Two competing populations with a common environmental resource

TL;DR

This paper extends feedback-evolving games to a two-population setting sharing a common environmental resource, introducing a responsible population that sustains the resource and an irresponsible population that may exploit it. Using coupled replicator dynamics and tipping-point resource dynamics, it derives stability conditions, identifies when the resource can persist or collapses, and computes the irresponsible population’s optimal consumption rate under those conditions. A hierarchical incentive design is analyzed via sensitivity to policy changes, showing that mutual cooperation incentives generally outperform unilateral ones in promoting resource longevity. The results offer a framework for hierarchical environmental policy in multi-population contexts and highlight the importance of coordinated incentives for resource sustainability.

Abstract

Feedback-evolving games is a framework that models the co-evolution between payoff functions and an environmental state. It serves as a useful tool to analyze many social dilemmas such as natural resource consumption, behaviors in epidemics, and the evolution of biological populations. However, it has primarily focused on the dynamics of a single population of agents. In this paper, we consider the impact of two populations of agents that share a common environmental resource. We focus on a scenario where individuals in one population are governed by an environmentally ``responsible" incentive policy, and individuals in the other population are environmentally ``irresponsible". An analysis on the asymptotic stability of the coupled system is provided, and conditions for which the resource collapses are identified. We then derive consumption rates for the irresponsible population that optimally exploit the environmental resource, and analyze how incentives should be allocated to the responsible population that most effectively promote the environment via a sensitivity analysis.

Figures (4)

• Figure 1: Diagram of our two-population model. Activities from both populations impact the shared environmental state, $n$. The rates $\theta_i$, $\alpha_i$ denote the restoration and degradation rates from the activities of population $i$, respectively.
• Figure 2: The set of all sustainable policies for population 1 is shown as the blue region. A sustainable policy maintains a stable, nonzero resource in the absence of other populations (Theorem \ref{['thm:PNAS']}). This region is the focus of Assumptions \ref{['assume:dgdn']} and \ref{['assume:responsible']}.
• Figure 3: (Left) This surface plot shows the resource level that results from the optimal consumption rate $\alpha_2^*$ detailed in Theorem \ref{['thm:consume']}. The red line indicates the bottom border of the set of feasible policies for population 1. We observe that $R(\alpha_2^*)$ is increasing in both $\delta_{RT0}^1$ and $\delta_{SP0}^1$. (Center) The optimal consumption rate detailed in Theorem \ref{['thm:consume']}. (Right) An example of the utility function $U_2(\alpha_2)$ under policy $(\delta_{SP0}^1,\delta_{RT0}^1) = (3,-0.5)$. For all simulations, the parameter values are: $\delta_{TR1} = 10$, $\delta_{PS1} = 6$, $\theta_1 = 0.75$, $\alpha_1 = 1$, $\epsilon = 0.1$.
• Figure 4: Sensitivity ratios. This series of plots shows the sensitivity ratio $\rho$ as the population 1 policy $(\delta_{SP0}^1,\delta_{RT0}^1)$ varies. The red lines indicate the boundaries of the set of feasible policies for population 1. In all plots, we observe the ratio is highest near the curve $C(\delta_{SP0}^1)$ (dashed red line), i.e. $u_s$ becomes effective, and indeed can be more effective than $u_r$ for lower values of $\delta_{PS1}^1$ (left and center plots). However, $u_s$ is generally not as effective relative to $u_r$. For higher values of $\delta_{PS1}^1$, $u_s$ will never be as effective as $u_r$, i.e. $\rho < 1$ for all policies (right plot). For all simulations, the parameter values are: $\delta_{TR1} = 10$, $\theta_1 = 0.75$, $\alpha_1 = 1$, $\epsilon = 0.1$.

Theorems & Definitions (6)

• Theorem 2.1: adapted from weitz2016oscillating
• Theorem 3.1
• Theorem 4.1: Optimal consumption rate
• Proposition 5.1
• proof : Proof of Theorem \ref{['thm:stability']}
• proof : Proof of Theorem \ref{['thm:consume']}