Simultaneous All-Pay Auctions with Budget Constraints

TL;DR

This paper analyzes the impact of budget constraints on Nash equilibria in all-pay auctions involving multiple heterogeneous items, revealing complex strategic effects and constructing equilibrium solutions for specific cases.

Contribution

It provides the first comprehensive characterization of Nash equilibria in budget-constrained all-pay auctions with multiple items, including explicit constructions for two- and three-item scenarios.

Findings

Nash equilibrium may not exist under budget constraints.

Characterization of equilibrium in single-item auctions with asymmetric budgets.

Explicit joint distribution equilibria for two- and three-item cases.

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.