We study a problem of trust in a distributed system in which a commonresource is shared by multiple parties. In such naturally information-limitedsettings, parties abide by a behavioral protocol that leads to fair sharing ofthe resource. However, greedy players may defect from a cooperative protocoland achieve a greater than fair share of resources, often without significantadverse consequences to themselves. In this paper, we study the role of a fewvigilante players who also defect from a cooperative resource-sharing protocolbut only in response to perceived greedy behavior. For a simple model ofengagement, we demonstrate surprisingly complex dynamics among greedy andvigilante players. We show that the best response function for thegreedy-player under our formulation has a jump discontinuity, which leads toconditions under which there is no Nash equilibrium. To study this property, weformulate an exact representation for the greedy player best response functionin 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 aNash equilibrium exists. We also illustrate that in the case when there is noNash equilibrium, then the discrete dynamic system generated from fictitiousplay will not converge, but will oscillate indefinitely as a result of the jumpdiscontinuity. The case of multiple vigilante and greedy players is studiednumerically. Finally, we explore the relationship between fictitious play andthe better response dynamics (gradient descent) and illustrate that thisdynamical system can have a fixed point even when the discrete dynamical systemarising from fictitious play does not.
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