CHAPTER ONE

Matrix Two-Person Games

If you must play, decide upon three things at the start: the rules of the game, the stakes, and the quitting time.

—Chinese proverb

Everyone has a plan until you get hit.

—Mike Tyson, Heavyweight Boxing Champ, 1986–1990, 1996

You can’t always get what you want.

—Mothers everywhere

1.1 The Basics

What is a game? We need a mathematical description, but we will not get too technical. A game involves a number of players1 N, a set of strategies for each player, and a payoff that quantitatively describes the outcome of each play of the game in terms of the amount that each player wins or loses. A strategy for each player can be very complicated because it is a plan, determined at the start of the game, that describes what a player will do in every possible situation. In some games, this is not too bad because the number of moves is small, but in other games, like chess, the number of moves is huge and so the number of possible strategies, although finite, is gigantic. In this chapter, we consider two-person games and give several examples of exactly what is a strategy.

Let’s call the two players I and II. Suppose that player I has a choice of n possible strategies and player II has a choice of m possible strategies. If player I chooses a strategy, say, strategy i, i = 1, ..., n, and player II chooses a strategy j, j = 1, ..., m, then they play the game and the payoff to each player is computed. In a zero sum game, whatever one player wins the other ...

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