We study a problem of trust in a distributed system in which a common resource is shared by multiple parties. In such naturally information-limited settings, parties are expected to abide by a behavioral protocol that leads to fair sharing of the resource. However, greedy players may defect from a cooperative protocol and achieve a greater than fair share of resources, often without significant adverse consequences to themselves. In this paper, we study the role of a few vigilante (attendant) players who also defect from a cooperative resource-sharing protocol but only in response to perceived greedy behavior. For a simple model of engagement, we demonstrate surprisingly complex dynamics among greedy and vigilante players. We show that the best response function for the greedy-player under our formulation has a jump discontinuity, which leads to conditions under which there is no Nash equilibrium. To study this property, we formulate an exact representation for the greedy player best response function in the case when there is one greedy player, one vigilante player and N − 2 cooperative players. We use this formulation to show conditions under which a Nash equilibrium exists. We also illustrate that in a case when there is no Nash equilibrium, the discrete dynamic system generated from fictitious play will not converge, but will oscillate indefinitely as a result of the jump discontinuity. The case of multiple vigilante and greedy players is studied numerically. Finally, we explore the relationship between fictitious play and the better-response dynamics (gradient descent) and illustrate that this dynamical system can have a fixed point even when the discrete dynamical system arising from fictitious play does not.
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