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

TL;DR

This paper generalizes Nash equilibrium to settings where only ordinal preferences are available, avoiding the need for precise utility elicitation, and explores its theoretical properties and practical computation.

Contribution

It introduces a new concept of context-ordinal Nash equilibrium, establishes its existence, and develops algorithms for learning and computing these equilibria.

Findings

Existence of context-ordinal Nash equilibria under mild conditions

Development of learning rules for computing equilibria

Application to human preference elicitation experiments

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.