On equilibrium uniqueness in generalized multi-prize nested lottery contests

The generalized multiple-prize nested lottery contest framework has been broadly applied to model noisy competitions that award prizes to multiple recipients. Equilibrium existence was not formally established until the recent contribution of Fu et al. (2022). This paper comprehensively examines the equilibrium uniqueness of this contest model. We first consider a multi-prize contest with identical players, a scenario that is commonly assumed in the literature. We verify that a symmetric equilibrium exists and that the equilibrium is unique, which lays a foundation for the numerous studies that adopt this model setting. We then proceed to an asymmetric setting in which players may differ in their prize valuations, impact functions, and/or cost functions. We show that the equilibrium uniqueness persists when players are weakly heterogeneous. However, equilibrium uniqueness may fail—and multiple equilibria may arise—when players are sufficiently heterogeneous.