Multi-dimensional Bidding in Two-player Auctions

초록

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.

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