Evolutionary Selection of Socially Sensitive Preferences in Random Matching Environments

Abstract

In this paper we study the evolutionary selection of socially sensitive preferences in the context of reference interaction settings such as coordination failure and cooperation. We refer to a specific class of socially sensitive preferences in which players weigh additively their own material payoff against the opponent with either a positive or negative coefficient (λ-players). Preference evolution is guided by replicator dynamics in a context of perfect observability of preferences types and stochastic pairwise matching. We take an indirect evolutionary approach, that is, the selection mechanism operates on the actual material payoffs earned by players, so that any instance of socially sensitive preference can be thought of as instrumentally maintained. We find that the evolutionary viability of socially sensitive preferences basically depends on whether or not they cause a substantial improvement in the achievement of socially efficient outcomes with respect to the case where only self-serving or unconditionally focused preference orientations are observed. Our results suggest that moderate pro-social preference orientations are likely to emerge from social selection even in the absence of an intrinsic motivational drive, whereas extremely pro-social orientations as well as competitive and anti-social ones may need a stronger motivational base.

Keywords:

• cooperation
• coordination
• evolution of preferences
• replicator dynamics
• socially sensitive preferences

An early version of this paper has circulated under the title “Evolutionary Dynamics with λ-Players.” While remaining fully responsible for the paper, we thank Angelo Antoci, as well as the editor and an anonymous referee, for useful suggestions. With the same caveat, we thank seminar audiences at the Florence-Konstanz colloquium on economic theory, Hamburg, Bologna, Verona, Trento, and Siena.

Notes

It is worth recalling that, although most of the literature focuses on pro-social examples of other-regarding preferences and consequent behaviors, in general one may also have the opposite case of anti-social (e.g., spiteful, malicious) preferences and behaviors (see, e.g., Fehr, Hoff, and Kshetramade, 2008). Both possibilities will be analyzed in the model below.

Notice how the passage from a (fixed) “green beard” approach to a “green beard (tag) chromodynamics” implies that the discriminating trait may be phenotypic only insofar as it is plastic enough to allow dynamic coding, for example, as in the color mutability of chameleons' skin.

This line of research somewhat parallels, at an analytical level and building up from preference structures, the social simulation literature that explores the evolutionary viability of strategies in specific interaction settings, in particular the prisoner's dilemma (see, e.g., Imhof, Fudenberg, and Nowak, 2007; Nowak, 2006b).

The fictional nature of the λ-preference rules out the difficulty that if two players are reciprocally and actually caring about the other's well being there is an infinite regress problem as far as |λ| ≥1, in that every player incorporates in her own utilitarian calculations the impact that her own happiness has on the happiness of the other, and so on.

Games in which B is strictly dominant (which requires a < c and b < d) yield analogous results.

In the whole paper we will only consider symmetric Nash equilibria when dealing with a symmetric game (like G).

A specific case of anticoordination game is briefly considered in the Appendix.

The relative proportion of types 1 and 4 players within y ₁ cannot be determined directly by this method; such a loss of information is however negligible since e ₁ and e ₄ are equivalent strategies. Clearly, with reference to the original four-strategy dynamic system, the equality holds for all t.

Observe that violating (3) and (4) requires λ < −1.

There also exists some non-generic x(0) such that x converges to a point in S ₃₄.

Binmore and Samuelson (1992) obtain an analogous result in the relatively different context of evolutionary games with finite automata.

Since b > a, (3) fails when λ > 0 is very large. Moreover, (4) is violated for λ = 0 and, since d > c, if fails also if λ <0.

Although trust can in some conditions extend to strangers through the weak ties channel (see, e.g., Macy and Skvoretz, 1998).

The proof of this claim is similar to the proofs in the Appendix. It turns out that (A, A) is the only NE of G _{br, lam} , (B, B) is the only NE of G _{lam, lam} and e ₂ is strictly dominated by e ₃, thus x ₂ → 0. In the remaining game, both e ₁ and e ₃ are weakly dominated by e ₄ and condition (7) of Lemma 2 in the Appendix applies; hence x ₁ and x ₃ tend to 0 and x → e ₄.

By definition, none of these NE can be asymptotically stable or ESS.

See chapter 9 in Van Damme (1991) for a basic study of the properties of ESSs. ESSs rarely arise in a model with discriminating players if the base game is a cooordination game or a dominance solvable game.