Rational Decision Making: Preferences, Utilities, and Constraints, Study notes of Computer Science

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Rational decisions

Chapter 16

Chapter 16

Outline

Rational preferences

Utilities

Money

Multiattribute utilities

Decision networks

Value of information

Chapter 16

Rational preferences

Rational preferences Idea: preferences of a rational agent must obey constraints.

behavior describable as maximization of expected utility

Constraints:

Orderability

( A  B ) ∨ ( B  A ) ∨ ( A ∼ B )

Transitivity

( A  B ) ∧ ( B  C ) ⇒ ( A  C )

Continuity

A

B

C

p

[

p, A

p, C

]

B

Substitutability

A

B

[

p, A

p, C

]

[

p, B

p, C

]

Monotonicity

A  B ⇒ ( p ≥ q ⇔ [

p, A

p, B

]

[

q, A

q, B

])

Chapter 16

Rational preferences contd.

away all its moneyFor example: an agent with intransitive preferences can be induced to give Violating the constraints leads to self-evident irrationality If

B

C

, then an agent who has

C

would pay (say) 1 cent to get

B

If

A

B

, then an agent who has

B

would pay (say) 1 cent to get

A

If

C

A

, then an agent who has

A

would pay (say) 1 cent to get

C

A

B

C

1c

1c

1c

Chapter 16

Utilities

Standard approach to assessment of human utilities: Utilities map states to real numbers. Which numbers?

compare a given state

A

to a

standard lottery

L

p

that has

“best possible prize”

u

with probability

p

“worst possible catastrophe”

u

⊥

with probability

p )

adjust lottery probability

p

until

A

L

p

L

0.000001 0.

instant death continue as before

pay $

Chapter 16

Utility scales

Normalized utilities

u

u

⊥

Micromorts

: one-millionth chance of death

useful for Russian roulette, paying to reduce product risks, etc.

QALYs

: quality-adjusted life years

useful for medical decisions involving substantial risk

Note: behavior is

invariant

w.r.t. +ve linear transformation

U

(^) ′ ( x

) =

k 1 U

(^) ( x

) +

k 2

where

k

1

ordinal utility With deterministic prizes only (no lottery choices), only

can be determined, i.e., total order on prizes

Chapter 16

Student group utility

For each

x

, adjust

p

until half the class votes for lottery (M=10,000)

p

$x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

0

500

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

Chapter 16

10

Decision networks

Add

action nodes

and

utility nodes

to belief networks

to enable rational decision making

U

Airport Site

Deaths Noise Cost

Litigation

Construction

Air Traffic

Algorithm:

For each value of action node

compute expected value of utility node given action, evidence

Return MEU action

Chapter 16

Strict dominance

Typically define attributes such that

U

is

monotonic

in each

Strict dominance

: choice

B

strictly dominates choice

A

iff

i X

i ( B ) ≥ X i ( A )

(and hence

U

B

U

A

1

X

2

X

A

B

C

D

1

X

2

X

A

B

C

dominates This region

A

Deterministic attributes

Uncertain attributes

Strict dominance seldom holds in practice

Chapter 16

Stochastic dominance

0

0.2 0.4 0.6 0. 1

-5.

-4.

-3.

-2.

Probability

Negative cost

S2S

0

0.2 0.4 0.6 0. 1

-5.

-4.

-3.

-2.

Probability

Negative cost

S2S

Distribution

p 1

stochastically dominates

distribution

p 2

iff

t

∫ t

−∞

p 1 ( x

) dx

∫ t

−∞

p 2 ( t ) dt

If

U

is monotonic in

x

, then

A

1

with outcome distribution

p

1

stochastically dominates

A

2

with outcome distribution

p 2 :

∫ ∞

−∞

p 1 ( x ) U

x

) dx

∫ ∞

−∞

p 2 ( x ) U

x

) dx

Multiattribute case: stochastic dominance on all attributes

optimal

Chapter 16

Label the arcs + or –

SocioEcon

Age

GoodStudent

ExtraCar

Mileage

VehicleYear

RiskAversion

SeniorTrain

DrivingSkill

MakeModel

DrivingHist

DrivQuality

Antilock

Airbag

CarValue

HomeBase

AntiTheft

Theft

OwnDamage

PropertyCost

LiabilityCost

MedicalCost

Cushioning

Ruggedness

Accident OtherCost

OwnCost

Chapter 16

Label the arcs + or –

SocioEcon

Age

GoodStudent

ExtraCar

Mileage

VehicleYear

RiskAversion

SeniorTrain

DrivingSkill

MakeModel

DrivingHist

DrivQuality

Antilock

Airbag

CarValue

HomeBase

AntiTheft

Theft

OwnDamage

PropertyCost

LiabilityCost

MedicalCost

Cushioning

Ruggedness

Accident OtherCost

OwnCost

+

Chapter 16

Label the arcs + or –

SocioEcon

Age

GoodStudent

ExtraCar

Mileage

VehicleYear

RiskAversion

SeniorTrain

DrivingSkill

MakeModel

DrivingHist

DrivQuality

Antilock

Airbag

CarValue

HomeBase

AntiTheft

Theft

OwnDamage

PropertyCost

LiabilityCost

MedicalCost

Cushioning

Ruggedness

Accident OtherCost

OwnCost

+

+

−

Chapter 16

Label the arcs + or –

SocioEcon

Age

GoodStudent

ExtraCar

Mileage

VehicleYear

RiskAversion

SeniorTrain

DrivingSkill

MakeModel

DrivingHist

DrivQuality

Antilock

Airbag

CarValue

HomeBase

AntiTheft

Theft

OwnDamage

PropertyCost

LiabilityCost

MedicalCost

Cushioning

Ruggedness

Accident OtherCost

OwnCost

+

+

−

−

Chapter 16