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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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) Felix’s strategies PP and PR are dominated. Oscar’s strategies FF and FS are dominated.
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.
(c) The strategic form follows:
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
(c) The extensive form is shown below:
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
(c) The cost of the trip to the owner of an orange is $250, but making the trip increases the
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
