A framework for modeling population strategies by depth of reasoning

This article presents a population-based cognitive hierarchy model that can be used to estimate the reasoning depth and sophistication of a collection of opponents' strategies from observed behavior in repeated games. This framework provides a compact representation of a distribution of complicated strategies by reducing them to a small number of parameters. This estimated population model can be then used to compute a best response to the observed distribution over these parameters. As such, it provides a basis for building improved strategies given a history of observations of the community of agents. Results show that this model predicts and explains the winning strategies in the recent 2011 Lemonade Stand Game competition, where eight algorithms were pitted against each other. The Lemonade Stand Game is a three-player game with simple rules that includes both cooperative and competitive elements. Despite its apparent simplicity, the fact that success depends crucially on what other players do gives rise to complex interaction patterns, which our new framework captures well.