Info

set, else the player would be able to infer from the set of actions available that some node(s) had or had not been reached.

In an extensive-form game, we will indicate that a collection of decision nodes constitutes an information set by connecting the nodes by a dotted line, as in the extensive-form representation of the Prisoners' Dilemma given in Figure 2.4.3. We will sometimes indicate which player has the move at the nodes in an information set by labeling each node in the information set, as in Figure 2.4.3; alternatively, we may simply label the dotted line connecting these nodes, as in Figure 2.4.4. The interpretation of Prisoner 2's information set in Figure 2.4.3 is that when Prisoner 2 gets the move, all he knows is that the information set has been reached (i.e., that Prisoner 1 has moved), not which node has been reached (i.e., what she did). We will see in Chapter 4 that Prisoner 2 may have a conjecture or belief about what Prisoner 1 did, even if he did not observe what she did, but we will ignore this issue until then.

As a second example of the use of an information set in representing ignorance of previous play, consider the following

Figure 2.4.4.

dynamic game of complete but imperfect information:

1. Player 1 chooses an action a\ from the feasible set A\ — {L,R}.

2. Player 2 observes a\ and then chooses an action a2 from the feasible set A2 = {L',R'}.

3. Player 3 observes whether or not (ai7a2) = (R, R') and then chooses an action from the feasible set A3 — {L",R"}.

The extensive-form representation of this game (with payoffs ignored for simplicity) is given in Figure 2.4.4. In this extensive form, player 3 has two information sets: a singleton information set following R b^t player 1 and R' by player 2, and a nonsin-gleton information set that includes every other node at which player 3 has the move. Thus, all player 3 observes is whether or not (aj,a2) = (R,R').

Now that we have defined the notion of an information set, we can offer an alternative definition of the distinction between perfect and imperfect information. We previously defined perfect information to mean that at each move in the game the player with the move knows the full history of the play of the game thus far. An equivalent definition of perfect information is that every information set is a singleton; imperfect information, in contrast, means that there is at least one nonsingleton information set.19 Thus, the extensive-form representation of a simultaneous-move game (such as the Prisoners' Dilemma) is a game of imperfect information. Similarly, the two-stage games studied in Section 2.2.A have imperfect information because the actions of players 1 and 2 are simultaneous, as are the actions of players 3 and 4. More generally, a dynamic game of complete but imperfect information can be represented in extensive form by using nonsingleton information sets to indicate what each player knows (and does not know) when he or she has the move, as was done in Figure 2.4.4.

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