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

TL;DR

This paper analyzes multi-dimensional contests with asymmetric players, identifying conditions for unique pure-strategy equilibria and characterizing optimal prize allocations that incentivize effort across battles.

Contribution

It provides a tight sufficient condition for equilibrium existence and characterizes the effort-maximizing prize rule in asymmetric multi-battle contests.

Findings

Unique pure-strategy equilibrium exists under certain conditions.

The optimal prize distribution favors the player leading by a margin in total victories.

Cross-battle externalities do not affect the optimal prize allocation rule.

Abstract

We study $n$-dimensional contests between two players with heterogeneous effort costs, where each dimension (battle) is modeled as a Tullock contest. Prize-allocation rules are identity-independent, budget-balanced, and weakly increasing in the number of victories. Players' costs can be separable across battles or exhibit cross-battle externalities. We identify a tight sufficient condition under which a unique equilibrium exists and is in pure strategies, for all admissible prize-allocation rules and all degrees of player asymmetry. Under this condition, we characterize the effort-maximizing prize-allocation rule: the entire prize goes to the player who wins more battles than the opponent by at least a prespecified margin, and is split equally if neither player meets this threshold. In the symmetric-player case, the majority rule is optimal if $n$ is odd. Interestingly, cross-battle cost externalities do not change the optimal prize allocation rule in our setting.