mba7020_09.ppt/july 25, 2005/page 1 georgia state university - confidential mba 7020 business...

MBA7020_09.ppt/July 25, 2005/Georgia State University - Confidential

MBA 7020

Business Analysis Foundations

Decision Tree & Bayes’ Theorem

July 25, 2005


Agenda

Bayes Theorem

Decision Tree Problems


Decision Trees

• A method of visually structuring the problem

• Effective for sequential decision problems

• Two types of branches– Decision nodes– Choice nodes– Terminal points

• Solving the tree involves pruning all but the best decisions

• Completed tree forms a decision rule


Decision Nodes

• Decision nodes are represented by Squares

• Each branch refers to an Alternative Action

• The expected return (ER) for the branch is – The payoff if it is a terminal node, or– The ER of the following node

• The ER of a decision node is the alternative with the maximum ER


Chance Nodes

• Chance nodes are represented by Circles

• Each branch refers to a State of Nature

• The expected return (ER) for the branch is – The payoff if it is a terminal node, or– The ER of the following node

• The ER of a chance node is the sum of the probability weighted ERs of the branches

– ER = P(Si) * Vi


Terminal Nodes

• Terminal nodes are optionally represented by Triangles

• The node refers to a payoff

• The value for the node is the payoff


Problem 1

• Jenny Lind is a writer of romance novels. A movie company and a TV network both want exclusive rights to one of her more popular works. If she signs with the network, she will receive a single lump sum, but if she signs with the movie company the amount she will receive depends on the market response to her movie.

• Jenny Lind – Potential Payouts

Movie company

Small box office - $200,000

Medium box office - $1,000,000

Large box office - $3,000,000

TV Network

Flat rate - $900,000

Questions:• How can we represent this problem?• What decision criterion should we use?


Jenny Lind – Payoff Table

Decisions

States of Nature

Small Box Office Medium Box Office Large Box Office

Sign with Movie Company

$200,000 $1,000,000 $3,000,000

Sign with TV Network $900,000 $900,000 $900,000


Jenny Lind – Decision Tree

Small Box Office

Medium Box Office

Large Box Office

Small Box Office

Medium Box Office

Large Box Office

Sign with Movie Co.

Sign with TV Network

$200,000

$1,000,000

$3,000,000

$900,000

$900,000

$900,000


Problem 2 – Solving the Tree

• Start at terminal node at the end and work backward• Using the ER calculation for decision nodes, prune branches (alternative

actions) that are not the maximum ER• When completed, the remaining branches will form the sequential decision

rules for the problem

Small Box Office

Medium Box Office

Large Box Office

Small Box Office

Medium Box Office

Large Box Office

Sign with Movie Co.


$200,000

$1,000,000

$3,000,000

$900,000

$900,000

$900,000

.3

.6

.1

.3

.6

.1

ER?

ER?

ER?


Jenny Lind – Decision Tree (Solved)

Small Box Office

Medium Box Office

Large Box Office

Small Box Office

Medium Box Office

Large Box Office

Sign with Movie Co.


$200,000

$1,000,000

$3,000,000

$900,000

$900,000

$900,000

.3

.6

.1

.3

.6

.1

ER900,000

ER960,000

ER960,000

Small Box Office

Medium Box Office

Large Box Office

Small Box Office

Medium Box Office

Large Box Office

Sign with Movie Co.


$200,000

$1,000,000

$3,000,000

$900,000

$900,000

$900,000

.3

.6

.1

.3

.6

.1

ER900,000

ER960,000

ER960,000


Decision Tree – Activation Test Source: Delta Airlines SkyMiles Program

SkyMiles Enrollment

Message A

Returned within xx days

Message B

Returned within xx days

Did not return within xx days

Message C


If Vc xx, send

Message D

Graduate to “SOW”


If Vc < xx, no more

messages

Graduate to “SOW”

If Vc xx, send

Message D

If Vc < xx, no more

messages


Probability

The Three Requirements of Probabilities:

1. All Probabilities must lie with the range of 0 to 1.

2. The sum of the individual probabilities equal to the probability of their union

3. The total probability of a complete set of outcomes must be equal to 1.


Direct Marketing Campaign Platform

ACQUIRE

RETAIN

REACTIVATE

“FIRE”

STORE DIFFERENT CHANNELS

A C T I V A T I O N P R O M O T I O NA C T I V A T I O N P R O M O T I O N

E-mail Address

Vehicles:

• Statements

• Newsletters

• Inserts

• Direct mail

• Personalized kits

• E-mail

• Telephone

Vc Cost to reactivateIf:

Vc < Cost to reactivateIf:

Ugly Postcard???

TestArea

• POS

• Partners

• Advertising

Vehicles:

• Direct Mail

• E-mail

• Statements

Triggered Promotions

highest value

customers

lowest value

customersdowngrade

trigger *

(for example)Days since last purchase = X

X = 30 days for PTNM

X = 60 days for GOLD

X = 120 days for CLUB

Direct Marketing Campaign Platform

PURCHASED

NO PURCHASE

PURCHASE

* < 1 purchase in last 12 mo

If : Time since inactive = X, and

Point balance > X


Communication “Variables”

Vehicles

= E-mail

= Kits

= Statement

= Telephone

= Direct Mail (USPS)

Message / Offer (incentive)

• Hurdle (SOW)

› trip x get y

• Next trip (Re-Activation)

› Rate of trip triggers

• Points (double/flat?)

• Miles (front & back-end)

•Other

Creative Execution

• Can test several executions tailored to clusters/segments

Timing/Frequency

• Monthly (statements)

• Repeat/Follow-up Mailings


“Measuring Effectiveness: Lift/Gains Chart

Percent of population targeted

Percent of potentialresponders captured

100

1000

90

45

45

Targeting

Random mailing


Example Direct Mail OptimizationSource: InterContinental Hotels Group Priority Club Rewards Program

• Using multivariate model we are able to maximize profit while minimizing costs

• In comparison to methodology used last year model savings = $XXX

– Savings attributable to reduced mailing to achieve last years result (variable cost savings).

• Other benefits - Customer Behavior, Planning Tool

CUMULATIVE ROI PREDICTED

0%

50%

100%

150%

200%

250%

300%

350%

400%

450%

40 37 34 31 28 25 22 19 16 13 10 7 4 1

Deciles

Hurdle 2000 Ranked List Q1 Last Year Mailed 798,313

Cost & Revenue assumptions Registered 92,523

FIXED COST 185,000$ (Last Year) % Last YearVC 0.41$ (Last Year) Planned Mail 691,951 -13.32%

Cost/Register 5.25$ (Last Year) Plan_Regis. 132,639 43.36%

Revenue/Register 23.79$ (Last Year) Equal Last Years Registration

Mailing 407,029

Multivariate Logistic Regression PREDICTIONS $ Savings 161,746$ PREDICTED PREDICTED Cumulative Cumulative Cumulative Cumulative Cumulative Cumulative Cumulative

Segments CUSTOMERS REGISTRATION RR % COST/REG REVENUE/REG PROFIT/REG ROI Predict RR % Registered Mailed

40 40,703 13,844 34.01% 14.58$ 18.54$ 3.97$ 27.2% 34.0% 13,844 40,703 39 40,703 12,282 30.17% 8.37$ 18.54$ 10.18$ 121.6% 32.1% 26,126 81,406 38 40,704 10,669 26.21% 6.40$ 18.54$ 12.14$ 189.8% 30.1% 36,795 122,110 37 40,700 9,674 23.77% 5.43$ 18.54$ 13.11$ 241.6% 28.5% 46,470 162,810 36 40,707 8,963 22.02% 4.86$ 18.54$ 13.69$ 282.0% 27.2% 55,433 203,517 35 40,702 8,383 20.60% 4.48$ 18.54$ 14.06$ 313.9% 26.1% 63,816 244,219 34 40,703 7,894 19.40% 4.22$ 18.54$ 14.32$ 339.2% 25.2% 71,710 284,922 33 40,705 7,472 18.36% 4.04$ 18.54$ 14.51$ 359.4% 24.3% 79,183 325,627 32 40,702 7,097 17.44% 3.90$ 18.54$ 14.65$ 375.6% 23.6% 86,279 366,329 31 40,700 6,755 16.60% 3.80$ 18.54$ 14.75$ 388.4% 22.9% 93,035 407,029

30 40,708 6,446 15.83% 3.72$ 18.54$ 14.82$ 398.5% 22.2% 99,480 447,737

29 40,703 6,154 15.12% 3.66$ 18.54$ 14.88$ 406.3% 21.6% 105,635 488,440 28 40,696 5,880 14.45% 3.62$ 18.54$ 14.92$ 412.2% 21.1% 111,514 529,136 27 40,711 5,627 13.82% 3.59$ 18.54$ 14.95$ 416.5% 20.6% 117,141 569,847 26 40,701 5,386 13.23% 3.57$ 18.54$ 14.97$ 419.5% 20.1% 122,527 610,548 25 40,702 5,162 12.68% 3.56$ 18.54$ 14.99$ 421.3% 19.6% 127,689 651,250 24 40,701 4,950 12.16% 3.55$ 18.54$ 14.99$ 422.2% 19.2% 132,639 691,951 23 40,707 4,749 11.67% 3.55$ 18.54$ 14.99$ 422.2% 18.8% 137,388 732,658 22 40,699 4,557 11.20% 3.56$ 18.54$ 14.99$ 421.6% 18.4% 141,945 773,357 21 40,709 4,373 10.74% 3.56$ 18.54$ 14.98$ 420.3% 18.0% 146,318 814,066 20 40,697 4,194 10.30% 3.58$ 18.54$ 14.97$ 418.5% 17.6% 150,512 854,763

.. .. .. .. .. .. .. .. .. .. ..5 40,695 2,393 5.88% 4.00$ 18.54$ 14.54$ 363.7% 13.5% 197,711 1,465,309 4 40,706 2,300 5.65% 4.04$ 18.54$ 14.51$ 359.3% 13.3% 200,012 1,506,015 3 40,709 2,196 5.39% 4.08$ 18.54$ 14.47$ 354.9% 13.1% 202,207 1,546,724 2 40,707 2,048 5.03% 4.12$ 18.54$ 14.43$ 350.3% 12.9% 204,255 1,587,431 1 40,705 1,509 3.71% 4.17$ 18.54$ 14.37$ 344.7% 12.6% 205,764 1,628,136

Totals 1,628,136 205,764 12.64% 4.17$ 18.54$ 14.37$ 344.7% 12.6% 205,764 1,628,136

0100

0

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 10 20 30 40 50 60 70 80 90 100


Agenda

Decision TreeBayes Theorem

Problems


Bayes' Theorem

• Bayes' Theorem is used to revise the probability of a particular event happening based on the fact that some other event had already happened.

Probabilities Involved• P(Event)

• Prior probability of this particular situation

• P(Prediction | Event)• Predictive power (Likelihood) of the information source

• P(Prediction Event)• Joint probabilities where both Prediction and Event occur

• P(Prediction)• Marginal probability that this prediction is made

• P(Event | Prediction)• Posterior probability of Event given Prediction

)(

)()|(

)(

)()|(

AP

BPBAP

AP

ABPABP


Bayes’ Theorem

• Bayes's Theorem begins with a statement of knowledge prior to performing the experiment. Usually this prior is in the form of a probability density. It can be based on physics, on the results of other experiments, on expert opinion, or any other source of relevant information. Now, it is desirable to improve this state of knowledge, and an experiment is designed and executed to do this. Bayes's Theorem is the mechanism used to update the state of knowledge to provide a posterior distribution. The mechanics of Bayes's Theorem can sometimes be overwhelming, but the underlying idea is very straightforward: Both the prior (often a prediction) and the experimental results have a joint distribution, since they are both different views of reality.


Bayes’ Theorem

• Let the experiment be A and the prediction be B. Both have occurred, AB. The probability of both A and B together is P(AB). The law of conditional probability says that this probability can be found as the product of the conditional probability of one, given the other, times the probability of the other. That is

P(A|B) ´ P(B) = P(AB) = P(B|A) ´ P(A)if both P(A) and P(B) are non zero.

Simple algebra shows that: P(B|A) = P(A|B) ´ P(B) / P(A) equation 1

• This is Bayes's Theorem. In words this says that the posterior probability of B (the updated prediction) is the product of the conditional probability of the experiment, given the influence of the parameters being investigated, times the prior probability of those parameters. (Division by the total probability of A assures that the resulting quotient falls on the [0, 1] interval, as all probabilities must.)


Bayes’ Theorem


Conditional Probability

The conditional probability of an event A assuming that B has occurred, denoted, equals

(1)

which can be proven directly using a Venn diagram. Multiplying through, this becomes

(2)

which can be generalized to

(3)

Rearranging (1) gives

(4)

Solving (4) for

and plugging in to (1) gives

(5)


Bayes' Theorem

Let A and be sets. Conditional probability requires that

(1)

where denotes intersection ("and"), and also that

(2)

Therefore,

(3)


Probability Information

• Prior Probabilities– Initial beliefs or knowledge about an event (frequently subjective

probabilities)

• Likelihoods– Conditional probabilities that summarize the known performance

characteristics of events (frequently objective, based on relative frequencies)


Circumstances for using Bayes’ Theorem

• You have the opportunity, usually at a price, to get additional information before you commit to a choice

• You have likelihood information that describes how well you should expect that source of information to perform

• You wish to revise your prior probabilities


Problem

• A company is planning to market a new product. The company’s marketing vice-president is particularly concerned about the product’s superiority over the closest competitive product, which is sold by another company. The marketing vice-president assessed the probability of the new product’s superiority to be 0.7. This executive then ordered a market survey to determine the products superiority over the competition.

• The results of the survey indicated that the product was superior to its competitor.

• Assume the market survey has the following reliability:– If the product is really superior, the probability that the survey will

indicate “superior” is 0.8.– If the product is really worse than the competitor, the probability that the

survey will indicate “superior” is 0.3.

• After completion of the market survey, what should the vice-president’s revised probability assignment to the event “new product is superior to its competitors”?


Joint Probability Table

P(Ai) P(B|Ai) P(Ai)* P(B|Ai) Revised Probability

P(Ai|B)

A1

Probability product is superior

0.7 0.8 0.56 0.56/0.65 = 0.86

A2

Probability product is not superior

0.3 0.3 0.09 0.09/0.65 = 0.14

1.0 P(B) = 0.65


Agenda

Decision TreeBayes Theorem

Problems


What kinds of problems?

• Alternatives known

• States of Nature and their probabilities are known.

• Payoffs computable under different possible scenarios


Basic Terms

• Decision Alternatives

• States of Nature (eg. Condition of economy)

• Payoffs ($ outcome of a choice assuming a state of nature)

• Criteria (eg. Expected Value)

Z


Example Problem 1- Expected Value & Decision Tree

States of NatureS1 S2 S3

Poor Average GoodDecision A 300 350 400Alternatives B -100 600 700

C -1000 -200 1200Probablilties 0.3 0.6 0.1


Expected Value

States of NatureS1 S2 S3

Poor Average Good EVDecision A 300 350 400 340 (300 x 0.3) + (350 x 0.6) + (400 x 0.1)Alternatives B -100 600 700 400 (-100 x 0.3) + (600 x 0.6) + (700 x 0.1)

C -1000 -200 1200 -300 (-1000 x 0.3) + (-200 x 0.6) + (1200 x 0.1)


Decision Tree

0.6 350

400-100

600

700

-1000

-200

1200

300

0.1

0.3

0.30.6

0.1

0.1

0.60.3

340

400

-300

A2400

A1

A2

A3


Example Problem 2- Sequential Decisions

S1-Low Economy

S2-Medium Economy

S3-High Economy

Favorable 20 60 70Unfavorable 80 40 30

100 100 100

• Would you hire a consultant (or a psychic) to get more info about states of nature?

• How would additional info cause you to revise your probabilities of states of nature occurring?

• Draw a new tree depicting the complete problem.

• Consultant’s Track Record

ZS1-Low Economy

S2-Medium Economy

S3-High Economy EV : Expected Values

A 300 350 400 340B -100 600 700 400C -1000 -200 1200 -300Probabilities 0.3 0.6 0.1


Example Problem 2- Sequential Decisions (Ans) Open MBA7020Joint_Probabilities_Table.xls

S1-Low Economy

S2-Medium Economy

S3-High Economy

Favorable 20 60 70Unfavorable 80 40 30

100 100 100

1. First thing you want to do is get the information (Track Record) from the Consultant in order to make a decision.

2. This track record can be converted to look like this:P(F/S1) = 0.2 P(U/S1) = 0.8P(F/S2) = 0.6 P(U/S2) = 0.4P(F/S3) = 0.7 P(U/S3) = 0.3

F= Favorable U=Unfavorable

3. Next, you take this information and apply the prior probabilities to get the Joint Probability Table/Bayles Theorum

ZS1-Low Economy

S2-Medium Economy

S3-High Economy Total

FAVorable 0.06 0.36 0.07 0.49UNFAVorable 0.24 0.24 0.03 0.51Prior Probabilities 0.3 0.6 0.1 1.00

FAVorable = 0.2 x 0.3 = 0.6 x 0.6 = 0.7 x 0.1 = 0.06 + 0.36 + 0.07UNFAVorable = 0.8 x 0.3 = 0.4 x 0.6 = 0.3 x 0.1 = 0.24 + 0.24 + 0.03Prior Probabilities Given Given Given = 0.3 + 0.6 + 0.1


Example Problem 2- Sequential Decisions (Ans) Open MBA7020Joint_Probabilities_Table.xls

4. Next step is to create the Posterior Probabilities (You will need this information to compute your Expected Values)

P(S1/F) = 0.06/0.49 = 0.122P(S2/F) = 0.36/0.49 = 0.735P(S3/F) = 0.07/0.49 = 0.143

P(S1/U) = 0.24/0.51 = 0.47P(S2/U) = 0.24/0.51 = 0.47P(S3/U) = 0.03/0.51 = 0.06

5. Solve the decision tree using the posterior probabilities just computed.

Z

mba7020_09.ppt/july 25, 2005/page 1 georgia state university - confidential mba 7020 business...

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