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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  
\[A_j\]
  instead.
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

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