On the robustness of learning in games with stochastically perturbed payoff observations
This work addresses the challenge of inaccurate feedback in practical game theory applications, providing robust learning guarantees that are incremental improvements over existing noise-free models.
The paper tackles the problem of learning in games with noisy payoff observations, showing that the proposed stochastic dynamics achieve no regret in single-player settings and ensure convergence to Nash equilibria in multi-player games, with specific results like strict Nash equilibria being stochastically stable and time averages converging to Nash equilibrium in zero-sum games.
Motivated by the scarcity of accurate payoff feedback in practical applications of game theory, we examine a class of learning dynamics where players adjust their choices based on past payoff observations that are subject to noise and random disturbances. First, in the single-player case (corresponding to an agent trying to adapt to an arbitrarily changing environment), we show that the stochastic dynamics under study lead to no regret almost surely, irrespective of the noise level in the player's observations. In the multi-player case, we find that dominated strategies become extinct and we show that strict Nash equilibria are stochastically stable and attracting; conversely, if a state is stable or attracting with positive probability, then it is a Nash equilibrium. Finally, we provide an averaging principle for 2-player games, and we show that in zero-sum games with an interior equilibrium, time averages converge to Nash equilibrium for any noise level.