Using Decision Trees to Solve Complex Problems: An Example
Consider the following problem
The Disney Corporation is thinking about taking its hit Broadway
musical, The Lion King on tour. It has made the following
calculations
 The tour would cost $12 million
 There is a .50 chance that it will earn $20 million and a .5
chance it will earn $10
Draw the decision tree for this problem using the following
conventions. (Let a box indicate a decision node and a circle
indicate a chance node).
$0
Don’t Produce
$8 million
Pr =.50
Produce
High
Pr=.50
Low
-$2 million
EV = .5x8-.5x2 = $3.0 million
Since the EV from the “Produce” node is greater than the EV from
the “Don’t Produce” node, they produce
But now consider the following complication
 Disney can do a preview showing of the production at some cost
$P
 If the audience is favorable, the show is certain to be a success
 If the audience is unfavorable, the show has only a .1 chance of
success.
 The prior belief is that there is a .50 chance that the preview
audience will react favorably.
To decide whether to do the preview, look at this decision tree.
Note that if they choose not to preview, they are essentially in the
previous situation, they go ahead with the production and expect to
earn $3.0 million.
No Preview
$3.0
Produce
Preview
Pr=.5
Favorable
EV = .5x8+.5x0-P
Pr =.5
$8.0-P
Not Produce
-P
Not Produce
-P
$8-P
Produce
Pr=.1
High
Pr =.9
Low
EV =.8-1.8-P=-1-P
-$2-P
 The “trick” is to work backwards towards the start of the tree.
Doing this, we see that
 If you get a favorable review, you produce ($8-P>-P).
 If get an unfavorable review, you don’t produce (-1-P<-P)
 Since the odds of getting a favorable review are .50, your
expected profit if you review is .5x8-.5x1-P=$4-P.
 Thus, if the preview costs less than $1 million, your better off
previewing.
Credit Analyst Dilemma
• You are trying to decide whether to offer credit to a population
of potential borrowers.
• Profit from good loan=$10
• Loss From Bad Loan = -$10
• Past Experience leads you to believe that the probability of
default is .260
Simple Decision Tree
No Loan
$0
EV=$4.80
Repay
Loan
$10
Pr=.74
Default
EV=$4.80
Pr = .26
-$10
More Complex Problem: Credit Reports
• Suppose that you can buy credit reports that are useful in
predicting a good pay
• Suppose you have some history with this reporting service and
that you see that for every 1000 reports purchased, the
customers fall into this classification
Credit Rating
Good Report
Bad Report
Total (Payment
History)
Payment History
Good Bad Pay
Pay
720
80
20
180
740
Total (Credit Rating)
800
200
260
 Note Pr (good pay given good report) = 720/800=.90
 Pr (good pay given bad report) = 20/200=.10
• How much would you pay for the report?
Decision Tree
EV = $8
Good Pay
$10
Pr=.90
EV=$8
Loan
Pr=.10
Bad Pay
-$10
Good Credit
$6.4
Pr=.8
No Loan
$0
$0
Pr=.20 No Loan
Bad Credit
$10
Good Pay
Pr = .10
Loan
Pr =.90
EV = 0
Bad Pay
EV = -$8.0
-$10
Computing the Value of Information
 Notice that the credit reports must have some value since
they change your behavior. (Without the reports, you offer
credit to 1000 customers, with the report, you offer credit to
800 customers).
 If you buy the credit reports, you expect to make $6.4. If you
don’t buy the report, you will get $4.80.
 One might reasonably conclude that the value of information
is
EV (with information) – EV (without information)=6.4-4.8=1.6
Bayes Rule and the Value of Information (Another Example)
• Suppose that you are interested in the absentee record of
workers and your concerned with measuring the benefits of an
anti-smoking program.
• A question would be how does smoking affect absenteeism.
• Imagine data collected on 1000 workers reveals the following
Absentee Record
Missed More Than One
Missed Less Than One
Week
Week
Smokers
300
200
Non-Smokers
100
400
• If we didn’t know whether a worker smoked, we would
conclude that Pr (Absent>One Week) = .4 (40% of all workers
miss more than one week).
• If a worker is a smoker, we know that Pr (Absent > One Week
given Smoker)=Pr (Absent and Smoker)/Pr(Smoker) = .3/.5=.6.
(60% of all smokers miss more than one week).
Bayes Rule and the Value of Information
(Formalities and General Application)
 Let “A” be interpretted as the some event (say, a worker absent
more than one week) and “S” be interpretted as another event
(the worker being a smoker).
 We’ve already seen that
Pr (A given S) = Pr (A and S)/Pr (S)
 By the same reasoning we could have written
Pr (S given A) = Pr (A and S)/Pr(A)
 Rearranging this expression gives
Pr (A and S) = Pr (S given A)xPr(A)
 Substituting this into our first expression, we can write
 Pr(S given A) 
Pr( A given S )  Pr( A) 

Pr(S )


 Think about what this expression is good for
 Pr(A) is your assessment of the prior probability (if you don’t
know whether S is true, it is how likely you think it is the event
A will occur) In the example, Pr (A) = .40

 The term in brackets tells you how to revise your assessment of
the liklihood of event A given the new information
 In the example Pr(S given A)=.75 (75% of those absent more
than one week are smokers). Pr (S) = .50 (50% of the
workforce smokes). Thus, the prior probability gets multiplied
by 75/50=1.5 to get the new revised probability.
Can information have some value even if it doesn’t raise the expected value?
 Modify the previous example by assuming that you are
evaluating an investment that has the same basic structure as
described above, except that it is a “one-shot” bet with the
odds described above.
o If you don’t have any added information, you will make
$10 million with pr = .74 or lose $10 million with pr
.26.
o This has an EV =4.8, but it is also quite risky
Variance = .5(10-4.8)2+.5(-10-4.8)2=76.96
 If you make the investment, your outcomes can be
represented with this tree
$10
Pr=.74
-$10
Pr=.26
EV = $4.8
 Now suppose that for exactly $2.6 million you can hire a
consultant who will recommend whether to make the
investment. The consultant is never wrong and so there is no
real decision: you do what you’re told.
 You also believe that there is a 74% chance that the
consultant will tell you it is a good investment.
 The tree for this problem looks like
Positive
Recommendation
$10-2.6=$7.4
Pr = .74
Pr=.26
Negative
Recommendation
-$2.6
EV = .74x7.4-.26*2.6=$4.8
 It would appear that this information isn’t worth the price
since the EV is the same when you buy it as when you don’t
($4.8 in each case).
 However, a risk averse investor might consider this to be a
good deal since it reduces the variability of the outcomes.
Variance = .74x(7.4-4.8)2+.26x(-2.6-4.8)2=19.24
Information “Puzzlers” (Revealing the Limits of Intuition)
• Guess the job
• Let’s Make a Deal
 Flight Insurance