Three essays in tournament contracts

This thesis consists of three related essays in tournament contracts. The first essay examines the optimal prize allocations in tournament games. Following Moldovanu and Sela (2001), I construct a general model with n players and k prizes where sabotage is considered as another variable. Players are heterogeneous with respect to their talents. It is shown that the optimal prize structure possesses a knife-edge property; namely, the optimal number of prizes is either 1 or k-1, regardless of whether costs are linear or convex. The solutions are shown to depend upon relative marginal costs between effort and sabotage. It is also shown that more prizes are offered when players become more risk averse.;The third essay examines the origin of property rights in the context of tournament game with production uncertainty. Under anarchy, capturing rival's output through a fight is permitted. Consequently, the winner is able to enjoy the lion-share of total outputs produced by all parties and this generates a diversification effect since the random outputs are polled together. Taking this effect along with the costs associated with a fight into account, I characterize the subgame perfect equilibrium for our two-stage game. Specifically, the emergence of property rights can be shown to depend on factors such as players' incentives to fight, variances, and market correlations. The model predicts that property rights are more likely to emerge when market correlations are sufficiently positive and when the volatility is sufficiently low.;The second essay reexamines the issue of prize allocation in the context of anarchy. By using a two-player-one-stage tournament game with n periods in the state of anarchy, I show that the allocation of resources is determined by the law of jungle and the winner-take-all reward scheme appears to be the equilibrium outcome when the game lasts only one period. When game lasts for infinitely many periods, it turns out that the winner-take-all scheme may not be optimal. That is, it may be in the interest of the winner to leave something to the loser. However, when the players are given an option to bargain, fighting can be avoided all together. In this case, both players are shown to equally divide their time endowment between farming and training---an outcome that is sub-optimal relative to the situation where players are fully protected by property rights.