Three Essays on Dynamic Contests

This dissertation consists of three essays studying the theory of dynamic contest. This analysis mainly focuses on how the outcome and the optimal design in a dynamic contest varies on contest technology, heterogeneous players, contest architecture, and bias instruments. The first chapter outlines the dissertation by briefly discussing the motivations, methods, and main findings in the following chapters. Chapter 2 considers a situation in which two groups compete in a series of battles with complete information. Each group has multiple heterogeneous players. The group who first wins a predetermined number of battles wins a prize which is a public good for the winning group. A discriminatory state-dependent contest success function will be employed in each battle. We found that in the subgame perfect Nash equilibrium (equilibria), the lower valuation players can only exert effort in earlier battles, while the higher valuation players may exert effort throughout the entire series of battles. The typical discouragement effect in a multi-battle contest is mitigated when players compete as a group. We also provide two types of optimal contest designs that can fully resolve the free-rider problem in group contests. Chapter 3 investigates optimal contest design with multiple heterogeneous players. We allow the contest designer to have one or multiple/mixed objectives, which includes the following parts: the total effort; the winner's effort; the maximal effort; and the winning probability of the strongest player. We provide a one-size-fits-all contest design that is optimal given any objective function. In the optimal contest, the designer will have one of the weaker players exhaust the strongest in the contest with infinite battles. We obtain the required conditions on different contest frameworks (e.g., all-pay auctions and lottery contests) and bias instruments (e.g., head starts and multiplicative bias). This means the contest designer has multiple alternatives to design the optimal contest. The last chapter investigates a situation where two players compete in a series of sequential battles to win a prize. A player can obtain certain points by winning a single battle, and the available points may vary across the battles. The player who first obtains predetermined points wins the prize. We fully characterize the subgame perfect Nash equilibrium by describing the indifference continuation value interval. We found that when two players are symmetric, they only compete in the separating battle. In the general case, we found that winning a battle may not create any momentum when the weight of the battle is small. A small enough adjustment of a battle's weight will not change both players' incentive to win the battle. Increasing (or decreasing) a battle's weight weakly increases (or weakly decreases) both players' incentive to win.