The Buck-Passing Game

Abstract

We consider situations where a finite number of agents want to transfer the responsibility of doing a job (the buck) to their neighbors in a social network. This can be seen as network variation of the public good model. The goal of each agent is to see the buck coming back as rarely as possible. We frame this situation as a game where players are the vertices of a directed graph and the strategy space of each player is the set of her out-neighbors. Nature assigns the buck to a random player according to a given initial distribution. Each player pays a cost that corresponds to the asymptotic expected frequency of times that she gets the buck. We consider two versions of the game. In the deterministic one each player chooses one of her out-neighbors once and for all at the beginning of the game. In the stochastic version a player chooses a probability distribution that determines which of her out-neighbors will be chosen when she passes the buck. We show that in both cases the game admits a generalized ordinal potential whose minimizers provide equilibria in pure strategies, even when the strategy set of each player is uncountable. We also show the existence of equilibria that are prior-free, in the sense that they do not depend on the initial distribution used to initially assign the buck. We provide different characterizations for the potential, we analyze fairness of equilibria, and, finally, we discuss a buck holding variant in which players want to maximize the frequency of times they hold the buck. As an application of the latter we briefly discuss the PageRank game.