An [imaginary time] Schr\"odinger approach to mean field games

TL;DR

This paper explores a novel approach to mean field games by drawing analogies with non-linear Schrödinger equations, enabling the transfer of physics-based techniques to socio-economic modeling.

Contribution

It introduces a Schrödinger-inspired framework for mean field games, connecting physics and economics to facilitate new analytical methods.

Findings

Established a mathematical analogy between MFG and non-linear Schrödinger equations

Demonstrated transfer of physics techniques to analyze MFGs

Provided a detailed example of coordinated players in the MFG context

Abstract

Mean Field Games (MFG) provide a theoretical frame to model socio-economic systems. In this letter, we study a particular class of MFG which shows strong analogies with the {\em non-linear Schr\"odinger and Gross-Pitaevski equations} introduced in physics to describe a variety of physical phenomena ranging from deep-water waves to interacting bosons. Using this bridge many results and techniques developed along the years in the latter context can be transferred to the former. As an illustration, we study in some details an example in which the "players" in the mean field game are under a strong incentive to coordinate themselves.