978-0393919684 Chapter 8 Solution Manual Part 3

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:

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

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 = =

–

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:

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

(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:

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

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

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

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

+ – – + – £

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.