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The Tragedy of the Commons in Multi-Population Resource Games

Yamin Vahmian, Keith Paarporn

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.

The Tragedy of the Commons in Multi-Population Resource Games

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.
Paper Structure (6 sections, 7 theorems, 32 equations, 6 figures, 1 table)

This paper contains 6 sections, 7 theorems, 32 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary: single population
  3. Model: Extraction game with multiple populations
  4. Analysis and Main Results
  5. Impact of equilibrium extraction
  6. Conclusion

Key Result

Theorem 2.1

The environmental policy $(\delta_{SP0},\delta_{RT0})$ determines the asymptotic properties of eq:1pop as follows. 1) Resource sustained: If $(\delta_{SP0},\delta_{RT0})$ satisfies then the fixed point $(x^*,n^*) = (\frac{\alpha}{\alpha + \theta}, \frac{g(x^*,0)}{-\frac{\partial g}{\partial n}(x^*)}) \in (0,1)\times[0,1)$ is the only asymptotically stable fixed point in the system. Here, $n^* = 0

Figures (6)

  • Figure 1: The shaded green region depicts the set of all policies $(\delta_{SP0},\delta_{RT0})$ that are responsible, as defined as in Definition \ref{['def:responsible']}.
  • Figure 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.
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 2.1: adapted from weitz2016oscillating
  • Definition 1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.1
  • ...and 3 more