Let $S := \{x \in \Bbb{R}^n \mid x \ge 0, \sum_{i=1}^n x_i = 1\}$ be the set of mixed strategies. For a bimatrix game with pay-off matrices $A$, $B$ we denote $C := \{ (u, v) \mid \exists (x,y)\in S\times S: u = x^TAy, v = x^TBy\}$ and refer to $C$ as the feasible set.
How do you prove that $C$ is convex? That is, for any convex combination of "achievable" values of expected pay-offs there exists a mixed strategy "implementing" this combined value?
Edit:
As Michael Greinecker pointed out (see comments below), this is not true in general case. I have read this statement in Game Theory by prof. Guillermo Owen, Chapter IX, page 201.
Having read it once again, it is formulated for correlated mixed strategies. Is it already true with this condition or still too general? Any ideas for a proof/counterxample?