Guilt in Games. P. Battigalli and M. Dufwenberg (AER, 2007) Presented by Luca Ferocino. March 21 st,2014

Transcription

1 Guilt in Games P. Battigalli and M. Dufwenberg (AER, 2007) Presented by Luca Ferocino March 21 st,2014 P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 1 / 29

2 ADefinitionofGuilt Guilt is a cognitive or an emotional experience that occurs when a person realizes or believes - accurately or not - that he or she has compromised his or her own standards of conduct or has violated a moral standard, and bears significant responsibility for that violation "Guilt." Encyclopedia of Psychology. 2nd ed. Ed. Bonnie R. Strickland. Gale Group, Inc., enotes.com December 2007 P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 2 / 29

3 What Is the Paper About? Framework The paper gives a contribution to the literature of psychological game theory, inwhichplayers utilitiesareallowedtodependonbeliefs Questions Addressed: 1 How can guilt be modeled? 2 How are human interaction and economic outcomes influenced? Aim of the paper Develop a portable theory of guilt aversion and show how to solve for sequential equilibria P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 3 / 29

4 Outline 1 Game Theoretic Preliminaries 2 Two Concepts of Guilt Aversion Simple Guilt Guilt from Blame 3 Equilibrium Analysis 4 Results and Examples P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 4 / 29

5 Idea of Guilt in a Game -Theoretical Framework Take an extensive game form which associates a monetary outcome with each end node Say that player i lets player j down if as a result of i s behavior, j gets a lower monetary payoffthan j expected to get before play started Player i s guilt may depend on how much he lets j down Player i s guilt may also depend on how much j believes i believes he lets j down We analyze equilibria when players are motivated, in part, by a desire to avoid guilt P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 5 / 29

6 Basic Settings We consider finite extensive game trees, mathematical structures given by: Γ= I, C, m, (Ā i, A i ( )) i I where: I is the set of players T the set of nodes with distinguished root t 0 Z is the set of terminal nodes X = T \ Z is the set of non-terminal histories Let X be partitioned into subsets X i of decision nodes i I and a set of chance nodes X c P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 6 / 29

7 Information Structure and Material Consequences Information Structure An information set is defined by a history, h. Theinformation structure of player i is given by the set H i 2 T (i.e. H i is a partition of T ) The information set containing node t is denoted H i (t): setof histories that are not prevented by node t The information structure H i satisfies perfect recall Material Consequences Material consequences are determined by a function m i : Z R, i I Material payoffs are denoted by m i = m i (z) Assume: m i (z ) = m i (z ) = H i (z ) = H i (z ): i observes his material payoff P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 7 / 29

8 Strategies Pure Strategies Apurestrategys i specifies a contingent choice h H i where i is active (h X i ) S i denotes the set of pure strategies of i S = S c Π i I S i, S i = S c Π j=i S j i, h, S i (h) denotes the set of i s strategies allowing h. Astrategyprofiles S yields an end node z(s) Behavior Strategies A behavior strategy for player i is an array of probability measures σ i ( h), h H i, h X i σ i (a h) is the probability of choosing action a at h By perfect recall, can compute the conditional prob. Pr σi (s i h), h H i P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 8 / 29

9 Beliefs h H i,α i ( h) (S i (h)) denotes player i s belief about the strategies of co-players (and chance) α i =(α i ( h)) h Hi is the system of first order beliefs of player i Player i also holds at each h H i asecondorderbeliefβ i (h) about the first order belief system α j of each co-player j, athirdorderbelief γ i (h) about second order beliefs, and so on... Beliefs at different histories are not mutually independent = They must satisfy Bayes Rule and common certainty that Bayes Rule holds P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 9 / 29

10 The Concept of Disappointment Given his strategy s j and initial first-order beliefs α j ( h 0 ),playerj forms expectations about his material payoff: E sj,α j (m j h 0 )= s j α j (s j h 0 )m j (z(s j, s j )) For any end node z consistent with with s j,themeasureofhowmuchj is let down is given by: D j (z, s j,α j )=max 0, E sj,α j [m j h 0 ] m j (z) P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 11 / 29

11 Simple Guilt If at the end of the game i knew the terminal node z, thestrategyprofile s i S i (z), andj s initial beliefs α j,thenhecouldderivehowmuchof D j (z, s j,α j ) is due to his behavior: G ij (z, s i,α j )=D j (z, s j,α j ) min s i D j (z(s i, s i ), s j,α j ) Definition Player i is affected by simple guilt toward player j if he has belief-dependent preferences represented by a utility function of the form: ui SG (z, s i,α i )=m i (z) θ ij G ij (z, s i,α j ) j=i where s i S i (z), θ ij 0. θ ij reflects i s guilt sensitivity We assume that i maximizes the expected value of ui SG,givenhisbeliefs P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 12 / 29

12 The Concept of Blame The concept of Guilt from Blame assumes that a player cares about others inferences regarding the extent to which he is willing to let them down Given s i and initial beliefs α i ( h 0 ) and β i (h 0 ), ex-ante i expects to let j down in the amount: G 0 ij (s i,α i,β i ) = E si,α i,β i [G ij h 0 ] = s i α i (s i h 0 )G ij (z(s i, s i ), s i,β 0 ij(h 0 )) where β 0 ij (h0 ) denotes the initial (point) belief of i about the initial belief α j ( h 0 ) P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 13 / 29

13 Guilt from Blame Suppose z Z is reached. The conditional expectation E αj,β j,γ j [G 0 ij H j(z)] measures j s inference regarding how much i intended to let j down or, alternatively, how much j blames i Definition Player i is affected by guilt from blame if he dislikes being blamed; i s preferences are represented by: ui GB (z,α i,β i,γ i )=m i (z) θ ij E αj,β j,γ j [Gij 0 H j (z)] j=i We assume that i maximizes the expected value of ui GB,givenhisbeliefs (up to the 4 th order) P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 14 / 29

14 Assessments and Consistency Assessment An assessment is a profile (σ, α, β,...) =(σ i,α i,β i,...) i I specifying behavior strategies and first- and higher-order beliefs Consistent Assessment An assessment is consistent if there is a strictly positive sequence σ k σ such that for all i I, h H i, s i S i (h), α i (s i h) = lim k Pr σc (s c )Π j=i Pr σ k (s j ) j s i S i (h) Pr σ c(sc)π j=i Pr σ k (s j j ) and higher-order beliefs at each information set are correct for all i I, h H i,β i (h) =α i,γ i (h) =β i,δ i (h) =γ i and so on P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 16 / 29

15 Equilibrium Definition Fix a profile of utility functions of the form u i (z, s i,α,β,...). Aconsistent assessment (σ, α, β,...) is a sequential equilibrium (SE) if each measure Pr σi ( h) assigns positive conditional probability only to conditional expected payoff maximizing strategies: Pr σi (s i h) > 0 = s i arg max s i S i (h) E s i,α i,β i,...[u i h] for all i I, h H i, s i S i (h) It can be proved that every psychological game with simple guilt, or guilt from blame, has a Sequential Equilibrium P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 17 / 29

16 AResultAboutSEWithSGandGB Proposition (1) In any two-player, simultaneous-move game form without chance moves, for any given parameter profile (θ ij ) i,j N,j=i, the pure strategy SE assessments of the psychological games with simple guilt and guilt from blame coincide Consider a material payoff game with utility functions u i m i.inany two-player game form without chance moves (no need simultaneous moves), for every pure strategy, consistent assessment (s,α,β,...), every i and s i, G 0 ij (s i,α i,β i ) = max{0, m j (z(s)) m j (z(s i, s i ))} = E αj,β j,γ j [G 0 ij H j (z(s i, s i ))] P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 19 / 29

17 AThree-Player Steal Game A Not Steal Steal B Not Steal Steal Not Steal Steal (0, 0, 2) (0, 2, 0) (2, 0, 0) (1, 1, 0) P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 20 / 29

18 SG Equilibria in Three-Player Steal Game Suppose A and B are symmetrically affected by guilt towards C: θ AC = θ BC = θ Suppose now that C correctly believes w.p. 1 that A and B are going to play (Not Steal, Not Steal) Then, if 1 <θ<2, (Not Steal, Not Steal) is a SE with SG but not with GB Why? Because C cannot be sure about whom to blame! P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 21 / 29

19 GB Equilibria in Three-Player Steal Game Not Steal A Steal Let α i = α C (a i = steal m C = 0) be the ex-post marginal prob. that i deviates, as assed by C Not Steal Steal B Not Steal Steal By consistency, C thinks that two deviations are infinitely less likely than one, i.e. (1 ˆα B ˆα C )=0 (0, 0, 2) (0, 2, 0) (2, 0, 0) (1, 1, 0) Therefore, ˆα B +ˆα C = 1, and α i 1/2 foratleastonei This player has no incentive to Steal only if 2 θ 2ˆα i 0 = θ 1/ˆα i 2 P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 22 / 29

20 SE in Material Payoff Games vs Psychological Games Proposition (2) In any simultaneous-move game form without chance moves, for any parameter profile (θ ij ) i,j I,j=i, all the pure strategy SE assessments of the material payoff game are also SE of the psychological games with simple guilt and guilt from blame Intuition: Fix a simultaneous game form and a SE (s,α,β,...) of the material payoff game If i deviates from s i, then he weakly decreases his payoff If no deviations occur, no player j is let down, while by deviating each player i who deviates increases in expectation the absolute value of each negative component of his psychological payoff function P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 23 / 29

21 Do Propositions Apply to Sequential Games? Ann Cont Bob cont Ann Share 3 3 Stop stop Grab x 2 0 y 6 0 Suppose 0 < x < 3, 0 < y 2. Then [(Stop, Grab), stop] is the only SE of the MP game, yielding (x, 2) This outcome is not supported by any SE of the psychological game with SG for high enough θ AB P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 24 / 29

22 Ann Cont Bob cont Ann Share 3 3 Stop stop Grab x 2 0 y 6 0 Suppose that Ann correctly guesses that Bob expects $2 At history (Cont., cont.), by choosing Grab, she would let Bob down in the amount: max{0, [2 0]} = 2 Her guilt will be given by: max{0, [2 0]} max{0, [2 3]} = 2 Ann would prefer to Share iff θ AB > 3/2 Anticipating this, Bob would continue after Cont. and Ann would deviate to Cont. at h 0 Proposition (2) does not extend to sequential game forms for SG P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 25 / 29

23 Ann Cont Bob cont Ann Share 3 3 Stop stop Grab x 2 0 y 6 0 Suppose that Ann correctly guesses that Bob expects $2 [(Stop, Grab), stop] is a SE of the game with guilt from blame Suppose at h 0 Ann deviates to Cont. = the blame by Bob on Ann is m B (Stop) m B (Cont., stop) =2 y > 0 The result holds independently of what happens afterwards, so Ann has no incentive to deviate Proposition (1) does not extend to sequential game forms P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 26 / 29

24 AResultAboutEfficiency Proposition (3) Fix a game form without chance moves and let z be a terminal node s.t. for all z Z \{z } there is some j I s.t. m j (z) < m j (z ).Thenfor sufficiently high guilt sensitivities (θ ij ) i,j I,j=i there is a SE of the game with SG that yields z with prob. one P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 27 / 29

25 Conclusions The paper provides the tools to analyze guilt and its impact on individual choices from a theoretical point of view It provides two different definitions of guilt: Simple Guilt Guilt from Blame It develops a general theory of guilt aversion and shows how to solve for SE in psychological games This framework is potentially useful for analyzing also other phenomena: disappointment regret anger... P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 28 / 29

26 Thank You! Q&A P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 29 / 29