91ÊÓƵ

Large tournament games

We consider a stochastic tournament game in which each player works toward accomplishing her goal and is rewarded based on her rank in terms of the time to completion. We prove existence, uniqueness and stability of the game with infinitely many players, and existence of approximate equilibrium with finitely many players. When players are homogeneous, the equilibrium has an explicit characterization. We find that the welfare may be increasing in cost of effort in its low range, as the cost reduces players’ eagerness to work too hard. The reward function that minimizes the expected time until a given fractionÌýαÌýof the population has reached the target, as well as the aggregate welfare, only depends on whether the rank is above or belowÌýα. However, that is no longer true when maximizing a function of the completion time. Numerical examples are also provided when players are inhomogeneous. (Joint work with Erhan Bayraktar and Jaksa Cvitanic)