Exercise 3.1, p.79
The strategic form game is shown in FIGURE SOL3.1.1. As the mugger has one information
set (the initial node) and three actions, he has three strategies.
Simon has two information sets; one is associated with the mugger showing a gun (w
Ex 7.2, p.212
ANSWER.
There is no Nash equilibrium in pure strategies. Let us conjecture that an
equilibrium has both the batter and pitcher randomizing. Let p denote the
frequency with which a batter sets up for a fast ball and q denote the frequency
wit
Ex. 6.7, p.176
ANSWER.
Examining player 1s payoff function, note that it is increasing in his strategy:
V1 ( x1 , x 2 )
x1
= 1 + 10 x 2 > 0
Hence, regardless of player 2s strategy, player 1 wants to set x1 at the highest
feasible value. We then have the
Ex. 6.8, p.176
ANSWER.
Country 1s optimal arms expenditure is that value for x1 for which the first order and
the 2nd order conditions hold. 1s best reply function is given by:
x1 = BR1 ( x2 ) = 6 x2 x2 . Note that, as a function of x2, BR1(x2) achieves m
Ex 7.1, p.212
ANSWER.
First, note that there are two pure strategy Nash equilibria.
Next, let p denotes the probability that Colleen chooses call and q the probability that
Winnie chooses call.
Winnie must randomize in such a way that Colleen is indiffere
Ex 7.6, p.213
ANSWER.
Note that c strictly dominates a and y strictly
dominates z. Thus, any Nash equilibrium in mixed
strategies must assign zero probability to the
dominated strategies a and z. We can then
eliminate them, so the reduced game is as shown
Ex 7.8, p.214
ANSWER.
Note typos. Every number in the payoff table should be preceded by a
decimal, as they all represent probabilities of scoring or no score.
Let l denote the probability that the offense chooses to run wide left, r the
probability it ru
Ex 8.1, p.244
a.
The strategic form games are shown in FIG SOL8.1.1.
The Nash equilibria are:
(DNR, DNC/DNC, torture), (R, C/C, torture), (R, C/DNC, torture),
(DNR, DNC/C, do not torture), (DNR, DNC/DNC, do not torture).
b.
In his last decision node (whic
Ex 8.6, p.249
ANSWER: Begin with Shieldss final four decision nodes.
If Lincoln proposes guns, then Shields will go forward with the duel, as his payoff
is 8, while it is 0 if he were to back down.
If Lincoln proposes guns with an apology, then Shields
Ex 8.9, p.251
ANSWER: Start with the final decision nodes for player 1. If player 1 chose a1 and
player 2 chose a2, then player 1 will optimally choose d1. If player 1 chose a1 and
player 2 chose b2, then player 1 will optimally choose e1. Substituting th
Ex 9.1, p.281
ANSWER: After solving for the two (proper) subgames, the extensive form game
is as shown in FIGURE SOL9.1.1:
and its strategic form is shown in FIGURE SOL9.1.2.
For player 2, the first action in his strategy is what he chooses when player 1
Ex 9.2, p.281
a. Consider the subgame between companies 2 and 3 associated with IBM having
developed OS/2 and company 1 having developed an application. The strategic form
of the game is shown in FIGURE SOL9.2.1. Develop is a dominant strategy for each
co
Ex 9.4, p.284
a.
b. There are five subgames in FIG SOL9.4.1. As with FIG PR9.3, in the subgames
in which Hermione is asked by Ron and either Victor asked and Hermione said no
or Victor did not ask, Hermiones optimal action is say yes. Now consider the
sub
Exercise 4.11, p.114
Answer.
Recall that there are 5 Nash eqa:
(A, A, A), (B, B, B), (C, C, C), (B, B, C), & (A, C, C).
For player 1, A weakly dominates C. This argues to eliminating eqm (C, C, C).
For player 2, B weakly dominates A & C, which argues to e
Exercise 4.4, p.113
Answer. a.
First observe that for player 1, strategy c is strictly dominated by strategy a. Delete
strategy c.
In the reduced game, note that for player 2, strategy x is strictly dominated by strategy y.
Delete strategy x.
What we h
Exercise 4.2, p.112
Answer. In addition to (drive left, drive left) and (drive right, drive right), (zigzag,
zigzag) is also a Nash equilibrium. Note that though drive right is a best reply for
driver 1 to driver 2’s choice of zigzag, (drive right, zigzag
Ex 9.6, p.285
a. ANSWER: It has a unique Nash equilibrium of (High, High) with each trading
company making profit of 57,500.
b. ANSWER: The Nash equilibria for the subgames will be summarized here.
If both managers are given profit-based contracts, then
Ex 9.11, p.286
ANSWER: The strategic form game of the subgame associated with the path no
Pitchout no suicide squeeze is shown in FIG SOL9.11.1.
This game has no pure strategy Nash eqm. Suppose p is the probability that LaRussa
chooses suicide squeeze, an
Ex. 10.1, p.313
A strategy for Marcia is a pair of actions: what to do if she likes Greg & he asks her
out, and what to do if she doesnt like Greg & he asks her out. Her optimal strategy
is fairly clear. If she likes Greg, then by accepting his invitation
Econ 2191 / Exercise Gaga / Answers
1.
3.
2
DNE
E
DNE
4,4
1,5
E
5,1
0,0
1
4.
B
Write 1
Write 2
Write 1
2,2
3,3
Write 2
3,3
4,4
A
2.
5.
Colin
I
DNI
S
0,h
w,w
W
wc,
wh
wc,
w
Roy
Econ 2191 / Exercise Gaga / Questions
1.
Firm A decides whether to enter (E) firm Bs industry or not (D). Firm B observes this
decision. If firm A enters, then the two firms simultaneously decide whether to advertise
(a). Otherwise, firm B alone decides w
Econ 2191 / Exercise Irina / Ques
1.
Write down the strategic (normal) form game for the following extensive form game.
2.
Write down the strategic (normal) form game for the following extensive form game.
3.
Write down an extensive form game for the foll
Econ 2191 / Spring 2012
MotherChild Game Due immediately before class starts on Tuesday, April 24
3-Point Extra Bonus, Question sheet
Put all your answers on the separate answer sheet provided by me. No other sheet will be accepted.
Consider a game betwee
Econ 2191 / Spring 2012
MotherChild Game
3-Point Extra Bonus, Answer sheet
Name _
(A)
Mother
Nash eqa and eqm payoffs:
P
NP
B
1 , 3
1 , 2 *
G
2 , 1
0,0
Child
(B)
_
_
_
The game tree:
SPNE and eqm payoffs:
_
_
(C)
Nash eqa and eqm payoffs
Mother
P/P
(alway
Exercise 3.4, p.80
Answer.
a. For player 1, a is strictly dominated by b. Neither b nor c is strictly dominated. For
player 2, z is strictly dominated by x. Player 1 plays either b or c and player 2 plays
either x or y.
b. By the assumption, we can go two
Ex 9.5, p.284
a. Consider the final subgame associated with the letter having been sent. The
strategic form is shown in FIG SOL9.5.1. Defect is a dominant strategy for both
Borodin and Melekhin, so there is a unique Nash eqm of (defect, defect).
Next, con