A mean field game approach for a class of linear quadratic discrete choice problems with congestion avoidance

Abstract

This paper approaches large multi-agent dynamic discrete choice problems, such as those arising in crowd evacuation or micro-robotic-based exploration of unknown terrain, via a linear quadratic mean field games framework (LQ-MFG). Two particular features distinguish the proposed LQ-MFG: (i) agents have to reach within a fixed finite time one of a predefined set of possible destinations, which depend on their initial position; (ii) agent running costs can become negative as agents accrue a reward the further they remain from other agents aiming for the same set of destinations, so as to simulate crowd avoidance with limited surroundings awareness. The desirable tractability of the LQ-MFG setup must be balanced in this case by the fact that feature (ii) of the model can lead to agents escaping to infinity in finite time. An upper bound on the time horizon is derived to guarantee that the finite escape time behavior is avoided. The existence of a Nash equilibrium for the infinite population is then established, and it is shown that this equilibrium remains ɛ-Nash for the more realistic case of a large but finite population of agents. Finally, the model behavior is explored via simulations for different scenarios.