Answers to Back-of-the-Chapter Problems
1. a. The expected values at points E, D, C, B, and A in the decision tree are
b. The manager is confused. Point D is a point of decision: The manager
2. a. Whether the parents should accept or reject the $500,000 settlement
depends on their assessment of the expected monetary award (net of
legal fees) in court. The parents would be wise to sketch a decision tree
and get input from their lawyers about the strength of their case, the
b. The defense lawyers should undertake a similar analysis on behalf of the
hospital. (In fact, there may be the involvement of a third party — the
insurance company obligated to pay part of any malpractice award. This
3. a. The expected value of continuing with its current software strategy is
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b. The “open” strategy is less risky in the sense of having a narrower range
4. According to the decision tree below, the consortium’s most profitable
5. a. The tree lists the six possible outcomes (in thousands of dollars) and
b. E(revenue) = (.2)(120,000) + (.3)(160,000) + (.5)(175,000) = $159,500.
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.
5
$25
6. a. The film would generate an expected loss so should not be launched.
b. The termination option is worth 4 – 0 = $4 million.
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6
$0
7. a. Let’s compute the expected costs (in $ billions) of the respective safety
programs. For the “standard” program, the expected cost is: .160 + (.01)
(10) = $.26 billion. For the “lax” program, the expected cost is: .040 +
b. At a judged 2 percent disaster risk, the (apparent) expected cost of the
“lax” policy is: .040 + (.02)(10) = $.24 billion, making it the appear to
8. The decision tree shows that the best strategy is to attempt a settlement,
accept the $400,000 settlement if this is offered, and otherwise go to
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9. a. If the customer response is weak, MD’s expected profit is: (.5)(20) +
(.5)(-100) = -$40 million. MD’s overall profit (averaging over strong
b. If the customer response is weak, the company does better by “pulling
the plug” a $20 million loss is better than an expected $40 million
10. a. The firm should not undertake the R&D development program. To do
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Court $.4 O.er
.5
.2
.3 $0
$1
$.6
.5
.6
(+.1)
Accept
Accept
Reject
$.95
$.65
.65
.55
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$0
.7 Do Not
c. i) In the joint venture, the chance that both programs fail is (.6)(.7) = .42.
Thus, according to the decision tree, developing both is a losing
proposition.
ii) Developing A alone is most profitable: (.4)(350) + (.6)(-200) = $20
11. The expected utility of pursuing the biochemical approach alone is:
The accompanying decision tree depicts the strategy of trying the
biogenetic approach first and then pursuing the biochemical approach
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48.1
Try
BioGen Pursue
BioChem
Quit
-$20
Success
Failure
Success
“Failure”
$60 (55)
$20 (32)
.7
.3
48.1
$180 (95)
.2
.8
57.5
12. a. From the given responses, we conclude that u($500) = 50, u($1,120) =
b. The function u = 2.24.1y = 2.24y.5, closely tracks the utility values in
part (a). For instance, if y = $500, then u = (2.24)(500.5) = 50.1. If y =
$1,530 then, then we find: u = (2.24)(1,530.5) = 87.6, and so on. In this
*13. The dealer must commit to ordering and selling some number of yachts
(say, Q) before knowing the course of the economy. Recall that the two
price equations are given by PG = 20 – .05Q, and PR = 20 – .1Q. Then,
the expected price required to sell Q yachts is: .6PG + .4PR = 20 – .07Q.
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Discussion Question
1. The case of olestra makes for an interesting discussion. Olestra was a
zero-fat cooking oil approved by the FDA to use in producing snack
foods: primarily chip snacks and pop corn. Eighty percent of subjects
cannot taste any difference between Olestra snacks and snacks made
2. PG&E’s story follows closely the lessons of the BP oil spill. In the push
to reduce costs and enhance revenues, it’s easy for a company to
knowingly, (or unknowingly) pay less attention to inspection and safety
3. Clearly, whether a big-budget Hollywood film will turn out to be a
blockbuster or bust is highly uncertain and unpredictable. Nevertheless,
Disney’s past film track record indicates that animated features (on
smaller budgets) are the studio’s “bread and butter” and far less risky
Bonus Problems: Risky Gambles and Risk Aversion
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1. Two Gambles
a. You are given $1,000 to keep. You then are offered a choice between
b. You are given $2,000 to keep. You then are offered a choice between
c. The majority of individuals choose the sure $500 in part (a) but select
the gamble in part (b). Show that this combination of choices is
Answer
b. If you are like the majority of subjects, you choose the 50/50 gamble.
c. Although they are described differently, the situations in parts (a) and
(b) offer identical choices. In part (a), you can choose to walk away
with a total of $1,500 for certain or to gamble on receiving either
2. Sell Now or Sell Later
Firm A is about to embark on a risky development project that offers
a .2 chance of a $4 million profit and a .8 chance of a $0 profit.
Unexpectedly, firm B proposes a joint venture. In return for its
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$4 (100)
$0 (0)
$0 (0)
$3 (u)
.2
.8
.75
.25
20
.25u
Alone
Partnership
Two years pass, and firm A has overcome most of the
development hurdles. The estimated chances of success are now .8.
Show that whatever firm A’s attitude toward risk, the pair of
decisions just outlined are contradictory. (Hint: There are only three
Answer
In the first decision (see tree), the company chooses to continue on its
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$4 (100)
$0 (0)
$3 (u)
.8
.2
80
Alone
Sell Out
In the subsequent decision, the company sells out, implying u > 80.
Answers to Back-of-the-Chapter Problems
1. a. The expected values at points E, D, C, B, and A in the decision tree are
b. The manager is confused. Point D is a point of decision: The manager
2. a. Whether the parents should accept or reject the $500,000 settlement
depends on their assessment of the expected monetary award (net of
legal fees) in court. The parents would be wise to sketch a decision tree
and get input from their lawyers about the strength of their case, the
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b. The defense lawyers should undertake a similar analysis on behalf of the
hospital. (In fact, there may be the involvement of a third party — the
insurance company obligated to pay part of any malpractice award. This
3. a. The expected value of continuing with its current software strategy is
b. The “open” strategy is less risky in the sense of having a narrower range
4. According to the decision tree below, the consortium’s most profitable
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5. a. The tree lists the six possible outcomes (in thousands of dollars) and
b. E(revenue) = (.2)(120,000) + (.3)(160,000) + (.5)(175,000) = $159,500.
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6. a. The film would generate an expected loss so should not be launched.
b. The termination option is worth 4 – 0 = $4 million.
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R = $120
.2
.
R = $160
.3
R = $175
.5
C = $150
C = $170
.6
.4
C = $150
C = $170
.6
.4
C = $150
C = $170
.6
.4
–$30
–$50
$10
–$10
$25
$5
–38
2
17
1.5
5
Do Not
.
6
-$70
$0
7. a. Let’s compute the expected costs (in $ billions) of the respective safety
programs. For the “standard” program, the expected cost is: .160 + (.01)
b. At a judged 2 percent disaster risk, the (apparent) expected cost of the
“lax” policy is: .040 + (.02)(10) = $.24 billion, making it the appear to
8. The decision tree shows that the best strategy is to attempt a settlement,
accept the $400,000 settlement if this is offered, and otherwise go to
court. (Note that the expected cost of rejecting an offer, $650,000, is the
expected value of going to court, plus the $50,000 cost of trying to
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9. a. If the customer response is weak, MD’s expected profit is: (.5)(20) +
b. If the customer response is weak, the company does better by “pulling
the plug” a $20 million loss is better than an expected $40 million
10. a. The firm should not undertake the R&D development program. To do
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Try to Se,le
Court $.4 O.er
.5
.5
$.9 O.er
.2
.3 $0
$1
$.6
.5
.6
(+.1)
Accept
Reject
$.45
Accept
Reject
$.95
$.65
$.65
.45
.65
.55
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B Fails
$0
.6 -$200
.7 Do Not
20
c. i) In the joint venture, the chance that both programs fail is (.6)(.7) = .42.
Thus, according to the decision tree, developing both is a losing
proposition.
ii) Developing A alone is most profitable: (.4)(350) + (.6)(-200) = $20
11. The expected utility of pursuing the biochemical approach alone is:
E(UChem) = .7U(80) + .3U(40) = (.7)(64) + (.3)(44) = 58.
The accompanying decision tree depicts the strategy of trying the
biogenetic approach first and then pursuing the biochemical approach
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-$20
Success
$60 (55)
.7
12. a. From the given responses, we conclude that u($500) = 50, u($1,120) =
b. The function u = 2.24.1y = 2.24y.5, closely tracks the utility values in
part (a). For instance, if y = $500, then u = (2.24)(500.5) = 50.1. If y =
$1,530 then, then we find: u = (2.24)(1,530.5) = 87.6, and so on. In this
*13. The dealer must commit to ordering and selling some number of yachts
(say, Q) before knowing the course of the economy. Recall that the two
price equations are given by PG = 20 – .05Q, and PR = 20 – .1Q. Then,
the expected price required to sell Q yachts is: .6PG + .4PR = 20 – .07Q.
Expected profit is simply expected revenue minus cost. This is
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Discussion Question
1. The case of olestra makes for an interesting discussion. Olestra was a
zero-fat cooking oil approved by the FDA to use in producing snack
foods: primarily chip snacks and pop corn. Eighty percent of subjects
cannot taste any difference between Olestra snacks and snacks made
2. PG&E’s story follows closely the lessons of the BP oil spill. In the push
to reduce costs and enhance revenues, it’s easy for a company to
knowingly, (or unknowingly) pay less attention to inspection and safety
3. Clearly, whether a big-budget Hollywood film will turn out to be a
blockbuster or bust is highly uncertain and unpredictable. Nevertheless,
Disney’s past film track record indicates that animated features (on
Bonus Problems: Risky Gambles and Risk Aversion
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1. Two Gambles
a. You are given $1,000 to keep. You then are offered a choice between
receiving an additional $500 for certain or taking a 50-50 gamble with
outcomes of $1,000 and $0. Which would you choose?
b. You are given $2,000 to keep. You then are offered a choice between
paying $500 for certain or taking a 50-50 gamble with outcomes of $0
and -$1,000. Which would you choose?
c. The majority of individuals choose the sure $500 in part (a) but select
the gamble in part (b). Show that this combination of choices is
inconsistent. What does this suggest about the way in which decision
makers should think about risks?
Answer
b. If you are like the majority of subjects, you choose the 50/50 gamble.
c. Although they are described differently, the situations in parts (a) and
(b) offer identical choices. In part (a), you can choose to walk away
with a total of $1,500 for certain or to gamble on receiving either
2. Sell Now or Sell Later
Firm A is about to embark on a risky development project that offers
a .2 chance of a $4 million profit and a .8 chance of a $0 profit.
Unexpectedly, firm B proposes a joint venture. In return for its
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.75
expertise, firm B will receive 25 percent of any profit. With firm B
aboard, the chance of success is expected to rise to .25. After much
deliberation, firm A decides to turn down the joint venture and
continue on its own.
Two years pass, and firm A has overcome most of the
development hurdles. The estimated chances of success are now .8.
Now firm C offers to pay firm A $3 million for all rights (and
profits) concerning the development project. Deciding to take the
sure $3 million, firm A now sells out.
Show that whatever firm A’s attitude toward risk, the pair of
decisions just outlined are contradictory. (Hint: There are only three
outcomes across the two decisions: $4 million, $0, and $3 million,
with the associated utilities 100, 0, and u, respectively.) Show that
no value of u is consistent with firm A’s pair of choices.
Answer
In the first decision (see tree), the company chooses to continue on its
own, implying 20 > .25u. Thus, u < 80.
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In the subsequent decision, the company sells out, implying u > 80.
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