Implementing Mediators with Asynchronous Cheap Talk

Lower Bounds Implementing Mediators in Asynchronous Systems

“…They also proved a matching lower bound [Abraham, Dolev, and Halpern 2008]. Abraham, Dolev, Geffner, and Halpern [2019] (ADGH from now on) extended this result to the asynchronous setting. They showed that if n > 4(k + t) and there exists a (k, t)-robust equilibrium σ + σ d in the mediator game Γ d , then there exists a (k, t)-robust equilibrium σ ACT in Γ ACT in an asynchronous setting such that, for all inputs, σ ACT and σ + σ d produce the same set of possible distributions over outputs (note that agents have no control over how long the messages take to be delivered, and this can affect the output).…”

“…The model used throughout this paper is the one used by ADGH [Abraham, Dolev, Geffner, and Halpern 2019], which consists of an asynchronous network in which there is a reliable, authenticated and asyn-chronous channel between all pairs of players. This means that all messages sent by player i to player j are guaranteed to be delivered eventually, and that j can identify that these messages were sent by i.…”

“…Theorem 5. [Abraham, Dolev, Geffner, and Halpern 2019] If σ + σ d is a (k, t)-robust strategy for an game Γ d that extends some game Γ and n > 4(k + t), then there exists a (k, t)-robust protocol σ ACT for Γ ACT that implements σ + σ d .…”

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Abraham, Dolev, Geffner, and Halpern [2019] 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.

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TRAP: The Bait of Rational Players to Solve Byzantine Consensus

“…This extension was indeed illustrated with an ǫ-(k, t)-robust secret sharing protocol where n > k + 2t. It was later shown [1] that mediators could be implemented with asynchronous cheap talks in ǫ-(k, t)-robust protocol when n > 3(k + t). This adaptation makes it impossible to devise even a 1-immune protocol that would solve the consensus problem [26] as the communication model becomes asynchronous [16].…”