Cooperation, Competition, and Common Pool Resources in Mean Field Games

TL;DR

This work extends mean-field game theory to CPR settings by introducing mixed mean field models that interpolate between fully non-cooperative and fully cooperative behavior at both the individual and population levels. It develops rigorous MF equilibrium concepts (MI-MFNE and MP-MFE), with Hamiltonian and forward-backward stochastic differential equation characterizations, and proves that MF equilibria yield approximate equilibria for large finite-agent games. The CPR extension yields extended state dynamics for resource stock $K_t$, and the fisheries application demonstrates existence and uniqueness of solutions via ODEs under small time, showing cooperators alleviate the tragedy of the commons. Numerically, varying altruism and population composition reveals that cooperation substantially improves CPR sustainability and mitigates TOTC, guiding decentralized design of incentives and regulation.

Abstract

Mean field games (MFGs) have been introduced to study Nash equilibria in very large population of self-interested agents. However, when applied to common pool resource (CPR) games, MFG equilibria lead to the so-called tragedy of the commons (TOTC). Empirical studies have shown that in many situations, TOTC does not materialize which hints at the fact that standard MFG models cannot explain the behavior of agents in CPR games. In this work, we study two models which incorporate a mix of cooperative and non-cooperative behaviors, either at the individual level or the population level. After defining these models, we study optimality conditions in the form of forward-backward stochastic differential equations and we prove that the mean field models provide approximate equilibria controls for corresponding finite-agent games. We then show an application to a model of fish stock management, for which the solution can be computed by solving systems of ordinary differential equations, which we prove to have a unique solution. Numerical results illustrate the impact of the level of cooperation at the individual and the population levels on the CPR.

Figures (4)

• Figure 1: Illustrative example. Evolution of the common pool resource (left), of the average change in effort (middle) and of the average effort (right). In the MFG, agents over-exploit the common pool resource.
• Figure 2: Mixed Individual MFG results under different model parameter, $\lambda$, choices. Left: CPR levels over time; Middle: Average control (i.e., average fishing effort change) in the population over time; Right: Average state (i.e., average fishing effort) over time.
• Figure 3: Mixed Population MFG results under different model parameter, $p$, choices. Left: CPR levels over time; Middle: Average control (i.e., average fishing effort change) for noncooperative (NC) and cooperative (C) fishers over time; Right: Average state (i.e., average fishing effort) for noncooperative (NC) and cooperative (C) fishers over time. Since $p=0$ and $p=1$ respectively correspond to fully cooperative and fully competitive populations, we only added the related lines in the middle and right subplots.
• Figure 4: Mixed Individual MFG vs. Mixed Population MFG results under different (but the same level of) model parameters. Left: CPR levels over time; Middle: Average control (i.e., average fishing effort change) in the fisher population over time; Right: Average state (i.e., average fishing effort) in the fisher population over time.

Theorems & Definitions (33)

• Definition 1
• Definition 2: MI-MFNE
• Remark 1
• Theorem 1: $\epsilon$-Nash for MI-MF
• Theorem 2: FBSDE for MI-MFNE
• Definition 3
• Definition 4: MP-MFE
• Remark 2
• Theorem 3: $\epsilon$-equilibrium for MP-MFE
• Theorem 4: FBSDE for MP-MFE