N-Player Prisoner's Dilemma in Multiple Groups: A Model of Multilevel Selection

Jeff Fletcher and Martin Zwick.

Proceedings of the Artificial Life VII Workshops, Portland, Oregon, 2000, Eilis Boudreau and Carlo Maley, eds.

Abstract

Simulations of the n-player Prisoner's Dilemma (PD) in populations consisting of multiple groups reveal that Simpson's paradox (1951) can emerge in such game-theoretic situations. In Simpson's paradox, as manifest here, the relative proportion of cooperators can decrease in each separate group, while the proportion of cooperators in the total population can nonetheless increase, at least transiently. The increase of altruistic behavior exhibited in these simulations is not based on reciprocal altruism (Trivers 1971), as there are no strategies (e.g. Tit-for-Tat) conditional on other players' past actions, nor does it depend on kin selection via inclusive fitness (Hamilton 1964), as there are no genomes. This model is very general in that it can represent both biological and social non-zero sum situations in which utility (fitness) depends upon both individual and group behavior. The two parameters of the PD in this model, which determine the gain in individual utility for defection and the dependence of utility on collective cooperation, are respectively analogous to within-group and between-group selective forces in multilevel selection theory. This work is more fully described in Fletcher and Zwick (2000).

The notion that a system (group) does better when it achieves cooperation among its parts (individuals), often against the self-interest of those parts, goes beyond just biological systems undergoing natural selection. It is applicable to hierarchical systems across a variety of fields. The non-zero sum nature of aggregation is general and optimization by subsystems often results in sub-optimization at a higher level. The PD is often used to model such non-zero sum situations. Like Simpson's paradox, the PD involves an anomaly of composition: individually-rational strategies, when aggregated, give a deficient collective outcome.

As Sober and Wilson (1998) have demonstrated, Simpson's paradox (even if not always identified as such) is important in understanding multilevel selection. These authors show (pp. 18-26) that this paradox can be derived from simple fitness functions for altruists and non-altruists in two populations. These functions amount to an n-player PD (see Appendix A), although Sober and Wilson do not call attention to this fact. In this paper and in Fletcher and Zwick (2000), we make the connection between the PD and Simpson's paradox explicit. Our main finding is that Simpson's paradox emerges transiently, but for a wide range of PD parameter values, when a minimal group structure is imposed on an n-player PD. This result is produced in a model which involves an implicit competition between two groups and a simple n-player PD in each. The model is based on only two parameters which correlate with the within-group and between-group selection components in multilevel selection theory.