Decision Analysis

       Structuring the Decision Problem Decision Making Without Probabilities Decision Making with Probabilities Expected Value of Perfect Information Decision Analysis with Sample Information Developing a Decision Strategy Expected Value of Sample Information Slide 1

Structuring the Decision Problem

     A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.

The decision alternatives are the different possible strategies the decision maker can employ.

The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

For each decision alternative and state of nature, there is an outcome. These outcomes are often represented in a matrix called a payoff table.

Slide 2

Decision Trees

    A decision tree is a chronological representation of the decision problem.

Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives. The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb. Slide 3

Decision Making Without Probabilities

   If the decision maker does not know with certainty which state of nature will occur, then he is said to be doing decision making under uncertainty.

Three commonly used criteria for decision making under uncertainty when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic approach • the conservative approach • the minimax regret approach. Slide 4

Optimistic Approach

   The optimistic approach would be used by an optimistic decision maker.

The decision with the largest possible payoff is chosen. If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

Slide 5

Conservative Approach

   The conservative approach would be used by a conservative decision maker. For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.) If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.) Slide 6

Minimax Regret Approach

    The minimax regret approach requires the construction of a regret table or an opportunity loss table. This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. Then, using this regret table, the maximum regret for each possible decision is listed. The decision chosen is the one corresponding to the minimum of the maximum regrets.

Slide 7

Example

Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits:

d

1 Decisions d 2

d

3 States of Nature

s

1

s

2

s

3 4 4 -2 0 3 -1 1 5 -3 Slide 8

Example

 Optimistic Approach An optimistic decision maker would use the optimistic approach. All we really need to do is to choose the decision that has the largest single value in the payoff table. This largest value is 5, and hence the optimal decision is d 3 .

Maximum choose d 3 Decision

d

1

d

2

d

3 Payoff 4 3 5 maximum Slide 9

Example

 Formula Spreadsheet for Optimistic Approach

1 2 3 4 5 6 7 8 9 A De cision Alte rna tive

d1 d2 d3

B

P A Y OFF TA B LE

C S ta te of Na ture

s 1 4 0 1 s 2 4 3 5 s 3 -2 -1 -3 B es t P ay off

D E

= M A X(E 5:E 7)

F M a x im um P a yoff

= M A X(B 5:D5) = M A X(B 6:D6) = M A X(B 7:D7)

Re com m e nde d De cision

= IF(E 5= $E $9,A 5,"") = IF(E 6= $E $9,A 6,"") = IF(E 7= $E $9,A 7,"") Slide 10

Example

 Spreadsheet for Optimistic Approach

1 2 3 4 5 6 7 8 9 A De cisio n Alte rn a tive

d1 d2 d3

B

P A Y O F F TA B LE

C

s 1 4 0 1

S ta te o f Na tu re

s 2 4 3 5 s 3 -2 -1 -3 B es t P ay off

D E M a x im u m P a yo ff

4 3 5

Re co m m e n d e d De cisio n d 3 5 F

Slide 11

Example

 Conservative Approach A conservative decision maker would use the conservative approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.

Minimum choose d 2 Decision

d

1

d

2

d

3 Payoff -2 -1 maximum -3 Slide 12

Example

 Formula Spreadsheet for Conservative Approach

1 2 3 4 5 6 7 8 9 A

P A Y OFF TA B LE

B De cision Alte rna tive

d1 d2 d3

C S ta te of Na ture

s 1 4 0 1 s 2 4 3 5 s 3 -2 -1 -3

Be st P a yoff D E F M inim um P a yoff

= M IN(B 5:D5) = M IN(B 6:D6) = M IN(B 7:D7)

Re com m e nde d De cision

= IF(E 5= $E $9,A 5,"") = IF(E 6= $E $9,A 6,"") = IF(E 7= $E $9,A 7,"") = M A X(E 5:E 7) Slide 13

Example

 Spreadsheet for Conservative Approach

1 2 3 4 5 6 7 8 9 A

P A Y O F F TA B LE

B De cisio n Alte rn a tive

d1 d2 d3

C

s 1 4 0 1

S ta te o f Na tu re

s 2 4 3 5 s 3 -2 -1 -3

Be st P a yo ff D E -1 F M in im u m P a yo ff

-2 -1 -3

Re co m m e n d e d De cisio n d 2

Slide 14

Example

 Minimax Regret Approach For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from 4; in the second column, subtract 4, 3, and 5 from 5; etc. The resulting regret table is:

s

1

s

2

s

3

d

1

d

2

d

3 0 1 1 4 2 0 3 0 2 Slide 15

Example

 Minimax Regret Approach (continued) For each decision list the maximum regret. Choose the decision with the minimum of these values.

choose d 1 Decision Maximum Regret

d d

2

d

3 1 1 minimum 4 3 Slide 16

Example

 Formula Spreadsheet for Minimax Regret Approach

1 2 A

PAYOFF TABLE

De cision B C Sta te of Na ture D 3 4 5 Alte rn.

d1 d2 d3 s1 4 0 1 s2 4 3 5 s3 -2 -1 -3

6 7 8 9 10 De cision Alte rn.

11 12 13 14

OPPORTUNITY LOSS TABLE d1 d2 d3 s1 =MA X($B$4:$B$6)-B4 =MA X($B$4:$B$6)-B5 =MA X($B$4:$B$6)-B6

Sta te of Na ture

s2 =MA X($C$4:$C$6)-C4 =MA X($C$4:$C$6)-C5 =MA X($C$4:$C$6)-C6 s3 =MA X($D$4:$D$6)-D4 =MA X($D$4:$D$6)-D5 =MA X($D$4:$D$6)-D6

Minim a x Re gre t Va lue E F Ma x im um Re gre t Re com m e nde d De cision

=MAX(B11:D11) =IF(E11=$E$14,A11,"") =MAX(B12:D12) =IF(E12=$E$14,A12,"") =MAX(B13:D13) =IF(E13=$E$14,A13,"") =MIN(E11:E13) Slide 17

Example

 Spreadsheet for Minimax Regret Approach

1 2

P A Y O F F TA B L E

D e c i si o n S ta te o f N a tu r e 3 4 5 6 7 8 9 1 0 A l te r n a ti v e

d 1 d 2 d 3 s 1 4 0 1 s 2 4 3 5 s 3 -2 -1 -3

1 1 1 2 1 3 1 4

O P P O R TU N ITY L O S S TA B L E

D e c i si o n A l te r n a ti v e

s 1

S ta te o f N a tu r e

s 2 s 3 d 1 d 2 d 3 0 4 3 1 2 0 1 0 2

M i n i m a x R e g r e t V a l u e M a x i m u m R e g r e t

1 4 3

1 R e c o m m e n d e d D e c i si o n d 1

Slide 18

Decision Making with Probabilities

 Expected Value Approach • If probabilistic information regarding he states of nature is available, one may use the expected value (EV) approach. • • Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. The decision yielding the best expected return is chosen.

Slide 19

Expected Value of a Decision Alternative

  The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative.

The expected value (EV) of decision alternative d defined as:

i

is

d i i

) )

j j

 1

j j

where: N = the number of states of nature P(s

j

) = the probability of state of nature s

j V ij

= the payoff corresponding to decision alternative d

i

and state of nature s

j

Slide 20

Example: Burger Prince

Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table for the three models is as follows: Average Number of Customers Per Hour

s

1 = 80 s 2 = 100 s 3 = 120

d

1

d

2

d

3 = Model A $10,000 $15,000 $14,000 = Model B $ 8,000 $18,000 $12,000 = Model C $ 6,000 $16,000 $21,000 Slide 21

Example: Burger Prince

 Expected Value Approach Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d 1 , d 2 , d 3 represent the decision alternatives of models A, B, C, and s 1 , s 2 , s 3 represent the states of nature of 80, 100, and 120.

Slide 22

Example: Burger Prince

 Decision Tree 1

d

1

d

2

d

3 2 3 4

s

1

s

2

s

3 .4

.2

.4

s

1

s

2

s

3 .4

.2

.4

s

1

s

2

s

3 .4

.2

.4

Payoffs 10,000 15,000 14,000 8,000 18,000 12,000 6,000 16,000 21,000 Slide 23

Example: Burger Prince

 Expected Value For Each Decision

d

1 2 EV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600 Model A 1 Model B

d

2 3 EV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600 Model C

d

3 4 EV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000 Choose the model with largest EMV -- Model C.

Slide 24

Example: Burger Prince

 Formula Spreadsheet for Expected Value Approach

4 5 6 7 8 9 A 1

PAYOFF TABLE

2 3 Decision B C State of Nature D Alternative

Model A Model B Model C

Probability

s1 = 80 s2 = 100 s3 = 120 10,000 8,000 6,000 0.4

15,000 18,000 16,000 0.2

14,000 12,000 21,000 0.4

Maximum Expected Value E Expected Value

=MAX(E5:E7)

F Recommended Decision

=$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"") =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"") =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"") Slide 25

Example: Burger Prince

 Spreadsheet for Expected Value Approach

1 2 3 4 5 6 7 8 9 A

P A Y OFF TA B LE

B De cision Alte rna tive

M odel A M odel B M odel C

P roba bility

8,000 6,000

C S ta te of Na ture

s 1 = 80 10,000 s 2 = 100 s 3 = 120 15,000 18,000 16,000

D

14,000 12,000 21,000 0.4

0.2

0.4

M a x im um Ex pe cte d V a lue E Ex pe cte d V a lue

12600 11600 14000

14000 F Re com m e nded De cision M ode l C

Slide 26

Expected Value of Perfect Information

   Frequently information is available which can improve the probability estimates for the states of nature. The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. The EVPI provides an upper bound on the expected value of any sample or survey information. Slide 27

Expected Value of Perfect Information

 EVPI Calculation • Step 1: Determine the optimal return corresponding to each state of nature.

• Step 2: Compute the expected value of these optimal returns.

• Step 3: Subtract the EV of the optimal decision from the amount determined in step (2).

Slide 28

Example: Burger Prince

 Expected Value of Perfect Information Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000 Slide 29

Example

: Burger Prince

 Spreadsheet for Expected Value of Perfect Information

1 2 3 A

P A Y O F F TA B L E

D e c i si o n 9 1 0 1 1 1 2 4 5 6 7 8 A l te rn a ti v e

d 1 = M o d e l A d 2 = M o d e l B d 3 = M o d e l C

P ro b a b i l i ty B C D S ta te o f N a tu re

s 1 = 8 0 1 0 , 0 0 0 8 , 0 0 0 6 , 0 0 0 0 . 4 s 2 = 1 0 0 1 5 , 0 0 0 1 8 , 0 0 0 1 6 , 0 0 0 0 . 2 s 3 = 1 2 0 1 4 , 0 0 0 1 2 , 0 0 0 2 1 , 0 0 0 0 . 4

M a x i m u m E x p e c te d V a l u e E x p e c te d V a l u e

1 2 6 0 0 1 1 6 0 0 1 4 0 0 0

1 4 0 0 0 M a x i m u m P a y o ff

1 0 , 0 0 0 1 8 , 0 0 0 2 1 , 0 0 0

E E V w P I 1 6 0 0 0 F R e c o m m e n d e d D e c i si o n d 3 = M o d e l C E V P I 2 0 0 0

Slide 30

Decision Analysis With Sample Information

    Knowledge of sample or survey information can be used to revise the probability estimates for the states of nature. Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities. With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. The outcomes of this analysis are called posterior probabilities.

Slide 31

Posterior Probabilities

 Posterior Probabilities Calculation • Step 1: For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator.

• Step 2: Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.

• Step 3: For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.

Slide 32

Expected Value of Sample Information

 The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. Slide 33

Expected Value of Sample Information

 EVSI Calculation • Step 1: Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature. Step 2: Compute the expected value of these optimal returns.

• Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).

Slide 34

Efficiency of Sample Information

  Efficiency of sample information is the ratio of EVSI to EVPI. As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.

Slide 35

Example: Burger Prince

 Sample Information Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are: P(favorable | 80 customers per hour) = .2

P(favorable | 100 customers per hour) = .5 P(favorable | 120 customers per hour) = .9 Should Burger Prince have the survey performed by Stanton Marketing?

Slide 36

Example: Burger Prince

 Posterior Probabilities Favorable Survey Results State Prior Conditional Joint Posterior 80 .4 .2 .08 .148

100 .2 .5 .10 .185

120 .4 .9 .36

.667

Total .54 1.000

P(favorable) = .54

Slide 37

Example: Burger Prince

 Posterior Probabilities Unfavorable Survey Results State Prior Conditional Joint Posterior 80 .4 .8 .32 .696

100 .2 .5 .10 .217

120 .4 .1 .04

.087

Total .46 1.000

P(unfavorable) = .46

Slide 38

Example: Burger Prince

 Formula Spreadsheet for Posterior Probabilities

1 2 A B

M a rk e t R e s e a rc h F a vo ra b le P rio r

C

C o n d it io n a l

1 0 1 1 1 2 1 3 1 4 1 5 3 4 5 6 7 8 9

S t a t e o f N a t u re s 1 = 8 0 s 2 = 1 0 0 s 3 = 1 2 0 P ro b a b ilit ie s 0 . 4 0 . 2 0 . 4 P ro b a b ilit ie s 0 . 2 0 . 5 0 . 9 P (F a vo ra b le ) = M a rk e t R e s e a rc h U n fa vo ra b le S t a t e o f N a t u re s 1 = 8 0 s 2 = 1 0 0 s 3 = 1 2 0 P rio r P ro b a b ilit ie s 0 . 4 0 . 2 0 . 4 C o n d it io n a l P ro b a b ilit ie s 0 . 8 0 . 5 0 . 1 P (U n fa vo ra b le ) =

D

Jo in t P ro b a b ilit ie s = B 4 * C 4 = B 5 * C 5 = B 6 * C 6 = S U M (D 4 : D 6 ) Jo in t P ro b a b ilit ie s = B 1 2 * C 1 2 = B 1 3 * C 1 3 = B 1 4 * C 1 4 = S U M (D 1 2 : D 1 4 )

E

P o s t erio r P ro b a b ilit ie s = D 4 / $ D$ 7 = D 5 / $ D$ 7 = D 6 / $ D$ 7 P o s t erio r P ro b a b ilit ie s = D 1 2 / $ D$ 1 5 = D 1 3 / $ D$ 1 5 = D 1 4 / $ D$ 1 5 Slide 39

Example: Burger Prince

 Spreadsheet for Posterior Probabilities

1 2 A B

M a rk e t R e s e a rc h F a vo ra b le P rio r

3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5

S t a t e o f N a t u re s 1 = 8 0 s 2 = 1 0 0 s 3 = 1 2 0 P ro b a b ilit ie s 0 . 4 0 . 2 0 . 4 M a rk e t R e s e a rc h U n fa vo ra b le S t a t e o f N a t u re s 1 = 8 0 s 2 = 1 0 0 s 3 = 1 2 0 P rio r P ro b a b ilit ie s 0 . 4 0 . 2 0 . 4

C

C o n d it io n a l P ro b a b ilit ie s 0 . 2 0 . 5 0 . 9 P (F a vo ra b le ) = C o n d it io n a l P ro b a b ilit ie s 0 . 8 0 . 5 0 . 1 P (F a vo ra b le ) =

D

J o in t P ro b a b ilit ie s 0 . 0 8 0 . 1 0 0 . 3 6 0 . 5 4 J o in t P ro b a b ilit ie s 0 . 3 2 0 . 1 0 0 . 0 4 0 . 4 6

E

P o s t e rior P ro b a b ilit ie s 0 . 1 4 8 0 . 1 8 5 0 . 6 6 7 P o s t e rior P ro b a b ilit ie s 0 . 6 9 6 0 . 2 1 7 0 . 0 8 7 Slide 40

Example: Burger Prince

 Decision Tree (top half) 1 I 1 (.54) 2

d

1

d

2

d

3 4 5 6

s

1 (.148)

s

2 (.185) $10,000 $15,000

s

3 (.667)

s

1 (.148) $14,000 $8,000

s

2 (.185)

s

3 (.667) $18,000

s

1 (.148)

s

2 (.185)

s

3 (.667) $12,000 $6,000 $16,000 $21,000 Slide 41

Example: Burger Prince

 Decision Tree (bottom half) 1

I

2 (.46) 3

d

1

d

2

d

3 7 8 9

s

1 (.696) $10,000

s

2 (.217)

s

3 (.087) $15,000

s s

2 1 (.696) (.217)

s

3 (.087) $14,000 $8,000 $18,000

s

1 (.696) $12,000 $6,000

s

2 (.217)

s

3 (.087) $16,000 $21,000 Slide 42

Example: Burger Prince

I

1 (.54) $17,855 2

d

1

d

2

d

3 4 EMV = .148(10,000) + .185(15,000) + .667(14,000) = $13,593 5 EMV = .148 (8,000) + .185(18,000) + .667(12,000) = $12,518 6 EMV = .148(6,000) + .185(16,000) +.667(21,000) = $17,855 1

I

2 (.46) 3 $11,433

d

1

d

2

d

3 7 EMV = .696(10,000) + .217(15,000) +.087(14,000)= $11,433 8 EMV = .696(8,000) + .217(18,000) + .087(12,000) = $10,554 9 EMV = .696(6,000) + .217(16,000) +.087(21,000) = $9,475 Slide 43

Example: Burger Prince

 Decision Strategy Assuming the Survey is Undertaken: • If the outcome of the survey is favorable, choose Model C. • If it is unfavorable, choose Model A.

Slide 44

Example: Burger Prince

 Question: Should the survey be undertaken?

 Answer: If the Expected Value with Sample Information (EVwSI) is greater, after deducting expenses, than the Expected Value without Sample Information (EVwoSI), the survey is recommended.

Slide 45

Example: Burger Prince

 Expected Value with Sample Information (EVwSI) EVwSI = .54($17,855) + .46($11,433) = $14,900.88

 Expected Value of Sample Information (EVSI) EVSI = EVwSI - EVwoSI assuming maximization EVSI= $14,900.88 - $14,000 = $900.88

Slide 46

Example: Burger Prince

 Conclusion EVSI = $900.88

Since the EVSI is less than the cost of the survey ($1000), the survey should not be purchased.

Slide 47

Example: Burger Prince

 Efficiency of Sample Information The efficiency of the survey: EVSI/EVPI = ($900.88)/($2000) = .4504

Slide 48

The End of Chapter 9

Slide 49