An adjusted payoff-based procedure for normal form games

TL;DR

This paper introduces an adaptive payoff-based method for N-player normal form games, analyzing its convergence properties and applying it to traffic scenarios, with insights into when convergence occurs or fails.

Contribution

It proposes a new payoff-based adaptive procedure for normal form games and analyzes its convergence behavior using stochastic approximation techniques.

Findings

Convergence to rest points is established under certain conditions.

The method converges with positive probability in some cases.

Examples show scenarios where convergence does not occur.

Abstract

We study a simple adaptive model in the framework of an N -player normal form game. The model consists of a repeated game where the players only know their own action space and their own payoff scored at each stage, not those of the other agents. Each player, in order to update her mixed action, computes the average vector payoff she has obtained by using the number of times she has played each pure action. The resulting stochastic process is analyzed via the ODE method from stochastic approximation theory. We are interested in the convergence of the process to rest points of the related continuous dynamics. Results concerning almost sure convergence and convergence with positive probability are obtained and applied to a traffic game. We also provide some examples where convergence occurs with probability zero.