Cross-Validation Equilibrium

2026-06-10

AI Summary

This work addresses the challenge that arises when players delegate belief formation to machine learning models: the endogeneity of training data induces mutual dependence between predictions and strategic behavior, rendering traditional equilibrium concepts inadequate. To capture this interaction, the paper introduces the Cross-Validated Equilibrium (CVE) framework in static Bayesian games, wherein each player’s agent selects a predictive model that minimizes out-of-sample squared error using endogenously generated data and forms beliefs accordingly to best respond. By integrating the cross-validation principle from machine learning into game theory, the framework reveals novel phenomena such as multiplicity of equilibria driven by endogenous model selection. The CVE successfully explains behavioral patterns unaccounted for by conventional models in settings including jury voting, speculative betting, and linear-quadratic payoff games, and identifies multiple equilibria in team effort scenarios.

Abstract

We study strategic interaction when players delegate belief formation to predictive machine learning (ML). In a static Bayesian game, each player's ML agent predicts a payoff-relevant outcome variable as a function of the player's type. The ML agent's training sample is endogenous: it is drawn from the outcome distribution generated by players' ML-guided behavior. In Cross-Validation Equilibrium (CVE), each player's ML agent selects a predictive model to minimize expected out-of-sample squared error, given its realized training sample, and each player best-replies to the belief generated by the model her ML agent selected. We analyze CVE and relate it to other equilibrium concepts. We apply CVE to jury voting, speculative betting, and games with linear-quadratic payoffs. E.g., in a team-effort game, endogenous model selection can give rise to multiple equilibria.

Problem

• strategic interaction
• machine learning
• Bayesian game
• endogenous training data
• belief formation

Innovation

• Cross-Validation Equilibrium
• machine learning delegation
• endogenous training data
• out-of-sample prediction
• Bayesian games