Game Theory

Strictly Determined Games Payoff Matrix is POV of row player $\left[\begin{array}{cc}3& -1\\ -1& -2\end{array}\right]$ So If the row player chooses the first row, they will wither get +3 or -1, and if they chose the bottom row, they will get -1 or -2

In a strictly determined game, the best strategy for each player is to chose the option with the best worst-case, or the maximin. For the row player, the minimums for the rows are -1, -2. So the Row player would chose to always play row one.

If a game has more than two options, first remove any rows/columns which are dominated. A row/column is dominated by another if every possible outcome in that row/column is less than the corresponding outcome in a particular alternative row/column.

How to know if a game is strictly determined?

A strictly determined game is one in which the maximin (the maximum of the row minimums) and the minimax (the minimum of the column maximums) are the same value. This value is called the saddle point of the game. The saddle point can be interpreted as the amount won per play by the row player

Games with mixed strategies $\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$ Here the minimax (of the columns) is 1, and the maximin (of the rows) is -2. Since these are not equal, the game is not strictly determined.

To determine the row players optimal mixed strategy, do this: $\left[\begin{array}{cc}p& 1-p\end{array}\right]\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}4p-3(1-p)& -2p+1(1-p)\end{array}\right]$ Then set the resulting columns equal to one another: $4p-3(1-p)=-2p+1(1-p)$ And solve for p: $p=0.4$ $p$ represents what fraction of the time the row player should play the first row

For the column player, do: $\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]\left[\begin{array}{c}q\\ 1-q\end{array}\right]$