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U8. (a) The extensive form of this game is shown below:
(b) eBay has an incentive to bluff, which in this case would consist of proposing a stingy
(c) The game table follows:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
AT&T
Accept Reject
S if i, S if g (SS) –20, 20
–320 • 0.25 = –80,
–20 • 0.75 + 300 • 0.25 = 60
Strategy GS and GG are dominated by both SS and SG. The truncated game table is thus:
AT&T
Accept Reject
There is no pure-strategy Nash equilibrium. The mixed-strategy Nash equilibrium occurs when eBay
U9. If k is the fraction of Kings in the deck, the expected payoffs from the game are as follows:
Oscar
Call Fold
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Bet if K, Bet if Q (BB) 4k – 2 , –4k + 2 1, –1
equilibria yield an expected payoff of (–1, 1), so this is not a fair game.
Notice that when 0 < k < 1, strategy FB and strategy FF are dominated by BB and BF,
When k ≥ 3/4 there is a unique pure-strategy Nash equilibrium: (BB, Fold), which has an
expected payoff of (1, –1). Thus whenever k ≥ 3/4 the resulting game is not fair.
( ) ( ) ( ) ( ) ( )
4 2 3 1 1 1 2 1 1 3 3
k
k p k p p k p p k
- + + - + - =- + - + - Þ = -
( ) ( ) ( ) ( ) ( )
2 2 2
4 2 1 1 3 1 2 1 1 3 3 3
k
k q q k q k q q k
-
- + - = - + - - Þ = =
-
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
The mixed-strategy Nash equilibrium occurs where Felix plays BB with probability p = k/(3 – 3k) and
Oscar plays Call with probability q = 2/3.
Felix’s expected payoff from the mixed-strategy equilibrium is given by
( ) ( ) ( ) ( )
2
2 1 3 4 2 3 4 1
4 2 1 3 1 2 1
3 3 3 3 3 3 3 3 3 3 3 3
24 33 9 (8 3)(3 3 )
8 3.
3 3 3 3
k k k k
k k k
k k k k
k k k k k
k k
- -
æ öæ ö æ öæ ö æ öæ ö æ öæ ö
- + + - + -
ç ÷ç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷ç ÷
- - - -
è øè ø è øè ø è øè ø è øè ø
- + - - -
= = = -
- -
Felix’s expected payoff is 0 when 8k – 3 = 0; that is, when k = 3/8. Stripped-Down Poker is a fair game
when the deck is composed of a mix of 3/8 Kings and 5/8 Queens.
U10. (a) The extensive form of the game is shown below:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Oscar
Call Fold
Bet if K, Bet if Q, Bet if J (BBB) 0, 0 1, –1
Bet if K, Bet if Q, Fold if J (BBF) 1/3, –1/3 1/3, –1/3
(e) The unique pure-strategy Nash equilibrium is (BBF, Call). That is, Felix Bets when he
(f) This is a semiseparating equilibrium: when Felix Folds he reveals his card type fully, but
(g) The expected payoff of the game is 1/3 for Felix, so on average, Felix wins 1/3 and Oscar
U11. (a) Let E1 and E2 represent the education level of types 1 and 2, respectively. If types are
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) If a type 2 worker were to take up the contract intended for a type 1, she would be paid
(c) The contract for a type 1 worker is the same: W1(E1) = E1. Let the contract for a type 2
To achieve separation, the incentive compatibility constraints are
The participation (or individual rationality) constraints for the two types are
Assume that IR1 binds, that is, E1 – (E12)/2 = 0. Then E1 = 0 or 2. Consider first the case where E1 = 0. The
incentive compatibility constraints become
Thus, when E1 = 0, employers can achieve separation with the contract:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
1
2 2
0 if 0
( ) if , where 2 3
E E
W E E E E Ew w w
= =
ì
=í= £ £
î
Now consider the case where E1 = 2. The incentive compatibility constraints become
Thus when E1 = 2 employers can achieve separation with the contract:
1
2
2 2
2 if 2
( ) 3 9 8
if , where 2 2
E E
W E
E E E E w w
w w
= =
ì
ï
=í+ -
= £ £
ï
î
(d) Employers will probably want to exploit the type 2 workers by offering a contract such
that ω is only slightly greater than 1 when they observe an education level of E2 ≥ 2ω. That way they can
(e) As seen in part (d), employers have an opportunity to gain from the information
Type 2 workers lose from this information asymmetry. They need to undertake more costly
Type 1 workers are paid their productivity and receive a surplus of zero in both the symmetric or
information cases. They neither win nor lose.
U12. Yes, the act of throwing away two years can demonstrate a commitment to a business career.
Stanford admits only smart people, but a corporation could also presumably determine who was smart.
A corporation will be willing to pay more to hire a highly committed person. This is because a
more committed applicant is likely to work harder when hired and not likely to shift careers after the
Now consider somebody who has (private) doubts about a career in business. He may well
consider learning about business to be more costly than the more committed applicant and expect that his
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U13. (a) As seen in Exercise S10, Wanda has a good year with probability 0.6, and a bad year with
probability 0.4. When she plays according to her equilibrium strategy, she always reports having a bad
( )
1
0.6 6
Pr good year | Wanda reports a bad year 0.2.
1
0.4(1) 0.6 6
æ ö
ç ÷
è ø
= =
æ ö
+ç ÷
è ø
In equilibrium Wanda is lying with 20% probability when she reports having had a bad year.
(b) The answer in part (a), 0.2, is smaller than the baseline probability of Wanda having a
good year, 0.6, because in equilibrium she reports having had a bad year much more often when she has
U14. (a) No matter how far the journey, an orange will complete the trip successfully. However, a
lemon will only complete the trip successfully with probability 1 – q, where q = m/(m + 500). Let fupdated be
( )
( ) ( )
updated
0.6 1 0.6 300
0.6 500
0.6 1 0.4 1 500
m
fmm
m
+
= = +
æ ö
+ -
ç ÷
+
è ø
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Substituting for fupdated above:
( ) ( ) ( )
2
0.6 300
10,000 6,000 0.5 12,500
0.6 500
10,000 0.6 300 0.5 0.6 500 6,500 0.6 500
0.3 1,850 250,000 0
138.23 6,028.43
mm
m
m m m
m m
m
+
æ ö+ - ³
ç ÷
+
è ø
Þ + - + ³ +
Þ - + - ³
Þ £ £
The smallest integer m that will prevent the market for oranges from collapsing is m = 139.
(b) If owners of lemons attempt the trip, they will succeed with probability 1 – q and receive
the market price pupdated minus a trip cost of 0.5m. However, with probability q they will fail and receive
( )
( ) ( )
( )
( )
updated
updated
3 2
10,000 6,000 0.5 1 6,000 2 6,000
10,000 0.5 1 2 0
0.6 300
10,000 0.5 1 2 0
0.6 500 500 500
1.2 1,150 2,875,000 1,500,00
f m q m q
f m q mq
m m m
m m
m m m
m m m
+ - - + - £
Þ - - - £
æ ö
+
æ ö æ ö æ ö
Þ - - - £
ç ÷ ç ÷ ç ÷
ç ÷
+ + +
è ø è ø è ø
è ø
Þ - - + + 0,000 0
1,397.05m
£
Þ ³
The minimum integer m that will induce owners of lemons to forgo the journey entirely—and thus create
a fully separating equilibrium—is m = 1,398.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
