On the performance of mildly greedy players in cut games

Abstract

We continue the study of the performance of mildly greedy players in cut games initiated by Christodoulou et al. in [14], where a mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves of a factor of more than 1+ε, for some ε≥0. Hence, in presence of mildly greedy players, the classical concepts of pure Nash equilibria and best-responses generalize to those of ε-approximate pure Nash equilibria and ε-approximate best-responses, respectively. We first show that the ε-approximate price of anarchy, that is the price of anarchy of ε-approximate pure Nash equilibria, is at least 1/2+ε and that this bound is tight for any ε. Then, we evaluate the approximation ratio of the solutions achieved after an ε-approximate one-round walk starting from any initial strategy profile, where an approximate one-round walk is a sequence of ε-approximate best-responses, one for each player. We improve the currently known lower bound on this ratio from min (equation present) up to min(equation present) and show that this is tight for any ε. © 2014 Springer International Publishing Switzerland.