A generalized Tullock contest and the existence of multiple equilibria

We construct a generalized Tullock contest under complete information where contingent upon winning or losing, the payoff of a player is a linear function of prizes, own effort, and the effort of the rival. This structure nests a number of existing contests in the literature. We characterize the equilibria and show that multiple equilibria might exist even under symmetric prize values. Finally, we introduce and characterize several new contests in which multiple equilibria may arise under very general conditions.