978-0393919684 Chapter 8 Solution Manual Part 2

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Oscar’s play

after Felix’s payoff when card draws are

Felix’s

expected

payoff

Hi Lo Hi Lo HiHi HiLo LoHi LoLo

P P F F 0 8 –8 0 0

P P S S 0 8 –8 0 0

P P S F 0 8 –8 0 0

The resulting payoff table follows:

Oscar

SS SF q-mix

In the mixed-strategy equilibrium, Felix plays each of RR and RP with probability 0.5, and Oscar

(d) Felix’s decision may reveal information (whether he drew High or Low), and he has an

S9. (a) The extensive form of Stripped-down Poker is shown below:

Solutions to Chapter 8 Solved Exercises 2 of 11

(b) Felix has two actions at each of two nodes, so he has 2 * 2 = 4 strategies. Oscar has two

actions at a single information set, so he has two strategies.

Feli

Oscar

Call Fold

Bet if K, Bet if Q (BB) 0, 0 1, –1

The payoffs in the table are the expected payoffs to the players. For example, if Felix plays BF

Solutions to Chapter 8 Solved Exercises 3 of 11

(d) FB is dominated by BB, and FF is dominated by BF. The reduced game table, with best

responses underlined, is

Feli

Oscar

Call Fold

The game has no equilibrium in pure strategies, but it has a mixed-strategy Nash equilibrium

where Felix plays BB with probability 1/3 and Oscar Calls with probability 2/3.

The expected payoff to Felix in equilibrium is a weighted average of the four possible payoffs in

S10. (a) If Wanda has a good year, reports a low income, and is then audited, she will have to pay

(b) Wanda has an incentive to bluff, which in this case would consist of her reporting a low

Solutions to Chapter 8 Solved Exercises 4 of 11

(d) Wanda has two actions at each of two nodes, so she has 2 * 2 = 4 strategies. The IRS has

two actions at a single information set, so it has two strategies.

(e) The strategic form follows:

IRS

A N

Strategy LH is dominated by strategy LL, and strategy HH is dominated by both LL and HL. The

smaller version of the game table, with best responses underlined, is thus:

Solutions to Chapter 8 Solved Exercises 5 of 11

IRS

A N

There is no pure-strategy Nash equilibrium. The mixed-strategy Nash equilibrium occurs when

Wanda plays LL with probability 1/6 and the IRS audits with probability 5/6.

Since Wanda always signals L when she has a bad year and sometimes—but not always—also

signals L when she has a good year, this game has a semiseparating equilibrium.

(f) When x is quite low, the IRS has a negative expected payoff from auditing, so that

strategy N dominates strategy A. There is then a pure-strategy Nash equilibrium where Wanda always

reports a low income and the IRS never audits. For example, when x is 0.1, the payoff table becomes

IRS

A N

Wanda always reports a low income in the pooling equilibrium of this game.

(g) For an arbitrary x, where x is the probability that Wanda has a good year, the reduced

payoff table follows:

IRS

A N

Solutions to Chapter 8 Solved Exercises 6 of 11

By the reasoning given in part (f), we need to find the range of values of x such that the payoff to

the IRS from not auditing is greater than or equal to its payoff from auditing when Wanda reports a low

income. That is, we need to find the set of x such that

So for all x between 0 and 0.2 (inclusive), Wanda always reports low income in equilibrium.

S11. The following is a much more involved discussion than the exercise calls for. As you can see, the

issue is quite complex. Even the outline below is not meant to be exhaustive; other issues will probably

occur to you.

Information Limitations and Asymmetries

Users (Potential and Actual Patients)

1. Know more about their own health risks (creates adverse selection) and control aspects of

2. Don’t know their own diagnosis, and don’t know the appropriate treatment (are poorly

3. Don’t know the providers’ quality or effort for sure (face adverse selection and moral

Among Providers

Insurers

Solutions to Chapter 8 Solved Exercises 7 of 11

and moral hazard).

Government

1. Can centralize information and reduce adverse selection and moral hazard.

Systemwide Objectives

1. Universal coverage

2. Quality of care

Different systems of insurance, charges to users, payment to providers, and so on affect the

Users’ Incentives

Conventional insurance, with 80% or more coverage, gives the users an incentive to overdemand services.

The problem would be much more severe for prepaid or HMO-type systems, because the patient’s

Solutions to Chapter 8 Solved Exercises 8 of 11

If consumers had to pay a much greater share of the cost of each use, this incentive problem

would be reduced. But then access would be limited by income or wealth even more than it is now, and

coverage would be even less universal.

Providers’ Incentives

Under the fee-for-service system, doctors and hospitals have the incentive to overprovide services. When

patients have insurance coverage, they offer no opposition. For items such as diagnostic tests, this is

Under a prepaid or HMO-type system, the providers can have the opposite incentive: to shirk or

to provide minimal care. This can be controlled by using reputation mechanisms in repeated relationships.

HMOs have more incentive to offer better preventive care. So should public (national health)

Doctors, hospitals, and equipment and drug makers have mixed incentives to innovate. Technical

Insurers’ Incentives

Adverse selection can lead to cream skimming. Each insurance company practices screening by offering a

contract that is attractive only to the healthiest individuals. Then a residual insurer, such as Blue Cross or

Solutions to Chapter 8 Solved Exercises 9 of 11

Cream skimming and the resulting loss of coverage or increase in cost to residual groups can be

avoided by requiring pooling: each plan has to have certain basic provisions and must accept all comers.

A nationalized system does this automatically, enrolling everyone into the same pool at birth.

S12. The most obvious actions include the gentleman’s trying to signal “great cook” or “romantic” or

“sophisticated” by serving a great dinner and cappuccino with dessert, the gentleman’s trying to

S13. We’re looking for the Prob(B conditional on Y). By Bayes’ rule, we have

Prob(B conditional on Y) = Prob(B and Y)/Prob(Y)

Enumerating all of the cases, of the 10,000 total people there are 100 who have the defect and 9,900 who

S14. (a) fupdated is the fraction of Citruses that successfully complete the trip and that are oranges.

That is, fupdated is the number of oranges divided by the number of Citruses that successfully complete the

fupdated is greater than f because half of the lemons have been weeded out from consideration (they broke

down during the trip). Therefore the fraction of oranges among this smaller pool of Citruses is larger.

Solutions to Chapter 8 Solved Exercises 10 of 11

(b) As seen in part (a), once a Citrus has successfully completed the trip, it is more likely to

be an orange. Thus the expected value of a random Citrus from this smaller pool will be higher for a

prospective buyer. A buyer will be willing to pay

price buyers are willing to pay to $13,500. If an owner of an orange doesn’t take the trip, the car will be

(d) If an owner of a lemon doesn’t take the trip, the car will be assumed to be a lemon by

prospective buyers. It will be sold for $6,000, and the owner will earn $3,000 surplus. A lemon that

(e) Since the owners of the two types of Citrus both decide to attempt the trip—sending the

same signal—this is technically a pooling equilibrium. However, since some of the lemons fail to

Solutions to Chapter 8 Solved Exercises 11 of 11