Introduction Model Results Examples and Discussion Biased-Belief Equilibrium Yuval Heller (Bar Ilan) and Eyal Winter (Hebrew University) Bar Ilan, Game Theory Seminar, May 2017 Heller & Winter Biased-Belief Equilibrium 1 / 43
Motivation Standard models of equilibrium behavior assume that: Players form beliefs that are consistent with reality. 1 Players maximize their material payoffs given their beliefs. 2 Several papers study the relaxation of the 2nd assumption: Social preferences: altruism, inequality aversion, ... (e.g., Levine, 1998; Fehr & Schmidt, 1999; Bolton & Ockenfels, 2000). Subjective preferences (e.g., Guth and Yaari, 1992; Dekel, Ely & Yilnakaya, 2007; Friedman and Singh, 2009; Herold and Kuzmics, 2009;); Psychological games (Geanakoplos, Pearce & Stacchetti, 1989; Rabin, 1993; Battigalli & Dufwenberg, 2007; 2009). Preferences are influenced by emotions (e.g., Winter, Garcia-Jurado & Mendez-Naya, 2017; Battigalli, Dufenberg & Smith, 2017)
Motivation We relax the 1st assumption, and study a model of distorted, yet structured, beliefs about the opponent’s behavior. In various applications people may have distorted beliefs, e.g., Wishful thinking (Babad & Katz, 1991; Budescu & Bruderman, 1995; Mayraz, 2013). Overconfidence (Forbes, 2005; Malmendier and Tate, 2005). Belief polarization (Lord et al., 1979; Ross & Anderson, 1982). Moral hypocrisy (Babcock & Loewenstein, 1997; Rustichini and Villeval; 2014) Distorted belief can have strategic advantage (commitment).
Introduction Motivation Model Highlights of the Model Results Brief Summary of Results Examples and Discussion Motivation Examples of mechanisms that can maintain distorted beliefs: Using biased source of information (e.g., reading a newspaper with a specific political orientation, Facebook feed). Following passionately religion / ideology / moral principle. Personality traits, such as narcissism or naivety. These mechanisms are likely to generate signals to the player’s counterpart about the biased beliefs. Heller & Winter Biased-Belief Equilibrium 4 / 43
Introduction Motivation Model Highlights of the Model Results Brief Summary of Results Examples and Discussion Highlights of the Model Biased-Belief Equilibrium (BBE) of a 2-stage game: Each player is endowed with a biased-belief function. 1 Each player chooses a best reply to the distorted belief about 2 the opponent’s strategy. Biased-belief functions are best replies to one another. If a player is endowed with a different biased-belief function, then he is outperformed in the induced biased game. Heller & Winter Biased-Belief Equilibrium 5 / 43
Brief Summary of Main Results Every Nash equilibrium is a BBE outcome. In some cases it can only be supported by biased beliefs. Characterizing BBE outcomes as those satisfying: (1) undominated strategies, and (2) payoffs above the “undominated” minmax payoffs. Necessary conditions in all games. Sufficient conditions in various classes of games. Wishful thinking in games with strategic complementarity. BBE of a “stubborn” player and a “rational” opponent.
Introduction Underlying Game Model Biased-Belief Function Results Biased-Belief Equilibrium (BBE) Examples and Discussion Underlying Two-Player Game G = ( S , π ) – normal-form two-player game. S = ( S 1 , S 2 ) , each S i is a convex closed set of strategies. In most applications either: Interval in R , or Simplex over a finite set of actions: S i = ∆( A i ) , π i is linear. π = ( π 1 , π 2 ) , each π i : S → R is a payoff function. � � π i is twice differentiable, and weakly concave in s i . s i , s j Notation: BR i : S j → S i Best Reply Correspondence ( BR − 1 : S i → S j ). i Heller & Winter Biased-Belief Equilibrium 7 / 43
Introduction Underlying Game Model Biased-Belief Function Results Biased-Belief Equilibrium (BBE) Examples and Discussion Definition (Biased-Belief Function ψ i : S j → S j ) A continuous function that assigns for each strategy of the opponent a (possibly distorted) belief about the opponent’s play. Blind belief - constant biased-belief function. Undistorted belief - ψ i is the identity function. Definition (Biased Game) ( G , ψ ) is a pair where G is an underlying game, and ψ = ( ψ 1 , ψ 2 ) is a pair of biased beliefs. Heller & Winter Biased-Belief Equilibrium 8 / 43
Introduction Underlying Game Model Biased-Belief Function Results Biased-Belief Equilibrium (BBE) Examples and Discussion Definition (Nash equilibrium s ∗ = ( s ∗ 1 , s ∗ 2 ) of biased game ( G , ψ ) ) Each s ∗ i is a best reply against the perceived strategy of the � � � ��� opponent, i.e., s ∗ s ∗ i = argmax s i ∈ S i π i s i , ψ i . j Let NE ( G , ψ ) ⊆ S 1 × S 2 denote the set of all Nash equilibria of the biased game ( G , ψ ) . Fact Any biased game ( G , ψ ) admits a Nash equilibrium. Heller & Winter Biased-Belief Equilibrium 9 / 43
Definition ( Biased-Belief Equilibrium ( BBE ) ) Pair (( ψ ∗ 1 , ψ ∗ 2 ) , ( s ∗ 1 , s ∗ 2 )) satisfying: (1) strategy profile is a Nash Eq. � � s ∗ i , s ∗ ∈ NE ( G , ψ ∗ ) , and (2) any agent who of the biased game: j chooses a different biased-belief function is outperformed in at least one equilibrium of the new biased game (i.e., ∀ i , ψ ′ i , � � � � �� � � � � s ′ i , s ′ ψ ′ i , ψ ∗ s ′ i , s ′ s ∗ i , s ∗ ∃ ∈ NE , s.t. π i ≤ π i . G , j j j j
Definition ( Biased-Belief Equilibrium ( BBE ) ) Pair (( ψ ∗ 1 , ψ ∗ 2 ) , ( s ∗ 1 , s ∗ 2 )) satisfying: (1) strategy profile is a Nash Eq. � � s ∗ i , s ∗ ∈ NE ( G , ψ ∗ ) , and (2) any agent who of the biased game: j chooses a different biased-belief function is outperformed in at least one equilibrium of the new biased game (i.e., ∀ i , ψ ′ i , � � � � �� � � � � s ′ i , s ′ ψ ′ i , ψ ∗ s ′ i , s ′ s ∗ i , s ∗ ∃ ∈ NE , s.t. π i ≤ π i . G , j j j j Definition ( strong BBE ) A BBE is strong if an agent who chooses a different biased-belief function is outperformed in all equilibria of the new biased game; � � � � �� � � � � s ′ i , s ′ ψ ′ i , ψ ∗ s ′ i , s ′ s ∗ i , s ∗ i.e., ∀ ∈ NE G , , π i ≤ π i . j j j j
Introduction BBE & Nash Equilibrium Model Characterization of BBE Outcomes Results Wishful Thinking & Strategic Complementarity Examples and Discussion BBE and Undominated Stackelberg Strategies Nash Equilibria and Distorted Beliefs In any BBE in which the outcome is not a Nash equilibrium, at least one of the players distorts the opponent’s strategy. Some Nash equilibria can be supported as BBE outcomes only with distorted beliefs. Example... Heller & Winter Biased-Belief Equilibrium 11 / 43
Example (NE supported only by distorted beliefs) Symmetric Cournot game G = ( S , π ) : S i = [ 0 , 1 ] . s i - quantity chosen by firm i . The price is determined by the linear demand function p = 1 − s i − s j ⇒ π i ( s i , s j ) = s i · ( 1 − s i − s j ) . Marginal cost is normalized to zero. j = 1 i = 1 Unique Nash equilibrium: s ∗ i = s ∗ 3 ⇒ π ∗ 9 . The NE cannot be supported by undistorted beliefs because a i ≡ 1 player would gain by deviating to blind belief ψ ′ 4 . � 1 �� 1 3 , 1 3 , 1 � �� The NE is the outcome of the strong BBE . , 3 3
Introduction BBE & Nash Equilibrium Model Characterization of BBE Outcomes Results Wishful Thinking & Strategic Complementarity Examples and Discussion BBE and Undominated Stackelberg Strategies Any Nash Equilibrium is a BBE outcome Proposition Let ( s ∗ 1 , s ∗ 2 ) be a Nash (strict) equilibrium. Then (( ψ ∗ 1 ≡ s ∗ 2 , ψ ∗ 2 = s ∗ 1 ) , ( s ∗ 1 , s ∗ 2 )) is a (strong) biased-belief equilibrium. Corollary Every game admits a biased-belief equilibrium. Heller & Winter Biased-Belief Equilibrium 13 / 43
Introduction BBE & Nash Equilibrium Model Characterization of BBE Outcomes Results Wishful Thinking & Strategic Complementarity Examples and Discussion BBE and Undominated Stackelberg Strategies Undominated Strategies Strategy is undominated if it is not strictly dominated. Let S u i ⊆ S i be the set of undominated strategies of player i . Observe: (1) BR − 1 0 iff s i ∈ S U i , and (2) S u ( s i ) � = / i is not necessarily convex. i Definition ( Undominated minmax payoff of player i ) The maximal payoff that player i can guarantee in the following process: (1) player j chooses an arbitrary undominated strategy, and (2) player i best replies to player j ’s strategy. Formally: M U i = min s j ∈ S U j (max s i ∈ S i π i ( s i , s j )) . Heller & Winter Biased-Belief Equilibrium 14 / 43
Introduction BBE & Nash Equilibrium Model Characterization of BBE Outcomes Results Wishful Thinking & Strategic Complementarity Examples and Discussion BBE and Undominated Stackelberg Strategies Necessary Conditions for BBE Outcome Proposition Let (( ψ ∗ 1 , ψ ∗ 2 ) , ( s ∗ 1 , s ∗ 2 )) be a BBE. Then, (1) each strategy s ∗ i is undominated, and (2) each player obtains at least his undominated minmax payoff, i.e., π i ( s ∗ 1 , s ∗ 2 ) ≥ M U i . Sketch of proof. 1 A dominated strategy is never a best reply. 2 A player with undistorted belief obtains at least M U in any i Nash equilibrium of any biased game. Heller & Winter Biased-Belief Equilibrium 15 / 43
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