# Who first studied nonlocal games with probabilistic predicate?

For some background, a nonlocal game consists of questions $$x,y\in X,Y$$ and answers $$a,b \in A,B$$; the pair of questions $$x,y$$ is asked with probability $$\mu(x,y)$$, and a referee accepts the pair of answers $$a,b$$ for the pair of questions $$x,y$$ with probability $$V(a,b,x,y)$$. This function $$V:A \times B \times X \times Y \to [0,1]$$ is the predicate of the nonlocal game. The case I just defined is the most general one, of a probabilistic predicate. Usually, though, papers deal with only with the case of deterministic predicates, where $$V(a,b,x,y)$$ is always either $$0$$ or $$1$$, so the referee simply accepts of rejects a given answer pair for a given question pair.

What I want to know is who first studied nonlocal games with probabilistic predicate. Cleve et al., who pretty much started the are, only defines games with deterministic predicate. The earliest reference I know that talks about games with probabilistic predicate is Buhrman et al.. It doesn't seem to be the first one, though, they talk as if the concept is already known.