Simultaneous All-Pay Auctions with Budget Constraints

2025-05-06

AI Summary

This paper investigates the existence and construction of Nash equilibria in all-pay auctions with multiple heterogeneous bidders under budget constraints and complete information. Addressing the limitation of conventional no-budget assumptions, the authors first systematically characterize the equilibrium structure for the asymmetric budget–valuation setting in single-item auctions. They then propose novel constructive methods for equilibria under joint value distributions for two- and three-item auctions. Crucially, they identify a new phenomenon: budget constraints can render Nash equilibria non-existent. Using game-theoretic modeling, nonlinear optimization, and tailored probability distribution design, the paper establishes a unified equilibrium theory framework for single- to three-item settings with budget constraints. It reveals how budgets structurally reshape bidding strategies and alter equilibrium existence conditions. These results provide foundational theoretical insights for mechanism design in resource-constrained environments.

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: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The all-pay auction, a classic competitive model, is widely applied in scenarios such as political elections, sports competitions, and research and development, where all participants pay their bids regardless of winning or losing. However, in the traditional all-pay auction, players have no budget constraints, whereas in real-world scenarios, players typically face budget constraints. This paper studies the Nash equilibrium of two players with budget constraints across multiple heterogeneous items in a complete-information framework. The main contributions are as follows: (1) a comprehensive characterization of the Nash equilibrium in single-item auctions with asymmetric budgets and valuations; (2) the construction of a joint distribution Nash equilibrium for the two-item scenario; and (3) the construction of a joint distribution Nash equilibrium for the three-item scenario. Unlike the unconstrained all-pay auction, which always has a Nash equilibrium, a Nash equilibrium may not exist when players have budget constraints. Our findings highlight the intricate effects of budget constraints on bidding strategies, providing new perspectives and methodologies for theoretical analysis and practical applications of all-pay auctions.

Problem

• Analyzing Nash equilibrium in all-pay auctions with budget constraints
• Characterizing equilibrium for asymmetric budgets and valuations in single-item auctions
• Exploring existence and construction of equilibrium in multi-item scenarios

Innovation

• Nash equilibrium with asymmetric budgets and valuations
• Joint distribution Nash equilibrium for two-item scenario
• Joint distribution Nash equilibrium for three-item scenario