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Abraham, Dolev, Geffner, and Halpern [ 1 ] proved that, in asynchronous systems, a (k, t)-robust equilibrium for n players and a trusted mediator can be implemented without the mediator as long as n > 4( k+t ), where an equilibrium is ( k, t )-robust if, roughly speaking, no coalition of t players can decrease the payoff of any of the other players, and no coalition of k players can increase their payoff by deviating. We prove that this bound is tight, in the sense that if n ≤ 4( k+t ) there exist ( k, t )-robust equilibria with a mediator that cannot be implemented by the players alone. Even though implementing ( k, t )-robust mediators seems closely related to implementing asynchronous multiparty ( k+t )-secure computation [ 6 ], to the best of our knowledge there is no known straightforward reduction from one problem to another. Nevertheless, we show that there is a non-trivial reduction from a slightly weaker notion of ( k+t )-secure computation, which we call ( k+t )-strict secure computation , to implementing ( k, t )-robust mediators. We prove the desired lower bound by showing that there are functions on n variables that cannot be ( k+t )-strictly securely computed if n ≤ 4( k+t ). This also provides a simple alternative proof for the well-known lower bound of 4 t +1 on asynchronous secure computation in the presence of up to t malicious agents [ 4 , 8 , 10 ]. 

Secure function computation has been thoroughly studied and optimized in the past decades. We extend techniques used for secure computation to simulate arbitrary protocols involving a mediator. The key feature of our notion of simulation is that it is bidirectional: not only does the simulation produce only outputs that could happen in the original protocol, but the simulation produces all such outputs. In asynchronous systems there are also new subtleties that arise because the scheduler can influence the output. Thus, these requirements cannot be achieved by the standard notion of secure computation. We provide a construction that is secure if n > 4t, where t is the number of malicious agents, which is provably the best possible. We also show that our construction is secure in the universal composability model and that it satisfies additional security properties even if 3t < n \le 4t. 

A mediator can help non-cooperative agents obtain an equilibrium that may otherwise not be possible. We study the ability of players to obtain the same equilibrium without a mediator, using only cheap talk, that is, nonbinding pre-play communication. Previous work has considered this problem in a synchronous setting. Here we consider the effect of asynchrony on the problem, and provide upper bounds for implementing mediators. Considering asynchronous environments introduces new subtleties, including exactly what solution concept is most appropriate and determining what move is played if the cheap talk goes on forever. Different results are obtained depending on whether the move after such ``infinite play'' is under the control of the players or part of the description of the game.