Products of synchronous games

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.