Game theory provides a mathematical framework for analyzing the decision-making processes and strategies of adversaries (or *players*) in different types of competitive situations. The simplest type of competitive situations are **two-person, zero-sum games**. These games involve only two players; they are called *zero-sum* games because one player wins whatever the other player loses.

### Example: Odds and Evens

Consider the simple game called **odds and evens**. Suppose that player 1 takes evens and player 2 takes odds. Then, each player simultaneously shows either one finger or two fingers. If the number of fingers matches, then the result is *even*, and player 1 wins the bet ($2). If the number of fingers does not match, then the result is *odd*, and player 2 wins the bet ($2). Each player has two possible strategies: show one finger or show two fingers. The *payoff matrix* shown below represents the payoff to player 1.

### Basic Concepts of Two-Person Zero-Sum Games

This game of odds and evens illustrates important concepts of simple games.

- A two-person game is characterized by the strategies of each player and the payoff matrix.
- The payoff matrix shows the gain (positive or negative) for player 1 that would result from each combination of strategies for the two players.
*Note that the matrix for player 2 is the negative of the matrix for player 1 in a zero-sum game.* - The entries in the payoff matrix can be in any units as long as they represent the
*utility (or value)*to the player. - There are two key assumptions about the behavior of the players. The first is that both players are
*rational*. The second is that both players are*greedy*meaning that they choose their strategies in their own interest (to promote their own wealth).