## Final Report Summary - OASIG (Ordered Algebraic Structures in Game Theory)

The main goal of this interdisciplinary project was to develop the methods and techniques of ordered algebraic structures for solving game-theoretic problems. Ordered algebras such as lattice ordered groups and Riesz spaces are abundant in most parts of pure and applied mathematics. In particular, various classes of partially ordered sets are used to model coalition structures in cooperative game theory. Lattice-ordered function spaces are the basis for abstract economic models and preference representation in decision theory. In the project we have put forward the detailed list of specific goals, each of which is related to a certain game-theoretic model (strategic, cooperative etc.) and a class of ordered algebras (MV-algebras, lattice groups, Riesz spaces). We will describe our work and the chief results of the project OASIG (Ordered Algebraic Structures in Game Theory).

In the area of cooperative games we achieved a description of extreme supermodular games. This class of coalitional games is important for modeling cooperative behavior, where players have incentive to form large coalitions. The problem of describing the generators of the polyhedral cone was mentioned already in the seminal paper by Shapley. Combinatorial explosion makes it difficult to analyze the geometrical structure of the cone already for a very low number of players. We have given a simple linear-algebraic criterion for deciding whether a given supermodular game generates an extreme ray. Moreover, we provided an in-depth comparison between our result and the description of extremality in the supermodular cone achieved by other researchers. The obtained results were based on the core solution of a coalitional game. In non-supermodular games the core solution need not depict the results of the game faithfully. Therefore we designed a new solution concept for transferable-utility coalitional games, the so-called intermediate set, which combines ideas from combinatorial optimization (Lovász extension) with methods of non-smooth analysis (generalized derivatives). It was shown that the intermediate set is a non-convex polyhedron containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. We computed the exact form of intermediate set for all games and proved its simplified characterization for the voting games and the class of glove games. In a related line of research we relaxed the assumption that the cooperation structure is a finite Boolean lattice and investigated a more general class of games over finite distributive lattices. This enabled us to show that there is a natural and a more general framework for supermodular games than the one based on the powerset algebra of coalitions.

We have also contributed to the subject of uncertainty modelling, which finds many applications in game theory. For example, probabilities are used in mixed strategy solutions or when dealing with beliefs of players. Our set up is genuinely algebraic and category-theoretic, making it possible to employ universal constructions in order to reason about properties of probabilities (or states). By introducing two distinct sorts we make a fundamental distinction between events, on the one hand, and probability degrees, on the other. At the same time both events and degrees of probabilty are naturally modelled by MV-algebras. A probability function is then conceived of as an operator between the two MV-algebras satisfying the standard axioms of finite additivity and normalization. Thus, generalised probabilities become unary operations in two-sorted algebraic structures that we call state algebras. We study free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice-groups. A non-Archimedean model of probability based on the Chang algebra was discussed. In summary, the two-sorted approach to probability is a promising platform for modeling a number of economic and game-theoretic phenomena, ranging from higher-order uncertainty representation to belief modelling.

In the area of cooperative games we achieved a description of extreme supermodular games. This class of coalitional games is important for modeling cooperative behavior, where players have incentive to form large coalitions. The problem of describing the generators of the polyhedral cone was mentioned already in the seminal paper by Shapley. Combinatorial explosion makes it difficult to analyze the geometrical structure of the cone already for a very low number of players. We have given a simple linear-algebraic criterion for deciding whether a given supermodular game generates an extreme ray. Moreover, we provided an in-depth comparison between our result and the description of extremality in the supermodular cone achieved by other researchers. The obtained results were based on the core solution of a coalitional game. In non-supermodular games the core solution need not depict the results of the game faithfully. Therefore we designed a new solution concept for transferable-utility coalitional games, the so-called intermediate set, which combines ideas from combinatorial optimization (Lovász extension) with methods of non-smooth analysis (generalized derivatives). It was shown that the intermediate set is a non-convex polyhedron containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. We computed the exact form of intermediate set for all games and proved its simplified characterization for the voting games and the class of glove games. In a related line of research we relaxed the assumption that the cooperation structure is a finite Boolean lattice and investigated a more general class of games over finite distributive lattices. This enabled us to show that there is a natural and a more general framework for supermodular games than the one based on the powerset algebra of coalitions.

We have also contributed to the subject of uncertainty modelling, which finds many applications in game theory. For example, probabilities are used in mixed strategy solutions or when dealing with beliefs of players. Our set up is genuinely algebraic and category-theoretic, making it possible to employ universal constructions in order to reason about properties of probabilities (or states). By introducing two distinct sorts we make a fundamental distinction between events, on the one hand, and probability degrees, on the other. At the same time both events and degrees of probabilty are naturally modelled by MV-algebras. A probability function is then conceived of as an operator between the two MV-algebras satisfying the standard axioms of finite additivity and normalization. Thus, generalised probabilities become unary operations in two-sorted algebraic structures that we call state algebras. We study free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice-groups. A non-Archimedean model of probability based on the Chang algebra was discussed. In summary, the two-sorted approach to probability is a promising platform for modeling a number of economic and game-theoretic phenomena, ranging from higher-order uncertainty representation to belief modelling.