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Cooperative stochastic differential games constitute a highly complex form of decision making under uncertainty. In particular, interactions between strategic behaviors, dynamic evolution, stochastic elements and solution agreement have to be considered simultaneously. This complexity leads to great difficulties in the derivation of dynamically stable solutions. Despite urgent calls for cooperation in the global economy, the lack of formal analyses has precluded rigorous analysis of this problem. In this paper, mechanisms for the derivation of dynamically stable solutions to cooperative stochastic differential games are presented. Games with transferable payoffs and those with nontransferable payoffs are considered. Numerical illustrations are also provided.

This paper presents dynamically stable solutions to a class of cooperative differential games of pollution management in collaborative abatement with uncertain future payoffs. Collaborative abatement leads to a decrease in cost due to reduction in duplicated efforts in the process of joint development and brings about an enhancement in the effectiveness of abatement activities via the sharing of knowledge from individual nation's research. It is believed by many researchers to be the key to effective pollution reduction. Uncertainties in future economic performance are prevalent in fast developing countries like China, Brazil and India. This type of uncertainties often hinders the reaching of cooperative agreements in joint pollution control initiatives. In dynamic cooperative games, a credible cooperative agreement has to be dynamically consistent. For dynamic consistency to hold, the agreed upon optimality principle must remain in effect at any instant of time throughout the game along the optimal state trajectory contingent upon the realization of specific random events. In this analysis, dynamically consistent cooperative solutions and analytically tractable payoff distribution procedures contingent upon specific random events are derived. This approach widens the application of cooperative differential game theory to environmental problems where future payoffs are not known with certainty.

In cooperative stochastic dynamic games a stringent condition — subgame consistency — is required for a dynamically stable solution. A cooperative solution is subgame consistent if an extension of the solution policy to a situation with a later starting time and any feasible state brought about by prior optimal behavior would remain optimal. This paper considers subgame consistent cooperative solutions in randomly furcating stochastic discrete-time dynamic games with uncertain horizon. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structure are derived. Novel forms of Bellman equations and Hamilton–Jacobi–Bellman equations for solving intertemporal problems with randomly furcating payoffs and random horizon are developed. This is the first time that subgame consistent solution for games with stochastic dynamics, uncertain future payoff structures and random horizon is obtained.

Cooperative adoption and development of clean technology play a key role to effectively solving the continual worsening industrial pollution problem. For cooperation over time to be credible, a subgame consistency solution which requires the agreed-upon optimality principle to remain in effect throughout the collaboration duration has to hold. In this paper, we present a cooperative dynamic game of collaborative environmental management with clean technology development. A subgame consistent cooperative scheme is derived. It is the first time that cooperative dynamic environmental games with clean technology development are analyzed. Given that there exist discrete choices of production techniques and switching to clean technology brings about cost savings and improved effectiveness, the group optimal solution cannot be obtained with standard differentiable optimization techniques. To overcome this problem the joint optimal solutions for all the possible patterns of production techniques are computed and the pattern with the highest joint payoff is then selected. The analysis widens the scope of study in collaborative environmental management.

Cooperative adoption and development of clean technology play a key role to effectively solving the continual worsening industrial pollution problem. For cooperation over time to be credible, a subgame consistency solution which requires the agreed-upon optimality principle to remain in effect throughout the collaboration duration has to hold. In this chapter, we present a cooperative dynamic game of collaborative environmental management with clean technology development. A subgame consistent cooperative scheme is derived. It is the first time that cooperative dynamic environmental games with clean technology development are analyzed. Given that there exist discrete choices of production techniques and switching to clean technology brings about cost savings and improved effectiveness, the group optimal solution cannot be obtained with standard differentiable optimization techniques. To overcome this problem the joint optimal solutions for all the possible patterns of production techniques are computed and the pattern with the highest joint payoff is then selected. The analysis widens the scope of study in collaborative environmental management.