Mean Field Games without Rational Expectations

TL;DR

This paper develops a framework for mean field games without assuming rational expectations, especially in low-dimensional coupling scenarios, simplifying the analysis by avoiding complex master equations.

Contribution

It introduces a non-rational expectations formulation for MFGs with low-dimensional coupling, enabling simpler finite-dimensional solutions instead of complex master equations.

Findings

Non-rational expectations can replace rational expectations in MFGs.

In low-dimensional coupling MFGs, the master equation can be avoided.

An adaptive learning model exemplifies non-rational expectations in practice.

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, it becomes necessary to solve the "Master 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.