Example 7. Verify that the strategies $y^{*}=(\frac{1}{2},\frac{1}{4},\frac{1}{4}),x^{*}=(\frac{1}{2},\frac{1}{4},\frac{1}{4})$ are optimal and $v=0$ is the value of the game $\overline{\Gamma}_{A}$ with matrix $$A= \begin{bmatrix} 1&-1&-1\\ -1&-1&3\\ -1&3&-1 \end{bmatrix}. $$
Adding a unity to all elements of the matrix $A$, we get the matrix
$$A^{'}= \begin{bmatrix} 2&0&0\\ 0&0&4\\ 0&4&0 \end{bmatrix}. $$
Each element of the matrix $A^{'}$ can be divided by $2$. The new matrix is of the form
$$A^{''}= \begin{bmatrix} 1&0&0\\ 0&0&2\\ 0&2&0\\ \end{bmatrix} $$ By the lemma we have $v_{A^{''}}=\frac{1}{2}v_{A^{'}}=\frac{1}{2}(v_{A}+1)$Verify the value of the game $\Gamma$ is equal to $\frac{1}{2}.$ Indeed, $K(x^{*},{y^{*}})=Ay^{*}=\frac{1}{2}.$ On the other hand, for each strategy $y \in Y,y=(\eta_{1},\eta_{2},\eta_{3})$ we have $K(x^{*},y)=\frac{1}{2}\eta_{1}+\frac{1}{2}\eta_{2}+\frac{1}{2}\eta_{3}=\frac{1}{2}\cdot 1=\frac{1}{2}$, and for all $x=(\xi_{1},\xi_{2},\xi_{3}),x \in X, K(x,y^{*})=\frac{1}{2}\xi_{1}+\frac{1}{2}\xi_{2}+\frac{1}{2}\xi_{3}.$ Consequently, the above-mentioned strategies $x^{*},y^{*}$ are optimal and $v_{A}=0.$
I don't know why $K(x^{*},y^{*})=Ay^{*}=\frac{1}{2}$, it should be $0$.