Decision Analysis

An insightful study of the decision-making process under uncertainty and risk

Decision TheoryA philosophy, articulated by a set of logical axioms and a methodology, for analyzing the complexities inherent in decision problems in order to identify the best alternative from among several alternatives.

I cannot decidewhat to do!

Decisions, based upon….

InstinctIntuitionSubjective JudgementExperienceQuantitative Analysis

Decision-makingin action!

Yes, sell all my oil and buy nuclear

power industries.

Problem complexity may be caused byLarge number of alternatives (combinatorial)more than one decision-maker (theory of games)multiple attributes (criteria) (AHP)risk and uncertaintysequential decisions - (decision trees)long time horizonshigh stakes

The Analysis of Decision Analysisi.e. the road ahead…

Decision making under uncertaintyDecision making under risk

without experimentationwith experimentation (sampling) (Bayesian)

Decision Trees (time permitting)

The Decision Table P1 P2 … Pn

S1 S2 … SN

A1 O1,1 O1,2 … O1,N

A2 02,1 O2,2 … O2,N

… . . … .

AM OM,1 OM,2 … OM,N

ALT

ER

NAT

IVE

S

STATES OF NATURE

PROBABILITIES

OUTCOMES, PAYOFFS OR UTILITIES

Decision Table Example

probability =

.2 .4 .3 .1

State of Nature

Demand is 0

Demand is 1

Demand is 2

Demand is 3

Stock 0 0 0 0 0 Stock 1 -10 5 5 5 Stock 2 -20 -5 10 10 Stock 3 -30 -15 0 15

Unit cost = $10Selling price = $15Unit profit = $ 5

Profit Matrix

Alte

rnat

ives

Decision Making under Uncertainty

Mom, what if you don’t know the probability

distribution for the states of nature? What do you

do then?

P1, P2, …, Pn are unknown

Decision-making without probabilitiesS1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

maximize payoff

states of nature

alte

rnat

ives

Pay attention! This is a

great decision matrix.

Dominance

If Ai is preferred over Aj for all States of Nature, then Ai dominates Aj and Aj can be eliminated as an alternative of choice.

Must investigate all possible pairwise comparisons. For M alternatives, there are

MC2 = M! / [(M-2)! 2!] comparisons.

If M = 6; then 6!/[4!2!] = 15 comparisons

Decision-making without probabilities

S1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

1. Dominance

A3 is always preferredover A4. ThereforeA4 is dominated by A3.

.Optimism

Maximax principle: Select the alternative that maximizes the maximum possible outcome

{ }i j

ijA SMax Max O

I am wild and adventurous. The best of all possible

outcomes will occur."It is demonstrable," said he, "that things cannot be otherwise than as they are; for as all things have been created for some end, they must necessarily be created for the best end.

- Voltaire’s Candide

Decision-making without probabilities

S1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

1. Dominance2. Optimism: maximize maximum payoff

Max

19

17

14

20

Pessimism

Maximin principle: Select the alternative that maximizes the minimum possible outcome

{ }i j

ijA SMax Min O

I am timid and unadventurous. The worst of all possible outcomes will occur.

Decision-making without probabilities

S1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

1. Dominance

2. Optimism: maximize maximum payoff

Max

19

17

14

20

3. Pessimism: maximize minimum payoff

Min

8

9

10

3

The Hurwicz Principle

Outlook is somewhere between extreme pessimism and extreme optimismThe degree of optimism (α) of the decision maker can be measured on a scale from 0 to 1.For each alternative i, the Hurwicz criterion is given by

Select the alternative that maximizes this criterion

( ){ } ( ){ }, ,max (1 ) minjj

i j i jSSO Oα α+ −

Note: The glass is neither half empty nor half full because it is bigger than it need be.

Professor Hurwicz

Leo Hurwicz received his LL.M. from Warsaw University - Poland in 1938. He teaches in the areas of theory, welfare economics, public economics, mechanisms and institutions, and mathematical economics. Professor Hurwicz's current research includes comparison and analysis of systems and techniques of economic organization, welfare economics, game-theoretic implementation of social choice goals, and modeling economic institutions.

Regents Professor EmeritusDepartment of EconomicsUniversity of Minnesota

Decision-making without probabilitiesS1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

1. Dominance

2. Optimism: maximize maximum payoff

Max

19

17

14

20

3. Pessimism: maximize minimum payoff

Min

8

9

10

3

Hurwicz(α = .8)

16.8

15.4

13.2

16.6

4. Hurwicz (coefficient of optimism)(α) maxi-max + (1- α) maxi-min

The degree of optimism

Hurwicz

02468

101214161820

0 0.2 0.4 0.6 0.8 1

Alpha

A1A2A3A4A5

Minimax RegretThe Savage principle:

Compute a regret matrix by finding for a given state of nature, the difference between each profit and the maximum profit.Then for each alternative, find the maximum regretSelect that alternative that minimizes the maximum regret

I get it. The regret is the difference in what we get versus what we

could have gotten if we had chosen the best alternative for that state of

nature.

Born: 20 Nov 1917 in Detroit, Michigan, USADied: 1 Nov 1971 in New Haven, Connecticut

Savage wrote on the foundations of statistics which led him into deep philosophical questions both about statistics and knowledge in general. The other main direction of his work was to study gambling as a source to stimulate problems in probability and decision theory. Savage's book The Foundations of Statistics (1954) is perhaps his greatest achievement. The book considers subjective probability and utility. It starts with six axioms, which are both motivated and discussed, and from these are deduced the existence of a subjective probability and a utility function. Another important work by Savage is How to gamble if you must : Inequalities for stochastic processes in 1965, written jointly with L Dubins. Other articles written by Savage relate to statistical inference, in particular the Bayesian approach. Heintroduced Bayesian hypothesis tests and Bayesian estimation.

Leonard Savage

Regret MatrixS1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A5 3 13 4 20

A1 0 5 5 1

A2 6 0 0 4

A3 5 3 1 9

A5 12 4 9 0

Regret:

max regret

5

6

9

12

Decision-making without probabilities

S1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

1. Dominance

2. Optimism: maximize maximum payoff

Max

19

17

14

20

3. Pessimism: maximize minimum payoff

Min

8

9

10

3

Hurwicz(α = .8)16.8

15.4

13.2

16.6

4. Hurwicz (coefficient of optimism)(α) maxi-max + (1- α) maxi-min

Regret

5

6

9

12

5. Savage: minimize maximumregret

Laplace Criterion (Principle of insufficient reason)It is sometimes suggested (initially by Laplace)

that in the absence of any evidence to the contrary, one might as well assume that all futures are equally likely.

Principle of Insufficient Reason: Assume all possible futures are of equal probability then select the alternative that maximizes expectation.

Laplace

Born: 23 March 1749 in Beaumont-en-Auge, Normandy, FranceDied: 5 March 1827 in Paris, France

The first edition of Laplace's Théorie Analytique des Probabilités was published in 1812. The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability, Bayes's rule (so named by Poincaré many years later), and remarks on moral and mathematical expectation. The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares, Buffon's needle problem, and inverse probability.Applications to mortality, life expectancy andthe length of marriages are given and finally Laplace looks at moral expectation and probabilityin legal matters.

Decision-making without probabilitiesS1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A4 9 8 9 6

A5 3 13 4 20

1. Dominance2. Optimism: maximize maximum payoff

Max

19

17

14

20

3. Pessimism: maximize minimum payoff

Min

8

9

10

3

Hurwicz(a = .8)16.8

15.4

13.2

16.6

4. Hurwicz (coefficient of optimism)(a) maxi-max + (1-a) maxi-min

5. Savage: minimize maximumregret

Regret

5

6

9

12

Laplace

13.5

13.75

11.75

10.0

6. Laplace: Assume equi-likelyoutcomes

p1 = p2 = p3 = p4 =.25

Which Alternative?

S1 S2 S3 S4

A1 15 12 8 19

A2 9 17 13 16

A3 10 14 12 11

A5 3 13 4 20

Max

19

17

14

20

Min

8

9

10

3

Hurwicz(a = .8)16.8

15.4

13.2

16.6

Regret

5

6

9

12

Laplace

13.5

13.75

11.75

10.0

The end of the first example and the beginning of the next…

That was a really great

example.

I can’t wait until the next

one.

ElectricBaseboard

HeatPump

CentralGas

SolarPanels

< 250 26-30 31-35 >350

300 240 125 90

380 270 110 70

250 175 130 100

450 200 90 50

avg winter temp

monthly heating cost

optim

ism

pess

imis

m

Hur

wic

z

Sav

age

Lapl

ace

90

70

100

50

300

380

250

450

a = .5

195

225

175

250

65

130

50

200

188.8

207.5

163.8

197.5

Regret Matrix< 250 26-30 31-35 >350

300 240 125 90

380 270 110 70

250 175 130 100

450 200 90 50Max6513050200

50 65 35 40130 95 20 200 0 40 50200 25 0 0

regret

What are the pitfalls? I see someproblems here.

Maxi-max or Not all is well!

S1 S2

A1 -$1000 $1000

A2 0 950

Max

1000

950

I want A1! No A2 ???

Maxi-min or more Not all is well!

S1 S2

A1 0 $1000

A2 $1 $1

Min

0

1

I want A2! No A1 ???

More Maxi-min -lack of column linearity

S1 S2min

A1 $ 5 $8 5A2 $10 $2 2

S1 S2min

A1 $ 5 $12 5A2 $10 $ 6 6

Say you receive an additional $4 if S2 occurs:

Hurwicz! What can be said about Hurwicz?S1 S2 S3 … SN

A1 1000 0 0 0

A2 0 1000 1000 … 1000

Max Min

1000 0

1000 0

Gosh! I am indifferentbetween A1 and A2regardless of the value of my coefficient of optimism.

What about regret? Will I regret that too?

S1 S2 S3

A1 1 6 4

A2 5 3 6

maxA1 4 0 2 4

A2 0 3 0 3

What about regret? Will I regret that too?

S1 S2 S3

A1 1 6 4

A2 5 3 6

maxA1 4 0 2 4

A2 0 3 0 3

S1 S2 S3

A1 1 6 4

A2 5 3 6

A3 4 8 1max

A1 4 2 2 4

A2 0 5 0 5

A3 1 0 5 5

There is always good old Laplace.Ignorance is little justification to assume all outcomesare equi-likely!

Try using subjective probabilitiesrather than a toss of the dice.

Equi-likely assumes least knowledge - most influenced byadditional information.

I didn’t know.

The Spreadsheet… and it came to the engineering

manager in a dream that the decision problem under uncertainty

can be analyzed using a spreadsheet.

I cannot wait to find out more about decision making?

This way to Decision Making under Risk…