$G(3,2)=\frac{1}{2}$.

*Proof.* Assume by contradiction that, for some family-set $F$ composed of 3-element sets, Red has a strategy that allows him to make two red elements in more than half the sets in $F$ (a "strategy" means a reply to every sequence of moves played by Green). Then, Green can make the first move arbitraily, and from then on, just copy Red's strategy (i.e, reply to each sequence of moves played by Red, in the same way Red would reply to that sequence if played by Green). This guarantees that there will be at least two green elements in more than half the sets in $F$.