BUILDING A MODEL OF A STRATEGIC SITUATION
� SITUATION: HAGGLING AT AN AUTO DEALERSHIP, I
Donna shows up at her local Lexus dealership looking to buy a car. Coming into the showroom and sauntering around a taupe sedan, a salesperson, Marcus, appears beside her. After chatting a bit, he leads the way to his cu- bicle to negotiate. To simplify the modeling of the negotiation process, sup- pose the car can be sold for three possible prices, denoted pL, pM, and pH, and suppose pH > pM > pL. (H is for “high,” M is for “moderate,” and L is for “low.”)
The extensive form game is depicted in FIGURE 2.4. Marcus initially decides which of these three prices to offer Donna. In response, Donna can either ac- cept the offer—in which case the transaction is made at that price—or reject it. If it is rejected, Donna can either get up and leave the dealership (thereby end- ing the negotiations) or make a counteroffer. In the latter case, Donna can re- spond with a higher price, but that doesn’t make much sense, so it is assumed
Accept Reject A R
A R A R A R
A R
A R
Leave
DonnaDonna Donna
Donna
Donna
Donna
Marcus
Marcus
Donna
0
Marcus Marcus Marcus
Marcus 0
0
0
0
0
0
0
0 0
2(pM � pL)
0
0 0
2(pM � pL)
0
00
2(pM � pL)
2(pH � pL)
pM � pL
0
pM � pL pM � pL 0
pM � pH
pL pM
pLLeavepL pM
pM
pH
FIGURE 2.4 Haggling at an Auto Dealership
2.2 Extensive Form Games: Perfect Information 25
that she selects among those prices which are lower than what she was ini- tially offered (and turned down). For example, if Marcus offers a price of pH, then Donna can respond by asking for a price of either pM or pL. If Donna has decided to counteroffer, then Marcus can either accept or reject her counterof- fer. If he rejects it, then he can counteroffer with a higher price (though it must be lower than his initial offer). This haggling continues until either Donna leaves, or an offer is accepted by either Donna or Marcus, or they run out of prices to offer.
In terms of payoffs, assume that both Marcus and Donna get a zero payoff if the game ends with no sale. (There is nothing special about zero, by the way. What is important is its relationship to the other payoffs.) If there is a trans- action, Marcus’s payoff is assumed to be higher when the sale price is higher, while Donna’s payoff is assumed to be lower. More specifically, in the event of a sale at a price p, Donna is assumed to receive a payoff of pM � p and Marcus gets a payoff of 2(p � pL). (Why multiply by 2? For no particular reason.)
Think about what this is saying. If Marcus sells the car for a price of pL, then his payoff is zero because 2(pL � pL) � 0. He is then indifferent between selling it for a price of pL and not selling the car. At a price of pM, his payoff is positive, which means that he’s better off selling it at that price than not selling it; and his payoff is yet higher when he sells it for pH. For Donna, she is indifferent between buying the car at a price of pM, and not buying it, since both give the same payoff (of zero). She prefers to buy the car at a price of pL, as that price gives her a payoff of pM � pL > 0; she is worse off (relative to not buying the car) when she buys it at a price of pH, since that gives her a payoff of pM � pH < 0. (Yes, payoffs can be negative. Once again, what is important is the ordering of the payoffs.) These payoffs are shown in Figure 2.4.
To be clear about how to interpret this extensive form game, consider what can happen when Marcus initially offers a price of pH. Donna can either accept—in which case Marcus gets a payoff of 2(pH � pL) and Donna gets a payoff of pM � pH—or reject. With the latter, she can leave or counteroffer with either pL or pM. (Recall that we are allowing her to counteroffer only with a price that is lower than what she has been offered.) If Donna chooses the counteroffer of pL, then Marcus can accept—resulting in payoffs of zero for Marcus and pM � pL for Donna—or reject, in which case Marcus has only one option, which is to counteroffer with pM, in response to which Donna can ei- ther accept or reject (after which there is nothing left to do). If she instead chooses the counteroffer pM, then Marcus can accept or reject it. If he rejects, he has no counteroffer and the game ends.
It is worth noting that this extensive form game can be represented alter- natively by FIGURE 2.5. Rather than have the same player move twice in a row, the two decision nodes are combined into one decision node with all of the available options. For example, in Figure 2.4, Donna chooses between accept and reject in response to an initial offer of pM from Marcus, and then, if she chooses reject, she makes another decision about whether to counteroffer with pL or leave. Alternatively, we can think about Donna having three options (branches) when Marcus makes an initial offer of pM: (1) accept; (2) reject and counteroffer with pL; and (3) reject and leave. Figure 2.5 is a representation equivalent to that in Figure 2.4 in the sense that when we end up solving these games, the same answer will emerge.