Stackelberg Equilibrium Sets in Polymatrix Mixed-Strategy Generalized Stackelberg Games

By modifying the principle of simultaneity in strategy/simultaneous games to a hierarchical/sequential principle according to which players select their strategies in a known order, we obtain other class of games called Generalised Stackelberg Games or simply Stackelberg Games. Such games are called sequential games to especially highlight a sequential process of decision making. At every stage of Stackelberg games one player selects his strategy. To ensure choosing of his optimal strategy he solves an optimization problem. For such games a Stackelberg equilibrium concept is considered as a solution concept (Von Stackelberg in Marktform und Gleichgewicht (Market Structure and Equilibrium). Springer, Vienna, XIV+134, 1934, [1], Chenet al., in IEEE Transactions on Automatic Control AC-17, 791–798, 1972, [2], Simaan and Cruz in Journal of Optimization Theory and Applications 11, 613–626, 1973, [3], Simaan and Cruz in Journal of Optimization Theory and Applications 11, 533–555, 1973, [4], Leitmann in Journal of Optimization Theory and Applications 26, 637–648, 1978, [5], Blaquiere in Une generalisation du concept d’optimalite et des certains notions geometriques qui’s rattachent, No. 1–2, Bruxelles, 49–61, 1976, [6], Basar and Olsder in Dynamic noncooperative game theory. SIAM, Philadelphia, [7], Ungureanu in ROMAI Journal 4(1), 225–242, 2008, [8], Ungureanu in Mathematical Modelling, Optimization and Information Technologies, International Conference Proceedings, ATIC, 181–189, Evrica, Chisinău, [9], Peters in Game theory: A multi-leveled approach, p. XVII+494, Springer, Berlin, 2015, [10], Ungureanu and Lozan in Mathematical Modelling, Optimization and Information Technologies, International Conference Proceedings, ATIC, 370–382, Evrica, Chisinău, 2016, [11], Korzhyk et al. in Journal of Artificial Intelligence Research, 41, 297–327, 2011, [12]). The set of all Stackelberg equilibria is described/investigated as a set of optimal solutions of an optimization problem. The last problem is obtained by considering results of solving a sequence of optimization problems that reduce all at once the graph of best response mapping of the last player to the Stackelberg equilibrium set. Namely the problem of Stackelberg equilibrium set computing in bimatrix and polymatrix finite mixed-strategy games is considered in this chapter. A method for Stackelberg equilibrium set computing is exposed.

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