The Tragedy of the Commons in Multi-Population Resource Games

Yamin Vahmian; Keith Paarporn

cs.GTFeb 24, 2026

The Tragedy of the Commons in Multi-Population Resource Games

Authors: Yamin Vahmian, Keith Paarporn

Organizations: Department of Computer Science, University of Colorado, Colorado Springs.

Abstract

Self-optimizing behaviors can lead to outcomes where collective benefits are ultimately destroyed, a well-known phenomenon known as the ``tragedy of the commons". These scenarios are widely studied using game-theoretic approaches to analyze strategic agent decision-making. In this paper, we examine this phenomenon in a bi-level decision-making hierarchy, where low-level agents belong to multiple distinct populations, and high-level agents make decisions that impact the choices of the local populations they represent. We study strategic interactions in a context where the populations benefit from a common environmental resource that degrades with higher extractive efforts made by high-level agents. We characterize a unique symmetric Nash equilibrium in the high-level game, and investigate its consequences on the common resource. While the equilibrium resource level degrades as the number of populations grows large, there are instances where it does not become depleted. We identify such regions, as well as the regions where the resource does deplete.

Figures & tables

Fig. 1 : The shaded green region depicts the set of all policies

(\delta_{SP0},\delta_{RT0})

that are responsible, as defined as in Definition 1 .

Fig. 2 : The resource extraction game is a strategic-form game among

M

players who each represent one of the greedy populations. Player

i\in\mathcal{G}

chooses extraction rate

\alpha_{i}\geq 0

for its population, where higher rates degrade the resource.

System Parameters
$\delta_{SP0},\delta_{RT0}$	Responsible policy in deplete state (see Figure 1 )
$\delta_{TR1},\delta_{PS1}$	Responsible policy in replete state (fixed, both positive)
$\alpha,\theta$	Consumption, restoration rates of responsible population
Decision Variables
$\alpha_{i}$	Consumption rate of greedy population $i=1,\ldots,M$
$\bar{\alpha}$	Total consumption rate of greedy populations

TABLE I : Parameters and variables of the multi-population system

Panel of Figure 4 — Fig. 3 : (a) Resource level $R^{*}_{M}$ under the symmetric equilibrium (Theorem 4.1 ) over the responsible policy space for $M=6$ greedy populations, with parameters $\theta=1.0,\ \alpha=0.40,\ \delta_{TR1}=2.1,\ \delta_{PS1}=2.0$ . The red dashed line indicates the curve $C(0;\delta_{SP0})$ . Figures (b), (c), and (d) show the resource levels $R^{*}_{M}$ vs. the number of greedy populations $M$ for selected parameter values (shown as dots in subplot (a)). In (b), $\delta_{RT0}=0.8$ is above $C(0;\delta_{SP0})$ , meaning that $\alpha^{*}=\theta/M$ for all $M\geq 1$ , leading to a constant value for $R^{*}_{M}$ . In (c), $\delta_{RT0}=0.2>0$ is below $C(0;\delta_{SP0})$ . $R^{*}_{M}$ decreases initially then stabilizes at a positive constant (Proposition 5.1 ). In (d) $\delta_{RT0}=-1.0<0$ . $R^{*}_{M}$ monotonically decreases toward zero, indicating complete resource depletion.

Social Dilemmas

Social dilemmas study how individually beneficial choices can undermine cooperation, shared resources, and group welfare.

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Learning to Harvest Without Collapse in a Regenerative Commons: A Lagrangian Framework

Jose Tupayachi, Xueping Li, Soham Das

Oak Ridge National Laboratory Department of Industrial and Systems Engineering, University of Tennessee, Knoxville · Department of Industrial and Systems Engineering, University of Tennessee, Knoxville

Constrained MARL Multi-Agent Control Constrained RL Multi-Agent Reinforcement Learning

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The tragedy of the commons poses a multi-agent safety problem: reward-seeking agents can deplete a shared resource, and cooperation among its users does not itself specify how much must be preserved. We make preservation an explicit requirement by formulating a regenerative commons as a constrained Markov game or a constrained multi-agent MDP with a designer-specified depletion budget. We develop a nonstationary Lagrangian framework that constructs a policy sequence from solutions of unconstrained games or cooperative control problems. Extending earlier time-average constructions, we introduce average-epoch solution concepts for reset episodes with discounted rewards and terminal costs. We prove a reward-independent feasibility certificate, cooperative feasibility and approximate optimality against feasible policy mixtures, and an extension to unbiased sampled costs. For self-interested agents, a constrained Nash certificate quantifies the price-dispersion term introduced by deviations that redistribute budget across epochs. Under the stated assumptions on solver accuracy and multiplier updates, these results give constrained policy-sequence guarantees using solutions of unconstrained problems. Experiments with constrained IPPO and MAPPO in a Gordon-Schaefer fishery examine how depletion budgets shape stock retention, harvest rewards, and price adaptation.

Abstract

Social Dilemmas

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