Approximate Best-Response Dynamics in Random Interference Games

TL;DR

This paper introduces a new approach to analyze the convergence of best-response dynamics in random interference games, showing that these dynamics lead to approximate Nash equilibria with high probability, supported by probabilistic bounds and simulations.

Contribution

The paper develops a novel probabilistic framework for convergence analysis of best-response dynamics in interference games, which are not potential games, using martingale theory and random player selection.

Findings

Convergence to approximate Nash equilibrium with high probability.

Sum-rate process is a submartingale, ensuring expected social welfare increases.

Probabilistic bounds on convergence time are established.

Abstract

In this paper we develop a novel approach to the convergence of Best-Response Dynamics for the family of interference games. Interference games represent the fundamental resource allocation conflict between users of the radio spectrum. In contrast to congestion games, interference games are generally not potential games. Therefore, proving the convergence of the best-response dynamics to a Nash equilibrium in these games requires new techniques. We suggest a model for random interference games, based on the long term fading governed by the players' geometry. Our goal is to prove convergence of the approximate best-response dynamics with high probability with respect to the randomized game. We embrace the asynchronous model in which the acting player is chosen at each stage at random. In our approximate best-response dynamics, the action of a deviating player is chosen at random among all the approximately best ones. We show that with high probability, with respect to the players' geometry and asymptotically with the number of players, each action increases the expected social-welfare (sum of achievable rates). Hence, the induced sum-rate process is a submartingale. Based on the Martingale Convergence Theorem, we prove convergence of the strategy profile to an approximate Nash equilibrium with good performance for asymptotically almost all interference games. We use the Markovity of the induced sum-rate process to provide probabilistic bounds on the convergence time. Finally, we demonstrate our results in simulated examples.