We consider nonlocal 2-player games which are "unique", in the sense that each player has a unique winning answer determined by a referee's question pair and the other player's answer. We show that such a game has a perfect vect-strategy (or quantum strategy) if and only if it has a perfect deterministic strategy, estimate some game values using C*-algebras and discuss games based on groups, which generalise the XOR games. This is joint work with Vern Paulsen.