Minimax Theorems for Games on Unit Square

We consider a class of infinite games with unbounded cores and establish the existence of their value. It is shown that a game with the core
$$ K(x,y)=\begin{cases} L(x,y),&x<y, \\ \varphi(x),&x=y, \\ M(x,y),&x>y, \end{cases} $$
where the functions $L$ and $M$ are defined and continuous on the triangles $0\leqq x\leqq y\leqq 1$, $0\leqq y\leqq x\leqq 1$, respectively, the function $\varphi$ is arbitrary and $L(0,0)\geqq M(0,0)$, $L(1,1)\leqq M(1,1)$, is a game with value.