Sequential competitions with nondecreasing levels of difficulty

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. The authors find the unqiue subgame-perfect equilibrium to obtain optimal strategies. First, assuming that a tie has no value, they solve explicitly the game with five or 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, the authors find that optimal strategies when there are three players depend critically on the utility of a tie. Based on these calculations, they assess the usefulness of a nondecreasing level of difficulty restriction in reducing inequalities in a sequential competition.