Abstract
The recent theory of sequential games and selection functions by Escardó & Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions. A normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved. Minimax strategies also generalize to the new class of games, and are computed by the Berardi–Bezem–Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice.
Highlights
The notion of optimization is common to many areas of applied mathematics, such as game theory and linear and nonlinear programming
If X and Y are sets, X → Y denotes the set of all functions with domain X and range Y
The quantifiers attained by these selection functions have closed graph only if n = 1. This game has the same equilibria as the equivalent game with multiple outcome spaces, but the Nash theorem cannot be proved in this way
Summary
The notion of optimization is common to many areas of applied mathematics, such as game theory and linear and nonlinear programming. A slightly more general notion of multi-valued quantifier is defined These concepts were introduced and applied to the theory of sequential games by Escardó & Oliva [1] in a series of papers summarized in [1]. For each player, the rules of the game define an outcome function mapping each play of the game to a real number called the utility of the play for that player This generalization is important for applications of game theory to economics, where utility often represents profit. By replacing R with an arbitrary set R and arg max with an arbitrary selection function ε : (X → R) → X, a rich theory of generalized games results, with deep connections to proof theory and theoretical computer science [4,5].
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