Existence and non-existence for time-dependent mean field games with strong aggregation

TL;DR

This paper studies the existence of solutions in time-dependent mean-field games with strong aggregation effects, showing solutions exist for small coupling strength but may fail for large coupling and long time horizons.

Contribution

It provides new existence results for classical solutions in quadratic mean-field games with strongly decreasing couplings, highlighting the impact of coupling strength and time horizon.

Findings

Existence of solutions for small coupling parameter $\sigma$.

Non-existence results for large $\sigma$ and long time horizons.

Conditions under which classical solutions fail to exist.

Abstract

We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $-\sigma m^\alpha$, $\alpha \ge 2/N$, where $m$ is the population density and $N$ is the dimension of the state space. We prove the existence of solutions under the assumption that $\sigma$ is small enough. For large $\sigma$, we show that existence may fail whenever the time horizon $T$ is large.