Generalized Multidimensional Contests with Asymmetric Players: Equilibrium and Optimal Prize Design

We study the $n$-dimensional contest between two asymmetric players with different marginal effort costs, with each dimension (i.e., battle) modeled as a Tullock contest. We allow general identity-independent and budget-balanced prize allocation rules in which each player's prize increases weakly in the number of their victories, e.g., a majority rule if $n$ is odd. When the discriminatory power of the Tullock winner-selection mechanism is no greater than $2/(n+1)$, a unique equilibrium arises where each player exerts deterministic and identical effort across all dimensions. This condition applies uniformly to all eligible prize allocation rules and all levels of players' asymmetry, and it is tight. Under this condition, we derive the effort-maximizing prize allocation rule: the entire prize is awarded to the player who wins more battles than his opponent by a pre-specified margin, and the prize is split equally if neither player does. When $n$ is odd, and players are symmetric, the majority rule is optimal.