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

Abstract

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.