Consider a two player matrix game with payoff matrix :
$$\begin{pmatrix}0 & 2 & -1\\ -2 & 0 & 1\ \\ 1 & -1 & 0\end{pmatrix}$$
I need to show that the game has no saddle point solution and find an optimal mixed strategy.
If simultaneously have a row minimum and a column maximum this is an example of a saddle point solution. As max(col1) = 1 , max(col2) = 2 , max(col3) = 1, min(row1) = -1 , min(row2) = 0 , min(row3) = -1 there is not a simultaneous row min and row max a saddle point does not exist.
How to find an optimal mixed strategy ?
From http://www.iun.edu/~mathiho/mathpol/fall00/chapter15.htm
An optimal mixed strategy for the row player is a mixed strategy for
which the lowest expected payoff (over all possible column player
mixed strategies) is as large as possible. An optimal mixed strategy
for the column player is a mixed strategy for which the highest
expected payoff (over all possible row player mixed strategies) is as
small as possible.
Do I just need to find the min of all rows and min of all cols ?
Best Answer
$A$ is antisymmetric so the game itself is symmetric. Thus if player 1 can guarantee at least $v$, then player 2 can guarantee at most $-v$. Therefore, $v=0$ so you just have to find strategies $x$ and $y$ such that $x^TA=(0,0,0)$ and $Ay=(0,0,0)^T$. You get $x=y=(1/4,1/4,1/2)^T$.