value of experimentation - university of...

Decision Analysis-1

Value of Experimentation

Decision Analysis-2

The Value of Experimentation

• Should we perform the experiment?– Imperfect information - outcome is not always “correct”

– What is the potential value of the experiment?

• Two ways to evaluate the value of information– Expected value of perfect information (EVPI)

• The value of having a “crystal ball”

• This is a quick preliminary calculation

• Provides an upper bound on the potential value of experimentation. If EVPI < Cost : don’t perform the experiment

– Expected value of experimentation (EVE)

• EVE is the difference between the expected payoff resulting from performing the experiment and the expected payoff without the experiment

Decision Analysis-3

Expected Payoff with Perfect Information

• Suppose the “crystal ball” could definitely tell us the true state of

nature. Then we will pick the action with the maximum payoff for this

true state of nature.

• However, we don’t know in advance which state will be identified. So

weigh the max payoff with prior probabilities.

• E[PI] = expected payoff with perfect information

=

State of Nature

Action Oil Dry

Drill for oil 700 -100

Sell the land 90 90

Maximum payoff

Prior probability 0.25 0.75

Decision Analysis-4

Expected Payoff with Perfect Information

• Suppose the “crystal ball” could definitely tell us the true state of

nature. Then we will pick the action with the maximum payoff for this

true state of nature.

• However, we don’t know in advance which state will be identified. So

weigh the max payoff with prior probabilities.

• E[PI] = expected payoff with perfect information

= (700× 0.25) + (90× 0.75) = 242.5

State of Nature

Action Oil Dry

Drill for oil 700 -100

Sell the land 90 90

Maximum payoff

Prior probability 0.25 0.75

700 90

Decision Analysis-5

Expected Value of Perfect Information

• Expected Value of Perfect Information:

EVPI = E[PI] – E[OI]

where E[OI] is expected payoff with original information

(i.e., without experimentation)

• EVPI for the Goferbroke problem:

Decision Analysis-6

Expected Value of Perfect Information

• Expected Value of Perfect Information:

EVPI = E[PI] – E[OI]

where E[OI] is expected payoff with original information

(i.e., without experimentation)

• EVPI for the Goferbroke problem = E[PI] – E[OI]

= 242.5 – 100

= 142.5

• Since EVPI is greater than the cost of the experiment,

142.5 > 30, we should compute the expected value of

the experiment

Decision Analysis-7

Expected Value of Experimentation

• We are interested in the value of the experiment. If the

value is greater than the cost, then it is worthwhile to

do the experiment.

• Expected Value of Experimentation:

EVE = E[EI] – E[OI]

where E[EI] is expected payoff with experimental

information

Decision Analysis-8

Goferbroke Example (cont’d)

• Expected Value of Experimentation:

EVE = E[EI] – E[OI]

• For the Goferbroke problem

E[EI] = E[payoff|USS]×P(USS) + E[payoff|FSS]×P(FSS)

= (90×0.7) + (300×0.3)

= 153

EVE = 153 - 100

= 53

Decision Analysis-9

Painting Problem

• Painting at an art gallery, you think is worth $12,000

• Dealer asks $10,000 if you buy today (Wed.)

• You can buy today (Wed.) or wait until tomorrow (Thurs.): if not sold

by then, it can be yours for $8,000

• Tomorrow (Thurs.) you can buy or wait until the next day (Fri.): if not

sold by then, it can be yours for $7,000

• In any day, the probability that the painting will be sold to someone

else is 50%

• What is the optimal policy?

Decision Analysis-10

Drawer Problem

• Two drawers

– One drawer contains three gold coins,

– The other contains one gold and two silver.

• Choose one drawer

• You will be paid $500 for each gold coin and $100 for each silver

coin in that drawer

• Before choosing, you may pay me $200 and I will draw a randomly

selected coin, and tell you whether it’s gold or silver and which

drawer it comes from (e.g. “gold coin from drawer 1”)

• What is the optimal decision policy? EVPI? EVE? Should you pay

me $200?


Utility Theory


Validity of Monetary Value Assumption

• Thus far, when applying Bayes’ decision rule, we

assumed that expected monetary value is the

appropriate measure

• In many situations and many applications, this

assumption is inappropriate

• For example, a decision maker’s optimal choice may

depend on his/her “utility” for money

• A decision maker’s utility is affected by his/her

willingness to take risks


An Example

• Imagine you just graduated from college and owe

$40,000 in educational loans to a bank. You have a rich

aunt who offers you the following choice:

– A 50-50 chance of winning $100,000 or nothing

(expected value=0.5*100,000+0.5*0=50,000).

– A gift of $40,000 with no uncertainty attached.

• Which one would you accept?

• So ... is Bayes’ expected monetary value rule invalid?

No - because we can use it with the utility for money

when choosing between decisions



Utility Examples

• Think of a capital investment firm deciding whether or

not to invest in a firm developing a technology that is

unproven but has high potential impact

• How many people buy insurance?

Is this monetarily sound according to Bayes’ rule?

• Treatment for a disease – quality of life

We’ll focus on utility for money, but in general it could be

utility for anything (e.g., consequences of a doctor’s actions)


Outline

• Types of utility functions (risk averse, risk neutral, risk

seeking)

• Decision analysis with utility functions, fundamental

property

• How to construct utility functions for decision makers

– Use fundamental property and answer “lottery” questions

– Use an exponential function for risk averse decision makers



A Typical Utility Function for Money

u(M)

M

4

3

2

1

0$100 $250 $500 $1,000

• What does this mean?The decision maker

values $500 only 3 times

as much as $100

• The utility function has

a decreasing slope at

the amount of money

increases

• decision maker has a

decreasing marginal

utility for money (risk

averse).


Types of Utility Functions

• Risk-averse

– Avoid risk

– Decreasing marginal utility for money

• Risk-neutral

– Monetary value = Utility

– Prizes money at its face value

– Linear utility for money

• Risk-seeking (or risk-prone)

– Seek risk

– Increasing marginal utility for money

• Combination of these

u(M)

M

u(M)

M

u(M)

M

u(M)

M

…


• Inclusion of utility theory in decision analysis is founded

in some key ideas

• Fundamental property:

The decision maker is indifferent between two alternative

courses of action that have the same expected utility

• An optimal action is one that maximizes expected utility

Utility Theory and Decision Analysis

Illustration of Fundamental Property• Imagine an individual with the following utility function.

• Suppose this individual has the opportunity to win $100,000 with probability p or nothing with probability 1-p. This person has the option of receiving a gift amount with certainty. Then the individual is indifferent between the following pairs of choices.

• As we shall see, this fundamental property can also be used to constructutility functions.


M 0 10,000 30,000 60,000 100,000

u(M) 0 1 2 3 4

p Guaranteed gift

amount

Expected utility

0.25 10,000

0.50 30,000

0.75 60,000

Illustration of Fundamental Property• Imagine an individual with the following utility function.

• Suppose this individual has the opportunity to win $100,000 with probability p or nothing with probability 1-p. This person has the option of receiving a gift amount with certainty. Then the individual is indifferent between the following pairs of choices.

• As we shall see, this fundamental property can also be used to constructutility functions.


M 0 10,000 30,000 60,000 100,000

u(M) 0 1 2 3 4

p Guaranteed gift

amount

Expected utility

0.25 10,000 4*0.25+0=1

0.50 30,000 4*0.50+0=2

0.75 60,000 4*0.75+0=3


Two Approaches to Constructing Utility

Functions

• Ask the decision makers a series of “lottery” questions

– Depends on the decision maker answering a series of difficult

questions

– Constructs utility function from the fundamental property

• Assume a mathematical form (typically exponential) of

the utility function

– The exponential utility function is for risk averse decision makers

– The decision maker only has to answer one question

– Constructs utility function by estimating one parameter


Choosing between ‘Lotteries’

• Assume you were given the option to choose from two

‘lotteries’– Lottery 1

50:50 chance of winning $100,000 or $0

– Lottery 2

Receive $50,000 for certain

• Which one would you pick?

• How about between these two?

– Lottery 1


– Lottery 2


$100,000

$0

0.5

0.5

$50,0001

$100,000

$0

0.5

0.5

$40,0001


What is Your Expected Utility?

• What is x so you are indifferent?

– Lottery 1


– Lottery 2

Receive x for certain

$100,000

$0

0.5

0.5

x1

• What is p so you are indifferent?

– Lottery 1

p:1-p chance of winning $100,000 or $0

– Lottery 2


$100,000

$0

p

1-p

$50,0001


Goferbroke Example (with Utility)

• We need the utility values for the following possible

monetary payoffs:

Monetary

Payoff Utility

-130

-100

60

90

670

700

M

u(M)45°


Constructing Utility FunctionsGoferbroke Example

• u(0) is usually set to 0. So u(0)=0

• Arbitrarily, set u(-130)=-150

• We ask the decision maker what value of p makes

him/her indifferent between the following lotteries:

• The decision maker’s response is p=0.2

• Solve for u(700)

u(700)

u(-130)

p

1-pu(0)

1



• u(0) is usually set to 0. So u(0)=0

• Arbitrarily, set u(-130)=-150

• We ask the decision maker what value of p makes



• Solve for u(700):

0.2*u(700) + 0.8*u(-130) = u(0)

u(700) = (0 – 0.8*(-150)) / 0.2 = 600

u(700)

u(-130)

p

1-pu(0)

1



• We now ask the decision maker what value of p makes



• Solve for u(90)

u(700)

u(0)

p

1-pu(90)

1







0.15*u(700) + 0.85*u(0)=u(90)

0.15*600 + 0.85*0 = 90 = u(90)

u(700)

u(0)

p

1-pu(90)

1






• Solve for u(60)

u(700)

u(0)

p

1-pu(60)

1







0.1*u(700) + 0.9*u(0)=u(60)

0.1*600 + 0 = 60 = u(60)

u(700)

u(0)

p

1-pu(60)

1



Monetary

Payoff Utility

-130 -150

-100 -105

60 60

90 90

670 580

700 600

-200

-100

0

100

200

300

400

500

600

700

800

-200 -100 0 100 200 300 400 500 600 700 800

u(M)

M

45°


Exponential Utility Functions

• One of the many mathematically prescribed forms of a “closed-

form” utility function

• It is used for risk-averse decision makers only

• Can be used in cases where it is not feasible or desirable for the

decision maker to answer lottery questions for all possible outcomes

• The single parameter R is approximately the one such that the

decision maker is indifferent between

R

-R/2

0.5

0.50

1and


Exponential Utility Functions

• Small R implies

significant risk

aversion

• Large R implies

small risk aversion

(close to risk neutral)

Decision Analysis-34Decision Analysis-34

Goferbroke Example (with Utility)

Decision Tree

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