# 3: Repeated Two-Person Zero-sum Games


If we are presented with a two-person zero-sum game we know that our first step is to look for an equilibrium point. If a game has an equilibrium point, then we know that our players should play the corresponding strategy pair. In this case the equilibrium pair and its payoff vector is the “solution” to the game. In this chapter we will explore games that do not necessarily have an equilibrium point. We will also try to determine what a player should do if they play the game repeatedly.

• 3.1: Introduction to Repeated Games
When playing the game several times, does it make sense for either player to play the same strategy all the time? Why or why not? Although we use the term “strategy” to mean which row (or column) a player chooses to play, we will also refer to how a player plays a repeated game as the player's strategy. In order to avoid confusion, in repeated games we will define some specific strategies.
• 3.2: Mixed Strategies: Graphical Solution
In this section we will learn a method for finding the maximin solution for a repeated game using a graph.
• 3.3: Using Sage to Graph Lines and Solve Equations
In this section we will use technology to graph lines and solve for the intersection point. In particular, we will use an open online resource called Sage.
• 3.4: Mixed Strategies: Expected Value Solution
In this section, we will use the idea of expected value to find the equilibrium mixed strategies for repeated two-person zero-sum games.
• 3.5: Liar's Poker
In this section, we will continue to explore the ideas of a mixed strategy equilibrium. We saw two different methods for finding an equilibrium. The first employed graphs in order to understand and find the maximin and minimax strategies, and hence the equilibrium mixed strategy. The second method employed the ideas of expected value to find the equilibrium strategy. We will continue to solidify these ideas with another game, a simplified variation of poker.
• 3.6: Augmented Matrices
In this section, we will see how to use matrices to solve systems of equations. In both the graphical method and the expected value method, you have had to solve a system of equations.
• 3.7: Undercut
This section requires you to be able to solve “large” systems of equations. You will be using the matrix techniques from Section 3.6. You are encouraged to use technology such as a graphing calculator or Sage. As we saw in Section 3.5, an important part of game theory is the process of translating a game to a form that we can analyze. As we saw in Section 3.5, an important part of game theory is the process of translating a game to a form that we can analyze.

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