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 $\frac{1}{2+\epsilon}$ 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 $\min\left\{\frac{1}{4+2\epsilon},\frac{\epsilon}{4+2\epsilon}\right\}$ up to $\min\left\{\frac{1}{2+\epsilon},\frac{2\epsilon}{(1+\epsilon)(2+\epsilon)}\right\}$ and show that this is tight for any ε.

This work was partially supported by the PRIN 2010–2011 research project ARS TechnoMedia: “Algorithmics for Social Technological Networks” funded by the Italian Ministry of University.