The Efficiency of Best-Response Dynamics

Best response (BR) dynamics is a natural method by which players proceed toward a pure Nash equilibrium via a local search method. The quality of the equilibrium reached may depend heavily on the order by which players are chosen to perform their best response moves. A deviator rule S is a method for selecting the next deviating player. We provide a measure for quantifying the performance of different deviator rules. The inefficiency of a deviator rule S with respect to an initial strategy profile p is the ratio between the social cost of the worst equilibrium reachable by S from p and the social cost of the best equilibrium reachable from p. The inefficiency of S is the maximum such ratio over all possible initial profiles. This inefficiency always lies between 1 and the price of anarchy.