The value of a game is the supremum of the expected value of winning the game over various natural families of probability densities. In this talk we initiate the study of finding such values of games when the densities are required to be synchronous densities arising from various families of quantum correlations.

Such densities are known to arise from various types of traces on C*-algebras, so finding the value of these games corresponds to maximizing the trace of an element of a certain C*-algebra that we call the ``value algebra" over a certain family of traces.

We are able to compute these numbers explicitly for some games based on graph colouring.

This work was supported by NSERC Discovery Grant 03784.