Lower Bounds on Implementing Robust and Resilient Mediators

We provide new and tight lower bounds on the ability of players to implement equilibria using cheap talk, that is, just allowing communication among the players. One of our main results is that, in general, it is impossible to implement three-player Nash equilibria in a bounded number of rounds. We also give the first rigorous connection between Byzantine agreement lower bounds and lower bounds on implementation. To this end we consider a number of variants of Byzantine agreement and introduce reduction arguments. We also give lower bounds on the running time of two player implementations. All our results extended to lower bounds on (k, t)-robust equilibria, a solution concept that tolerates deviations by coalitions of size up to k and deviations by up to t players with unknown utilities (who may be malicious).