The way this situation is represented is that Guy doesn’t know ex- actly where he is in the game: Is he at node III or node IV?
In any extensive form game, a player who is to act always has an informa- tion set representing what he knows. So what about when Vivica moves? What is her information set? It is just node II; in other words, she knows exactly where she is in the game. If we want to be consistent, we would then put a box around node II to represent Vivica’s information set. So as to avoid unneces- sary cluster, however, singleton information sets (i.e., an information set with a single node) are left unboxed. In Figure 2.6, then, Guy has two information sets; one is the singleton composed of the initial node (denoted I), and the other comprises nodes III and IV.
Returning to Vivica, since she is modeled as moving before Guy decides about Orlando, she makes her decision without knowing what has hap- pened or will happen to Orlando. Do you notice how I’m unclear about the timing? Does Vivica move chronologically before, after, or at the same time as Guy? I’ve been intentionally unclear because it doesn’t matter. What mat- ters is information, not the time of day at which someone makes a decision. What is essential is that Vivica does not know whether Guy has released or killed Orlando when she decides whether to pay ransom and that Guy does not know whether Vivica has paid the ransom when he decides whether to release or kill Orlando. In fact, FIGURE 2.7 is an extensive form game equiv- alent to that in Figure 2.6. It flips the order of decision making between Vivica and Guy, and the reason it is equivalent is that we haven’t changed the information that the players have when they move. In both games, we’ll say that Vivica and Guy move simultaneously (with respect to the ransom and release-or-kill decisions), which is meant to convey the fact that their
FIGURE 2.7 Extensive Form Equivalent to Figure 2.6
Guy
Guy 3
Vivica 5 Kill
Guy
4
1
2
2
5
3
1
4
Do not kidnap
Release
Kidnap
Vivica
Pay ransom
Do not pay ransom
Pay ransom
Do not pay ransom
2.3 Extensive Form Games: Imperfect Information 29
information is the same as when they make their decisions at the exact same time.
An extensive form game in which all information sets are singletons— such as the games in Figures 2.1–2.5—is referred to as a game of perfect information, since players always know where they are in the game when they must decide. A game in which one or more information sets are not singletons, such as the game in Figure 2.6, is known as a game of imper- fect information.