Does randomization matter in dynamic games?

TL;DR

This paper explores the role of mixed strategies in dynamic games, showing they can be beneficial in some cases but are generally unnecessary in two-player zero-sum games with nature, and establishing the existence of pure-strategy equilibria.

Contribution

It proves that mixed strategies are ineffective in two-player zero-sum dynamic games with nature and demonstrates the existence of pure-strategy subgame-perfect equilibria in such games.

Findings

Mixed strategies can yield higher payoffs in some dynamic games.

In two-player zero-sum games with nature, mixed strategies are unnecessary.

Pure-strategy subgame-perfect equilibria exist in these games.

Abstract

This paper investigates mixed strategies in dynamic games with perfect information. We present an example to show that a player may obtain higher payoff by playing mixed strategy. By contrast, the main result of the paper shows that every two-player dynamic zero-sum game with nature has the no-mixing property, which implies that mixed strategy is useless in this most classical class of games. As for applications, we show the existence of pure-strategy subgame-perfect equilibria in two-player zero-sum games with nature. Based on the main result, we also prove the existence of a universal subgame-perfect equilibrium that can induce all the pure-strategy subgame-perfect equilibria in such games. A generalization of the main result for multiple players and some further results are also discussed.