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Abstract: We study the convergence time of the best response dynamics inplayer-specific singleton congestion games. It is well known that this dynamicscan cycle, although from every state a short sequence of best responses to aNash equilibrium exists. Thus, the random best response dynamics, which selectsthe next player to play a best response uniformly at random, terminates in aNash equilibrium with probability one. In this paper, we are interested in theexpected number of best responses until the random best response dynamicsterminates.As a first step towards this goal, we consider games in which each player canchoose between only two resources. These games have a natural representation asmulti-graphs by identifying nodes with resources and edges with players. Forthe class of games that can be represented as trees, we show that thebest-response dynamics cannot cycle and that it terminates after On^2 stepswhere n denotes the number of resources. For the class of games represented ascycles, we show that the best response dynamics can cycle. However, we alsoshow that the random best response dynamics terminates after On^2 steps inexpectation.Additionally, we conjecture that in general player-specific singletoncongestion games there exists no polynomial upper bound on the expected numberof steps until the random best response dynamics terminates. We support ourconjecture by presenting a family of games for which simulations indicate asuper-polynomial convergence time.



Author: Heiner Ackermann, Heiko Roeglin

Source: https://arxiv.org/







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