Effort comparisons for a class of four-player tournaments

We propose a novel tournament design that incorporates some properties of a round-robin tournament, a Swiss tournament, and a race. The new design includes an all-play-all structure with endogenous scheduling and a winning threshold. Considering a standard round-robin tournament as a baseline model, we first characterize the equilibrium strategies in round-robin tournaments with exogenous and endogenous schedules. Afterward, following an equilibrium analysis of the new tournament design, we compare thirty-six tournament structures inherent in our model with round-robin tournaments on the basis of expected equilibrium effort per battle. We show that a round-robin tournament with an endogenous schedule outperforms all the other tournament structures considered here. We further note that if expected total equilibrium effort is used as a comparison criterion instead, then the new tournament design has a potential to improve upon round-robin tournaments.