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Ordered Algebraic Structures in Game Theory (OASIG)
Start date: 01 Nov 2014, End date: 31 Oct 2016 PROJECT  FINISHED 

"The main goal of this project is to study ordered algebraic structures with regard to their applications in game theory. Theory of Riesz spaces, groups, and MV-algebras provide the tools necessary for modeling important aspects of mathematical games, such as economic rationality, strategic invariance, or fairness. This is already witnessed by many parts of game theory: for example, Aumann-Shapley value is a positive equivariant linear operator into the Riesz space of measures and MV-algebras model many coalition games. Our objective is to employ the methods, which were developed for solving deep algebraic and logical problems, in several important game-theoretic problems. Specifically, we will study the class of piecewise linear continuous strategic games by using the polyhedral representation of free MV-algebras and Baker-Beynon duality for unital free vector lattices, with the goal to show the existence of finitely-supported mixed strategy equilibria. We will investigate MV-algebras and their measures in order to model coalition games and their solutions. The motivation is to generalize the Danilov-Koshevoy representation of core by the convex Minkowski combination of simplices for larger classes of games. This inevitably leads to building dual representations of games based on the notion of generalized Moebius transform and the space of filters of an MV-algebra. Each facet of this project is transdisciplinary: the emphasis on algebra and order pervades all the selected game-theoretic scenaria. The presented proposal is an opportunity for the applicant to acquire new mathematical skills under the guidance of a scientist in charge - the specialist in ordered algebraic structures - and achieve thus a unique position in his own research field (game theory, many-valued logics)."
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