Optimal Dynamic Contests

We study the design of optimal incentives in a two-player dynamic contest. Two players continuously spend costly effort to attain a score lead, which is also affected by noise. The first player to reach a predetermined score difference (finish line) wins a prize. We focus on the choice of the optimal prize for the winner and on the optimal scoring rule, which penalizes or boosts the leader at each point in the game. We solve for a Symmetric Markov Perfect Equilibrium of the contest, and use it to evaluate a few possible principal's objectives. We find that equilibrium effort is always positive, increasing or hump-shaped in own lead, and the leader always exerts more effort than the follower. These results replicate in our continuous time, continuous state space setting those obtained by Harris and Vickers (1987) in a different discrete time, discrete state model. Our model is more tractable and affords our main innovation, the normative analysis of this game. Due to the strategic interaction, the optimal prize that maximizes expected total agents' output is finite even if effort costs and the value of the prize are of no concern to the principal. Too large a prize and the strategic complementarities of agents' efforts generate an initial war phase, followed by low effort thereafter and whenever the lead is not very small. We conjecture that the optimal scoring rule entails penalizing the leader to keep the laggard from giving up, despite the adverse ex ante incentives of this rule. We show how to solve numerically for the optimal scoring rule.