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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].

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an efficient scheme for encoding simple outerplanar maps.

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. Even though implementing (k, t)-robust mediators seems closely related to implementing asynchronous multiparty (k + t)-secure computation [Ben-Or, Canetti, and Goldreich 1993], 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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