A sharp characterization of equilibria in a tournament with performance and distinction standards

The objective of this paper is to provide a characterization of the equilibrium of an academic tournament model which highlights the relationship between minimum performance standards (graduation requirements) and distinction standards (academic prizes). We consider a generic effort game among a finite number of students that differ by ability in a single academic institution. The incentive mechanism combines a minimum performance standard with a tournament for prizes awarded based on relative performance rankings. We show that a high graduation requirement is necessary for the existence of a pure strategy equilibrium. It is characterized by the highest ability students obtaining the prizes. With a wide enough ability distribution, students in the tails of the distribution put forth low effort. For a lower graduation standard, equilibrium is in mixed strategies, but with just two students strictly randomizing in their effort choices. Interestingly, a top ability student exploits their ability advantage by randomly shirking, a lower ability student randomizes to perhaps beat out that student, and students of abilities between put in high effort to insure a prize. Finally, we consider a variety of extensions of the model.