# Solving genereal zero sum games

Suppose I have this payoff matrix for a zero sum game

\begin{array}{ccc} 8 & 3 & 4 & 1 \\ 4 & 7 & 1 & 6 \\ 0 & 3 & 8 & 5 \end{array}

Since it has no saddle point and no domination, I tried to solve it by equalizing payoffs which led me to the answer that player I(row)'s strategy is $(3/8, 3/8, 1/4)$ and the value of the game $V=2/9$.

However, plugging this into a solver gave me the following results.

The value is $4.08333$.
An optimal strategy for Player I is: $(0.32292,0.375,0.30208)$
An optimal strategy for Player II is: $(0.33333,0,0.25,0.41667)$.

Aftwerwards, I found out that equalizing payoffs only works if all the probablities assigned to each move are positive which is not the case here. So, how then would one solve this problem by hand?

The linear programming formulation (for the row player's moves) of an $m \times n$ zero-sum game with payoff matrix $(a_{ij})$:
\eqalign{\text{maximize}\ z & \ \text{subject to}\cr \sum_{i=1}^m a_{ij} x_i &\ge z \ \text{for}\ j = 1 \ldots n\cr \sum_{i=1}^m x_i &= 1\cr x_i &\ge 0\ \text{for } i=1\ldots m} At $5$ constraints and $4$ variables it's a bit large for solving by hand, but doable.