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
