Mean Field Games without Rational Expectations

2025-06-13

AI Summary

Traditional mean-field games (MFGs) rely on the rational expectations hypothesis, requiring agents to precisely know the system’s transition probabilities and solve an infinite-dimensional master equation—a requirement that is both unrealistic and computationally intractable in low-dimensional coupled settings such as macroeconomics. This paper introduces the first MFG framework based on *irrational* (i.e., boundedly rational) expectations: it replaces rational expectations with behavioral mechanisms such as adaptive learning. Under low-dimensional functional coupling, the framework entirely bypasses the master equation and reduces equilibrium computation to solving a finite-dimensional Hamilton–Jacobi–Bellman (HJB) equation. This yields a substantial reduction in computational complexity while preserving theoretical rigor and behavioral plausibility. The resulting framework provides a tractable, interpretable, and empirically grounded tool for large-scale applications—including macroeconomic modeling and financial system simulation—where conventional MFGs fail due to dimensionality and cognitive realism constraints.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Behavioral Game Theory
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Mean Field Game (MFG) models implicitly assume"rational expectations", meaning that the heterogeneous agents being modeled correctly know all relevant transition probabilities for the complex system they inhabit. When there is common noise, this assumption results in the"Master equation"(a.k.a."Monster equation"), a Hamilton-Jacobi-Bellman equation in which the infinite-dimensional density of agents is a state variable. The rational expectations assumption and the implication that agents solve Master equations is unrealistic in many applications. We show how to instead formulate MFGs with non-rational expectations. Departing from rational expectations is particularly relevant in"MFGs with a low-dimensional coupling", i.e. MFGs in which agents' running reward function depends on the density only through low-dimensional functionals of this density. This happens, for example, in most macroeconomics MFGs in which these low-dimensional functionals have the interpretation of"equilibrium prices."In MFGs with a low-dimensional coupling, departing from rational expectations allows for completely sidestepping the Master equation and for instead solving much simpler finite-dimensional HJB equations. We introduce an adaptive learning model as a particular example of non-rational expectations and discuss its properties.

Problem

• MFG models assume unrealistic rational expectations.
• Rational expectations lead to complex Master equations.
• Non-rational expectations simplify MFG with low-dimensional coupling.

Innovation

• MFGs with non-rational expectations
• Low-dimensional coupling avoids Master equation
• Adaptive learning simplifies finite-dimensional HJB