On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games

TL;DR

This paper analyzes the expected convergence time of random best response dynamics in player-specific singleton congestion games, showing polynomial bounds for certain graph structures and conjecturing super-polynomial bounds in general.

Contribution

It provides the first polynomial bounds on convergence time for tree and cycle graph structures and introduces a conjecture on super-polynomial bounds in general cases.

Findings

Best response dynamics terminates after O(n^2) steps in tree-structured games.

Expected termination time is also O(n^2) in cycle-structured games.

Simulations suggest super-polynomial convergence times in general games.

Abstract

We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n^2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n^2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time.