5. Nash and Stackelberg Equilibrium

We provide an approach to locating the Nash equilibrium in this chapter. The technique depends on identifying a scalar \(\lambda ^{*}\) and the associated strategies \(d^{*}(\lambda ^{*})\) fixing particular boundaries (min and max) that belong to the Pareto front. Bounds refer to limits placed by the player over the Pareto front that form a specific decision region where the strategies can be selected. We first use a nonlinear programming issue to illustrate the Pareto front of the game, introducing a set of linear constraints for the Markov chain game based on the c-variable technique. We suggest using the Euler method and a penalty function with regularization to solve the strong Nash equilibrium issue. The convergence to a single (strong) equilibrium point is ensured using Tikhonov’s regularization method. The subsequent single-objective restricted problems that result from using the regularized functional of the game were then solved using a nonlinear programming technique. We use the gradient approach to resolve the first-order optimality requirements in order to accomplish the aim. The approach solves an optimization issue by adding linear constraints necessary to identify the best strong strategy, d (lambda d), starting from a utopia point (Pareto optimum point) given an initial lambda of the individual objectives. We demonstrate that the game’s functional in the regularized issue decreases and ultimately converges, demonstrating the presence and exclusivity of strong Nash equilibrium (Pareto-optimal Nash equilibrium). We also provide a method for calculating the Markov chain games’ strong Stackelberg/Nash equilibrium. The minimization of the \(L p-\)norm, which shortens the distance to the utopian point in Euclidian space, is taken into consideration while solving the cooperative n-leaders and m-followers Markov game. Next, we formulate a Pareto-optimal solution to the optimization issue. For finding the strong Lp-Stackelberg/Nash equilibrium, we use a bi-level programming technique that is carried out through extraproximal optimization.