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

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.

- Entire Paper (pdf / postscript)
- Alife page