Nash without Numbers: A Social Choice Approach to Mixed Equilibria in Context-Ordinal Games

2026-05-08

AI Summary

This work addresses the limitation of classical Nash equilibrium, which relies on precise cardinal utilities, despite the fact that players’ preferences in real-world settings are often available only in ordinal form. The paper introduces, for the first time, social choice theory into game-theoretic analysis under a setting where agents express context-dependent ordinal preferences. It proposes a novel solution concept—contextual ordinal Nash equilibrium—and constructs a mixed-strategy equilibrium framework that dispenses entirely with cardinal utility assumptions. By integrating preference aggregation, regularization, and approximation techniques, the authors establish the existence of such equilibria under mild conditions, develop a learnable approximation algorithm, and empirically validate its applicability and effectiveness using human-subject experimental data on real ordinal preferences.

Abstract

Nash equilibrium serves as a fundamental mathematical tool in economics and game theory. However, it classically assumes knowledge of player utilities, whereas economics generally regards preferences as more fundamental. To leverage equilibrium analysis in strategic scenarios, one must first elicit numerical utilities consistent with player preferences, a delicate and time-consuming process. In this work, we forgo precise utilities and generalize the Nash equilibrium to a setting where we only assume a player is capable of providing an ordinal ranking of their actions within the context of other players' joint actions. The key technical challenge is to rethink the definition of a best-response. While the classical definition identifies actions maximizing expected payoff, we naturally look towards social choice theory for how to aggregate preferences to identify the most preferred actions. We define this generalized notion of a context-ordinal Nash equilibrium, establish its existence under mild conditions on aggregation methods, introduce notions of regularization, approximation, and regret, explore complexity for simple settings, and develop learning rules for computing such equilibria. In doing so, we provide a generalization of Nash equilibrium and demonstrate its direct applicability to elicited preferences in human experiments.

Problem

• Nash equilibrium
• ordinal preferences
• context-ordinal games
• social choice theory
• utility elicitation

Innovation

• context-ordinal games
• Nash equilibrium
• social choice theory
• preference aggregation
• ordinal preferences