proposition 5 and the definition of a monotone BBE imply thatmin (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) ≤ s ∗ i ≤ max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . The previous inequalities jointly imply thatmin (cid:16) BR (cid:16) ψ ∗ i (cid:16) s ∗ j (cid:17)(cid:17)(cid:17) = s ∗ i = max (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) ∈ { min ( S i ) , max ( S i ) } , .7 Proof of Proposition 6 s ∗ i / ∈ { min ( S i ) , max ( S i ) } . Lemma 4.
Let G be a game with positive externalities and strategic substitutability of the payoffof player i (i.e., ∂ π i ( s i ,s j ) ∂s i · ∂s j > for each s i , s j ). Then s j < s j implies that min (cid:16) BR (cid:16) s j (cid:17)(cid:17) ≥ max ( BR ( s j )) with an equality only if min (cid:16) BR (cid:16) s j (cid:17)(cid:17) = min ( BR ( s j )) ∈ { min ( S i ) , max ( S i ) } .Proof. The proof is analogous to the proof of Lemma 1, and is presented for completeness. Theinequality s j < s j and the strategic substitutability of the payoff of player i implies that ∂π i ( s i ,s j ) ∂s i > ∂π i ( s i ,s j ) ∂s i for each s i ∈ S i , which implies that whenever min (cid:16) BR (cid:16) s j (cid:17)(cid:17) / ∈ { min ( S i ) , max ( S i ) } , thenmin (cid:16) BR (cid:16) s j (cid:17)(cid:17) = min s ∗ i ∈ S i | ∂π i (cid:16) s i , s j (cid:17) ∂s i = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i > max ( s ∗ i ∈ S i | ∂π i ( s i , s j ) ∂s i = 0 (cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i ) = max ( BR ( s j )) . This shows that the strict inequality holds whenever min (cid:16) BR (cid:16) s j (cid:17)(cid:17) / ∈ { min ( S i ) , max ( S i ) } . Itremains to show that the weak inequality (namely, min (cid:16) BR (cid:16) s j (cid:17)(cid:17) ≥ max ( BR ( s j ))) holds whenmin (cid:16) BR (cid:16) s j (cid:17)(cid:17) ∈ { min ( S i ) , max ( S i ) } . If min (cid:16) BR (cid:16) s j (cid:17)(cid:17) = max ( S i ) then this is immediate. As-sume that min (cid:16) BR (cid:16) s j (cid:17)(cid:17) = min ( S i ). Then:min ( S i ) = min s ∗ i ∈ S i | ∂π i (cid:16) s i , s j (cid:17) ∂s i ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i ≥ max ( s ∗ i ∈ S i | ∂π i ( s i , s j ) ∂s i ≥ (cid:12)(cid:12)(cid:12)(cid:12) s i = s ∗ i ) ≥ max ( BR ( s j )) . F.7 Proof of Proposition 6
The proof is analogous to the proof of Proposition 4, and is presented for completeness.
Part 1:
Proposition 3 implies (I) and (II). It remains to show (III). Let (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ ))be a BBE. We begin by showing overinvestment of player 2. Assume to the contrary that s ∗ < min (cid:16) BR (cid:16) s ∗ j (cid:17)(cid:17) . Consider a deviation of player 2 to a blind belief that the opponent always playsstrategy s ∗ (i.e., ψ ≡ s ∗ ). Let ( s , s ) ∈ P N E (cid:16) G ( ψ ∗ ,ψ ) (cid:17) be a plausible equilibrium of the newbiased game. Observe first that s ∈ BR ( ψ ( s )) = BR ( s ∗ ) . This implies that s > s ∗ , and, thus,due to the monotonicity of ψ ∗ , we have : ψ ∗ ( s ) ≥ ψ ∗ ( s ∗ ). We consider two cases:1. If ψ ∗ ( s ) > ψ ∗ ( s ∗ ), then the strategic complementarity of player 1’s payoff implies that s ≥ min ( BR ( ψ ∗ ( s ))) ≥ max ( BR ( ψ ∗ ( s ∗ ))) ≥ s ∗ , and, this, in turn, implies that player 2 strictlygains from his deviation: π ( s , s ) ≥ π ( s , s ∗ ) > π ( s ∗ , s ∗ ), a contradiction.2. If ψ ∗ ( s ) = ψ ∗ ( s ∗ ), then ( s ∗ , s ) ∈ P N E (cid:16) G ( ψ ∗ ,ψ ) (cid:17) and π ( s ∗ , s ) > π ( s ∗ , s ∗ ), which con-tradicts that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE.6 F PROOFS
Next we show underinvestment of player 1. Assume to the contrary that s ∗ > max ( BR ( s ∗ )).Consider a deviation of player 1 to a blind belief that the opponent always plays strategy s ∗ (i.e., ψ ≡ s ∗ ). Let ( s , s ) ∈ P N E (cid:16) G ( ψ ,ψ ∗ ) (cid:17) be a plausible equilibrium of the new biased game. Observefirst that s ∈ BR ( ψ ( s )) = BR ( s ∗ ) . This implies that s < s ∗ and, thus, due to the monotonicityof ψ ∗ , we have: ψ ∗ ( s ) ≤ ψ ∗ ( s ∗ ). We consider two cases:1. If ψ ∗ ( s ) < ψ ∗ ( s ∗ ), then the strategic substitutability of player 2’s payoff implies that s ≥ min ( BR ( ψ ∗ ( s ))) ≥ max ( BR ( ψ ∗ ( s ∗ ))) ≥ s ∗ , and this, in turn, implies that player 1 strictlygains from his deviation: π ( s , s ) ≥ π ( s , s ∗ ) > π ( s ∗ , s ∗ ), a contradiction.2. If ψ ∗ ( s ) = ψ ∗ ( s ∗ ), then ( s , s ∗ ) ∈ P N E (cid:16) G ( ψ ,ψ ∗ ) (cid:17) and π ( s , s ∗ ) > π ( s ∗ , s ∗ ), which con-tradicts that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE. Part 2:
Assume that strategy profile ( s ∗ , s ∗ ) satisfies I, II, and III. Let s e = min (cid:0) BR − ( s ∗ ) (cid:1) . Foreach strategy s < s ∗ define X ( s ) as the set of strategies s for which player 2 is worse off (relativeto π ( s ∗ , s ∗ )) if he plays strategy s , while player 1 plays a best-reply to s . Formally: X s ∗ ( s ) = n s ∈ S | π ( s , s ) ≤ π (cid:16) s ∗ i , s ∗ j (cid:17) ∀ s ∈ BR (cid:0) s (cid:1)o . The assumption that π ( s ∗ , s ∗ ) > ˜ M U implies that X s ∗ ( s ) is nonempty for each s ∈ S . Theassumption of strategic complements of player 1’s payoff implies that X s ∗ ( s ) is an interval startingat min ( S ). Let φ s ∗ ( s ) = sup ( X s ∗ ( s )). The assumption that the payoff function is continuouslytwice differentiable implies that φ s ∗ ( s ) is continuous. The assumption that s e = min (cid:0) BR − ( s ∗ ) (cid:1) implies that lim s % s ∗ ( φ s ∗ ( s )) = s e . These observations imply that there exists a monotone biasedbelief ψ ∗ satisfying (1) ψ ∗ ( s ) = s e and (2) ψ ∗ ( s ) ≤ φ s ∗ ( s ) for each s < s ∗ with an equality onlyif φ s ∗ ( s ) = min ( S ).Let s e = max (cid:0) BR − ( s ∗ ) (cid:1) . For each strategy s > s ∗ define X ( s ) as the set of strategies s ∈ S for which player 1 is worse off (relative to π ( s ∗ , s ∗ )) if he plays strategy s , while player 2plays a best-reply to s . Formally: X s ∗ ( s ) = (cid:8) s ∈ S | π ( s , s ) ≤ π ( s ∗ , s ∗ ) ∀ s ∈ BR (cid:0) s (cid:1)(cid:9) . The assumption that π ( s ∗ , s ∗ ) > ˜ M U implies that X s ∗ ( s ) is nonempty for each s ∈ S . Theassumption of strategic substitutes of player 2’s payoff implies that X s ∗ ( s ) is an interval endingat max ( S ). Let φ s ∗ ( s ) = inf ( X s ∗ ( s )). The assumption that the payoff function is continuouslytwice differentiable implies that φ s ∗ ( s ) is continuous. The assumption that s e = max (cid:0) BR − ( s ∗ ) (cid:1) implies that lim s & s ∗ ( φ s ∗ ( s )) = s e . These observations imply that there exists a monotone biasedbelief ψ ∗ satisfying (1) ψ ∗ ( s ) = s e and (2) ψ ∗ ( s ) ≥ φ s ∗ ( s ) for each s > s ∗ with an equality onlyif φ s ∗ ( s ) = max ( S ).We now show that these properties of ( ψ ∗ , ψ ∗ ) imply that (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE. Considera deviation of player 2 into an arbitrary biased belief ψ . For each s ≥ s ∗ , and each ( s , s ) ∈ P N E (cid:16) G ( ψ ∗ ,ψ ) (cid:17) , the fact that ψ ∗ ( s ) = ψ ∗ ( s ∗ ) implies that s = s ∗ , and due to assumption (III)of the overinvestment of player 2 and the concavity of the payoff function, we have π ( s , s ) = π ( s , s ∗ ) ≤ π ( s ∗ , s ∗ ). For each s < s ∗ , and each ( s , s ) ∈ N E (cid:16) G ( ψ ∗ ,ψ ) (cid:17) , the fact that ψ ∗ ( s ) ≤ .8 Proof of Corollary 6 (Pessimism in Games with Strategic Opposites) φ s ∗ ( s ) with an equality only if φ s ∗ ( s ) = min ( S ) implies that π ( s , s ) ≤ π ( s ∗ , s ∗ ). This showsthat player 2 cannot gain from his deviation.Finally, consider a deviation of player 1 to an arbitrary biased belief ψ . For each s ≤ s ∗ , and each ( s , s ) ∈ P N E (cid:16) G ( ψ ,ψ ∗ ) (cid:17) , the fact that ψ ∗ ( s ) = ψ ∗ ( s ∗ ) implies that s = s ∗ ,and due to assumption (III) of the underinvestment of player 1 and the concavity of the payofffunction, we have π ( s , s ) = π ( s , s ∗ ) ≤ π ( s ∗ , s ∗ ). For each s > s ∗ , and each ( s , s ) ∈ N E (cid:16) G ( ψ ,ψ ∗ ) (cid:17) , the fact that ψ ∗ ( s ) ≥ φ s ∗ ( s ) with an equality only if φ s ∗ ( s ) = max ( S ) impliesthat π ( s , s ) ≤ π ( s ∗ , s ∗ ). This shows that player 1 cannot gain from his deviation, which impliesthat (( ψ ∗ , ψ ∗ ) , ( s ∗ , s ∗ )) is a BBE. F.8 Proof of Corollary 6 (Pessimism in Games with Strategic Opposites)
The proof is analogous to the proof of Corollary 3, and is presented for completeness.Assume to the contrary that ψ ∗ i (cid:16) s ∗ j (cid:17) > s ∗ j for some player i . Assume first that ψ ∗ ( s ∗ ) > s ∗ ; thenLemma 4 implies that max ( BR ( ψ ∗ ( s ∗ ))) ≤ min ( BR ( s ∗ )) with an equality only ifmax ( BR ( ψ ∗ ( s ∗ ))) ∈ { min ( S ) , max ( S ) } . Part 1 of Proposition 6 and the definition of a monotone BBE imply thatmax ( BR ( ψ ∗ ( s ∗ ))) ≥ s ∗ ≥ min ( BR ( s ∗ )) . The previous inequalities jointly imply thatmax ( BR ( ψ ∗ ( s ∗ ))) = s ∗ = min ( BR ( s ∗ )) ∈ { min ( S ) , max ( S ) } , which contradicts the assumption that s ∗ / ∈ { min ( S ) , max ( S ) } .We are left with the case of ψ ∗ ( s ∗ ) > s ∗ ; then Lemma 2 implies that min ( BR ( ψ ∗ ( s ∗ ))) ≥ max ( BR ( s ∗ )) with an equality only ifmin ( BR ( ψ ∗ ( s ∗ ))) ∈ { min ( S ) , max ( S ) } . Part 1 of Proposition 6 and the definition of a monotone BBE imply thatmin ( BR ( ψ ∗ ( s ∗ ))) ≤ s ∗ ≤ max ( BR ( s ∗ )) . The previous inequalities jointly imply thatmin ( BR ( ψ ∗ ( s ∗ ))) = s ∗ = max ( BR ( s ∗ )) . ∈ { min ( S ) , max ( S ) } , which contradicts the assumption that s ∗ / ∈ { min ( S ) , max ( S ) } .8 F PROOFS
F.9 Proof of Proposition 9
Recall that we assume the payoff function π i to be continuously twice differentiable. This impliesthat π i is Lipschitz continuous. Let K i > π i withrespect to its first parameter, i.e., K i satisfies (cid:13)(cid:13) π i ( s , s ) − π i (cid:0) s , s (cid:1)(cid:13)(cid:13) ≤ K i · (cid:13)(cid:13) s − s (cid:13)(cid:13) . Assume that ( s ∗ , s ∗ ) is undominated and π i ( s ∗ , s ∗ ) > M Ui for each player i . Let 0 < D i = π i ( s ∗ , s ∗ ) − M Ui . For each player j , let s pj be an undominated strategy that guarantees that player i obtains, atmost, his minmax payoff M Ui , i.e., s pj = argmin s j ∈ S Uj (max s i ∈ S i π i ( s i , s j )) . The strict concavity of π i ( s i , s j ) with respect to s i implies that the best-reply correspondence is a continuous one-to-onefunction. Thus, BR − ( s i ) is a singleton for each s i , and we identify BR − ( s i ) with the uniqueelement in this singleton set.Let (cid:15) > (cid:15) < min (cid:16) D i K i , D j K j (cid:17) . For each δ ∈ [0 ,
1] definefor each player i : s δi = (cid:15) − δ(cid:15) · s ∗ i + δ(cid:15) · s pi . Let ψ (cid:15)i be defined as follows: ψ (cid:15)i (cid:16) s j (cid:17) = BR − (cid:18) s | s j − s j | i (cid:19) (cid:12)(cid:12)(cid:12) s j − s j (cid:12)(cid:12)(cid:12) < (cid:15)BR − ( s pi ) (cid:12)(cid:12)(cid:12) s j − s j (cid:12)(cid:12)(cid:12) ≥ (cid:15). Note that ψ (cid:15)i is continuous. We now show that (( ψ (cid:15) , ψ (cid:15) ) , ( s ∗ , s ∗ )) is a strong BBE. Observe firstthat the definition of ( ψ (cid:15) , ψ (cid:15) ) immediately implies that ( s ∗ , s ∗ ) ∈ N E (cid:16) G ( ψ (cid:15) ,ψ (cid:15) ) (cid:17) . Next, consider adeviation of player i to an arbitrary biased belief ψ i . Consider any equilibrium of the new biasedgame (cid:16) s i , s j (cid:17) ∈ N E (cid:18) G ( ψ i ,ψ (cid:15)j ) (cid:19) . If | s i − s i | ≥ (cid:15) , then the definition of ψ (cid:15)j ( s i ) implies that s pj = s j ,and that player i achieves a payoff of at most M Ui < π i ( s ∗ , s ∗ ). If s i = s ∗ i , then it is immediate that s j = s ∗ j and that player i does not gain from his deviation. If 0 < | s i − s i | < (cid:15) , then the definitionof ψ (cid:15)j ( s i ) implies that π i (cid:16) s i , s j (cid:17) = π i (cid:18) s i , s | s i − s i | j (cid:19) = π i (cid:18) s i , (cid:15) − | s i − s i | (cid:15) · s ∗ j + | s i − s i | (cid:15) · s pj (cid:19) ≤ (cid:15) − | s i − s i | (cid:15) · π i (cid:16) s i , s ∗ j (cid:17) + | s i − s i | (cid:15) · π i (cid:16) s i , s pj (cid:17) ≤ (cid:15) − | s i − s i | (cid:15) · π i (cid:16) s i , s ∗ j (cid:17) + | s i − s i | (cid:15) · M Ui ≤ (cid:15) − | s i − s i | (cid:15) · π i (cid:16) s ∗ i , s ∗ j (cid:17) + K i · (cid:12)(cid:12) s i − s i (cid:12)(cid:12) + | s i − s i | (cid:15) · M Ui = (cid:15) − | s i − s i | (cid:15) · π i (cid:16) s ∗ i , s ∗ j (cid:17) + K i · (cid:12)(cid:12) s i − s i (cid:12)(cid:12) + | s i − s i | (cid:15) · (cid:16) π i (cid:16) s ∗ i , s ∗ j (cid:17) − D i (cid:17) = π i (cid:16) s ∗ i , s ∗ j (cid:17) + (cid:15) − | s i − s i | (cid:15) · K i · (cid:12)(cid:12) s i − s i (cid:12)(cid:12) − | s i − s i | (cid:15) · D i ≤ π i (cid:16) s ∗ i , s ∗ j (cid:17) + K i · (cid:12)(cid:12) s i − s i (cid:12)(cid:12) − | s i − s i | (cid:15) · D i = .9 Proof of Proposition 9 π i (cid:16) s ∗ i , s ∗ j (cid:17) + (cid:12)(cid:12) s i − s i (cid:12)(cid:12) · (cid:18) K i − D i (cid:15) (cid:19) < π i (cid:16) s ∗ i , s ∗ j (cid:17) , where the first inequality is due to the convexity of π i ( s i , s j ) with respect to s j , the second inequalityis due to π i (cid:16) s i , s pj (cid:17) ≤ M Ui , the third inequality is due to the Lipschitz continuity, the penultimateinequality is implied by (cid:15) − | s i − s i | (cid:15) <
1, and the last inequality is due to defining (cid:15) to satisfy (cid:15) < min (cid:16) D i K i , D j K j (cid:17) . This proves that player i cannot gain from his deviation, and that (( ψ (cid:15) , ψ (cid:15) ) , ( s , s2