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MBA7020_09.ppt/July 25, 2005/Page 1Georgia State University - Confidential
MBA 7020
Business Analysis Foundations
Decision Tree & Bayes’ Theorem
July 25, 2005
MBA7020_09.ppt/July 25, 2005/Page 2Georgia State University - Confidential
Agenda
Bayes Theorem
Decision Tree Problems
MBA7020_09.ppt/July 25, 2005/Page 3Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 4Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 5Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 6Georgia State University - Confidential
Terminal Nodes
• Terminal nodes are optionally represented by Triangles
• The node refers to a payoff
• The value for the node is the payoff
MBA7020_09.ppt/July 25, 2005/Page 7Georgia State University - Confidential
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?
MBA7020_09.ppt/July 25, 2005/Page 8Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 9Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 10Georgia State University - Confidential
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.
Sign with TV Network
$200,000
$1,000,000
$3,000,000
$900,000
$900,000
$900,000
.3
.6
.1
.3
.6
.1
ER?
ER?
ER?
MBA7020_09.ppt/July 25, 2005/Page 11Georgia State University - Confidential
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.
Sign with TV Network
$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.
Sign with TV Network
$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
MBA7020_09.ppt/July 25, 2005/Page 12Georgia State University - Confidential
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
Did not return within xx days
If Vc xx, send
Message D
Graduate to “SOW”
Did not return within xx days
If Vc < xx, no more
messages
Graduate to “SOW”
If Vc xx, send
Message D
If Vc < xx, no more
messages
MBA7020_09.ppt/July 25, 2005/Page 13Georgia State University - Confidential
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.
MBA7020_09.ppt/July 25, 2005/Page 14Georgia State University - Confidential
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
• Telephone
Vc Cost to reactivateIf:
Vc < Cost to reactivateIf:
Ugly Postcard???
TestArea
• POS
• Partners
• Advertising
Vehicles:
• Direct 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
MBA7020_09.ppt/July 25, 2005/Page 15Georgia State University - Confidential
Communication “Variables”
Vehicles
= 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
MBA7020_09.ppt/July 25, 2005/Page 16Georgia State University - Confidential
“Measuring Effectiveness: Lift/Gains Chart
Percent of population targeted
Percent of potentialresponders captured
100
1000
90
45
45
Targeting
Random mailing
MBA7020_09.ppt/July 25, 2005/Page 17Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 18Georgia State University - Confidential
Agenda
Decision TreeBayes Theorem
Problems
MBA7020_09.ppt/July 25, 2005/Page 19Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 20Georgia State University - Confidential
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.
MBA7020_09.ppt/July 25, 2005/Page 21Georgia State University - Confidential
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.)
MBA7020_09.ppt/July 25, 2005/Page 22Georgia State University - Confidential
Bayes’ Theorem
MBA7020_09.ppt/July 25, 2005/Page 23Georgia State University - Confidential
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)
MBA7020_09.ppt/July 25, 2005/Page 24Georgia State University - Confidential
Bayes' Theorem
Let A and be sets. Conditional probability requires that
(1)
where denotes intersection ("and"), and also that
(2)
Therefore,
(3)
MBA7020_09.ppt/July 25, 2005/Page 25Georgia State University - Confidential
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)
MBA7020_09.ppt/July 25, 2005/Page 26Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 27Georgia State University - Confidential
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”?
MBA7020_09.ppt/July 25, 2005/Page 28Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 29Georgia State University - Confidential
Agenda
Decision TreeBayes Theorem
Problems
MBA7020_09.ppt/July 25, 2005/Page 30Georgia State University - Confidential
What kinds of problems?
• Alternatives known
• States of Nature and their probabilities are known.
• Payoffs computable under different possible scenarios
MBA7020_09.ppt/July 25, 2005/Page 31Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 32Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 33Georgia State University - Confidential
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)
MBA7020_09.ppt/July 25, 2005/Page 34Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 35Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 36Georgia State University - Confidential
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
MBA7020_09.ppt/July 25, 2005/Page 37Georgia State University - Confidential
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