Open Loop Robust Equilibria in Uncertain Discrete Time Games
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Open Loop Robust Equilibria in Uncertain Discrete Time Games Fernando Guerrero Vélez, Manuel Jiménez Lizárraga*, and Celeste Rodriguez Carreon Abstract: This paper tackles the problem of a discrete time N players game affected by some sort of time-varying uncertain perturbation. The philosophy of the uncertainty worst case scenario with respect to the i-player is developed to derive necessary and sufficient conditions for the existence of an Open Loop Robust Nash Equilibria. Such conditions are presented in terms of the solvability of a set of discrete time Riccati type equations with some boundary conditions. As an illustration of the solution, a simulation of the coordination of a two-echelon supply chain with uncertain seasonal fluctuations in the demand is developed. Keywords: Discrete time game, robust Nash equilibria, two echelon supply chains.
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INTRODUCTION
The studies of dynamic games increase importantly in the last decades because of the realization that they suitable represent strategic behavior between different agents (or players), needed in situations of competitions, conflicts, or partially conflict scenarios. These situations often arises in a variety of fields as engineering, economics, biology, among others (see [10], [18], and [19]). One of the most pervasive solution, when the players are not allowed to cooperate, is the so-called Nash Equilibria. In this kind of non-cooperative equilibrium, two or more players interact, and each player wants to maximize or minimize his individual payoffs through a cost function (see [10] and [20]). The best response of player i with respect to the set of Nash equilibrium actions of the rest of players, leads to a situation where he has no incentives to abandon his strategy. If all the players calculate the best response as player i, then all the participants in the dynamic game achieve an "equilibrium of forces" that is called a Nash Equilibria. Despite the notion of "robustness of the strategies" is such an important feature in the practice because of the inevitable presence of uncertainties or noises that we find in most real application, there are not many studies of dynamic games that are affected by some sort of uncertainties or disturbances, particularly in discrete time dynamic games the works are scarce. As mentioned by [3], [4], [5], robust control systems deal with model inaccuracies, sensors noises, system perturbations, nonlinearities, and in general, the lack of modeled dynamics. Some recent developments related to this, manly in continuous time games, can be mentioned. In [13], it is presented a no-
tion of Open Loop Nash Equilibrium (OLNE) where the parameters of the game are within a finite set and the solution is given in terms of the worst-case scenario, that is, the result of the application of certain control input (in terms of the cost function value) is associated with the worst or least favorable value of the unknown parameter. The article of [6] shows also an OLNE and de
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