Perfect strategies for non-signalling games

TL;DR

This paper unifies and extends results on non-signalling games by introducing new game families, characterizing perfect strategies via C*-algebras, and exploring algebraic and quantum properties of these games.

Contribution

It introduces reflexive, imitation, and mirror games, and characterizes perfect strategies using C*-algebraic methods, extending known results in non-local game theory.

Findings

Characterization of perfect strategies via C*-algebras.

Introduction of new game families: reflexive, imitation, and mirror games.

Connection between approximate quantum strategies and amenable traces.

Abstract

We unify and consolidate various results about non-signall-ing games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are {\it reflexive games,} which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce {\it imitation games,} in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and {\it unique} games. We associate a C*-algebra $C^*(\mathcal{G})$ to any imitation game $\mathcal{G}$, and show that the existence of perfect quantum commuting (resp.\ quantum, local) strategies of $\mathcal{G}$ can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call {\it mirror games,} and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.