Simplifying the Extensive Form of the Haggling Game
Accept Reject
A Leave
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DonnaDonna Donna
A R
Marcus
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Marcus Marcus Marcus 0
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0pM � pH
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A R
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pM � pL
pL pM
pM
pL A LeavepL pM
pH
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2(pM � pL) 2(pH � pL)
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0pM � pL
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pM � pL
2(pM � pL)
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2(pM � pL)
Consider a two-player game in which a father chooses between actions yes, no, and maybe. His daughter moves second and chooses between stay home and go to the mall. The payoffs are as follows:
Outcome Father’s Payoff Daughter’s Payoff
yes and stay home 8 3
yes and go to the mall 5 9
no and stay home 4 1
no and go to the mall 1 5
maybe and stay home 7 2
maybe and go to the mall 2 7
2.1 CHECK YOUR UNDERSTANDING*
Write down the extensive form game for this strategic situation.
*All answers to Check Your Understanding are in the back of the book.
2.3 Extensive Form Games: Imperfect Information 27
2.3 Extensive Form Games: Imperfect Information RETURNING TO THE KIDNAPPING SCENARIO, suppose we want to model Guy (the kid- napper) and Vivica (the victim’s kin) as making their decisions without knowl- edge of what the other has done. The extensive form game in Figure 2.1 as- sumes that Guy learns whether ransom has been paid prior to deciding what to do with Orlando (the victim). An alternative specification is that Guy de- cides what to do with Orlando at the same time that Vivica decides about the ransom. You could imagine Guy deciding whether to release Orlando some- where in the city while Vivica is deciding whether to leave the ransom at an agreed-upon location. How do you set up an extensive form game with that feature?
The essential difference between these scenarios is information. In Figure 2.1, Guy knew what Vivica had done when it was time for him to make his decision, whereas Vivica did not know what was to happen to Orlando when she had to decide about paying the ransom. Vivica’s lack of knowledge was represented by having Vivica move before Guy. Now we want to suppose that at the time he has to decide about killing or releasing Orlando, Guy is also lacking knowledge about what Vivica is to do or has done. Well, we can’t make a decision tree in which Vivica moves after Guy and Guy moves after Vivica.
To be able to represent such a situation, the concept of an information set was created. An information set is made up of all of the decision nodes that a player is incapable of distinguishing among. Every decision node belongs to one and only one information set. A player is assumed to know which in- formation set he is at, but nothing more. Thus, if the information set has more than one node, then the player is un- certain as to where exactly he is in the game. All this should be clearer with an example.
FIGURE 2.6 is a reformulation of the Kid- napping game with the new assumption that Guy doesn’t get to learn whether Vivica has paid the ransom when he decides what to do with Orlando. In terms of nodes and branches, the trees in Figures 2.1 and 2.6 are identical. The distinctive element is the box drawn around the two decision nodes associ- ated with Guy choosing whether to release or kill Orlando (which are denoted III and IV). The nodes in that box make up Guy’s infor- mation set at the time he has to decide what to do with Orlando. Guy is assumed to know that the game is at either node III or node IV, but that’s it; he doesn’t know which of the two it is. Think about what this means. Not to know whether the game is at node III or node IV means that Guy doesn’t know whether the sequence of play has been “kid- nap and ransom is paid” or “kidnap and ran- som is not paid.” Well, this is exactly what we wanted to model; Guy doesn’t know whether
FIGURE 2.6 Kidnapping Game When the Exchange Is Simultaneous. The Box Around Nodes III and IV Represents the Information Set at the Point That Guy Has to Decide What to Do with Orlando
Guy
II
I II I
I I II I I IVIV
Guy 3
Vivica 5 Pay ransom
Vivica
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Do not kidnap
Do not pay ransom
Kidnap
Guy
Kill KillRelease Release