171 NAME:
Econ (Grossman) Spring 2010
Exam 2
May 13
You have 75 minutes to take this exam. Please answer all 4 questions, each of which is worth
15 points. Point subtotals are indicated. Show your work to obtain full credit.
1. Consider the extensive-form game shown in Figure 1.
1
L
l1
1, 2
2
R
r1
l2
2, 1
0, 3
2
r2
1
l
2, 2
r
1, 4
Figure 1: The game from question 1.
(a) [5] List the strategies for each player.
(b) [5] Find the Nash Equlibria.
(c) [5] Apply backwards induction and state the resulting SPNE.
1
1
A
B
2
C
5, 5
D
1
E
F
6, 6
E
4, 0 0, 4
F
4, 4
Figure 2: The game from question 2.
2. Consider the game shown in Figure 2.
(a) [5] Find all the pure-strategy Nash equilibria of this game.
(b) [5] Which of these are subgame perfect?
(c) [5] Challenge question : There is a concept called forward induction that can be
used to argue that one of the SPNE is unreasonable. We havent studied it in
class, but the name kind of gives you a hint as to how to apply it. Look at the
SPNE and try to come up with an argument for why one of them should be eliminated. Briey explain your argument.
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3. Players 1 and 2 play a three-period alternating-oer bargaining game over a pie of size
2. They have discount factor , with 0 < < 1. In the rst period, player 1 makes
an oer, which 2 can either accept or reject. If 2 rejects, she gets to make an oer in
the second period. If 1 rejects this oer, then in the third period the pie shrinks to size
1 player and 1 can make one nal oer. If this oer is rejected, then both players get
nothing.
(a) [5] In the unique SPNE of this game, what oer does player 1 make in period
3? Does player 2 accept or reject it? (Write the oer as (x, y ), where x is the
amount that player 1 would get and y is the amount that player 2 would get.)
(b) [5] What oer does player 2 make in period 2? Does player 1 accept or reject it?
(c) [5] What oer does player 1 make in period 1? Does player 2 accept or reject it?
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4. Consider two players who own a rm and want to dissolve their partnership. Each owns
half of the rm. The value of the rm for players A and B is va and vB , respectively,
where vA > vB > 0. They have agreed to proceed as follows.
Player A sets a price p for half of the rm. Player B then decides whether to sell his
share or to buy As share at this price.
If B decides to sell his share, then A will own the entire rm and pays p to B.
This yields payos vA p and p, for players A and B, respectively.
If B decides to buy, then B owns the entire rm and pays p to A, yielding payos
p and vB p for A and B, respectively.
(a) [5] Find Bs best response, as a function of the p chosen by A.
(b) [5] Is it the case that B strictly prefers one of the actions (buy or sell) in equilibrium? Why or why not? Which action must B actually choose in equilibrium?
Why cant B choose the other action?
(c) [5] Fully describe the subgame-perfect Nash equilibrium of this game.
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