Chapters 19 – 20:Dealing With Uncertainties

• TPS Example: Uncertain outcome• Decision rules for complete uncertainty• Utility functions• Expected value of perfect information• Statistical decision theory• The value-of-information procedure

TPS Decision Problem 7OS Development Options:• Option B-Conservative (BC)– Sure to work (cost=$600K)– Performance = 4000 tr/sec

• Option B-Bold (BB)– If successful, Cost=$600K

Perf.= 4667 tr/sec– If not, Cost=$100K

Perf.= 4000 tr/secWhich option should we choose?

Operating System Development Options

Option BB (Bold)

Successful Not Successful

Option BC (Conservative)

Option A

Performance (tr/sec) 4667 4000 4000 2400

Value ($0.3K per tr/sec) 1400 1200 1200 720

Basic cost 600 600 600 170

Total cost 600 700 600 170

Net value NV 800 500 600 550

NV relative to Option A 250 -50 50 0

Payoff Matrix for OS Options BB, BC

State of Nature

Alternative Favorable Unfavorable

BB (Bold) 250 -50

BC (Conservative)

50 50

Decision Rules for Complete Uncertainty

• Maximin Rule. Determine minimum payoff for each option. Choose option maximizing the minimum payoff.

• Maximax Rule. Determine max-payoff for each option. Choose option maximizing max-payoff.

• Laplace Rule. Assume all states of nature equally likely. Choose option with maximum expected value.

BB = 0.5(250) + 0.5(-50) = 100 BC = 0.5(250) + 0.5(50) = 50

Difficulty with Laplace Rule

Favorable U1 U2 Expected Value

BB 250 -50 -50 50

BC 50 50 50 50

U1 : performance failure

U2 : reliability failure

Breakeven Analysis• Treat uncertainty as a parameter

P= Prob (Nature is favorable)• Compute expected values as f(P)

EV (BB) = P(250) + (1-P)(-50) = -50 + 300PEV (BC) = P(50) + (1-P)(50) = 50

• Determine breakeven point P such thatEV (BB) = EV (BC): -50 + 300P = 50, P=0.333

If we feel that actually P>0.333, Choose BB P<0.333, Choose BC

Utility Functions

Is a Guaranteed $50K

The same as

A 1/3 chance for $250K

and A 2/3 chance for -$50K?

Utility Function for Two Groups of Managers

Utility

1 2 3 4 5-1-2-3

10

20

-10

-20

Payoff ($M)

Group 1

Group 2

Possible TPS Utility Function

10

20

-10

-20

-100 100 200 300

(-80,-25)

(-50,-20)

(50,5) (100,10)(170,15)

(250,20)

Utility Functions in S/W Engineering

• Managers prefer loss-aversion– EV-approach unrealistic with losses

• Managers’ U.F.’S linear for positive payoffs– Can use EV-approach then

• People’s U.F.’S aren’t– Identical– Easy to predict– Constant

TPS Decision Problem 7

•Available decision rules inadequate

•Need better information

Info. has economic value

State of Nature

Alternative Favorable Unfavorable

BB (Bold) 250 -50

BC (Conservative) 50 50

Expected Value of Perfect Information (EVPI)

Build a Prototype for $10K– If prototype succeeds, choose BB

Payoff: $250K – 10K = $240K– If prototype fails, choose BC

Payoff: $50K – 10K = $40KIf equally likely,

Ev = 0.5 ($240K) + 0.5 ($40K) = $140K

Could invest up to $50K and do better than before – thus, EVPI = $50K

However, Prototype Will Give Imperfect Information

That is,P(IB|SF) = 0.0

Investigation (Prototype) says choose bold state of nature: Bold will fail

P(IB|SS) = 1.0Investigation (prototype) says choose bold

state of nature: bold will succeed

Suppose we assess the prototype’s imperfections asP(IB|SF) = 0.20, P(IB|SS) = 0.90

And Suppose the states of nature are equally likelyP(SF) = 0.50 P(SS) = 0.50

We would like to compute the expected value of using the prototypeEV(IB,IC) = P(IB) (Payoff if use bold)

+P(IC) (Payoff if use conservative)

= P(IB) [ P(SS|IB) ($250K) + P(SE|IB) (-$50K) ] + P(IC) ($50K)

But these aren’t the probabilities we know

How to get the probabilities we need

P(IB) = P(IB|SS) P(SS) + P(IB|SF) P(SF)

P(IC) = 1 – P(IB)

P(SS|IB) = P(IB|SS) P(SS) (Bayes’ formula)

P(IB)

P(SF|IB) = 1 – P (SS|IB)

P(SS|IB) = Prob (we will choose Bold in a state of nature where it will succeed)

Prob (we will choose Bold)

Net Expected Value of PrototypePROTO

COST, $KP (PS|SF) P (PS|SS) EV, $K NET EV,

$K

0 60 0

5 0.30 0.80 69.3 4.3

10 0.20 0.90 78.2 8.2

20 0.10 0.95 86.8 6.8

30 0.00 1.00 90 0

8

4

010 20 30

NE

T E

V, $

K

PROTO COST, $K

Conditions for Successful Prototyping (or Other Info-Buying)

1. There exist alternatives whose payoffs vary greatly depending on some states of nature.

2. The critical states of nature have an appreciable probability of occurring.

3. The prototype has a high probability of accurately identifying the critical states of nature.

4. The required cost and schedule of the prototype does not overly curtail its net value.

5. There exist significant side benefits derived from building the prototype.

Pitfalls Associated by Success Conditions

1. Always build a prototype or simulation– May not satisfy conditions 3,4

2. Always build the software twice– May not satisfy conditions 1,2

3. Build the software purely top-down– May not satisfy conditions 1,2

4. Prove every piece of code correct– May not satisfy conditions 1,2,4

5. Nominal-case testing is sufficient– May need off-nominal testing to satisfy conditions

1,2,3

Statistical Decision Theory: Other S/W Engineering Applications

How much should we invest in:– User information gathering– Make-or-buy information– Simulation– Testing– Program verification

How much should our customers invest in:– MIS, query systems, traffic models, CAD,

automatic test equipment, …

The Software Engineering Field Exists Because

Processed Information Has Value