On the Impact of Fair Best Response Dynamics

TL;DR

This paper characterizes how the frequency of player participation in congestion game dynamics influences the speed of reaching efficient states, revealing fairness as a key factor for fast convergence.

Contribution

It provides tight bounds on convergence times based on player participation frequency and demonstrates that fairness is both necessary and sufficient for rapid convergence in congestion games.

Findings

States with approximation ratio O(β) times the price of anarchy are reached after T log log n best responses.

Fairness among players is essential for guaranteeing fast convergence to efficient states.

In symmetric games, convergence to efficient states occurs in polynomial time regardless of individual move frequencies.

Abstract

In this work we completely characterize how the frequency with which each player participates in the game dynamics affects the possibility of reaching efficient states, i.e., states with an approximation ratio within a constant factor from the price of anarchy, within a polynomially bounded number of best responses. We focus on the well known class of congestion games and we show that, if each player is allowed to play at least once and at most $\beta$ times any $T$ best responses, states with approximation ratio $O(\beta)$ times the price of anarchy are reached after $T \lceil \log \log n \rceil$ best responses, and that such a bound is essentially tight also after exponentially many ones. One important consequence of our result is that the fairness among players is a necessary and sufficient condition for guaranteeing a fast convergence to efficient states. This answers the important question of the maximum order of $\beta$ needed to fast obtain efficient states, left open by [9,10] and [3], in which fast convergence for constant $\beta$ and very slow convergence for $\beta=O(n)$ have been shown, respectively. Finally, we show that the structure of the game implicitly affects its performances. In particular, we show that in the symmetric setting, in which all players share the same set of strategies, the game always converges to an efficient state after a polynomial number of best responses, regardless of the frequency each player moves with.