Solving Maxmin Optimization Problems via Population Games

Abstract

We introduce a class of population games which we call ``tester games''. These games are not intended as models for situations in biology or economics. Instead, they serve a computational purpose: Nash equilibria of tester games provide maxmin solutions for certain types of multicriteria decision problems. The relation between Nash equilibria and maxmin solutions via tester games is different from the traditional connection via two-person zero-sum games. Using tester games to compute maxmin solutions of multicriteria problems is convenient in particular when transformations can be found such that optimization of a weighted sum of transformed criteria is a relatively easy problem. The transformations applied to individual criteria should be monotonic, but are not subject to convexity requirements and can be different for different criteria. A special case of this technique has already been known in numerical analysis for a long time under the name of IRLS (Iteratively Reweighted Least Squares). We illustrate the use of the generalized IRLS method in applications to collective investment and to decision making under model uncertainty.

Keywords: Maxmin optimization, population games, collective decision, Nash equilibrium, Pareto map

JEL Classification: C61, C73, D70, D81, G11