Lower Bounds Implementing Mediators in Asynchronous Systems
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Abraham, Dolev, Geffner, and Halpern 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 <cit.>, 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 4t+1 on asynchronous secure computation in the presence of up to t malicious agents.
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