  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.
