SITUATION: HAGGLING AT AN AUTO DEALERSHIP, II
This game is more complicated than the ones we have considered thus far. Consider the version in Figure 2.5. Marcus has four information sets: (1) the ini- tial node, (2) the node associated with his having offered pM and Donna’s having rejected it and made a counteroffer of pL, (3) the node associated with his having offered pH and Donna’s having rejected it and made a counteroffer of pM, and (4) the node associated with his having offered pH and Donna’s having rejected it and made a counteroffer of pL. Marcus’s strategy template is then as follows:
At the initial node, offer _____ [fill in pL, pM, or pH].
If I offered pM and Donna rejected it and offered pL, then _____ [fill in accept or reject].
If I offered pH and Donna rejected it and offered pM, then _____ [fill in accept or reject].
If I offered pH and Donna rejected it and offered pL, then _____ [fill in accept or reject and offer pM].
If you write them all out, you will see that there are 24 distinct strategies for Marcus—in other words, 24 different ways in which to fill out those four blanks.
Donna has four information sets, and her strategy template is the following:
If Marcus offered pL, then _____ [fill in accept or reject].
If Marcus offered pM, then _____ [fill in accept, reject and offer pL, or reject and leave].
If Marcus offered pH, then _____ [fill in accept, reject and offer pL, reject and offer pM, or reject and leave].
If Marcus offered pH, I rejected and offered pL, and Marcus rejected and offered pM, then _____ [fill in accept or reject].
Donna has 48 strategies available to her. These are a lot of strategies, but keep in mind the complete nature of a strategy: no matter where Donna finds her- self in the game, her strategy tells her what to do.
Suppose Marcus and Donna chose the following pair of strategies. For Marcus:
At the initial node, offer pH.
If I offered pM and Donna rejected it and offered pL, then reject.
If I offered pH and Donna rejected it and offered pM, then accept.
42 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
If I offered pH and Donna rejected it and offered pL, then reject and offer pM.
For Donna:
If Marcus offered pL, then accept.
If Marcus offered pM, then accept.
If Marcus offered pH, then reject and offer pL.
If Marcus offered pH, I rejected and offered pL, and Marcus rejected and offered pM, then accept.
With this strategy pair, let us determine the sequence of play that logically fol- lows and thereby the associated payoffs. At the initial node, Marcus offers a price of pH, as prescribed by his strategy. According to Donna’s strategy, she rejects the offer and counters with a price of pL. In response to that offer, Marcus’s strategy tells him to reject it and counteroffer with pM (reading from the bottom line of his strategy). Finally, Donna’s strategy has her accept the offer of pM. The path of play that emerges is then as follows: Marcus offers a price of pH, Donna rejects the offer and proposes a price of pL, Marcus rejects and counters with a price of pM, and Donna accepts. The transaction is then made at a price of pM. For this strategy pair, the associated payoffs are (pM � pM) or zero, for Donna and 2(pM � pL), for Marcus.
2.7 Going from the Strategic Form to the Extensive Form ALTHOUGH EVERY EXTENSIVE FORM GAME has a unique strategic form game associ- ated with it, the same strategic form game can be associated with more than one extensive form game. This means that when we move from the extensive form to the strategic form, we lose some information, but, as we’ll explain, the lost information is irrelevant.
Shown in FIGURE 2.15 are two extensive form games, both of which gener- ate the strategic form game in Figure 2.11. In the game in Figure 2.15(a),
FIGURE 2.15 Two Extensive Form Games That Generate the Same Strategic Form Game
(a) (b)
Scarpia
Tosca
Scarpia
2
2
Real
4
1
1
4
3
3
Blanks
Tosca
Stab StabConsent Consent
2
2
4
1
1
4
3
3
Tosca
Scarpia
Tosca
Scarpia
Real Blanks Real Blanks
Stab Consent
2.8 Common Knowledge 43
Scarpia moves first and then Tosca moves, but Tosca has only one information set, which indicates that she doesn’t know what Scarpia chose when she de- cides between stab and consent. By this time, it ought to be straightforward to show that this extensive form game produces the strategic form game depicted in Figure 2.11.
The game in Figure 2.15(b) is the same as that in Figure 2.15(a), except that the sequencing of players has been reversed; still, it produces the same strate- gic form game, and it makes sense that it does. We’ve pre- viously argued that what matters is not the chronological order of moves, but rather what players know when they act. In both of these extensive form games, Scarpia doesn’t know Tosca’s move when he acts; in the game in Figure 2.15(a), it is because he moves first, and in the game in Figure 2.15(b), it is because his information set includes both of Tosca’s ac- tions. Similarly, in both games, Tosca doesn’t know what Scarpia has told the firing squad when she makes her choice.
2.8 Common Knowledge JACK AND KATE ARE TO meet at the French restaurant Per Se in New York City. Jack has since learned that the restaurant is closed today, so he e-mails Kate, suggesting that they meet at 7 P.M. at Artie’s Delicatessen, their second-favorite place. Kate receives the e-mail on her BlackBerry and e-mails back to Jack, saying she’ll be there. Jack receives her confirmation. Kate shows up at Artie’s at 7 P.M. and Jack is not there. She wonders whether Jack received her reply. If he didn’t, then he might not be sure that she had received the message, and thus he may have gone to Per Se with the anticipation that she would go there. It’s 7:15, and Jack is still not there, so Kate leaves to go to Per Se. It turns out that Jack was just delayed, and he’s surprised to find that Kate is not at Artie’s when he arrives there.
The problem faced by Jack and Kate is what game theorists call a lack of common knowledge. Jack knows that Per Se is closed. Kate knows that Per Se is closed, because she received Jack’s message telling her that. Jack knows that Kate knows it, since he received Kate’s confirmation, and obviously, Kate knows that Jack knows it. But Kate doesn’t know that Jack knows that Kate knows that Per Se is closed, because Kate isn’t sure that Jack received her confirming mes- sage. The point is that it need not be enough for Jack and Kate to both know that the restaurant is closed: they may also need to know what the other knows.
To be a bit more formal here, I’m going to define what it means for an event (or a piece of information) to be common knowledge. Let E denote this event. In the preceding example, E is “Per Se is closed and meet at Artie’s.” E is com- mon knowledge to players 1 and 2 if
■ 1 knows E and 2 knows E.
■ 1 knows that 2 knows E and 2 knows that 1 knows E.
■ 1 knows that 2 knows that 1 knows E and 2 knows that 1 knows that 2 knows E.
■ 1 knows that 2 knows that 1 knows that 2 knows E and 2 knows that 1 knows that 2 knows that 1 knows E.
■ And so on, and so on.
Write down the strategic form game for the Mugging game in Figure 2.8.
2.4 CHECK YOUR UNDERSTANDING
44 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
Are we there yet? No, because this goes on ad infinitum. Common knowl- edge is like the infinite number of reflections produced by two mirrors facing each other. Here, the “reflection” is what a player knows. Common knowledge, then, is much more than players knowing something: it involves them know- ing what the others know, and knowing what the others know about what the others know, and so forth.
The concept of common knowledge is quite crucial because an underlying assumption of most of what we do in this book is that the game is common knowledge to the players. Each player knows that the game that is being played, each knows that the others know that the game that is being played, and so on.