Periodic Strategies: A New Solution Concept and an Algorithm for NonTrivial Strategic Form Games

TL;DR

This paper introduces a new solution concept called periodicity for strategic form games, providing insights into non-trivial games and showing that every finite non-trivial game has at least one periodic strategy, with implications for various game types.

Contribution

The paper defines and studies periodic strategies, proving their existence in all finite non-trivial games and exploring their properties in mixed and pure strategy contexts.

Findings

Periodic strategies yield Nash equilibrium payoffs in mixed strategies.

Every non-trivial finite game has at least one periodic strategy.

Periodic strategies can outperform Nash strategies in certain scenarios.

Abstract

We introduce a new solution concept, called periodicity, for selecting optimal strategies in strategic form games. This periodicity solution concept yields new insight into non-trivial games. In mixed strategy strategic form games, periodic solutions yield values for the utility function of each player that are equal to the Nash equilibrium ones. In contrast to the Nash strategies, here the payoffs of each player are robust against what the opponent plays. Sometimes, periodicity strategies yield higher utilities, and sometimes the Nash strategies do, but often the utilities of these two strategies coincide. We formally define and study periodic strategies in two player perfect information strategic form games with pure strategies and we prove that every non-trivial finite game has at least one periodic strategy, with non-trivial meaning non-degenerate payoffs. In some classes of games where mixed strategies are used, we identify quantitative features. Particularly interesting are the implications for collective action games, since there the collective action strategy can be incorporated in a purely non-cooperative context. Moreover, we address the periodicity issue when the players have a continuum set of strategies available.