  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 player's 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.
