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
$A_i$
is said to be (strictly) dominated by
$A_j$
if the payoff for A for strategy
$A_i$
is always (less than) less than or equal to the payoff to A for strategy
$A_j$
, whatever the strategy of B. If
$A_i$
is dominated by
$A_j$
, then A will never play
$A_i$
, preferring
$A_j$
In the payoff matrix above,
$A_1$
is dominated by
$A_2$
since ever element in
$A_1$
is less than the corresponding element in
$A_2$
. We can eliminate strategy
$A_1$
.
 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
$B_3$
are greater than for strategy
$B_1$
. We can therefore eliminate strategy
$B_3$
.
 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
$B_4$
than strategy
$B_2$
. We can eliminate strategy
$B_4$
leaving the payoff matrix
 A\B $B_1$ $B_2$ $A_2$ 1 3 $A_3$ 2 -4