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). ∗Part of the work was done while the author visited Cornell university. The work was funded in part by ISF, ISOC, NSF, CCR, and AFOSR. †Supported in part by NSF under grants CCR-0208535, ITR-0325453, and IIS-0534064, by ONR under grant N00014-0110-511, by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grant FA9550-05-1-0055. ar X iv :0 70 4. 36 46 v2 [ cs .G T ] 6 D ec 2 00 7