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The main objective of this work is to give conditions for the existence of Nash equilibria for a nonzero-sum constrained stochastic differential game with additive structure and Markovian switchings. In this type of game, each player is interested only in maximizing their finite-horizon total payoff when an additional cost function of the same type is required to be dominated above by another function (in particular, by a constant). The dynamic system for this game is controlled by two players and evolves according to a Markov-modulated diffusion (also known as switching diffusions or piecewise diffusion or diffusion with Markovian switchings). Given that, each player has to solve an optimization problem with constraints. The existence of a Nash equilibrium is thus proved using the Lagrange multipliers approach combined with standard dynamic programming arguments. The Lagrange approach allows the transformation of a constrained game into an unconstrained game. Therefore, this work gives conditions under which a Nash equilibrium for the unconstrained stochastic differential game is also a Nash equilibrium for the corresponding nonzero-sum constrained stochastic differential game. The theory developed here is illustrated by a pollution accumulation problem with two players. Therein, the evolution is governed by a linear stochastic differential equation with Markovian switching, and the decay pollution rate depends on a Markov chain.

In this paper, we consider the continuous-time nonzero-sum constrained stochastic games with the discounted cost criteria. The state space is denumerable and the action space of each player is a general Polish space, while the transition rates and cost functions are allowed to be unbounded from below and from above. The strategies for each player may be history-dependent and randomized. Models with these features seemingly have not been handled in the previous literature. By constructing a sequence of continuous-time finite-state game models to approximate the original denumerable-state game model, we prove the existence of constrained Nash equilibria for the constrained games with denumerable states. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete-time constrained average stochastic games with independent state processes. The state space of each player is denumerable and one-stage cost functions can be unbounded. In these game models, each player chooses an action each time which influences the transition probability of a Markov chain controlled only by this player. Moreover, each player needs to pay some costs which depend on the actions of all the players. First, we give an existence condition of stationary constrained Nash equilibria based on the technique of average occupation measures and the best response linear program. Then, combining the best response linear program and duality program, we present a non-convex mathematic program and prove that each stationary Nash equilibrium is a global minimizer of this mathematic program. Finally, a controlled wireless network is presented to illustrate our main results.

In this paper, we consider discrete-time $$N$$ -person constrained stochastic games with discounted cost criteria. The state space is denumerable and the action space is a Borel set, while the cost functions are admitted to be unbounded from below and above. Under suitable conditions weaker than those in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261-285, ) for bounded cost functions, we also show the existence of a Nash equilibrium for the constrained games by introducing two approximations. The first one, which is as in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261-285, ), is to construct a sequence of finite games to approximate a (constrained) auxiliary game with an initial distribution that is concentrated on a finite set. However, without hypotheses of bounded costs as in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261-285, ), we also establish the existence of a Nash equilibrium for the auxiliary game with unbounded costs by developing more shaper error bounds of the approximation. The second one, which is new, is to construct a sequence of the auxiliary-type games above and prove that the limit of the sequence of Nash equilibria for the auxiliary-type games is a Nash equilibrium for the original constrained games. Our results are illustrated by a controlled queueing system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete-time constrained average stochastic games with independent state processes. The state space of each player is denumerable and one-stage cost functions can be unbounded. In these game models, each player chooses an action each time which influences the transition probability of a Markov chain controlled only by this player. Moreover, each player needs to pay some costs which depend on the actions of all the players. First, we give an existence condition of stationary constrained Nash equilibria based on the technique of average occupation measures and the best response linear program. Then, combining the best response linear program and duality program, we present a non-convex mathematic program and prove that each stationary Nash equilibrium is a global minimizer of this mathematic program. Finally, a controlled wireless network is presented to illustrate our main results. [ABSTRACT FROM AUTHOR]