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

2026-02-24

AI Summary

This study investigates equilibrium behavior and optimal reward mechanism design in multi-dimensional Tullock contests involving two participants with asymmetric marginal costs of effort. Under reward rules satisfying identity independence and budget balance, the authors employ game-theoretic and optimization techniques to characterize equilibrium strategies. The main contributions include the first proof that a unique equilibrium exists when the contest’s discriminatory power does not exceed \(2/(n+1)\), with each player exerting identical deterministic effort across all dimensions. Furthermore, the optimal reward allocation rule that maximizes total effort under this condition is derived: the entire prize is awarded to the winner if the margin of victory is non-zero, otherwise it is split equally. Notably, in symmetric settings with an odd number of dimensions, the majority-win rule is shown to be optimal.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• multidimensional contests
• asymmetric players
• Tullock contest
• prize design
• equilibrium

Innovation

• multidimensional contests
• asymmetric players
• Tullock contest
• optimal prize design
• equilibrium uniqueness