Econ 100C Practice Midterm #2  Answer Key
Question 1
Consider the following sequential game with two stages:
A single seller o ers a good to
a single buyer. In the
rst stage of the game, the buyer can choose between three di erent
strategies: o er $20 for the good, o er $40, or o er $60. In the second stage, the seller can
accept or refuse the o er. The seller values the good at $32. If she does not sell the good,
her payo
is zero. The buyer values the good at $45, and his payo
is zero if the seller rejects
his o er.
a) Draw the game tree.
First, we need to
gure out the payo s for the game.
If a sale occurs, the buyer’s pay
o
will be his value of the item minus the price he pays. The seller’s payo
will be the price
she receives minus her value of the item. If a sale does not occur, both players’ payo s are
zero. We can now draw the game tree as below:
b) Find the subgame perfect Nash Equilibrium of the game.
We can solve for the subgame perfect Nash Equilibrium by backwards induction.
Start
with the seller’s decision. The seller’s response to each of the buyer’s three potential o ers
is a subgame. Since the seller is just comparing her payo
from accepting the o er to her
payo
from rejecting the o er, solving each subgame is very simple.
Subgame 1: buyer o ers $20.
In this subgame, the seller’s payo
from accepting is $12
dollars while her payo
from rejecting is $0. Therefore, the seller will reject the o er in this
case, getting a payo
of $0. This means that the buyer can expect a payo
of $0 if he o ers
$20.
Subgame 2: buyer o ers $40. In this subgame, the seller’s payo
from accepting is $8 dollars
while her payo
from rejecting is $0. Therefore, the seller will accept the o er in this case,
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
getting a payo
of $8. This means that the buyer can expect a payo
of $5 if he o ers $40.
Subgame 3:
buyer o ers $60.
In this subgame, the seller’s payo
from accepting is $28
dollars while her payo
from rejecting is $0. Therefore, the seller will accept the o er in this
case, getting a payo
of $28. This means that the buyer can expect a payo
of $15 if he
o ers $60.
Initial decision: how much to o er? Now that we know the buyer’s payo s from each o er he
might make, we can select the best one. Therefore, the buyer o ers $40, the seller accepts the
o er, and the buyer earns $5. That means of subgame perfect Nash Equilibrium is [(O er
$40), (Reject if o er = $20, Accept if o er = $40, Accept if o er = $60)].
c) Now, assume that the order of the moves is reversed. In the
rst stage, the seller can o er
the good for $20, $40 or $60, and in the second stage, the buyer can accept or refuse the
o er. Draw the new game tree and
nd the subgame perfect Nash Equilibrium.
Note that the possible outcomes, and therefore the payo s of the game are unchanged from
the previous parts.
To construct a new game tree, all we need to do is switch the player
order.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Ohly
 Economics, Harshad number, BR Firm B, Industry MC, BR Firm, Qcoll Qcour

Click to edit the document details