How can the tragedy of the commons be prevented?: Introducing Linear Quadratic Mixed Mean Field Games

TL;DR

This paper introduces two linear quadratic mean field game models incorporating altruism to prevent the tragedy of the commons, providing explicit solutions and proving existence and uniqueness of equilibria.

Contribution

It develops novel mixed altruistic and non-cooperative mean field game models within a linear quadratic framework, with explicit solutions and theoretical guarantees.

Findings

Explicit solutions via ordinary differential equations

Existence and uniqueness of equilibria proven

Models demonstrate how altruism can prevent tragedy of the commons

Abstract

In a regular mean field game (MFG), the agents are assumed to be insignificant, they do not realize their effect on the population level and this may result in a phenomenon coined as the Tragedy of the Commons by the economists. However, in real life this phenomenon is often avoided thanks to the underlying altruistic behavior of (all or some of the) agents. Motivated by this observation, we introduce and analyze two different mean field models to include altruism in the decision making of agents. In the first model, mixed individual MFGs, there are infinitely many agents who are partially altruistic (i.e., they behave partially cooperatively) and partially non-cooperative. In the second model, mixed population MFGs, one part of the population behaves cooperatively and the remaining agents behave non-cooperatively. Both models are introduced in a general linear quadratic framework for which we characterize the equilibrium via forward backward stochastic differential equations. Furthermore, we give explicit solutions in terms of ordinary differential equations, and prove the existence and uniqueness results.