Monotone Perfection

2025-09-01

AI Summary

This paper addresses the lack of trembling-hand robustness in standard monotone equilibria of Bayesian games. It introduces and formally defines the concept of *perfect monotone equilibrium*. The authors demonstrate that classical single-crossing and quasi-supermodularity conditions are insufficient to guarantee existence, and instead propose stronger *increasing differences* and *supermodularity* conditions—refining and extending the frameworks of Athey (2001), McAdams (2003), and Reny (2011). Using trembling-hand perturbation analysis, monotone comparative statics, and constructive counterexamples, they establish both sufficiency and necessity of the new conditions. The theory is validated across canonical models—including multi-unit auctions, first-price auctions, all-pay auctions, and Bertrand competition—confirming existence and applicability. By ensuring robustness to infinitesimal strategic errors, this work provides a more rigorous equilibrium foundation for robust mechanism design.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper introduces the concept of perfect monotone equilibrium in Bayesian games, which refines the standard monotone equilibrium by accounting for the possibility of unintended moves (trembling hand) and thereby enhancing robustness to small mistakes. We construct two counterexamples to demonstrate that the commonly used conditions in Athey (2001), McAdams (2003) and Reny (2011) - specifically, the single crossing condition and quasi-supermodularity - are insufficient to guarantee the existence of a perfect monotone equilibrium. Instead, we establish that the stronger conditions of increasing differences and supermodularity are required to ensure equilibrium existence. To illustrate the practical relevance of our findings, we apply our main results to multiunit auctions and further extend the analysis to first-price auctions, all-pay auctions, and Bertrand competitions.

Problem

• Refining monotone equilibrium with trembling hand robustness
• Demonstrating insufficiency of single crossing and quasi-supermodularity conditions
• Establishing stronger supermodularity requirements for equilibrium existence

Innovation

• Introduces perfect monotone equilibrium concept
• Replaces single crossing with stronger supermodularity conditions
• Applies results to multiunit and first-price auctions