An Adjusted Payoff-Based Procedure for Normal Form Games

• Published in: Mathematics of operations research Vol. 41; no. 4; pp. 1469 - 1483
• Main Author: Bravo, Mario
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01-11-2016; Institute for Operations Research and the Management Sciences
• Subjects: adaptive dynamics; Analysis; Approximations; Game theory; learning; normal form games; Probability distribution; stochastic approximation; Stochastic models; Stochastic processes; Studies; Vector space; United States
• ISSN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2016.0785

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.