Lecture 3 - Decision Analysis

8/10/2019 Lecture 3 - Decision Analysis

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Decision Analysis


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Example

Consider the following problem with threedecisionalternativesand threestates of naturewith the followingpayoff table representing profits:

States of Nature

s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1d3 1 5 -3

Which decisiondo you choose?


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Decision Making without Probabilities

Three commonly used criteria for decision makingwhen probability information regarding the likelihoodof the states of nature is unavailable are:

the optimistic approach the conservative approach

the minimax regret approach.


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Example

Consider the following problem with three decisionalternatives and three states of nature with the followingpayoff table representing profits:

States of Nature

s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1d3 1 5 -3


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Example: Optimistic Approach

An optimistic decision maker would use theoptimistic (maximax) approach. We choose the decisionthat has the largest single value in the payoff table.

MaximumDecision Payoff

d1 4

d2 3

d3 5

Maximaxpayoff

Maximaxdecision


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Conservative Approach

The conservative approach would be used by aconservative decision maker.

For each decision the minimum payoff is listed andthen the decision corresponding to the maximum ofthese minimum payoffs is selected. (Hence, theminimum possible payoff is maximized.)

If the payoff was in terms of costs, the maximum costswould be determined for each decision and then the

decision corresponding to the minimum of thesemaximum costs is selected. (Hence, the maximumpossible cost is minimized.)


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Example


States of Nature

s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1d3 1 5 -3


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Example: Conservative Approach

A conservative decision maker would use theconservative (maximin) approach. List the minimumpayoff for each decision. Choose the decision with themaximum of these minimum payoffs.

Minimum

Decision Payoff

d1 -2

d2 -1

d3 -3

Maximindecision Maximinpayoff


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Minimax Regret Approach

The minimax regret approach requires theconstruction of a regret or opportunity loss table.

Regret: for each state of nature (column), regretfor a

decision is the difference between that payoff and thelargest one.

Regret is with respect to decisions, not states ofnature.

Minimax regret: Find maximum regret for each decision.

Choose decision with the smallest maximum regret


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Example


States of Nature

s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1d3 1 5 -3


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Example: Minimax RegretApproach

For the minimax regret approach, first compute aregret table by subtracting each payoff in a columnfrom the largest payoff in that column. In this

example, in the first column subtract 4, 0, and 1 from4; etc. The resulting regret table is:

s1 s2 s3

d1 0 1 1

d2 4 2 0

d3 3 0 2


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Example: Minimax RegretApproach

For each decision list the maximum regret.Choose the decision with the minimum of these values.

Maximum

Decision Regret

d1 1

d2 4

d3 3

Minimaxdecision

Minimaxregret


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Decision Making with Probabilities


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Decision Making with Probabilities

Expected Value Approach

If probabilistic information regarding the states ofnature 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 payoffunder each state of nature and the probability of

the respective state of nature occurring. The decision yielding the best expected return is

chosen.


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Payoff Table

Average Number of Customers Per Hour

s1= 80 s2= 100 s3= 120

Model A

10,000

15,000

14,000Model B 8,000 18,000 12,000

Model C 6,000 16,000 21,000

probabilities 0.4 0.2 0.4


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Expected Value Approach

Calculate the expected value for each decision.


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Expected value of each decision

Average Number of Customers Per Hour

s1= 80 s2= 100 s3= 120

Model A

10,000

15,000

14,000 EV=

12,600Model B 8,000 18,000 12,000 EV=11,600

Model C 6,000 16,000 21,000 EV=14,000

probabilities 0.4 0.2 0.4

E.g., EV for Model A = .4(10,000)+.2(15,000)+.4(14,000)

= 4,000 + 3,000 + 5,600 = 12,600.

Choose Model C: highest EV.


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Expected Value with Decision Trees

The same calculation can be done with a decision tree

(next slide).

Here d1, d2, d3 represent the decision alternatives ofmodels A, B, C, and s1, s2, s3 represent the states of

nature of 80, 100, and 120.


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Decision Trees

A decision tree is a chronologicalrepresentation of the decision problem.

Branches leaving round nodes correspond todifferent states of nature

Branches leaving square nodes correspond todifferent decision alternatives.

At the end of each limb of the tree (each leaf) is thepayoff from that series of branches.


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Expected Value (EV) for Each Decision

Choose the model with largest EV, Model C.

3

d1

d2

d3

EV = .4(10,000) + .2(15,000) + .4(14,000)= 12,600

EV = .4(8,000) + .2(18,000) + .4(12,000)= 11,600

EV = .4(6,000) + .2(16,000) + .4(21,000)

=

14,000

Model A

Model B

Model C

2

1

4


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Sensitivity Analysis

Sensitivity analysis can be used to determine howchanges to the following inputs affect the recommendeddecision alternative:

probabilities for the states of nature values of the payoffs

If a small change in the value of one of the inputs causesa change in the recommended decision alternative, extra

effort and care should be taken in estimating the inputvalue.


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EVPI example

EV if omniscient= .4(10,000)+.2(18,000)+.4(21,000) = 16,000

EV of Model C (best alternative) =14,000

Expected value of perfect information =2,000

If it cost3,000 to do a study to clarify whether you were most likely to get 80,

100, or 120 customers/hour, would the study be worthwhile?- No. Even perfect information could only add an expected value of

2,000.

What if it cost1,000?

- Maybe. Yes, if your information would be very good; no, if it doesntimprove your probability estimates enough to justify the cost.


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Expected Value of Sample Information

The expected value of sample information (EVSI) isthe additional expected profit possible through

knowledge of the sample or survey information.

Similar to expected value of perfect information(EVPI), only not perfect. ;)


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Expected Value of Sample Information

EVSI Calculation Determine the optimal decision and its expected return,

for the possible outcomes of the sample, using theposterior probabilities for the states of nature.

Compute the expected value of these optimal returns. Subtract the EV of the optimal decision based on the

information you have now.

This difference is the (expected) value of the(imperfect) information you could gain.

Like EVPI, but with sample information rather thanomniscience.


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Efficiency of Sample Information

Efficiency of sample information is the ratio EVSI/EVPI.

As the EVPI provides an upper bound for the EVSI,

efficiency is always a number between 0 and 1.


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Posterior ProbabilitiesSuppose you expect the survey to be favorable(high demand) with

probability 0.54, unfavorablew.p. 0.46.

Suppose the posterior probabilities are:Favorable case: Pr( 80 | favorable) = 0.148

Pr(100 | favorable) = 0.185 Check: these sum to 1! Pr(120 | favorable) = 0.667

Unfavorable case: Pr( 80 | unfavorable) = 0.696 Pr(100 | unfavorable) = 0.217 Check: these sum to 1! Pr(120 | unfavorable) = 0.087

Check that the posteriors match the priors (0.4, 0.2, 0.4) Pr(80)= Pr(favorable)*Pr(80 | favorable) + Pr(unfavorable)*Pr(80 | unfavorable)

= 0.54*0.148 + 0.46*0.696 = 0.40. Good! (Otherwise you hold contradictory beliefs.) Check Pr(100), Pr(120) similarly.


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Decision Tree

top half (case where survey is favorable)

s1(.148)

s1 (.148)

s1(.148)

s2(.185)

s2(.185)

s2(.185)

s3(.667)

s3 (.667)

s3(.667)

10,000

15,000

14,0008,000

18,000

12,0006,000

16,000

21,000

I1

(.54)

d1

d2

d3

2

4

5

6

1


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Decision Tree

bottom half (case where survey is unfavorable)

s1(.696)

s1

(.696)

s1(.696)

s2(.217)

s2(.217)

s2(.217)

s3(.087)

s3 (.087)

s3(.087)

10,000

15,000

18,000

14,0008,000

12,000

6,00016,000

21,000

I2(.46) d1

d2

d3

7

9

83

1


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I2(.46)

d1

d2

d3

EV = .696(10,000) + .217(15,000)+.087(14,000)= 11,433

EV = .696(8,000) + .217(18,000)+ .087(12,000) = 10,554

EV = .696(6,000) + .217(16,000)+.087(21,000) = 9,475

I1(.54)

d1

d2

d3

EV = .148(10,000) + .185(15,000)+ .667(14,000) = 13,593

EV = .148 (8,000) + .185(18,000)+ .667(12,000) = 12,518

EV = .148(6,000) + .185(16,000)

+.667(21,000) = 17,855

4

5

6

7

8

9

2

3

1

17,855

11,433

Decision Tree

If the outcome of the survey is "favorable, choose C. Unfavorable, choose A.


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Decision Tree

If the outcome of the survey is "favorable, choose C. Unfavorable, choose A.

Expected value with sample information =.54(17,855) + .46(11,433) =14,900.88

This is how much we expect to get if we do the survey, wait for the results, thenchoose an alternative.

Without the survey, our best option was Model C. Recall thatEV of Model C =14,000

This is how much we get if we choose an alternative without the survey.

Expected value of sample information:EVSI =14,900.88 -14,000 =900.88

Since this is less than the cost of the survey (1,000), the survey should not bepurchased.


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Efficiency of Sample Information

The efficiency of the survey:

EVSI/EVPI = (900.88)/(2000) = .4504

The survey gives 45% of the extra value that perfectinformation would give.


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Utility and multiple objectives


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Profit[millions of pounds]

15-5

utility

0.00

1.00

0.50

5

0.25

0.75

0 10

uRA

uRN

uRS

Risk neutral (uRN):

Risk averse (uRA):

Risk seeking (uRS):

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Utility: Risk Attitude


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long termsustainability

profitabilitysize of

business

flexibility

shortterm

longterm

marketshare growth

Multi-criteria decision analysis (MCDA)


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Developing Value Functions

Value Function - Flexibility

Attribute: Degree of flexibility

provided by the alternative

Score

Easy to diversify to similar product 100

Diversification is possible, but itrequires some adaptation 60

Diversification is possible, but hard toimplement 40

Inflexible, very hard to diversify 0

47


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Making Trade-Offs

Associate a swing weightwith each attribute

The value of a decision alternative is the sum of the utilities for eachattribute, weighted by the swing weights Suppose alternative A had 30% market share (value 100) and flexibility

possible (value 40).

Suppose market share is 2 times as important to you as flexibility,leading you to choose swing weights of 2 and 1.

The value of decision A would be 2*100 + 1*40, plus similar contributionsfrom the other attributes.

The best decision is the one with the largest weighted utility.

If uncertainty is involved, the best decision is the one maximizing theexpected weighted utility.

E l ti O ti (M ki T d ff )


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long termsustainability

profitability size ofbusiness

flexibilityshortterm

longterm

marketshare growth

100

c

50%44%

6%34% 66% 62% 38%

0

594746

Overall Performances

Evaluating Options (Making Tradeoffs)

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Other multi-criteria decision-making methods

In mathematical optimization, the goal is to maximize afunction f subject to constraints.

Suppose you wish to maximize two functions, f and g.

Their maxima are likely to occur at different points.

If you give relative weights a and b to the twoattributes, you could maximize a*f+b*g.

If f is more important to you, and you only want tomaximize g if it results in no loss in f, you could trymaximizing f+0.001*g, for example (this is similar togoal programming in the books next chapter).

You might try to put everything on a common scale,utility or (more simple-mindedly), $$.


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Other multi-criteria decision-making methods

In mathematical optimization, the goal is to maximize afunction f subject to constraints.

Suppose you wish to maximize two functions, f and g.

Their maxima are likely to occur at different points.

If there are just 2 attributes (2 dimensions), plot theefficient frontier (the points maximizing f for a giveng).

That is, discard points where you can do better inboth f and g.

Pick the trade-off you like after seeing thepossibilities.


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Game theory MA402 game theory

OR409 auctions and game theory

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Lecture 3 - Decision Analysis

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