Building a Model of a Strategic Situation
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is introduced in Section 2.3, and this concept provides the foundation for de- scribing the strategic form of a game in Section 2.4—the second type of scaf- folding. Though more abstract than the extensive form, the strategic form is more concise and easier to work with. Common knowledge is a concept perti- nent to both methods of modeling a strategic situation and is covered in Section 2.5. Common knowledge deals with what a person knows about what others know.
Before we move forward, let me remind you that this chapter is about building a game. Solving a game will begin with the next chapter, so be pre- pared for some delayed gratification.
2.2 Extensive Form Games: Perfect Information IN SPITE OF ITS NAME, game theory can deal with some fairly dire subjects, one of which is the criminal activity of kidnapping for ransom. This is a sufficiently serious and persistent problem in some countries—such as Colombia, Mexico, and Russia—that companies have taken out insurance against their executives being held for ransom. Building a model of kidnapping can involve factoring in a great many considerations. The focus of our task, however, is not so much on gaining insight into kidnapping, but on learning how to construct a game- theoretic model.
Because the objective of game theory is to derive implications about be- havior, a model should focus on those individuals who have decisions to make. Our attention will accordingly be on the kidnapper, whom we’ll call Guy, and the victim’s wife, Vivica, who has been contacted to pay ransom. Although the victim (whom we’ll name Orlando) is surely affected by what transpires, we are presuming that the victim has no options. In describing the situation, our model should address the following questions: When do Guy and Vivica get to act? What choices are available when they get to act? What do they know
when they get to act? More information will be needed to derive predictions about behav- ior, but the information obtained by answer- ing these questions is sufficient for starters.
The model is represented by what is known as a decision tree, such as that shown in FIGURE 2.1. A decision tree is read from top to bottom. (It can also be depicted to be read from left to right.) Each of the dots is called a decision node, which represents a point in the game at which someone has a decision to make. Coming out of a decision node is a series of branches, where each branch represents a different action avail- able to the decision maker. Choosing a branch is equivalent to choosing an action.
At the top of the decision tree, Guy is to make the initial decision, and his choices are kidnap (Orlando) and do not kidnap.* If he
18 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
FIGURE 2.1 Extensive Form of the Kidnapping Game
Guy
Guy 3
Vivica 5 Pay ransom
Vivica
4
1
5
3
2
2
1
4
Do not kidnap
Do not pay ransom
Kidnap
Guy
Kill KillRelease Release
Guy
*The name of an action or strategy will typically be italicized in this book.