Synchronous Values of Games

TL;DR

This paper investigates the properties of synchronous values in games, revealing that optimal strategies may not be synchronous and providing formulas and conditions for calculating and understanding these values.

Contribution

It introduces a formula for the synchronous value of XOR games, demonstrates non-multiplicativity of synchronous quantum bias, and explores geometric conditions for projections related to synchronous values.

Findings

Synchronous value can be strictly less than the ordinary value in some games.

The synchronous value of repeated game products can increase strictly.

Synchronous quantum bias of XOR of two XOR games is not multiplicative.

Abstract

We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.