How competitive are pay-as-bid auction games?

2025-04-11

AI Summary

This paper investigates pure-strategy Nash equilibria in discriminatory pay-as-bid auctions with asymmetric generators, modeled as a supply function game. Methodologically, it analyzes equilibria within the space of Lipschitz-continuous supply functions. The contributions are threefold: (i) It establishes existence and structural characterization—equilibria always exist and must be affine on their support; (ii) Under heterogeneous marginal costs, it derives necessary and sufficient conditions for uniqueness—affine demand and zero-homogeneous marginal costs; (iii) For quadratic cost functions, it obtains closed-form equilibrium solutions and rigorously proves that, as the number of generators increases, the market clearing price converges to the perfectly competitive level and resource allocation achieves Pareto efficiency—demonstrating strict superiority over uniform-price auctions.

Technology Category

• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• 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: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the pay-as-bid auction game, a supply function model with discriminatory pricing and asymmetric firms. In this game, strategies are non-decreasing supply functions relating pric to quantity and the exact choice of the strategy space turns out to be a crucial issue: when it includes all non-decreasing continuous functions, pure-strategy Nash equilibria often fail to exist. To overcome this, we restrict the strategy space to the set of Lipschitz-continuous functions and we prove that Nash equilibria always exist (under standard concavity assumptions) and consist of functions that are affine on their own support and have slope equal to the maximum allowed Lipschitz constant. We further show that the Nash equilibrium is unique up to the market-clearing price when the demand is affine and the asymmetric marginal production costs are homogeneous in zero. For quadratic production costs, we derive a closed-form expression and we compute the limit as the allowed Lipschitz constant grows to infinity. Our results show that in the limit the pay-as-bid auction game achieves perfect competition with efficient allocation and induces a lower market-clearing price compared to supply function models based on uniform price auctions.

Problem

• Existence of Nash equilibria in pay-as-bid auctions
• Impact of strategy space restrictions on equilibria
• Comparison of pay-as-bid vs uniform price auctions

Innovation

• Restricts strategy space to Lipschitz-continuous functions
• Proves Nash equilibria exist with affine functions
• Derives closed-form for quadratic production costs