Simplifying a Zero Sum Game Payoff Matrix

We can often simplify a payoff matrix for a zero sum game by eliminating strategies that a rational player will never use. We can find 'dominated' strategies. Any duplicate strategies can also be eliminated.

A\B	\[B_1\]	\[B_2\]	\[B_3\]	\[B_4\]
\[A_1\]	0	-1	2	-4
\[A_2\]	1	3	3	6
\[A_3\]	2	-4	5	1

The strategy    is said to be (strictly) dominated by    if the payoff for A for strategy    is always (less than) less than or equal to the payoff to A for strategy  , whatever the strategy of B. If    is dominated by  , then A will never play  , preferring    instead.
In the payoff matrix above,    is dominated by    since ever element in    is less than the corresponding element in  . We can eliminate strategy  .

A\B	\[B_1\]	\[B_2\]	\[B_3\]	\[B_4\]
\[A_2\]	1	3	3	6
\[A_3\]	2	-4	5	1

Similarly for B. B wants to minimise his own losses - the gain made by A. We can eliminate any strategy for B (the columns) if the entries in any column are greater than the corresponding entry in another column. The gains for A (Loews to B) for strategy    are greater than for strategy  . We can therefore eliminate strategy  .

A\B	\[B_1\]	\[B_2\]	\[B_4\]
\[A_2\]	1	3	6
\[A_3\]	2	-4	1

Now notice that the payoff to A (and losses to B) are greater for strategy    than strategy  . We can eliminate strategy    leaving the payoff matrix

A\B	\[B_1\]	\[B_2\]
\[A_2\]	1	3
\[A_3\]	2	-4