Can a Weaker Player Win? Adaptive Play in Repeated Games

TL;DR

This paper investigates whether a weaker player can achieve positive gains in repeated two-player games by using adaptive strategies, analyzing optimal policies and asymptotic behaviors.

Contribution

It introduces a dynamic programming approach to identify parameter regimes where a weaker player can still win and characterizes the asymptotic gain behavior.

Findings

Optimal adaptive policies can yield positive gains for weaker players in certain regimes.

Structural conditions determine when the gain remains negative or becomes nonnegative.

Asymptotic analysis shows the gain converges to 0 or -1 depending on game fairness and styles.

Abstract

Consider a two-player game repeated N times. Player 1 can choose between two styles (for interpretability, offensive and defensive), whereas Player 2 uses a single fixed style. Let X N\,:= \#wins -\#losses for Player 1 after N games, and define the match gain as E[sign(X N )], with sign(0) = 0. We assume Player 1 is weaker in the sense that each pure style is losing in expectation. Our objective is to identify under which parameter regimes Player 1 can nevertheless achieve a positive gain under an optimal adaptive policy. Using dynamic programming, we solve the finite-horizon control problem and numerically identify parameter regimes in which the optimal gain is strictly positive at some horizon N $\star$ . We also derive structural conditions guaranteeing that g $\star$ N is always negative, and regimes (notably with fair (D)) where g $\star$ N is nonnegative for all N and can be strictly positive for every N $\ge$ 2. We then characterize the asymptotic behavior as N $\rightarrow$ $\infty$ for a weak player. In the safe case, where the defensive style induces a sure draw, the limiting gain varies continuously with the parameters and may take any value in [0, 1]. In the non-safe case, the limiting gain converges to -1 when both styles are strictly losing, and to 0 when (D) is fair (and non-safe).