Sequential competitions with nondecreasing levels of difficulty

Abstract

Acting in a predetermined order, players select nonincreasing probabilities and perform Bernoulli trials with these success probabilities. The winner is the last successful player, and a ‘tie’ occurs if all players fail. This game is a single-round idealization of athletic competitions such as high jump or pole vault. We find the unique subgame-perfect equilibrium to obtain optimal strategies. First, assuming that a tie has no value, we solve explicitly the game with five of fewer players, and report on numerical calculations for larger games. The properties of these games are surprising in several ways — for instance, the win probabilities under optimal play can decrease for players later in the sequence. In a separate calculation, we find that optimal strategies when there are three players depend critically on the utility of a tie. Based on these calculations, we assess the usefulness of a nondecreasing level of difficulty restriction in reducing inequities in a sequential competition.