🤖 AI Summary
This work provides a unified categorical characterization of solution concepts in strategic games, such as Nash equilibria and Pareto-efficient outcomes. By constructing a category of games whose morphisms are mappings between player sets and strategy profiles, and by introducing presheaves valued in the category of sets that assign to each game its solution set, the study formally captures these solution concepts using tools from category theory. It establishes, for the first time, the precise order-theoretic conditions under which such presheaves are well-defined: the Nash equilibrium presheaf is valid when strategy mappings are either order-reflecting or order-embedding and morphisms are relational; the Pareto-efficient presheaf requires strategy mappings to be order-embedding. This framework offers a cohesive categorical semantics for game-theoretic solutions.
📝 Abstract
We define a category of strategic games in which a morphism is a pair of a map between sets of players and a map between sets of strategy profiles with specific properties, and we show that this category is well-defined. Three cases are considered for the maps between sets of strategy profiles: they may be order-preserving, order-reflecting or order-embedding with respect to each player's preference relation. We define a presheaf on the category of games valued in a category of sets that sends each strategic game to a set of strategy profiles, and we present equivalent conditions for this presheaf to be well-defined. Two cases are considered for the morphisms in the category of sets: they may be relations or maps. We define a presheaf that sends each strategic game to the set of Nash equilibria (resp.\ Pareto efficient strategy profiles), and we show that this presheaf is well-defined if and only if the maps between sets of strategy profiles are order-reflecting or order-embedding (resp.\ the maps between sets of strategy profiles are order-embedding), and the morphisms in the category of sets are relations.