HO27-PracticeFinal2WS

MS&E 252 November 29th, 2001 Handout #25, page 1 of 25

SAMPLE FINAL: EES&OR 252 Final Examination (1997-1998)

Please do not begin the exam until you are instructed to do so.

Name (printed clearly): _______________________________________________

1. Count the number of pages in this exam. There should be 34 pages including this one. The examhas 15 problems.

2. This exam is closed-book and closed-notes. You may use a calculator, a foreign languagedictionary, and one 8.5" x 11" sheet of paper with notes on one side if you wish. Please sit inalternate seats if possible.

3. The exam has 15 problems worth a total of 200 points. The value of each problem is stated atthe beginning of the problem.

4. Write your answers in this exam and turn in the entire exam at the end of testing period. If youmake a false start please remember to erase or scratch out your work.

5. Please write clearly and show your work. You will not receive credit for illegible work or if you donot show your work. Partial credit will be given where appropriate. For problems with numericalanswers, please draw a box around the final answer; this will facilitate our work.

6. The exam lasts three hours. After time is called, you will have five minutes to turn in your exam.We will not accept any exams that are turned in after this deadline.

7. In the spirit of the Honor Code, the TA's will not remain in the classroom during the exam. Ifyou have any questions, come see us outside. Clarifications will be announced to the entire class.

8. Unless stated otherwise the characters in the exam prefer more money to less and follow the FiveRules of Actional Thought.

9. Do not make any additional assumptions.10. Good luck and make good decisions.

In recognition and in the spirit of the Honor Code, I certify that I will neither receive nor giveunpermitted aid on this exam and that I will report, to the best of my ability, all Honor Codeviolations observed.

Name (signed): _______________________________________________

For grading use only.

1 2 3 4 5 6 7 8 9 10

/30 /4 /10 /4 /10 /10 /18 /15 /7 /15

11 12 13 14 15

/10 /16 /15 /16 /20

MS&E 252 November 29th, 2001 Handout #25, page 2 of 25 1. Definitions and Distinctions (30 points)

Part I (10 points)

Consider the following terms. Briefly define the terms that are appropriate distinctions for use inDecision Analysis. If the term is not an appropriate distinction for Decision Analysis, explain whyand list an appropriate distinction that we use instead.

a) (2 points) prospect

b) (2 points) Clairvoyant c) (2 points) dependence d) (2 points) u-value e) (2 points) risk attitude

Part II (10 points)

Please provide short answers to the following questions.

a) (2 points) What is the definition of a decision? b) (2 points) What constitutes the decision basis? c) (2 points) What is the clarity test and why is it important? d) (2 points) What is a distinction and why is it important in decision analysis? e) (2 points) What do we mean when we say that a distinction is material?

Part III (10 points)

Short answer questions – please be concise, we are looking for the main idea, not the details.

a) (2 points) Define sunk cost. Give a brief example. b) (2 points) Define certain equivalent. c) (2 point) Briefly explain the concept of e-value. How would you calculate an e-value or a deal? d) (2 points) Explain the relationship, if any, between preference probabilities and u-values. e) (2 points) What is deterministic sensitivity analysis (tornado diagrams, or sensitivity to range)

and why do we use it?


$1500

-$500

0.8

0.2

$500

0.8

0.2

2. The Five Rules (4 points)

a) (2 points) Scott says that the Five Rules allow us to multiply prospect dollar values timesprobabilities. Do you agree with him? Explain why or why not. (3 points)

b) (2 points) Oz has just spent five hours assessing Randy's u-curve. Unfortunately, Randy

knocked over his drink and all the u-values washed away. Oz is left with the following u-curve:

-100 -50 0 50 100

wealth

u

Randy needs to determine his PISP for an uncertain deal within the next few minutes. What wouldyou recommend Oz do? (3 points)

3. u-curves (10 points)

Angus satisfies the delta-property and has an initial wealth of $500. Angus states that he isindifferent between a sure $500 and the following deal: (Assume he has no other deals relevant toany of the deals in this problem)

a) (2 points) Provide the simplest form of Angus' u-curve. b) (2 points) What is Angus' risk tolerance? c) (2 points) What probability, p, would make Angus indifferent between a sure $0 and a deal where

he could win $100 with probability p or lose $100 with probability 1-p? d) (2 points) What is Angus' certain equivalent for the following deal:


e) (2 points) Malcolm is a risk preferring delta-person. Will his risk odds (in any unit) be greater orless than Angus'?

4. Probability (4 points)

An unrelated man and woman each have two children. The man says, "My oldest child is a boy."The woman says, "At least one of my children is a boy."Assume: {having a boy | &} = {having a girl | &} = 0.5

The sex of the first child is not relevant to the sex of the second child.

a) (2 points) What is the chance that the man has two boys?

b) (2 points) What is the chance that the woman has two boys?

5. Probability Assessments (10 points)

You need probability assessments from two experts: Jim from The Weather Channel and Mary fromCNN weather news. You want to assess their beliefs about the chances of rain during the upcomingweekend. After some deliberation with each of them, you obtained their individual assessments interms of the following four events (assume that the distinction "Rain on a given day" has passed theclarity test):

• A: Rain on Saturday, given &• B: Rain on Sunday, given &• C: Rain on Saturday, or Sunday, or both days, given &• D: Rain on Saturday and Sunday, given &

a) (5 points) Is it possible for the probability assessments of Jim and Mary to differ from eachother? Why or why not?

The following are the probabilities that each of the experts assigned to the four events happening,given their background state of information:Jim: Mary:A: 60 percent A: 50 percentB: 60 percent B: 60 percentC: 80 percent C: 75 percentD: 24 percent D: 35 percent

b) (5 points) As a decision analyst, would you feel comfortable using Jim’s, Mary’s, both, or neitherprobability assessment? Why?

6. Understanding Clairvoyance (10 points)

a) (5 points) For a non-delta person, explain how you would calculate the value of clairvoyance.Support your answer. (Use a picture if necessary)

b) (5 points) The value of clairvoyance for a delta person can be calculated as the difference

between the value of the deal with free clairvoyance and the value of the deal with noclairvoyance. Clearly explain what is it about the delta property that allows us to do this?


7. The Interesting Clairvoyant (18 points)

Dominique, a risk-neutral decision-maker, is sailing solo across the Pacific; she stops at an islandwhere she faces the following decision:

I

II

0.5

0.5

0.5

0.5

$100

$0

$20

$80

A

A’

A

A’

The island is inhabited by a clairvoyant. Unfortunately, the clairvoyant, when dealing with a specificperson, either always lies or always tells the truth.

If Dominique assigns probability p to the clairvoyant always telling her the truth, for what range ofvalues of p is she willing to pay $20 for the clairvoyant’s information on A?

8. Troublemakers (15 points)

Ted has become suspicious that his teammate Brian is a troublemaker.Ted separates all people into two types: troublemakers and non-troublemakers. Troublemakers comelate to meetings more often that non-troublemakers. Assume Ted has defined this distinction so thatit passed the clarity test.

Over the past three meetings, Brian has been late to the first and third meeting, and on-time for thesecond. Ted makes the following statement: "If you know a person is (or is not) a troublemaker, thenhis lateness to one meeting tells you nothing about lateness to other meetings."

a) (5 points) Draw a relevance diagram representing Ted’s state of information. Draw the diagramwith as few arrows as possible.

Ted assesses the following probabilities.• If a person is a troublemaker, he or she will be late 80% of the time.• If a person is a non-troublemaker, he or she will be late 20% of the time.

The world is full of troublemakers; Ted assigned a prior probability of 60% Brian was atroublemaker.

MS&E 252 November 29th, 2001 Handout #25, page 6 of 25b) (10 points) In the light of Brian's attendance record, what probability should Ted assign to

Brian's being a troublemaker?

9. Relevance Diagram (7 points)

Examine the following relevance diagram.

A B

C D E

Check all statements implied by the diagram, given you can make any necessary, but ‘legal’operations. (‘Legal’operations, i.e. an operation or a sequence of operations allowed within the rulesof relevance diagrams)

A and B are not relevant given D and &

A and B are relevant given &

A and B are not relevant given &

D and B are relevant given A and &

D and B are not relevant given A and &

D and E are not relevant given &

D and E are relevant given B and &

MS&E 252 November 29th, 2001 Handout #25, page 7 of 2510. Decision Diagram (15 points)

Examine the following decision diagram.

FirstPurchase

SecondPurchase

Location ValueFirst

ReportSecondReport

AlphaIndication

BetaIndication Weather

ab

cd

e

f

l p

q r

s

g h i j k

n o

m

Part I (8 points)

Give the name of each arrow in the above diagram. Your choices are information (which includesnon-forgetting), functional, direct value, and relevance arrows.a.b.c.d.e.f.g.h.i.j.

k.l.m.n.o.p.q.r.s.

Part II (3 points)Explain briefly what arrows a, h, and o mean.

Part III (4 points)What if I add a new uncertainty node to the diagram? Call it node A. I tell you the following:• The decision maker will have clairvoyance on A before the first decision is made.• The value function also includes A.• A is not relevant to the two indications given &• A is not relevant to the weather given the indications and &.

If I put node A into the diagram, what arrow or arrows do I need to add? (Add the fewest number ofarrows necessary.)

MS&E 252 November 29th, 2001 Handout #25, page 8 of 2511. Research and Development Decision (10 points)

R&DEffort Value

Technical

SuccessMarketSuccess

a) (4 points) In many R&D decision situations, you will be able to observe the technical success ofa new product before you decide on the level of marketing effort. Market success is relevant tothe level of technical success and is influenced by the marketing effort decision. Redraw thedecision diagram with a “Marketing Effort” decision node added.

b) (2 points) Explain in words what the arrow from the “R&D Effort” decision node to the“Technical Success” uncertainty node means.

c) (4 points) Draw a decision tree representing the R&D decision problem (with the marketing

effort decision). Assume that each uncertain distinction has two degrees: High and Low. Foreach of the decisions, you have two alternatives: Large Effort and Small Effort.

12. Let's Make a Deal (16 points)

Tommy D. is a contestant on the game show "Let’s Make a Deal" and does not follow the deltaproperty. Assume he faces no other uncertain deals. On stage there are three boxes (A,B, and C):one contains $100; the other two are empty. The rules of the game are as follows:

Tommy D. first chooses one of the three boxes. Then Monty Hall, the game show host who knowswhere the prize is, shows that one of the two boxes that Tommy D. did not choose is indeed empty.(If Tommy D. happened to select the box with the prize, Monty is equally likely to open either ofthe other two.)Monty then gives Tommy D. the opportunity to switch his choice to the remaining unopened box,or stick with the box he originally chose. After this decision, Tommy D. gets what is inside the boxhe has: either the prize or nothing.

a) (3 points) Draw the decision diagram, representing Tommy D.’s decision opportunity, inassessed form.

b) (2 points) Draw the decision diagram, representing Tommy D.’s decision opportunity, in

inferential form. Please do not put this diagram in canonical form, i.e., leave the influencearrows in.

Suppose Tommy D. believes that the three boxes are equally likely to contain the prize. Tommy D.states that his PISPs for deals One, Two, and Three are $25, $30, and $50 respectively.


$100

$0

1/3

2/3

$100

$0

1/2

1/2

One Two

$100

$0

2/3

1/3

Three

c) (5 points) After making an initial selection and being shown an empty box, should Tommy D.switch to the unopened box or stick with his original selection?What is Tommy's chance of winning?

d) (2 points) What box should Tommy D choose initially? e) (4 points) Monty Hall offers to tell Tommy D. which box the prize is in before he makes his

initial choice. What is Tommy's PIBP for this information?

13. Trixie and Howy Baby (15 points)

Trixie (T) makes Howy Baby (HB) a proposition. She has a box full of certificates that entitle theowner to call a coin toss heads or tails, and if the call is correct, the owner wins some quantity ofmoney. Half of the certificates entitle the owner to $20 upon calling the coin correctly, and the restof the certificates entitle the owner to $200 upon calling the coin correctly. In either case, if theowner calls the coin incorrectly, the payment is $0.

T is charging $x for the right to draw a certificate and $15 to flip the coin. If HB pays the $x he canreach in and pick a certificate. After he knows which certificate he has drawn, HB can choosewhether or not to continue by paying the $15 and then flipping the coin.

Over the range of prospects possible in this deal, HB's u-curve is given by:

u(x) = -42.6 + .169x

HB believes that he is equally likely to draw either certificate and{Heads|Certificate Drawn, &} = {Heads|&} = 0.5.

a) (7 points) What is HB's PIBP for T’s proposition? (What is the largest $x HB should pay?)

b) (2 points) We now know that T is charging $35 for the right to draw a certificate. The $15charge to flip the coin still applies. Should he play?

c) (6 points) Suppose that the Clairvoyant offers to tell HB how the coin will land if he plays and

chooses to flip the coin and he must pay the Clairvoyant before he draws the certificate. What isHB’s PIBP for the Clairvoyant’s information?

MS&E 252 November 29th, 2001 Handout #25, page 10 of 25 14. A Shot in the Arm (16 points)

The flu season just started, and Bill is wondering if he should get a flu shot. Bill assigns a PISP of -$1000 to getting the flu at some point during the flu season. Bill believes there is a 20% chance hewill become sick if he does not get the flu shot. He also believes that if he gets a flu shot he is certainnot to get sick. You may assume for this problem that Bill is risk neutral.

a) (3 points) Draw the decision diagram representing Bill's decision.

Unfortunately, Bill also has a fear of needles; getting a shot is a very traumatic experience. Beforeinterviewing Bill about his PISP for getting a shot, you want to get some idea about its sensitivity.You assume that Bill's PISP will lie between $0 and -$300.

b) (4 points) Draw a sensitivity analysis chart, showing how the CE of each alternative changeswith Bill's PISP of getting a flu shot (we call these one-way sensitivities of CE to PISP of the shot).Put both alternatives on the same graph. What PISP would Bill need to assign to getting a flu shotfor him to be indifferent between the two alternatives?

In an interview, you have assessed Bill’s PISP for getting a shot; it equals -$150. However, he hasbecome less sure of his assessment of the probability of getting the flu.

c) (3 points) Draw a sensitivity to probability chart, showing how the CE of each alternativechanges with Bill’s probability of getting the flu with no shot. What probability would Bill needto assign for him to be indifferent between the two alternatives?

After a short discussion, Bill goes back to his original probability assessment of 20%.

d) (3 points) Draw the decision tree for Bill's decision. What is the best alternative? What is itsCE?

However, a thought strikes Bill during the interview; if someone in his working group gets the flu, hischances of getting the flu will increase. He wishes to understand the value of clairvoyance on this newdistinction (a teammate getting the flu).

e) (3 points) Draw a decision diagram for Bill's decision to buy clairvoyance. Use a report node.

15. Norma Vestor (20 points)

Norma Vestor has a risk averse u-curve and follows the delta property over the prospects in thisproblem. Her risk tolerance equals $20, and she likes to invest in companies. Norma believes theprobability that the price of oil will rise equals fifty percent. She also believes the probability that theprice of sugar will rise equals fifty percent, and furthermore, she believes that the price of sugar andthe price of oil are irrelevant given her background state of information.

Norma currently owns no deals relevant to the price of oil or to the price of sugar, and none of hercurrently owned deals relate to the deals below in any way. She intends to buy shares of one or moreof the following companies.A share of the Kane Company, for which an increase in sugar price gives Norma a $10 profit fromthat share. Otherwise she gets $0 profit from that share.A share of Exxoff, for which an increase in oil price gives Norma a $10 profit from that share.Otherwise she gets $0 profit from that share.A share of Glacier Incorporated, for which an increase in oil price gives Norma a $0 profit from thatshare. Otherwise she gets $10 profit from that share.

a) (4 points) Find Norma's personal indifferent buying price for one share of Kane, assuming thatshe buys no other shares of any company.

MS&E 252 November 29th, 2001 Handout #25, page 11 of 25 b) (2 points) Find Norma's personal indifferent buying price for one share of Exxoff, assuming that

she buys no other shares of any company c) (2 points) Find Norma's personal indifferent buying price for one share of Glacier, assuming that

she buys no other shares of any company.

Norma has purchased one share of Exxoff. She intends to buy one more share of some company inaddition to her first Exxoff share. She will choose from a share of Kane, a share of Glacier, andanother share of Exxoff (the second share will pay out the same as the first).

d) (4 points) Given that Norma owns one share of Exxoff, find Norma's personal indifferent buyingprice for a share of Kane.

e) (4 points) Given that Norma owns one share of Exxoff, find Norma's personal indifferent

buying price for a second share of Exxoff.

f) (4 points) Given that Norma owns one share of Exxoff, find Norma’s personal indifferent pricefor a share of Glacier.

MS&E 252 November 29th, 2001 Handout #25, page 12 of 25Final Exam Solutions

1. Definitions and Distinctions (30 points)

Part I (10 points)a) A prospect is the combination of an alternative that the decision maker chooses and a possibilitythat may be realized in the future of the decision maker's life.b) The Clairvoyant is a person who can answer any question about events in the past, present, orfuture without exercising judgment.c) Dependence is not an appropriate distinction for Decision Analysis. Instead we use thedistinction relevance which is defined such that two distinctions are relevant if knowledge about onedistinction changes the probability distribution about another distinction.d) u-value is a scaled preference probability.e) Risk attitude is the decision maker's attitude or willingness towards risk and is reflected througha u-curve. A person can be risk neutral, risk averse, or risk preferring.

Part II (10 points)a) A decision is an irrevocable allocation of resources.b) The decision basis consists of the following:

Alternatives: What we can doInformation: What we knowPreferences: What we want

c) To pass the clarity test, the clairvoyant must be able to answer a question about a distinctionwithout exercising judgment. It is a important tool in facilitating communication.d) A distinction divides the world into mutually exclusive an collectively exhaustive events.Distinctions are important to the understanding of decision situations.e) A distinction is material if it has the possibility of changing the decision.

Part III (10 points)a) A sunk cost is a past irrevocable allocation of resources that has no relevance to the evaluationof the future prospect. An example would be the money spent on a pair of non refundablesymphony tickets.b) The certain equivalent is equal to the decision maker's personal indifferent selling price of anuncertain deal that he/she owns.c) The e-value of a deal is the probability weighted average of the dollar measures associated to theprospects of the deal. An e-value is calculated by multiplying the probability of a possible outcometimes the dollar measure associated to that outcome and summing over all possible outcomes.d) u-values are scaled preference probabilities.e) Deterministic sensitivity precedes probabilistic analysis in the evaluation phase of theDecision Analysis cycle. It is used to determine the most sensitive variables or uncertainties thatneed to be included in the probabilistic analysis.

2. The Five Rules (4 points)a) No. The Five Rules only allow us to multiply probabilities and preference probabilities.b) Oz should either add a new scale to the u-axis or approximate Randy's CE graphically. He

does not have time or need to reassess the u-curve.

3. u-curves (10 points)a) We can subtract $500 from each prospect and we are left with the following deal:

$1000

-$1000

0.8

0.2$0 ~


From this we see that Angus' risk odds for +/- $1000 (r1000) = 4. Therefore, his u-curve can berepresented by:

u(x) = a + br1000-x/1000

where x is measured in dollars. Now we need to select a and b. The simplest form is a littlesubjective. I like to u(0) = 0. This means a = -b. So let's let a = 1 and b = -1. We know have:

u(x) = 1 - r1000-x/1000

b) risk tolerance = 1/ln(risk odds) = 1/ln(41/1000) = $721.c) r100 = r1000

100/1000 = r10001/10 = 1.149. p100 = r100/(1+ r100) = 0.53.

d)

e) Less, because his p will be less.

4. Probability (4 points)There are four possibilities: M1M2 M1F2

F1M2 F1F2

a) We know that the first child is a boy. So we are left with: M1M2 and M1F2. Thus, {M1M2 |First Child is a boy,&} = 1/2.

b) We know that at least on child is a boy. So we are left with: M1M2, M1F2, and F1M2. Thus,

{M1M2|At least one boy,&} = 1/3.

5. Probability Assessments (10 points)a) Yes. Probability assessments reflect an individual's state of information rather than a state ofnature.b) Jim's assessment of the four events is inconsistent whereas Mary's is consistent. Therefore adecision analysts should feel comfortable with Mary's assessment, but not Jim's.

6. Understanding Clairvoyance (10 points)a) The value of clairvoyance for a non-delta person can be calculated through iteration. If we definethe price of clairvoyance as b, we begin by setting this price to zero and calculate the certainequivalent of the deal having paid b for clairvoyance. Continue this process, increasing b until thedecision maker is indifferent between the deal without clairvoyance and the deal with clairvoyance.b) In calculating the difference between the value of the deal with free clairvoyance and the dealwith no clairvoyance, you are calculating the personal indifference selling price of clairvoyance.However, if the decision maker is a delta person, we know that adding (or subtracting) a fixedamount, delta, from all the prospects of a deal lead to an increase (or decrease) in the certainequivalent of the deal by the same amount, delta. Therefore, the buying price for clairvoyance isequal to the selling price of clairvoyance.

7. The Interesting Clairvoyant

$1500 - $1000 = $500

-$500 - $1000 = -$1500

0.8

0.2$500 - $1000 = -$500 ~


I

II

p

1-p

p

1-p

$100

$0

$20

$80

Truth

Lies

Lies

I

II

p

1-p

p

1-p

$0

$100

$80

$20

Truth

Lies

Truth

Lies

Truth

100-100p

60p+20

80-60p

100p

“A”

“ A’ ”

0.5

0.5

CE without clairvoyance = $50CE with clairvoyance = 0.5[max(100p, 80-60p)] + 0.5[max(100-100p, 20+60p)]

By setting the two possible values equal to each other, it is easy to see that Dominique is indifferentbetween the two alternatives when p = 0.5

100p = 80-60p 100-100p = 20+60pp = 0.5 p = 0.5

When p > 0.5, CE with clairvoyance = 0.5[100p] + 0.5[20+60p] = 80p+10When p < 0.5, CE with clairvoyance = 0.5[80-60p] + 0.5[100-100p] = 90-80p

To determine which values of p Dominique will be willing to pay $20 for clairvoyance, must look atthe case when p > 0.5 and p < 0.5 separately. We know that the CE without clairvoyance = $50 sothat the CE with clairvoyance must be equal to or greater than $50+$20=$70. Thus we have thefollowing equations:

p > 0.5 80p+10 = 70p = 3/4

MS&E 252 November 29th, 2001 Handout #25, page 15 of 25p < 0.5 90-80p = 70

p = 1/4

From this we can see that after paying $20 for clairvoyance, the CE with clairvoyance will be greateror equal to the CE without clairvoyance when p > 3/4 and when p < 1/4.

8. Troublemakersa)

Trouble-makers

1st meetingon time

2nd meetingon time

3rd meetingon time

b) 0.857. To get this answer, you need to draw a tree with the following distinctions: TroubleMaker, Arrival to 1st meeting, Arrival to 2nd meeting, Arrival to 3rd meeting and flip it to get a treein the following order: Arrival to 1st meeting, Arrival to second meeting, Arrival to 3rd meeting,Trouble Maker. You do not need to find the probabilities for every possibility. Using Bayes rulesyou can find the probability you need with only a few calculations.

9. Relevance Diagram (7 points)The only statements implied by the diagram are the first and the sixth:A and B are not relevant given D and &D and E are not relevant given &

10. Decision Diagrams (15 points)

Part I (8 points)a. direct value k. direct valueb. informational l. functionalc. informational m. functionald. informational n. functionale. direct value o. functionalf. informational p. direct valueg. functional q. relevanceh. informational r. relevancei. functional s. relevancej. informational

Part II (3 points)a. A direct value arrow from a decision to the value node indicates which value model should be usedbased on which alternatives were chosen.b. An informational arrow indicates that the result of the uncertainty or decision is known beforethe decision it points to is madec. A functional arrow points into a deterministic node from the uncertainties or decisions that areused in the calculation of the deterministic node.


Part III (4 points)

FirstPurchase

SecondPurchase

Location ValueFirst

ReportSecondReport

AlphaIndication

BetaIndication Weather

ab

cd

e

f

l p

q r

s

g h i j k

n o

m

A

11. Research and Development Decision (10 points)a)

R&DEffort Value

Technical

SuccessMarketSuccess

MarketingEffort

b) This is an influence arrow. The probability distribution of Technical Success may changedepending on which alternative is chosen.c)R&D Effort Tech. Market Market

Suc. Effort Success


H

L

Large

H

L

Small

H

L

Large

H

L

Small

H

L

Large

H

L

Small

H

L

Large

H

L

Small

H

L

H

L

Large

Small

12. Let’s Make a Deal (16 points)a.

InitialChoice

FinalChoice

$

Box MHShows

Box PrizeIs In

b.


InitialChoice

FinalChoice

$

Box MHShows

Box PrizeIs In

c. Let's draw the probability trees in assessed form, conditioned on Tommy D's (TD's) initial choice.Choose A

A

B

C

“C”

“B”

1/3

1/3

1/3

“B”

“C”

1/2

1/2

1/6

1/6

1/3

1/3

Box Prize Is In Box MH Shows

Choose B

A

B

C

“C”

“A”

1/3

1/3

1/3

“A”

“C”

1/2

1/2

1/6

1/6

1/3

1/3


Choose C


A

B

C

“B”

“A”

1/3

1/3

1/3

“A”

“B”

1/2

1/2

1/6

1/6

1/3

1/3


Now we need to place these in inferential form - flip the tree.

Choose A

“B”

“C”

1/2

1/2

A

B

C

1/3

2/3

0

1/6

0

1/3

A

B

C

1/3

0

2/3

1/6

1/3

0

Box MH Shows Box Prize Is In

After MH show TD the empty box, the probability the prize is in the unopened unchosen box is 2/3.So Tommy should switch and his chance of winning will be 2/3 - not 1/2 or 1/3.

Choose B


“A”

“C”

1/2

1/2

A

B

C

0

2/3

1/3

0

1/6

1/3

A

B

C

2/3

0

1/3

1/3

1/6

0


Same logic as above.Choose C

“A”

“B”

1/2

1/2

A

B

C

0

1/3

2/3

0

1/3

1/6

A

B

C

2/3

1/3

0

1/3

0

1/6


Same logic as above.d. It does not matter which box Tommy D initially picks - he always trades and the probability ofwinning is always 2/3.Here is an even better solution. When Tommy D. makes his initial choice there is a 1/3 chance hehas picked the prize. There is 2/3 chance the prize is in one of the other two boxes. You couldthink of these other two boxes with one big box with two compartments - 1 & 2. MH tells TommyD. that the prize is not in compartment 1. This does not change the chance it is in the large box, sothe probability it is in compartment 2 is 2/3.

MS&E 252 November 29th, 2001 Handout #25, page 21 of 25e. Before talking to MH,TD is facing a deal with a 2/3 chance of $100 and and 1/3 chance of $0.His PISP for this is $50. After TD gets MH's information he will have a sure $100. Therefore, hisPIBP for the information is $100 - $50 = $50. Notice, this result is true even though TD does notfollow the delta-property.

13. Trixie and Howy Baby (15 points)

Draw

Don't Draw

200

20

1/2

1/2

Flip

Don't Flip

1/2

1/2

Win

Lose

200-15 = 185

0-15 = -15

0

Flip

Don't Flip

1/2

1/2

Win

Lose

20-15 = 5

0-15 = -15

0

85

042.5

0

85

-5

The above diagram assumes that the cost to draw is zero. Since HB follows the delta-property we cansee that the most he should pay to play is $42.50.b. $35 is less than HB's PIBP -- he should play. His CE is $42.50 - $35 = $7.50.c. Let's leave the amount HB pays as a variable and call it x. Here is the tree conditioned the

clairvoyant saying "H", the "T" tree is the same.

Draw

Don't Draw

200

20

1/2

1/2

Flip

Don't Flip

200-15-x= 185-x

-x

Flip

Don't Flip

20-15-x = 5-x

-x

185-x

5-x95-x

0

85

-5

After talking to the clairvoyant, HB's CE is 95-x. If x is greater than 42.5 HB would chose not todraw without the clairvoyant's information. Let's consider the following cases:

0 < x <= 42.5 CE with Clairvoyance = 95-x CE w/o = 42.5-x VOC = 52.542.5 < x <= 95 CE with Clairvoyance = 95-x CE w/o = 0 VOC = 95-x

MS&E 252 November 29th, 2001 Handout #25, page 22 of 25x > 95 CE with Clairvoyance = 0 CE w/o = 0 VOC = 0

Here is a plot of the above:

Therefore, if the HB has to pay $35 to draw the VOC = $52.5.

14. A Shot in the Arm (16 points)a)

ShotDecision

Acquire Fluthis season

Value

b)

$0

-$200

-$300 $0

No Shot

Get Shot

PISP of shot

-$300

-$100

CE of alternatives

-$200

c)

52.5

42.5 95 x

VOC

In this range the VOC isindependent of how much HBhas to pay to draw!

In this range HB is choosing toplay even though the gamecosts more than the PIBP youfound in part a.


$0

-$150

-$1000

0.0 1.0p

No Shot

Get Shot

p=0.15

CE of alternatives

d)Flu

No Flu

Flu

No Flu

Get Shot

No Shot

0

1

0.2

0.8

-$1150

-$150

-$1000

$0

CE=-$150

e)

BuyClairvoyance

ShotDecision ValueReport

FluIndication

Flu | NoShot

15. Norma Vestor (20 points)

Recognize first that Norma follows the delta property, and therefor her PIBP for a deal she does not own equals herPISP of that deal if she did own it.

Create the following distinctions for the rest of this problem:

MS&E 252 November 29th, 2001 Handout #25, page 24 of 25• Sugar Price S = Sugar price rises

¬S = Sugar price does not rise• Oil Price O = Oil price rises

¬O = Oil price does not rise

We will use the following u-function to represent Norma's u-curve.

u x ex

( ) = −−

1 20

a) The following tree represents the deal of owning one share of Kane:

S

.5

.5

¬S

$10

$0

u = .20PISP = $4.38

0.39

0

u-values

b) Norma's PIBP equals $4.38.c) Norma's PIBP equals $4.38.Parts b and c have similar trees to part a: a fifty-fifty chance for ten dollars versus zero dollars. Thus they all havethe same PISP (and PIBP).d) Norma's PIBP for a share of Kane equals $4.38.The following tree represents the deal of owning a share of Exxoff and a share of Kane.

O

.5

.5

¬O

$20

$10

0.63

0.39

u-values

O

.5

.5

¬O

$10

$0

0.39

0

S

.5

.5

¬S

0.51

0.20

u = .35PISP = $8.76

Norma's PISP for the share of Kane equals the PISP of both shares minus the PISP of the Exxoff share. In diagramform, letting s equal the PISP of the Kane share:

NormaExxoff shareKane share

NormaExxoff share

+ sIndifferent

So, her PISP equals $8.76 - $4.38 = $4.38. Therefore, her PIBP also equals $4.38 (the same as a share of Kanewithout the Exxoff share).

e) Norma's PIBP for a second share of Exxoff equals $3.22.

MS&E 252 November 29th, 2001 Handout #25, page 25 of 25The following tree represents the deal of owning two shares of Exxoff.

O

.5

.5

¬O

$20

$0

u = .32PISP = $7.60

0.63

0

u-values

So, her PIBP for the second Exxoff share equals $7.60 - $4.38 = $3.22 (lower that the first Exxoff share).

f) Norma's PIBP for a share of Glacier equals $5.62.

The following tree represents the deal of owning a share of Exxoff and a share of Glacier.

O

.5

.5

¬O

$10

$10

u = .39PISP = $10

0.39

0.39

u-values

Note that Norma will always get a profit of $10, no matter how oil prices rise or fall. Her PIBP for the Glacier shareequals $10 - $4.38 = $5.62 (higher than the share of Glacier without the Exxoff share)

HO27-PracticeFinal2WS

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