Subgame Consistent Research Articles

Two-person cooperative uncertain differential game with transferable payoffs

Uncertain differential game models conflicts and interests among players in the context of an uncertain dynamic system. However, cooperative behavior in uncertain differential game is an unexplored terrain. This paper proposes a spectrum of a cooperative uncertain differential game with transferrable payoffs. First, group rationality is achieved by maximizing the coalitional payoff through an uncertain optimal control method. Second, the concept of imputation is introduced as a solution, and a stability condition called subgame consistency condition for an imputation is proposed as well. Furthermore, the derivation of payoff distribution procedure for subgame consistent imputation is discussed. In addition, two optimal principles, i.e., Nash bargaining solution and Shapley value, are extended to this kind of model and are proved to be subgame consistent imputations, and their payoff distribution procedures are derived analytically. Finally, a two-person resource extraction game is studied for illustrating purpose.

Subgame Consistent Cooperative Behavior in an Extensive form Game with Chance Moves

We design a mechanism of the players’ sustainable cooperation in multistage n-person game in the extensive form with chance moves. When the players agreed to cooperate in a dynamic game they have to ensure time consistency of the long-term cooperative agreement. We provide the players’ rank based (PRB) algorithm for choosing a unique cooperative strategy profile and prove that corresponding optimal bundle of cooperative strategies satisfies time consistency, that is, at every subgame along the optimal game evolution a part of each original cooperative trajectory belongs to the subgame optimal bundle. We propose a refinement of the backwards induction procedure based on the players’ attitude vectors to find a unique subgame perfect equilibrium and use this algorithm to calculate a characteristic function. Finally, to ensure the sustainability of the cooperative agreement in a multistage game we employ the imputation distribution procedure (IDP) based approach, that is, we design an appropriate payment schedule to redistribute each player’s optimal payoff along the optimal bundle of cooperative trajectories. We extend the subgame consistency notion to extensive-form games with chance moves and prove that incremental IDP satisfies subgame consistency, subgame efficiency and balance condition. An example of a 3-person multistage game is provided to illustrate the proposed cooperation mechanism.

Cooperative Dynamic Games with Control Lags

Controls with lags are control strategies with prolonged effects lasting for more than one stage of the game after the controls had been executed. Lags in controls yielding adverse effects often make the negative impacts more significant. Cooperation provides an effective means to alleviate the problem and obtains an optimal solution. This paper extends the existing paradigm in cooperative dynamic games by allowing the existence of controls with lag effects on the players’ payoffs in subsequent stages. A novel dynamic optimization theorem with control lags is developed to derive the Pareto optimal cooperative controls. Subgame consistent solutions are derived to ensure sustainable cooperation. In particular, subgame consistency guarantees that the optimality principle agreed upon at the outset will remain effective throughout the game and, hence, there is no incentive for any player to deviate from cooperation scheme. A procedure for imputation distribution is provided to formulate a dynamically stable cooperative scheme under control lags. An application in cooperative environmental management is presented. This is the first time that cooperative dynamic games with control lags are studied.

A Cooperative Dynamic Environmental Game of Subgame Consistent Clean Technology Development

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.

Subgame consistent cooperative solution for NTU dynamic games via variable weights

In cooperative dynamic games a stringent condition–subgame consistency–is required for a dynamically stable solution. In particular, a cooperative solution is subgame consistent if the optimality principle agreed upon at the outset remains in effect in any subgame starting at a later stage with a state brought about by prior optimal behavior. Hence the players do not have incentives to deviate from the previously adopted optimal behavior. For the first time, subgame consistent solutions in cooperative dynamic games with non-transferable payoffs/utility (NTU) using a variable payoff weights scheme is analyzed. A solution mechanism for characterizing subgame consistent solutions is derived. The use of a variable payoff weights scheme allows the derivation of subgame consistent solutions under a wide range of optimality principles. The analysis widens the application of NTU cooperative dynamic games through the provision of a dynamically stable solution.

SUBGAME CONSISTENT COOPERATIVE SOLUTIONS FOR RANDOMLY FURCATING STOCHASTIC DYNAMIC GAMES WITH UNCERTAIN HORIZON

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.

Subgame Consistent Cooperative Provision of Public Goods

The provision of public goods constitutes a classic case of market failure which calls for cooperative optimization. However, cooperation cannot be sustainable unless there is guarantee that the agreed-upon optimality principle can be maintained throughout the planning duration. This paper derives subgame consistent cooperative solutions for public goods provision by asymmetric agents with transferable payoffs in a stochastic differential game framework. In particular, subgame consistency ensures that as the game proceeds agents are guided by the same optimality principle, and hence they do not possess incentives to deviate from the previously adopted optimal behavior. A “payoff distribution procedure” leading to subgame-consistent solutions is derived and illustrative examples are presented. This is the first time that subgame consistent cooperative provision of public goods with transferable payoffs is analyzed.

Subgame-consistent cooperative solutions in randomly furcating stochastic dynamic games

In the analysis of 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 subgame starting at a later time with a 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. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structures are derived. In computer modeling and operations research discrete-time analysis often proved to be more applicable and compatible with actual data than continuous-time analysis. This is the first time that a subgame consistent solution for randomly-furcating stochastic dynamic games has been obtained. It widens the application of cooperative dynamic game theory to discrete-time problems where the evolution of the state and future payoff structures are not known with certainty.

SUBGAME CONSISTENT SOLUTION FOR COOPERATIVE STOCHASTIC DYNAMIC GAMES WITH RANDOM HORIZON

In cooperative stochastic dynamic games a stringent condition — that of subgame consistency — is required for a dynamically stable cooperative solution. In particular, a cooperative solution is subgame consistent if an extension of the solution policy to a subgame starting at a later time with a state brought about by prior optimal behavior would remain optimal. This paper extends subgame consistent solutions to cooperative stochastic dynamic (discrete-time) games with random horizon. In the analysis new forms of the stochastic Bellman equation and the stochastic Isaacs–Bellman equation in discrete time are derived. Subgame consistent cooperative solutions are obtained for stochastic dynamic games. Analytically tractable payoff distribution mechanisms which lead to the realization of these solutions are developed. This is the first time that subgame consistent solutions for cooperative stochastic dynamic games with random horizon are presented.

Subgame Consistent Cooperative Solution of Dynamic Games with Random Horizon

In cooperative dynamic games, a stringent condition—that of subgame consistency—is required for a dynamically stable cooperative solution. In particular, under a subgame-consistent cooperative solution an extension of the solution policy to a subgame starting at a later time with a state brought about by prior optimal behavior will remain optimal. This paper extends subgame-consistent solutions to dynamic (discrete-time) cooperative games with random horizon. In the analysis, new forms of the Bellman equation and the Isaacs–Bellman equation in discrete-time are derived. Subgame-consistent cooperative solutions are obtained for this class of dynamic games. Analytically tractable payoff distribution mechanisms, which lead to the realization of these solutions, are developed. This is the first time that subgame-consistent solutions for cooperative dynamic games with random horizon are presented.

International Cooperation on Pollution Control Between Asymmetric Countries

The purpose of this study was to investigate international cooperation on pollution control between asymmetric countries under uncertainty. We considered any two countries that are asymmetric with respect to their technological level and the amount of environmental damage they incur. Production processes result in the emission of pollutants that are added to the existing pollution stock common to both countries. The stock of pollution evolves stochastically, according to the geometric Brownian motion. The governments of the two countries set their pollutant emission levels at the Markov perfect equilibrium to maximize their expected net present value of social welfare. In addition, they set their pollutant emission levels at cooperative equilibrium to maximize the sum of their expected net present value of social welfare.In cooperative stochastic differential games, a credible cooperative agreement must be subgame consistent. Subgame consistency ensures extension of an optimal policy to a later starting time, and any possible state brought about by prior optimal behaviour of the governments remains optimal.We considered a cooperative game in which the governments of two countries agree to maximize and divide the sum of their expected net present value of social welfare in a way that shares the gain from their cooperation proportional to the relative sizes of their expected non-cooperative net present value of social welfare at every instant of time. In conclusion, the country with a higher technological level or that incurs lower environmental damage obtains a larger instantaneous payoff at the subgame consistent solution.JEL Classification: F18, L13, Q58

Subgame Consistent Solutions for Cooperative Stochastic Dynamic Games

In cooperative games over time with uncertainty, a stringent condition (subgame consistency) is required for a dynamically stable solution. In particular, 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 derives an analytically tractable payoff distribution procedure leading to the realization of subgame consistent solutions in cooperative stochastic dynamic games. This is the first time that subgame consistent solutions in discrete-time dynamic games under uncertainty are provided.

A cooperative stochastic differential game of transboundary industrial pollution

Though cooperation in environmental control holds out the best promise of effective actions, limited success has been observed because existing multinational joint initiatives fail to satisfy the property of subgame consistency. A cooperative solution is subgame consistent if the solution optimality principle is maintained in any subgame which starts at a later time with any feasible state brought about by prior optimal behaviors. This paper presents a cooperative stochastic differential game of transboundary industrial pollution with two novel features. The first feature is that industrial production creates short-term local impacts and long-term global impacts on the environment. Secondly, a subgame consistent cooperative solution is derived in this stochastic differential game together with a payment distribution mechanism that supports the subgame consistent solution. This is the first time that pollution management is analyzed in a cooperative stochastic differential game framework under these novel features.

Subgame Consistent Solutions of a Cooperative Stochastic Differential Game with Nontransferable Payoffs

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by the prior optimal behavior of the players would remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In this paper, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs. The previously intractable subgame consistent solution for games with nontransferable payoffs is rendered tractable.

Subgame Consistent Cooperative Solutions in Stochastic Differential Games

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that: (i) the extension of the solution policy to a later starting time and to any possible state brought about by the prior optimal behavior of the players would remain optimal; (ii) all players do not have incentive to deviate from the initial plan. In this paper, we develop a mechanism for the derivation of the payoff distribution procedures of subgame consistent solutions in stochastic differential games with transferable payoffs. The payoff distribution procedure of the subgame consistent solution can be identified analytically under different optimality principles. Demonstration of the use of the technique for specific optimality principles is shown with an explicitly solvable game. For the first time, analytically tractable solutions of cooperative stochastic differential games with subgame consistency are derived.