Products of synchronous games

TL;DR

This paper establishes a mathematical framework connecting the *-algebra of product synchronous games with tensor products, and characterizes their perfect strategies and algebraic structures.

Contribution

It proves the *-algebra of product games is the tensor product of individual game algebras and characterizes perfect strategies via C*-algebra isomorphisms.

Findings

Product game *-algebra equals tensor product of individual game *-algebras

Perfect C*-strategy exists for the product if and only if for each game

Examples of supermultiplicative synchronous game values provided

Abstract

We show that the *-algebra of the product of two synchronous games is the tensor product of the corresponding *-algebras. We prove that the product game has a perfect C*-strategy if and only if each of the individual games does, and that in this case the C*-algebra of the product game is *-isomorphic to the maximal C*-tensor product of the individual C*-algebras. We provide examples of synchronous games whose synchronous values are strictly supermultiplicative.