Existence Of Open-loop Nash Equilibrium Research Articles

New results on the existence of open loop Nash equilibria in discrete time dynamic games via generalized Nash games

We address the problem of finding conditions which guarantee the existence of open-loop Nash equilibria in discrete time dynamic games (DTDGs). A classical approach to DTDGs involves analyzing the problem using optimal control theory. Sufficient conditions for the existence of open-loop Nash equilibria obtained from this approach are mainly limited to linear-quadratic games (Basar and Olsder in Dynamic noncooperative game theory, 2nd edn, SIAM, Philadelphia, 1999). Another approach of analysis is to substitute the dynamics and transform the game into a static game. But the substitution of state dynamics makes the objective functions of the resulting static problems extremely hard to analyze. We introduce a third approach in which the dynamics are not substituted, but retained as constraints in the optimization problem of each player, resulting thereby in a generalized Nash game. Using this, we give sufficient conditions for the existence of open-loop Nash equilibria for a class of DTDGs where the cost functions of players admit a quasi-potential function. Our results apply with nonlinear dynamics and without stage additive cost functions, and allow constraints on state and actions spaces, and in some cases, yield a generalization of similar results from linear-quadratic games.

An Approach Based on Generalized Nash Games and Shared Constraints for Discrete Time Dynamic Games

In this paper, we provide new results on conditions for the existence of open-loop Nash equilibria in discrete time dynamic games (DTDGs) using the theory of generalized Nash games. We use the state conjecture formulation of DTDGs introduced in Abraham and Kulkarni (Under Rev Math Methods Oper Res, 2017) for the analysis, which transforms a DTDG into an equivalent generalized Nash game. The existence of a Nash equilibrium for a generalized Nash game is intractable in general, without strong assumptions in the problem structure. We introduce a new and related game called the DTDG with consistent conjectures, which has a shared constraint structure unlike the original game. We show that an open-loop Nash equilibrium of the original DTDG is a Nash equilibrium of this new game, and hence, the existence of a Nash equilibrium for the DTDG with consistent conjectures becomes necessary for the existence of an open-loop Nash equilibrium in the original DTDG. Then we show that when the cost function of the game admits a potential function, the existence of a Nash equilibrium in the DTDG with consistent conjectures follows under mild assumptions. Finally, we show a reverse relation, wherein under certain conditions, a Nash equilibrium of the DTDG with consistent conjectures is shown to be an \(\epsilon \)-Nash equilibrium of the original game. We use an example from dynamic duopoly competition with sticky prices to illustrate the concepts and results.

N-person differential games governed by semilinear stochastic evolution systems

In this paper the problem ofN-person infinite-dimensional stochastic differential games governed by semilinear stochastic evolution control systems is discussed. First the minimax principle which is the necessary condition for the existence of open-loop Nash equilibrium is proved. Then the necessary and sufficient conditions of open-loop and closed-loop Nash equilibrium for linear quadratic infinite-dimensional stochastic differential games are derived.