Guilt aversion in trust games

Transcription

1 Guilt aversion in trust games Elena Manzoni Università degli Studi di Milano Bicocca MMRG, 5 May 01 E. Manzoni Bicocca 5/5 1 / 31

2 Plan of the talk Trust games Experimental results Psychological games Attanasi G., P. Battigalli and E. Manzoni 01, Incomplete information models of guilt aversion in the trust game - Role-dependent guilt aversion with complete information - Role-dependent guilt aversion with incomplete information - Role-independent guilt aversion E. Manzoni Bicocca 5/5 / 31

3 Trust games Ann In Bob C Out D Figure: Trust game E. Manzoni Bicocca 5/5 3 / 31

4 Trust games Ann In Bob C Out D By backward induction: at stage Bob defects; at stage 1 Ann goes Out. In a traditional analysis of the game, where utilities depend only on material payoffs, Out, D is the only equilibrium of the game. E. Manzoni Bicocca 5/5 4 / 31

5 Trust games and experiments Several experiments show that player in role of player A choose In with positive probability, and that players in the role of player B choose C with positive probability. Berg, Dickhaut and McCabe 1995; Ortmann, Fitzgerald and Boeing 000; Koford 1998; Charness and Dufwenberg 006; Attanasi, Battigalli and Nagel 01. E. Manzoni Bicocca 5/5 5 / 31

6 Berg, Dickhaut and McCabe 1995 E. Manzoni Bicocca 5/5 6 / 31

7 Trust games and experiments Possible explanations: Guilt: disutility from disappointing other players Charness and Dufwenberg 006, Attanasi et al. 01 Reciprocity: tendency of responding with a positive action to a positive action of other players Attanasi et al. 01 Here we focus on guilt aversion as a possible explanation for the experimental behavior in the trust game. E. Manzoni Bicocca 5/5 7 / 31

8 Guilt Psychologists. if people feel guilt for hurting their partners... and for failing to live up to their expectations, they will alter their behavior to avoid guilt in ways that seem likely to mantain and strengthen the relationship Baumeister et al, Psychological Bulletin, We say that player i disappoints player j if as a result of i s choice of strategy j gets a lower material payoff than j initially expected. Simple guilt. Agent i s guilt may depend on how much i believes he lets j down. Guilt from blame. Agent i s guilt may also depend on how much i believes j believes i intended to let j down. E. Manzoni Bicocca 5/5 8 / 31

9 Psychological games In order to model guilt in games, we need to have a utility function that depends on beliefs. Therefore classical game theory U i = U i actions is not enough. Psychological game theory: Geanakoplos, Pearce and Stacchetti 1989: Static games, U i = U i beliefs i, actions; Battigalli and Dufwenberg 009: Dynamic Games, U i = U i cond. bel. i, cond. bel. i, actions. E. Manzoni Bicocca 5/5 9 / 31

10 Attanasi G., P. Battigalli and E. Manzoni, Incomplete information models of guilt aversion in the trust game We provide an analysis of trust games with guilt aversion and incomplete information. We follow Battigalli and Dufwenberg 009. We consider four models, depending on the following characteristics: Type of guilt aversion: role-dependent only Bob is guilt averse or role-independent; Type of subjective beliefs: homogeneous or heterogeneous. E. Manzoni Bicocca 5/5 10 / 31

11 Attanasi G., P. Battigalli and E. Manzoni, Incomplete information models of guilt aversion in the trust game Why is it relevant? We show how to model incomplete information with non-standard preferences. Useful for experiments which focus on non-standard preferences where it is unlikely that preferences are known. Let s start from the analysis of the role-dependent guilt aversion case with complete information. E. Manzoni Bicocca 5/5 11 / 31

12 Role-dependent guilt aversion in trust games Ann s disappointment Ann In Bob C Out D Figure: Trust game monetary payoffs The disappointment of Ann after In, D, given her belief α is D Ann In, D, α = αm A In, C m A In, D + 1 α 0 = α E. Manzoni Bicocca 5/5 1 / 31

13 Role-dependent guilt aversion in trust games Complete information Ann In Bob C Out D αθ Figure: Trust game with role-dependent guilt aversion The psychological utility of Bob, given his guilt parameter θ, is u Bob In, D, α = m Bob In, D θd Ann In, D, α = 4 αθ E. Manzoni Bicocca 5/5 13 / 31

14 Role-dependent guilt aversion in trust games Complete information Ann In Bob C Out D αθ Out, D is an equilibrium for every value of θ: if Ann believes that Bob will choose D it is optimal for her to choose Out; with α = 0 it is optimal for Bob to choose D. E. Manzoni Bicocca 5/5 14 / 31

15 Role-dependent guilt aversion in trust games Complete information Ann In Bob Coop. Out Def αθ In, C is an equilibrium for θ > 1: if Ann believes that Bob will choose C it is optimal for her to choose In; in this case α = 1 therefore, given that θ > 1 it is optimal for Bob to choose C. E. Manzoni Bicocca 5/5 15 / 31

16 Role-dependent guilt aversion in trust games Incomplete information: types In order to model the incomplete information in this game we need to introduce a description of types and beliefs. Types: We consider structures of Harsanyi types where the set of types T i can be factorized in a set of payoff types Θ i ; a set of epistemic types identified with the interval [0, 1], that is t i = θ i, e i Θ i [0, 1] = T i. When we model role-dependent guilt aversion we assume Θ a = Θ B, where Θ A is a singleton and Θ B = {θ L, θ H }, with θ H > θ L = 0. Therefore T A = [0, 1] and T B = {θ L, θ H } [0, 1]. E. Manzoni Bicocca 5/5 16 / 31

17 Role-dependent guilt aversion in trust games Incomplete information: beliefs Beliefs: We call exogenous a belief about an exogenous variable or a parameter; we call endogenous a belief about a variable we try to explain with the strategic analysis of the game. In our case the relevant beliefs are: exogenous beliefs: we consider the beliefs of A ont B, τ A and the beliefs of B on T A, τ B ; endogenous beliefs: the endogenous beliefs that are relevant for our analysis are Ann s first order endogenous belief α A and Bob s second order conditional belief β B where α A = Pr Ann [C ] β B = E Bob [α A In] E. Manzoni Bicocca 5/5 17 / 31

18 Role-dependent guilt aversion in trust games Incomplete information: exogenous beliefs We assume that in A s view Bob s epistemic component e B is independent of θ B. A-subjects have heterogeneous beliefs about θ B, parametrized by t A, and common beliefs about e B given by a cdf F. τ A t A [θ B = θ H, e B y] = t A F y B-subjects have heterogeneous beliefs about A s type parametrized by e B ; such beliefs are distributed according to a perturbation of the uniform distribution on [0, e B ]. τ B θ B, e B [t A x] = 1 ε min 1, x + εx e B E. Manzoni Bicocca 5/5 18 / 31

19 Bayesian perfect equilibrium A Bayesian perfect equilibrium is a pair of decision functions σ A : T A {In, Out}, σ B : T B {C, D} such that: for each player i = A, B and type t i T i, choice σ i t i maximizes i s expected psychological utility, given the beliefs of type t i about the co-player choice and the endogenous beliefs induced by σ i ; in the case of i =B, the maximization is conditional on In, with conditional beliefs computed by Bayes rule, if possible. E. Manzoni Bicocca 5/5 19 / 31

20 Role-dependent guilt aversion Optimal strategies σ B θ L, e B = D; given θ L = 0, if Bob s payoff type is θ L he has standard preferences and he maximizes his monetary payoff. σ A t A = In if and only if t A > x; the higher is t A, the higher is the probability that Bob is guilt averse, the higher are Bob s incentives to cooperate and therefore Ann s incentives to choose In. σ B θ H, e B = C if and only if e B > y; the higher e B, the higher is Bob s distribution on Ann s types, the higher is E Bob [α In], and therefore his incentives to cooperate. E. Manzoni Bicocca 5/5 0 / 31

21 Role-dependent guilt aversion with incomplete information Endogenous beliefs Ann s first order endogenous belief: α A t A = τ A t A [θ B = θ H, e B > y] = t A 1 F y. Bob s second order endogenous belief: Note that every type θ B, e B assigns positive probability to In: eb x τ B θ B, e B [σ A = In] = 1 ε max, 0 + ε 1 x > 0. e B Hence we obtain a conditional second-order belief by Bayes rule: β B θ B, e B = E θb,e B [α A In] 1 F y 1+x, if e B x 1, = 1 F y 1 εe B x +εe B1 x if 0 [1 εe B x+εe B x < e 1 x] B. E. Manzoni Bicocca 5/5 1 / 31

22 Role-dependent guilt aversion Ann s incentive conditions Since the expected payoff of In for type t A is α A t A = t A 1 F y, the incentive conditions for x are 0 < x < 1 x1 F y = 1, 1 x = 1 1 F y 1. E. Manzoni Bicocca 5/5 / 31

23 Role dependent guilt aversion Bob s incentive conditions Taking into account that all types θ B, e B with e B y or θ B = θ L defect, and that the conditional expected payoff of defection for type θ B, e B minus the payoff of cooperation is 0 < y < 1 E θb,e B [u B In, D] = θ B β B θ B, e B, we obtain the incentive conditions for threshold y: 1 F y 1+x = x y = 1 1 F y < 1 θ H., if x > y, θ H, if x y, θ H 1 F y 1 εy x +εy1 x [1 εy x+εy1 x] = 1 E. Manzoni Bicocca 5/5 3 / 31

24 Role-dependent guilt aversion with incomplete information Equilibria If we assume F y = y, from the above incentive conditions we derive the following equilibria: for θ H [ 4 3, ] : { x = θh 4 θ H y = 3 θ H for θ H > : { x = 1 y = θh 1 θ H for any value of θ H we find the equilibrium Out, D, that is: { x = 1 y = 1 E. Manzoni Bicocca 5/5 4 / 31

25 Role-independent guilt aversion Ann In Bob C Out 1 θa D B Out, α B, e B 1 D 0 4 α A θ B Here we have: θ i is the guilt parameter of player i; α A = Pr Ann [C ] is Ann s first order exogenous belief; α B = E Bob [In] is Bob s first order exogenous belief. E. Manzoni Bicocca 5/5 5 / 31

26 Role-independent guilt aversion Ann In Bob C Out 1 θa D B Out, α B, e B 1 D 0 4 α A θ B Bob s disappointment is: D B Out, α B, s B = max {0, m B α B, s B m B Out} { max0, αb 1, if s = B = C max0, 4α B 1, if s B = D E. Manzoni Bicocca 5/5 6 / 31

27 Role-independent guilt aversion Complete information Ann In Bob C Out 1 θa D B Out, α B, e B 1 D 0 4 α A θ B There are three possible equilibria in pure strategies: Out, D is an equilibrium for every pair θ A, θ B ; In, C is an equilibrium when θ A 1; In, D is an equilibrium when θ B 1 3. E. Manzoni Bicocca 5/5 7 / 31

28 Role-independent guilt aversion in trust games Incomplete information: types Now T A = T B = {θ L, θ H } [0, 1]. We assume that it is correctly and commonly believed that θ i and e i are independent and that the marginal distribution of e i is given by cdf F : [0, 1] [0, 1], a continuous and strictly increasing function. e i [0, 1], x, τ i e i [θ i = θ H, e i x] = e i F x. E. Manzoni Bicocca 5/5 8 / 31

29 Role-independent guilt aversion in trust games Incomplete information: optimal strategies σ B θ L, e B = D; given θ L = 0, if Bob s payoff type is θ L he has standard preferences and he maximizes his monetary payoff. σ A θ L, e A = In if and only if e A > x;ann is not guilt averse, therefore the higher is t A, the higher is the probability that Bob is guilt averse, the higher are Bob s incentives to cooperate and therefore Ann s incentives to choose In. E. Manzoni Bicocca 5/5 9 / 31

30 Role-independent guilt aversion in trust games Incomplete information: optimal strategies σ B θ H, e B = C if and only if e B < y; the more Bob believes that Ann is guilt averse the higher e B, the lower E Bob [α A In], the lower Bob s incentives to cooperate. σ A θ H, e A : mathematically it can be both increasing or decreasing; we conjecture that it has one of the following shapes: either σ A θ H, e A In if and only if e A > z or σ A θ H, e A = In. E. Manzoni Bicocca 5/5 30 / 31

31 Conclusion We showed how to analyse incomplete information games when there are non-standard preferences. In the comparison between the complete information and the incomplete information analysis we can observe that in the incomplete information case we have less polarized behavior and beliefs. E. Manzoni Bicocca 5/5 31 / 31