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