Decisional Diffie-Hellman assumption vs decisional bilinear Diffie-Hellman assumption
I don't know if it can helps. But actually, you can win the DDH game if you are using pairings. 

Suppose you are given $X = g^x$ and $Y = g^y$, you can apply $e(X,Y) = e(g,g)^{xy}$. 

Given $Z$, your goal is to know if $(X,Y,Z)$ is a DDH triplet. 

You can test, using useful properties of pairings, if $e(g,Z) = e(X,Y)$ and so you can distinguish from a random element. 

So I think that's why we consider 3 numbers to distinguish. 

You can also think about Key-exchange Diffie-Hellman and Joux "Tripartie DH" as described in this PDF.