We consider a game in which players are the vertices of a directed graph. Initially, Nature chooses one player according to some fixed distribution and gives her a buck, which represents the request to perform a chore. After completing the task, the player passes the buck to one of her out-neighbors in the graph. The procedure is repeated indefinitely and each players cost is the asymptotic expected frequency of times that she receives the buck. We consider a deterministic and a stochastic version of the game depending on how players select the neighbor to pass the buck. In both cases we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. We then use the price of anarchy and price of stability to measure fairness of these equilibria. We also study a buck-holding variant of the game in which players want to maximize the frequency of times they hold the buck, which includes the PageRank game as a special case.
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