# Chapter 3Learning in Two-Player Matrix Games

## 3.1 Matrix Games

In this chapter, we will examine the two-player stage game or the matrix game problem. Now, we have two players each learning how to play the game. In some cases they may be competing with each other, or they may be cooperating with other. In this section, we will introduce the class of game that we will investigate in this chapter. In fact, almost every child has played some version of these games. We will focus on three different games: matching pennies, rock-paper-scissors, and prisoners' dilemma. These are all called matrix games or stage games because there is no state transition involved. We will limit how far we delve into game theory and focus on the learning algorithms associated with these games. The idea is for the agents to play these games repetitively and learn their best strategy. In some cases one gets a pure strategy; in other words the agent will choose the same particular action all the time, and in some cases it is best to pick an action with a particular probability, which is known as a mixed strategy.

In the prisoners' dilemma game, two prisoners who committed a crime together are being interrogated by the police. Each prisoner has two choices; one choice is to cooperate with the police and defect on his accomplice, and the other is to cooperate with his accomplice and lie to the police. If both of them cooperate with each other and do not confess to the crime, then they will get just a few ...

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