Testing for Monotone Equilibrium Strategies in Games of Incomplete Information

2026-04-07

AI Summary

This study addresses the problem of testing monotonicity of Bayesian Nash equilibrium strategies with respect to unobserved types in games of incomplete information. Under the assumption of symmetric independent private values, it establishes an equivalence between the monotonicity of unobserved strategies and that of identified pseudo-inverse strategies derived from observable behavior. The paper develops a unified testing framework by reformulating the problem as a system of unconditional moment inequalities that depend solely on observable data, flexibly accommodating covariates and heterogeneity across games. To handle the resulting infinite-dimensional inequality constraints, the authors propose a joint test based on a Cramér–von Mises–type statistic with critical values obtained via the bootstrap. Monte Carlo simulations demonstrate favorable finite-sample performance, and the method is successfully applied to procurement auction data to detect collusive bidding behavior.

Abstract

This paper develops a unified framework for testing monotonicity of Bayesian Nash equilibrium strategies in unobserved types in games of incomplete information. We show that, under symmetric independent private types, monotonicity of differentiable equilibrium strategies is equivalent to monotonicity of a quasi-inverse strategy identified from observed actions. This allows the problem to be reformulated as testing a countable set of moment inequalities involving unconditional expectations. We propose a Cramer-von Mises-type statistic with bootstrap critical values. The method accommodates covariates and game heterogeneity. Monte Carlo simulations demonstrate finite-sample performance, and an application to procurement auctions illustrates cartel detection.

Problem

• monotonicity
• Bayesian Nash equilibrium
• incomplete information games
• unobserved types
• moment inequalities

Innovation

• monotonicity testing
• Bayesian Nash equilibrium
• moment inequalities
• Cramér–von Mises statistic
• unobserved types