Optimal Timing in Competition for Advantage: A Two-Stage Contest

Abstract

We study a two-stage contest between two players who differ in ability, with a prize awarded in the second stage. In the first stage, players compete, and the winner enjoys a reduced effort cost in the second stage. The second-stage contest is simultaneous, while the first-stage contest can be simultaneous or sequential. We investigate how a sequential first stage affects the catching-up and discouragement effects between players. Additionally, we explore the optimal first-stage timing structure of contest designers with different objectives, that is, maximizing effort in the second stage, total effort across both stages, or the winner's total effort. Interestingly, a designer focused on maximizing second-stage effort prefers a simultaneous first-stage contest, contrary to the conventional intuition that the stronger player should lead.