The Third Place Game

We study an elimination tournament with four contestants, each of whom has either a high value of winning (a strong player) or a low value of winning (a weak player) and these values are common knowledge. Each pairwise match is modeled as an all-pay auction. The winners of the first stage (semifinal) compete in the second stage (final) for the first prize, while the losers of the first stage compete for the third prize. We examine whether or not the game for the third prize is profitable for the designer who wishes to maximize the total effort of the players. We demonstrate that if the players are asymmetric and there are at least two strong players, then there is always a seeding of the players such that the third place game is not profitable. On the other hand, if there are at least two weak players, then there is always a seeding of the players such that the third place game is profitable.