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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.