Lower Bounds on Implementing Robust and Resilient Mediators

TL;DR

This paper establishes fundamental lower bounds on the ability to implement robust mediators in multi-player games through communication protocols, considering various assumptions about the game and communication environment.

Contribution

It provides matching lower bounds on the implementation of (k,t)-robust mediators via cheap talk under diverse conditions, extending understanding of the limits of secure multi-party computation.

Findings

Lower bounds depend on k, t, and n relationships.

Implementation feasibility varies with knowledge of utilities.

Cryptography and communication channels influence bounds.

Abstract

We consider games that have (k,t)-robust equilibria when played with a mediator, where an equilibrium is (k,t)-robust if it tolerates deviations by coalitions of size up to k and deviations by up to $t$ players with unknown utilities. We prove lower bounds that match upper bounds on the ability to implement such mediators using cheap talk (that is, just allowing communication among the players). The bounds depend on (a) the relationship between k, t, and n, the total number of players in the system; (b) whether players know the exact utilities of other players; (c) whether there are broadcast channels or just point-to-point channels; (d) whether cryptography is available; and (e) whether the game has a $k+t)-punishment strategy; that is, a strategy that, if used by all but at most $k+t$ players, guarantees that every player gets a worse outcome than they do with the equilibrium strategy.