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Seoul Journal of Economics
[ Article ]
Seoul Journal of Economics - Vol. 39, No. 1, pp.1-34
ISSN: 1225-0279 (Print)
Print publication date 28 Feb 2026
Received 04 Aug 2025 Revised 05 Feb 2026 Accepted 06 Feb 2026
DOI: https://doi.org/10.22904/sje.2026.39.1.001

Multi-dimensional Bidding in Two-player Auctions

Hanjoon Michael Jung
Hanjoon Michael Jung, College of Business Management, Hongik University hanjoonjung8@gmail.com

JEL Classification Number: D44, D72, C72.

Abstract

We introduce a general model of auctions involving two players. In this model, each player selects a monetary bid and an effort level; a general score function then combines these inputs to determine the winner. This framework unifies standard all-pay contests and winner-pay auctions by allowing a continuous tradeoff between effort and bidding. We prove the existence of a Nash equilibrium under mild technical conditions on the score function. In symmetric contests, we derive explicit expressions for equilibrium bids and efforts, which reveal how the degree of substitutability between effort and bidding influences strategic allocations. We characterize the equilibrium structure in asymmetric settings. Furthermore, we examine how complementarity between effort and bid inputs can amplify or attenuate competitive pressures. Our theoretical contributions include a constructive equilibrium solution in closed form and new insights into multi-dimensional bidding strategies under general score rules.

Keywords:

All-pay auction, Bidding strategy, Contest theory, Conditional investment, Score function

Acknowledgments

The author would like to thank the editor and an anonymous referee for their helpful and insightful comments and suggestions, which significantly improved this paper.

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