Reciprocity and Trust under an Evolutionary Context

Transcription

1 Reciprocity and Trust under an Evolutionary Context Francisco Poggi UdeSA September 24 Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 1 / 33

2 Motivation The economist is usually not interested in choices or preferences in themselves; (...) There is little investigation of the utility function in itself, and what assumptions are made about it are of a very general nature. The sociologist and the social psychologist, on the other hand, are interested as much in the choices themselves as in the consequences which flow from a general theory of decision-making. Kenneth Arrow, 1958 Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 2 / 33

3 Motivation Typical economic problem Exogenous preferences + restriction + rationality concept rational choice Game Theory problem Exogenous preferences + solution concept equilibrium Mechanism design problem Exogenous possible preference profiles + solution concept implementation game Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 3 / 33

4 Motivation Virtually any behavior, no matter how bizarre, can be explained after the fact by simply assuming a taste for it. Thus the chief attraction of the present-aim model turns out also to be its biggest liability. Because it allows to explain everything, we end up explaining nothing.(...) Our dilemma is how to expand our view of human motives without at the same time becoming vulnerable to Stingler s crankcase oil objection. Robert H. Frank, 1997 Microeconomics and Behavior Solution: tastes are not arbitrarily given, but forged by the pressures of natural selection. Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 4 / 33

5 Motivation Some tastes are simple (useful independently of whether others share the same taste) but some other preferences are more complex (in the sense that the usefulness of having them depends on the fraction of other individuals who share them). Strategic preference. Hawks and doves Morality Love Anger and irrationality Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 5 / 33

6 Aim of this paper Studying a particular social dilemma (a trust game) I wonder which preferences could survive under an evolutionary context. I found that, under some circumstances, an heterogeneous equilibrium might arise with both reciprocal and materialistic individuals. This allows for cooperation in equilibrium, but also generates situations of betrayal. Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 6 / 33

7 Introduction Where do preferences come from? At first sight: pure fitness. However, this does not hold true if we consider social interaction. Two possible reasons that might lead to pro-social behavior: assortative matching observability of types Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 7 / 33

8 Introduction Assortative matching People interact relatively more often with people of the same type. Hamilton (1975) Toro and Silio (1986) Bergstrom (2003) Sethi and Somanathan (2001) and Alger and Weibull (2013) use this approach. Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 8 / 33

9 Introduction Observability of types The logic is similar to Vickers(1985) where a firm whose manager maximizes income has a greater benefit than if it was directly maximizing benefits. If you could commit to an non-optimal behavior can lead you to a better outcome. Preferences, when somewhat observable, can serve as a commitment devise. Guth and Yaari (1992), Heifetz, Shannon and Spiegel (2007) use the observability approach. Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 9 / 33

10 The Trust Game Two players play a sequential game as follows: The first one has to decide whether to trust the other (T ) or not ( T ). When the first player chooses T, the game ends and both players get the outcome x. When trusted, the second has to decide whether to betray (B) or to serve (S). If the second player betrays, he gets outcome z and the first player gets outcome l. However, if the second player serves, both players get outcome y. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

11 The Trust Game Expanded form Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

12 The Trust Game Utility and Material Payoff Lets call Σ the set of possible outcomes (Σ = {l, x, y, z}) There is an objective profit function q : Σ R Assumption :q(z) > q(y) > q(x) > q(l) = 0 An subjective utility functions u i : Σ R individuals maximize their expected utility and differ only by their utility functions. As usual, profit drives the evolutionary process while utility determines how agents behave. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

13 Model We start from a continuum of individuals with a measure normalized to one. Let Θ be the set of all possible types. Definition A state (θ, τ, π) is a population with individuals of type θ, τ Θ where π [0, 1] represents the proportion of individuals of type θ. A state contains all the relevant demographic data and it will always be public information. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

14 Model Types It will be useful to distinguish types of individuals into two disjoint sets. Definition A type θ is reliable if and only if prefers the outcome Y to Z. In other words, θ is reliable if never betrays. Definition A type τ is opportunistic if and only if τ is not reliable. In other words, τ is opportunistic if always betrays. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

15 Model Types: Reciprocal and materialistic individuals Definition individual i is reciprocal if her type R Θ reliable and behaves as maximizing expected profit when plays first. Definition individual i is materialistic if her type M Θ is opportunistic and behaves as maximizing expected profit when plays first. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

16 Model Observability Individuals will be randomly paired to play the trust game. First movers receive a imperfect signal s {R, O} of the reliability of the type of the second mover. The signal tells the true with probability p and with probability (1 p) takes a random value with the population ratio. Thus, Pr(s = R t j = R) = p + (1 p)π Pr(s = R t j = O) = (1 p)π p is the realization of a random variable with cumulative function F (p) differentiable and a density function denoted by f (p). Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

17 Model Timing individuals are randomly matched into ordered pairs. for each couple p is realized and observed. signal s is also realized and observed. First player decides whether to trust or not. If the first player trust, the second player decides if betray or serve. Individuals receive their respective payoff. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

18 M-R Equilibrium Trust Condition Let s a = X Y Proposition When the proportion of reliable is low (π a), first movers trust others when the signal is positive (s = R) and they know the other player enough (p a π 1 π ). When the proportion of reliable is high enough (π > a), first movers trust others when (A) the signal is positive (s = R), or when (B) the signal is negative (s = O) but they don t know a lot about the other player (p < π a π ). Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

19 M-R Equilibrium Probability of Being Trusted I Proposition b 1 = a π 1 π andb 2 = π a π When π < a, 1 P(trust t j = R, π) = π[1 F (b 1 )] + (1 π) f (p)p dp b 1 1 P(trust t j = M, π) = π[1 F (b 1 )] π f (p)p dp b 1 Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

20 M-R Equilibrium Probability of Being Trusted II Proposition When π > a, b2 P(trust t j = R, π) = π + (1 π)[e(p) + F (b 2 ) f (p)p dp 0 b2 P(trust t j = M, π) = π πe(p) + (1 π)f (b 2 ) + π f (p)p dp 0 Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

21 M-R Equilibrium Probability of Being Trusted III Proposition P(trust t j = R, π) P(trust t j = M, π) for every π (0, 1), being equal only when (a) π < a and Pr(p > b 1 ) = 0, or (b) π > a and Pr(p > b 2 ) = 0. For these cases, P(trust t j = R, π) = P(trust t j = M, π) = 0. Proposition P(trust t j, π) is continuous in π for t j = R, M. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

22 M-R Equilibrium Example: probability of getting each possible outcome Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

23 M-R Equilibrium Expected material payoff P T (π) = w W wp wt (π) Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

24 Evolutionary Equilibrium Definition A proportion π is an evolutionary equilibrium of context (θ, τ) Ω 2 if P θ (π ) = P τ (π ) if or if π = 0 and P θ (0) P τ (0) π = 1 and P θ (1) P τ (1) Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

25 Evolutionary Equilibrium Stability Definition An evolutionary equilibrium π is evolutionary stable in context (θ, τ) Ω 2 if. dp θ (π ) dπ > dp τ (π ) dπ Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

26 Results Heterogeneity Proposition If Pr(p > a) > 0 then π = 0 is not an evolutionary equilibrium of context (R, M). Proposition If 1 Z Y 1 a pf (p) dp < Z X context (R, M). then π = 1 is not an evolutionary equilibrium of Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

27 Results Existence, Unicity and Stability Theorem (existence) If Pr(p > a) > 0 and 1 Z Y 1 a pf (p) dp < Z X exists at least one π (0, 1) such that P R (π ) = P O (π ). Theorem (unicity and stability) If p U(0, 1) and 1 Z Y 1 a pf (p) dp < Z X exists only one evolutionary equilibrium π. This equilibrium is stable. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

28 Example Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

29 Example Parameters: q(x) = 0.66 q(y) = 1 q(z) = 1.09 q(l) = 0 p U[0, 1] π = 0.83 probability of betrayal = 0.09 Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

30 M-R justification Evolutionary dominance Definition We say that a preference relationship θ Θ evolutionary dominates θ Θ if and only if P θ (π, θ, θ ) P θ (π, θ, θ ) π [0, 1] θ Θ being the relationship strict for some (π, θ ) [0, 1]xΘ. We denote this relationship with an arrow. Theorem preference θ is maximal in the partially ordered set < Θ, > if and only if individuals with preferences θ behave always as reciprocal or materialistic. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

31 Conclusions Payoff-driven evolution is a strong theory and a powerful tool that economy should borrow in order to understand preference formation and explain certain phenomena. Of course the use of these tools does not replace other possible mechanisms to explain cooperation such as repeated games, but it is a different approach that might help us understand endogenously preference formation and some stylized behavior. I have shown that, when types are imperfectly observable, an heterogeneous situation with a proportion of reciprocal individual might be evolutionary stable, even under random matching. I have also illustrated the formal conditions about observability and payoffs that guarantees an heterogeneous equilibrium. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

32 Conclusions The intuition is quite simple: reciprocal individuals are reliable, and this characteristic is somewhat observable. This gives them a greater probability to achieve cooperation, and thus a relative advantage against materialistic individuals. However, if the proportion of reciprocal individuals is relatively high rational agents are more confident and this gives a beneficial opportunity to materialistic individuals. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

33 David K. Levine (1998), Modeling Altruism and Spitefulness in Experiments. Review of Economic Dynamics, 1, Ingela Alger and Jorgen W. Weibull (2013), Homo Moralis - Preference Evolution under Incomplete Information and Assortative Matching. Econometrica, 81-6, Bergstrom, T (2003), The Algebra of Assortative Encounters and the Evolution of Cooperation. International Game Theory Review, 5, Guth, W. and Yaari, M. (1992), An evolutionary approach to Explain Reciprocal Behavior in a Simple Strategic Game. Explaining Process and Change - Approaches to evolutionary economics, ed. by U. Witt. Ann Harbor: University of Michigan Press. Hamilton, W. D. (1975), Innate Social Aptitudes in Man, an Approach From Evolutionary Genetics. Biosocial Antropology, ed. by R. Fox. London: Malaby. Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

34 Heifetz, A. Shannon, C. and Spiegel, Y (2007), The Dynamic Evolution of Preferences. Economic Theory, 32, Toro, M. and Silio, L. (1986), Assortment of Encounters in the Two-Strategy Game. Journal of Theoretical Biology, 123, Vickers, J. (1985), Delegation and the Theory of the Firm. Economic Journal. Sethi, R. and Somanathan, E. (2001), Preference Evolution and Reciprocity. Journal of Economic Theory, Bergstrom, T. C. and Stark, O. (1993), How Altruism can Prevail in an Evolutionary Environment. Papers and proceedings of the hundred and fifth annual meeting of the american economic association, Bolton, G. E. and Ockenfels, A (2000), ERC: A Theory of equity, reciprocity and competition. American Economic Review, Dekel, E. Ely, J. and Yilankaya, O. (2007), Evolution of preferences. Review of Economic Studies, Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

35 Dufwenberg, M. and Kirchsteiger, G. (2004), A theory of Sequential Reciprocity. Games and Economic Behavior, Frank, R. (1987) If Homo Economicus could choose his own Utility Function, Would He Choose One With a Conscience?. American Economic Review, Guth, W. and Peleg, B. (2001), When Will Payoff Maximization Survive? An Indirect Evolutionary Analysis. Journal of Evolutionary Economics, Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33

36 Agradecimientos Muchas gracias! Francisco Poggi (UdeSA) Reciprocity and Trust... September / 33