## Dominating Strategies in Two Player Games

Players obviously want to maximising their winnings – or minimize their losses – when playing a game. If player 1 has a choice of strategies, but the winnings from strategy A, which depend on the strategy adopted by the player 2, are at least as great as the winnings of strategy B, which also depend on the strategy adopted by player 2, for every strategy adopted by player 2, then strategy A is said to dominate strategy B. Player 1 will never play strategy B since the winnings from playing strategy A are at least as great.

For example, the payoff matrix for player 1 is given in the table below.

Player 2 | ||||

Player 1 | 1 | 2 | 3 | |

1 | 5 | 2 | 8 | |

2 | 2 | 5 | 7 | |

3 | 8 | 3 | 8 |

Strategy 3 for player 1 dominates strategy 1, since the winnings for player 1 from playing strategy 3 are always at least as great as the winnings from playing strategy 1.

The row corresponding to strategy 1 can be removed to give the table below.

Player 2 | ||||

Player 1 | 1 | 2 | 3 | |

2 | 2 | 5 | 7 | |

3 | 8 | 3 | 8 |

If pl;ayer 1 is less lucky, he is making losses. The table shows the losses for player 1 for each combination of strategies.

Player 2 | ||||

Player 1 | 1 | 2 | 3 | |

1 | -5 | -1 | -7 | |

2 | -2 | 0 | -7 | |

3 | 0 | -3 | -8 |

The losses for player 1 for strategy 2 are at most equal to the losses for playing strategy 1, so player 1 always plays strategy 2 in preference 1to 1. Strategy 2 is said to dominate strategy 1.

The row corresponding to strategy 1 can be removed to give the table below.

Player 2 | ||||

Player 1 | 1 | 2 | 3 | |

2 | -2 | 0 | -7 | |

3 | 0 | -3 | -8 |