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Sustainable cooperation on the hybrid pollution-control game with heterogeneous players

Yilun Wu, Anna Tur, Peichen Ye

TL;DR

This work analyzes a three-player hybrid pollution-control differential game with two farsighted and one myopic actor under seasonal regime switches. It derives open-loop strategies under all feasible coalition structures using Pontryagin's Maximum Principle, proving that the pollution stock $z(t)$ converges to a unique hybrid limit cycle for each coalition, with the grand coalition minimizing pollution. A sustainably-cooperative optimality principle is developed within the imputation framework and a time-consistent imputation distribution procedure is proposed to sustain cooperation over time. The results have implications for international environmental agreements and multi-agent coalition formation in engineering systems, highlighting the value of time-consistent cooperation under regime shifts.

Abstract

This paper considers a hybrid pollution-control differential game with two farsighted players and one myopic player. Both the seasonal regime shifts in the state dynamics and the players' heterogeneous preferences are introduced into the model. The strategies under cooperative, noncooperative and partially cooperative scenarios are obtained by utilizing the Pontryagin's Maximum Principle. Under all feasible coalition structures, the convergence of the state variable is proved. A new sustainably--cooperative optimality principle is proposed according to the coalition structures, which belongs to the imputation set. The prerequisite for the existence of time-consistency in the sustainably-cooperative optimality principle is explicitly obtained. The seasonal imputation distribution procedure (IDP) is designed to maintain the time-consistentcy (dynamic stability) of cooperation over time.

Sustainable cooperation on the hybrid pollution-control game with heterogeneous players

TL;DR

This work analyzes a three-player hybrid pollution-control differential game with two farsighted and one myopic actor under seasonal regime switches. It derives open-loop strategies under all feasible coalition structures using Pontryagin's Maximum Principle, proving that the pollution stock converges to a unique hybrid limit cycle for each coalition, with the grand coalition minimizing pollution. A sustainably-cooperative optimality principle is developed within the imputation framework and a time-consistent imputation distribution procedure is proposed to sustain cooperation over time. The results have implications for international environmental agreements and multi-agent coalition formation in engineering systems, highlighting the value of time-consistent cooperation under regime shifts.

Abstract

This paper considers a hybrid pollution-control differential game with two farsighted players and one myopic player. Both the seasonal regime shifts in the state dynamics and the players' heterogeneous preferences are introduced into the model. The strategies under cooperative, noncooperative and partially cooperative scenarios are obtained by utilizing the Pontryagin's Maximum Principle. Under all feasible coalition structures, the convergence of the state variable is proved. A new sustainably--cooperative optimality principle is proposed according to the coalition structures, which belongs to the imputation set. The prerequisite for the existence of time-consistency in the sustainably-cooperative optimality principle is explicitly obtained. The seasonal imputation distribution procedure (IDP) is designed to maintain the time-consistentcy (dynamic stability) of cooperation over time.

Paper Structure

This paper contains 15 sections, 3 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Players' strategies under different coalition structures
  4. Cooperative scenario: $\pi_1=\{1,2,3\}$
  5. Noncooperative scenario: $\pi_2=\{ \{1\},\{2\},\{3\}\}$
  6. Partial-cooperative scenario
  7. Two farsighted players cooperate: $\pi_3=\{\{1,2\},\{3\}\}$
  8. Farsighted player $1$ cooperates with myopic player: $\pi_{41}=\{\{1,3\},\{2\} \}$
  9. Farsighted player $2$ cooperate with myopic player: $\pi_{42}=\{ \{2,3\}, \{1\} \}$
  10. Dynamics of the state under different coalition structures
  11. Stability of cooperation
  12. Sustainably-cooperative optimality principle
  13. Time-consistency of the sustainably-cooperative optimality principle
  14. Numerical illustration
  15. Conclusion

Key Result

Theorem 1

Under coalition structure $\pi =\pi_1,\ \pi_2,\ \pi_3,\ \pi_{41}, \ \pi_{42}$, for an arbitrary initial state $z^{\pi}(0)=z^{\pi}_0\ge 0$, the state variable $z^{\pi}(t)$ driven by $\{v_{i}^{\pi}\}_{1}^{3}$ exponentially converges to a unique hybrid limit cycle $\bar{z}^{\pi}(t)$ as $t\to \infty$.

Figures (2)

  • Figure 1: Strategies and the corresponding state under different coalition structures.
  • Figure 2: Payment to each player.

Theorems & Definitions (13)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Definition 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • ...and 3 more