# 2: Two-Person Zero-Sum Games


In this chapter, we will look at a specific type of two-player game. These are often the first games studied in game theory as they can be straightforward to analyze. All of our games in this chapter will have only two players. We will also focus on games in which one player's win is the other player's loss.

• 2.1: Introduction to Two-Person Zero-Sum Games
In all of the examples from the last section, whatever one player won, the other player lost. A two-player game is called a zero-sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the terms (or coordinates) in each payoff vector must add up to the same value for each payoff vector. Such games are sometimes called constant-sum games instead.
• 2.2: Dominated Strategies
Recall that in a zero-sum game, we know that one player's win is the other player's loss. Furthermore, we know we can rewrite any zero-sum game so that the player's payoffs are in the form (a,-a). Note, this works even if a is negative; in which case, -a is positive.
• 2.3: Probability and Expected Value
Many games have an element of chance. In order to model such games and determine strategies, we should understand how mathematicians use probability to represent chance. Consider a standard deck of 52 playing cards. What is the chance of drawing a red card? What is the probability of drawing a red card? Is there a difference between chance and probability? Yes! The probability of an event has a very specific meaning in mathematics.
• 2.4: A Game of Chance
Now that we have worked with expected value, we can begin to analyze some simple games that involve an element of chance.
• 2.5: Equilibrium Points
In this section, we will try to gain a greater understanding of equilibrium strategies in a game. In general, we call the pair of equilibrium strategies an equilibrium pair, while we call the specific payoff vector associated with an equilibrium pair an equilibrium point.
• 2.6: Strategies for Zero-Sum Games and Equilibrium Points
Throughout this chapter, we have been trying to find solutions for two-player zero-sum games by deciding what two rational players should do. In this section, we will try to understand where we are with solving two-player zero-sum games. The exercises in this section are intended to review the concepts of dominated strategies, equilibrium points, and the maximin/minimax strategies.
• 2.7: Popular Culture: Rationality and Perfect Information
In this section, we will look at applications of rationality and perfect information in popular culture. We present films with connections to game theory and suggest some related questions for essays or class discussion. The movie Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb (1964) depicts the cold war era with the USA and the Soviet Union on the brink of atomic war.

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