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Resilience Informatics for Innovation Classical Decision Theory

RERC/TMIKazuo FURUTA

Decision-making and rationality

What is decision-making?n Methodology for making a choicen The quality of decision-making determines success

or failure of innovation and business.

Rational decision-makingn No means exist that certainly lead us to a correct

decision due to uncertainties in the world.n Anyhow, we need to decide rationally in some

sense, with no regret. ⇒ Max. use of information

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Classical decision-making

Normative model of decision-makingn Choosing one option from manyn Utility is what one wants to make maximum in the

choice.

Option 1Option 2Option 3

…… Option i

Assess options

Collectinformation

Predefined options

ChoiceUtility of options(u1, u2, u3, ....)

Choice and preference

Preferencen An act to chose one option from manyn Basic unit of classical decision

Order of preference in a dyada is preferred to b.a is equivalent to b.a is preferred or equivalent to b.

ba ≈ba f

ba f

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Utility and utility function

Utilityn What one wants to make maximum in the choicen Subjective value of each option evaluated by the

decision-maker

Utility function u(x)n Mapping from the set of options to real numbers

n Utility function is a tool to deal with the utility mathematically

)()( buauba ≥⇔f

Condition for rational decision

In order that the preference is mathematically consistent, it must be a weak order relation.

n Completeness or for any pair of a and b

n Transitivity

ba f ba p

cacbba fff ⇒,

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Expected utility

Definition of expected utility

si : possible state pi : occurrence probability of si u(si) : utility of si

Expected utility hypothesisn The utility of option a under uncertainty is

represented by its expected utility E(a).

∑=

=n

iii psuE

1)(

Example of expected utility

Win.prob. Prize ConsolationLottery 1 0.01 $15,000 noneLottery 2 0.02 $8,000 noneLottery 3 0.01 $10,000 $50

E(L1) = 0.01 x $15,000 = $150E(L2) = 0.02 x $8,000 = $160E(L3) = 0.01 x $10,000 + 0.99 x $50 = $149.5

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Choice under uncertainty

Uncertain situation si : Possible state pi : Occurrence probability of si

aj : Option of action uij : Utility when si obtains after aj has

been taken

Various decision criteria (1)

Expected utility criterion

n If the occurrence probability distribution is unknown, suppose pi = 1/n (Laplace criterion)

Max-min criterion

n Maximize the utility for the worst case

max)(s.t.*1

→== ∑=

n

iiijjj puaUaa

maxmin)(s.t.* →== ijijj uaUaa

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Various decision criteria (2)

Hurwitz criterion

n α = 0 and α = 1 correspond to complete pessimism (max-min criterion) and complete optimism.

Regret criterion (Minimum opportunity loss)

n Opportunity loss rij is the degree of regret compared with the maximum utility obtainable with perfect foresight.

maxmin)1(max)(s.t.* →−+== ijiijijj uuaUaa αα

ijijjij

ijijj

uurraLaa

−=→==

maxminmax)(s.t.*

Example of choice under uncertainty (1)

You are a street food stall owner. Which sells well depends on the weather: ice cream or hot dog. Which will you stock more for tomorrow’s business.

Weather Sunny Cloudy RainyProbability 0.6 0.3 0.1Ice cream 1,000 500 200Half & half 800 700 450Hot dog 650 600 500

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Example of choice under uncertainty (2)

Weather Hurwitz* RegretIce cream 600 300Half & half 625 200Hot dog 575 350

Criterion Expected Laplace Max-minIce cream 770 567 200Half & half 735 650 450Hot dog 620 583 500

* α = 0.5

Conditional probability

U : Sample spaceE, F : Subset of U| X | : Size of X

Conditional probability

UE F

)()(

||||||||

||||)|(

FPFEP

UFUFE

FFEFEP ∩

=∩

=∩

=

)|()()|()()( FEPFPEFPEPFEP ==∩

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Example of conditional probability

Two invisible bins A and BA : 3 silver & 1 gold coinsB : 2 silver & 4 gold coins

n You drew a silver coin from either of the two bins by chance. Which bin did you choose?

692.0139

6/18/38/3

)()()|( ≈=

+=

∩=

SPSAPSAP

A

start

B

1/2G

G

S1/2

3/4 S

1/4

2/6

4/6

3/8

1/8

1/6

2/6

Independent events

When event E and F satisfy the following conditions, they are independent.

The following is the necessary and sufficient condition for that E and F are independent.

)()|()()|( FPEFPEPFEP ==

)()()( FPEPFEP =∩

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Bayes theorem

General form of Bayes theoremn H1, H2, … Hn are exclusive each other and

)()()|()|(

EPFPFEPEFP =

UHHH n =∪∪∪ L21

)|()()|()()|()()|(

11 nn

iii HEPHPHEPHP

HEPHPEHP++

=L

Coins-in-bins problem revisited

Two invisible bins A and BA : 3 silver & 1 gold coinsB : 2 silver & 4 gold coins

n You drew a silver coin from either of the two bins by chance. Which bin did you choose?

139

2/13/12/14/32/14/3

)()|()()|()()|()|( =

⋅+⋅⋅

=+

=BPBSPAPASP

APASPSAP

A

start

B

1/2G

G

S1/2

3/4 S

1/4

2/6

4/6

3/8

1/8

1/6

2/6

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Bayesian inference (1)

H : Hypothesis

E : Evidence, testimony, or symptom

P(H) : Prior probability with no evidence

P(H|E) : Posterior probability after evidence E has been obtained

Bayesian inference (2)

After E has been obtained, how we should modify the probability of H?

Modification of odds in terms of E

)()()|()|(

EPHPHEPEHP =

)()()|()|(

EPHPHEPEHP =

)()()(

)|()|(

)|()|()|( HO

HPHP

HEPHEP

EHPEHPEHO Eλ===

Prior oddsPosterior odds

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Bayesian inference (3)

After another evidence F has been obtained,

)()()(

)|()|(

)|()|(

)|()|()|(

HOHPHP

HFPHFP

HEPHEP

FEHPFEHPFEHO

FEλλ==

∩∩

=∩

Modification factor for each evidence

Hasty doctor example

You received a cancer test. The doctor said that 1 out of 1,000 is the average rate of cancer at your age. The test is accurate and it gives a correct result for 99% of cancer patients and 97% of non cancer patients. Your test result was positive, and the doctor recommended you hospitalization ASAP.

301

999.003.0001.099.0

)()(

)()(

)( ≈××

=++

=+CPCP

CPCP

CO

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Three prisoners problem(Monty Hall problem)

Out of three prisoners, two will be executed and one will be freed tomorrow. Prisoner A heard from the jailor that B will be executed. A was delighted that his alive probability has increased from 1/3 to 1/2 with this information, since either A or C will be freed.

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13/103/12/13/12/13/1

)|()()|()()|()()|()()|(

=⋅+⋅+⋅

⋅=

++=

CbPCPBbPBPAbPAPAbPAPbAP

Component failure example

n Failure statistics of a particular component are collected for every 100 days of operation, and the following data were obtained. Evaluate the failure rate.

Interval 1 2 3 4

Times of failures 6 3 5 6

Average = (6+3+5+6)/4/100 = 0.05 day-1

STD = 0.02 day-1

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Probability density function

Cumulative distribution function F(x)n Probability that a random variable X takes

a value not greater than x

Probability density function f(x)

)()(

)()()()(

dxxXxPdxxf

dxxfxFdx

xdFxf x

+≤<≈

== ∫ ∞−

)()( xXPxF ≤<−∞=

Bayesian inference on distribution

: Prior density function of θ: Posterior density function of θ

Bayesian update of the density function after event A has been observed

)(θf)|( Af θ

∫=

θθθθθθ

dfAPfAPAf

)()|()()|()|(

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Application of Bayesian approach to component failure example

Poisson distribution

λ : Failure rateT : Interval of observation (100 days)k : Times of failures observed

n Uniform distribution in [0, 0.1] is assumed for f (λ) at the beginning.

Tk

ekTkP λλλ −=!)()|(

Bayesian update of failure rate

0

10

20

30

40

0.00 0.02 0.04 0.06 0.08 0.10

01 (k = 6)2 (k = 3)3 (k = 5)4 (k = 6)

λ (day-1)

P(λ|

A)

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Concepts of probability (1)

Normative definition (Pascal)n Fraction of number of cases among the

sample space

Statistical definition (d’Alembert)n Asymptotic value of event frequency

)/(lim nfp nn ∞→=

||/|| UAp =

Concepts of probability (2)

Subjective definition (Bayes)n Degree of individual confidence on

occurrence of the event

Informatics definition (Shannon)n Amount of information that implies

occurrence of the event

pS 2log−=