## 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 |

\[A_i\]

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 |

\[B_3\]

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 |

\[B_4\]

than strategy \[B_2\]

. We can eliminate strategy \[B_4\]

leaving the payoff matrixA\B | \[B_1\] | \[B_2\] |

\[A_2\] | 1 | 3 |

\[A_3\] | 2 | -4 |