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Outline

Probability Review

Introduction and Terminology

Decision Making under Uncertainty

Maximax and Maximin

Laplace and Hurwicz

Decision Making under Risk

Maximum Likelihood

Baye’s Decision Rule

Decision Trees

New Information

Utility Theory

Chapter 9

Outline

Probability Review

Introduction and Terminology

Decision Making under Uncertainty

Maximax and Maximin

Laplace and Hurwicz

Decision Making under Risk

Maximum Likelihood

Baye’s Decision Rule

Decision Trees

New Information

Utility Theory

Chapter 9

Discrete Random Variables

Suppose we have an experiment whose outcome depends on chance

The outcome of the experiment is called random variable

The sample space of the experiment is the set of all possible outcomes.

A Random Variable, X, is said to be discrete if it can take on at most a countable (finite) number of values

That is, the sample space is finite

A subset of the sample space is called event

Discrete Random Variables

Rolling a standard six-sided die

6 possible outcomes

Sample space, S={1,2,3,4,5,6}

X is the outcome we get

X is a random variable

Probability of having 4 after we roll is 1/6

event

Discrete Random Variables

Rolling a standard six-sided die

What is the probability of having an even number?

6 possible outcomes after we roll

Our event E={2,4,6}  P(E)=1/6+1/6+1/6=1/2

Discrete Random Variables

P(X=x) = “Probability that X takes on the value x”

0 ≤ P(X=x) ≤ 1

S P(X=x) =1

Probability of an event is the sum of the probabilities of the outcomes in this event

Probability distribution function (pdf) = f(X)

Cumulative distribution function (cdf) = F(X)

Discrete Random Variables

E[X] = “expected value of the X”

E[X] = S x P(X=x)

V[X] = “variance of the X”

V[X] =S x2 P(X=x) – E[X]2

V[X]= S P(X=x)(x – E[X])2

SD[X] = “standard deviation of X”

SD[X] = √V[X]

Discrete Random Variables

Rolling a die

Discrete Random Variables

Three coins are tossed. Let X be the number of heads obtained. Construct a probability distribution for X and find its mean and standard deviation.

X	P(X)
0	1 / 8
1	3 / 8
2	3 / 8
3	1 / 8

E[X]= 0 * (1/8) + 1 * (3/8) + 2 * (3/8) + 3 * (1/8) = 1.5

V[X]= (0 - 1.5) 2 * (1/8) + (1 - 1.5) 2 * (3/8) + (2 - 1.5) 2 * (3/8) + (3 - 1.5) 2 * (3/8)= 1.11

Expectation

Recall that

E[X] = S x P(X=x)

We have the following properties

E[X+c]=E[X]+c where c is a constant

E[aX]=aE[X] where a is a constant

E[aX+bY]=aE[X]+bE[Y] where Y is also a random variable

Variance

Recall that

V[X] =S x2 P(X=x) – E[X]2

We have the following properties

V[X+c]=V[X] where c is a constant

V[aX]=a2V[X] where a is a constant

V[aX+bY]= a2V[X]+ b2V[Y] if X and Y are independent variables

Expectation and Variance

Multiplication of two random variables

Suppose Z = XY,

X and Y are independent random variables

That is, if P(X Y) = P(X)P(Y)

THEN Z is a random variable with

E[Z] = E[X]E[Y]

V[Z] = V[X]E[Y]2 + V[Y]E[X]2 + V[X]V[Y]

Conditional Probability

The conditional probability of an event A is the probability that the event will occur given the knowledge that an event B has already occurred

P(A|B), notation for the probability of A given B

What is the probability of having a 4 after you roll a die and you know that you have an even number?

P(X=4| even number)=1/3

Conditional Probability

) is the probability of A and B

EA={4}, EB={2,4,6} then EAnB={4}

Discrete Distributions

Bernoulli (yes-no) distribution:

Two outcomes: whether an event occurs or not

Binomial Distribution:

Describes the number of times an event occurs in a fixed number of trials (e.g., number of heads in 10 flips of a coin)

For each trial, only two outcomes possible

Trials are independent

Probability remains the same for each trial

Discrete Distributions

Geometric Distribution:

Describes the number of trials until an event occurs (e.g., number of times to roll a die until you get 6)

Same probability for each trial

Continue until succeed, unlimited trials

Negative Binomial Distribution:

Describes the number of trials until an event occurs n times

Similar to geometric (when n=1, you have geometric distribution)

Discrete Distributions

Poisson Distribution:

Describes the number of times an event occurs during a given period of time or space

Occurrences are independent

Any number of events is possible

Discrete Uniform Distribution:

Each outcome is equally likely

Rolling a dice, each outcome is ½ probability

Read Chapter 13.7

Continuous Random Variables

A random variable is called continuous if it can assume all possible values in the possible range of the random variable.

The interarrival times

The age of a bulp

The weight of a fish caught

The heat in a day

Continuous Random Variables

Let X be a continuous random variable

Then it has a probability density function, f(X)

Let X be a continuous random variable between a and b

where

Continuous Distributions

Uniform Distribution

Triangular Distribution

Normal Distribution

Exponential Distribution

Gamma Distribution

Erlang Distribution

Lognormal Distribution

Read Chapter 3.17

Continuous Distributions

Uniform distribution example

Outline

Probability Review

Introduction and Terminology

Decision Making under Uncertainty

Maximax and Maximin

Laplace and Hurwicz

Decision Making under Risk

Maximum Likelihood

Baye’s Decision Rule

Decision Trees

New Information

Utility Theory

Chapter 9

Decision Analysis: Introduction

Managers often need to make decisions under uncertainty

Introducing a new product

A financial firm’s investment decisions

Agricultural firm’s mix of crop planning

Oil company’s drilling decisions

Storage Wars!!!! 

Me preparing exams!!! Which questions to ask?

Decision Analysis: Example

The Goferbroke Company develops oil wells in unproven territory

A consulting geologist has reported that there is a one-in-four chance of oil on a particular tract of land.

Drilling for oil on this tract would require an investment of about $100,000.

If the tract contains oil, it is estimated that the net revenue generated would be approximately $800,000.

Another oil company has offered to purchase the tract of land for $90,000.

Question: Should Goferbroke drill for oil or sell the tract?

Decision Analysis: Example

Prospective profits

	Profit
Status of Land	Oil	Dry
Alternative
Drill for oil	$700,000	–$100,000
Sell the land	90,000	90,000
Chance of status	1 in 4	3 in 4

Decision Analysis: Terminology

The decision maker is the individual or group responsible for making the decision.

The alternatives are the options for the decision to be made.

The outcome is affected by random factors outside the control of the decision maker. These random factors determine the situation that will be found when the decision is executed. Each of these possible situations is referred to as a possible state of nature.

The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities.

Each combination of a decision alternative and a state of nature results in some outcome. The payoff is a quantitative measure of the value to the decision maker of the outcome. It is often the monetary value.

Decision Analysis: Prior Probability

The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities.

State of Nature	Prior Probability
The tract of land contains oil	0.25
The tract of land is dry (no oil)	0.75

Decision Analysis: Payoff Table

Each decision alternative can have different values (payoffs) under different states of nature

	State of Nature
Alternative	Oil	Dry
Drill for oil	$700K	$–100K
Sell the land	$90K	$90K
Prior probability	0.25	0.75

Outline

Probability Review

Introduction and Terminology

Decision Making under Uncertainty

Maximax and Maximin

Laplace and Hurwicz

Decision Making under Risk

Maximum Likelihood

Baye’s Decision Rule

Decision Trees

New Information

Utility Theory

Chapter 9

Decision Analysis: Decisions?

When we have certainty, we can use

Linear programming, integer programming,

Binary programming, nonlinear programming

What if we do not have certainty?

Uncertainty vs. Risk

Uncertainty: Probabilities are not known

Risk (Stochastic): Probabilities are known

We can still use mathematical programming for both cases!!

Maximax Criterion

The maximax criterion is the decision criterion for the eternal optimist.

It focuses only on the best that can happen.

Plan for the best worst case

Procedure:

Identify the maximum payoff from any state of nature for each alternative.

Find the maximum of these maximum payoffs and choose this alternative.

Maximax Criterion

Drill for oil

	State of Nature
Alternative	Oil	Dry	Maximum in Row
Drill for oil	700	–100	700  Maximax
Sell the land	90	90	90

Maximax Criterion

Another example…

Future state	F1	F2	F3
Alternative		(Payoff)
A1	100	100	400
A2	-200	150	600
A3	0	200	500
A4	100	300	200

Best Best Case

400 (j=3)

600 (j=3)

500 (j=3)

300 (j=2)

Risky

Payoff

Maximin Criterion

The maximin criterion is the decision criterion for the total pessimist.

It focuses only on the worst that can happen.

Plan for the best worst case

Procedure:

Identify the minimum payoff from any state of nature for each alternative.

Find the maximum of these minimum payoffs and choose this alternative.

Maximin Criterion

Sell the land

Note that the focus is on payoffs…

Do not confuse with cost vs. profits

	State of Nature
Alternative	Oil	Dry	Minimum in Row
Drill for oil	700	–100	–100
Sell the land	90	90	90  Maximin

Maximin Criterion

Another example…

Future state	F1	F2	F3
Alternative		(Payoff)
A1	100	100	400
A2	-200	150	600
A3	0	200	500
A4	100	300	200

Best Worst Case

100 (j=1 or 2)

-200 (j=1)

0 (j=1)

100 (j=1)

Conservative

Payoff

Laplace Criterion

Assume that probability of each occurrence is identical

Probability	N/A	N/A	N/A
Future State	F1	F2	F3
Alternative		(Payoff)
A1	100	100	400
A2	-200	150	600
A3	0	200	500
A4	100	300	200

1/3 1/3 1/3

Average Payoff

200=(100+100+400)/3

183=(-200+150+600)/3

233=(0+200+500)/3 .

200=(100+300+200)/3

Hurwicz (Realism) Criterion

Combination of maximin maximax

What if = 0?  Maximin

What if = 1?  Maximax

How optimistic you are

Hurwicz (Realism) Criterion

(Adapted From: Blanchard and Fabrycky, “System Engineering and Analysis, Prentice Hall, 1998)

Values for the Hurwicz Rule for Four Alternatives

$600

400

200

$600

400

200

-200

0 0.2 0.4 0.6 0.8 1.0

i=1

i=3

i=2

Risky

Conservative

Decision Making Under Uncertainity

When you have no information about probabilities

Maximax

Minimax

Laplaca

Hurwicz

“Cost-stable truck scheduling at a cross-dock facility with unknown truck arrivals” Konur and Golias (2013)

You do not know when the trucks will arrive

Schedule to minimize maximum costs?

Stability: range between maximum and minimum possible costs

Minimize the range while minimizing the average costs

Decision Making under Risk

Now suppose that you know some information about the probabilities

You can still use maximax, minimax, laplace, hurwicz

Risk has three primary components:

an event

a probability of occurrence of that event

the impact of that event

You can utilize the probabilities that you know

Maximum likelihood criterion

Baye’s decision rule

Outline

Probability Review

Introduction and Terminology

Decision Making under Uncertainty

Maximax and Maximin

Laplace and Hurwicz

Decision Making under Risk

Maximum Likelihood

Baye’s Decision Rule

Decision Trees

New Information

Utility Theory

Chapter 9

Maximum Likelihood Criterion

The maximum likelihood criterion focuses on the most likely state of nature.

Procedure:

Identify the state of nature with the largest prior probability

Choose the decision alternative that has the largest payoff for this state of nature.

	State of Nature
Alternative	Oil	Dry
Drill for oil	700	–100	–100
Sell the land	90	90	90  Step 2: Maximum
Prior probability	0.25	0.75
		
	Step 1: Maximum

Maximum Likelihood Criterion

Practice…

Probability	0.3	0.2	0.5
Future State	F1	F2	F3
Alternative		(Payoff)
A1	100	100	400
A2	-200	150	600
A3	0	200	500
A4	100	300	200

Which alternative will you choose based on

Maximum likelihood criterion?

Baye’s Decision Rule

Bayes’ decision rule directly uses the prior probabilities.

Procedure:

For each decision alternative, calculate the weighted average of its payoff by multiplying each payoff by the prior probability and summing these products. This is the expected payoff (EP).

Choose the decision alternative that has the largest expected payoff.

Baye’s Decision Rule

	State of Nature
Alternative	Oil	Dry
Drill for oil	$700K	$–100K
Sell the land	$90K	$90K
Prior probability	0.25	0.75

EP=700*1/4-100*3/4=100

EP=90*1/4+100*3/4=90

Baye’s Decision Rule

Another example…

E[X] =  x p(x)

A1: 100(0.3) + 100(0.2) + 400(0.5) = $250

A2: -200(0.3) + 150(0.2) + 600(0.5) = $270

A3: 0(0.3) + 200(0.2) + 500(0.5) = $290

A4: 100(0.3) + 300(0.2) + 200(0.5) = $190

Must choose Alternative 3!

Probability	0.3	0.2	0.5
Future State	F1	F2	F3
Alternative
A1	100	100	400
A2	-200	150	600
A3	0	200	500
A4	100	300	200

Payoffs

Baye’s Decision Rule

Features of Bayes’ Decision Rule

It accounts for all the states of nature and their probabilities.

The expected payoff can be interpreted as what the average payoff would become if the same situation were repeated many times. Therefore, on average, repeatedly applying Bayes’ decision rule to make decisions will lead to larger payoffs in the long run than any other criterion.

Criticisms of Bayes’ Decision Rule

There usually is considerable uncertainty involved in assigning values to the prior probabilities.

Prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures.

It ignores typical aversion to risk. By focusing on average outcomes, expected (monetary) payoffs ignore the effect that the amount of variability in the possible outcomes should have on decision making.

Other criteria

Aspiration-Level Criterion - desired/undesired level of achievement

Suppose that you do not want to have a possibility of losing money, then you will sell the land

Mean-Variance Criterion - based on “average” outcome and “variance”

You may want to have variance of an alternative to be less than a specific value

Decision Trees

A decision tree can apply Bayes’ decision rule while displaying and analyzing the problem graphically.

A decision tree consists of nodes and branches.

A decision node, represented by a square, indicates a decision to be made.

The branches represent the possible decisions.

An event node, represented by a circle, indicates a random event.

The branches represent the possible outcomes of the random event.

Decision Trees

Decision tree of the Goferbroke Company

E[Payoff] = 700(.25) – 100(.75) = $100

E[Payoff] = 90(1) = $90

Sensitivity Analysis

At what probability of oil are you indifferent?

E[Payoff of Drill] = E[Payoff of Sell]

700p -100(1-p) = $90

 p = .2375

Value of More Information

Might it be worthwhile to spend money for more information to obtain better estimates?

What if we knew for sure whether or not there was oil?

We can make the decision after we learn the true state

A quick way to check is to pretend that it is possible to actually determine the true state of nature (“perfect information”).

Value of More Information

EP (with perfect information) = Expected payoff if the decision could be made after learning the true state of nature.

EP (without perfect information) = Expected payoff from applying Bayes’ decision rule with the original prior probabilities.

The expected value of perfect information is then EVPI = EP (with perfect information) – EP (without perfect information).

Value of More Information

For the Goferbroke Company

When we have perfect information on the state, we automatically select the best option for that state

If Dry, we will sell

If Oil, we will drill

Value of More Information

For the Goferbroke Company

EP(without perfect information)=100

EP(with perfect information)=242.5

Expected value of perfect information:

EVPI= EP(with perfect information)- EP(without perfect information)

EVPI=242.5-100=142.5

Value of More Information

Let’s say you can pay $C to have perfect information

A seismic survey to obtain better estimates

EPVI>C, it might be worthwhile to do the survey

EPVI<C, it is not worthwhile to do the survey

Posterior Probabilities

The prior probabilities of the possible states of nature often are quite subjective in nature.

They may only be rough estimates.

It is frequently possible to do additional testing or surveying (at some expense) to improve these estimates.

The improved estimates are called posterior probabilities.

Using the New Information

Goferbroke can obtain improved estimates of the chance of oil by conducting a detailed seismic survey of the land, at a cost of $30,000.

Possible findings from a seismic survey:

FSS: Favorable seismic soundings; oil is fairly likely.

USS: Unfavorable seismic soundings; oil is quite unlikely.

P(finding | state) =Probability that the indicated finding will occur, given that the state of nature is the indicated one.

Using the New Information

Prior Probabilities

P(Oil) = 0.25

P(Dry) = 0.75

Conditional Probabilities

	P(finding \| state)
State of Nature	Favorable (FSS)	Unfavorable (USS)
Oil	P(FSS \| Oil) = 0.6	P(USS \| Oil) = 0.4
Dry	P(FSS \| Dry) = 0.2	P(USS \| Dry) = 0.8

Using the New Information

Each combination of a state of nature and a finding will have a joint probability determined by the following formula:

P(state and finding) = P(state) P(finding | state)

P(Oil and FSS) = P(Oil) P(FSS | Oil) = (0.25)(0.6) = 0.15.

P(Oil and USS) = P(Oil) P(USS | Oil) = (0.25)(0.4) = 0.1.

P(Dry and FSS) = P(Dry) P(FSS | Dry) = (0.75)(0.2) = 0.15.

P(Dry and USS) = P(Dry) P(USS | Dry) = (0.75)(0.8) = 0.6.

Using the New Information

Given the joint probabilities of both a particular state of nature and a particular finding, the next step is to use these probabilities to find each probability of just a particular finding, without specifying the state of nature.

P(finding) = P(Oil and finding) + P(Dry and finding)

P(FSS) = 0.15 + 0.15 = 0.3.

P(USS) = 0.1 + 0.6 = 0.7.

Using the New Information

The posterior probabilities give the probability of a particular state of nature, given a particular finding from the seismic survey.

P(state | finding) = P(state and finding) / P(finding)

P(Oil | FSS) = 0.15 / 0.3 = 0.5.

P(Oil | USS) = 0.1 / 0.7 = 0.14.

P(Dry | FSS) = 0.15 / 0.3 = 0.5.

P(Dry | USS) = 0.6 / 0.7 = 0.86.

Calculating the Posterior Probabilities

The formula for a posterior probability (Baye’s Theorem)

P(state | finding)=

P(state)P(finding | state)

P(oil)P(finding | oil)+P(dry)P(finding | dry)

	P(state \| finding)
Finding	Oil	Dry
Favorable (FSS)	P(Oil \| FSS) = 1/2	P(Dry \| FSS) = 1/2
Unfavorable (USS)	P(Oil \| USS) = 1/7	P(Dry \| USS) = 6/7

Decision Trees

Decision tree with the survey option

Decision Trees

And the payoffs

(0.7)

Decision Making

Sell

Drill

Do survey

Decision Making

Best decision

Do the seismic survey

If the result is unfavorable, sell the land

If the result is favorable, drill for oil

The expected payoff is 123

Outline

Probability Review

Introduction and Terminology

Decision Making under Uncertainty

Maximax and Maximin

Laplace and Hurwicz

Decision Making under Risk

Maximum Likelihood

Baye’s Decision Rule

Decision Trees

New Information

Utility Theory

Thus far, when applying Bayes’ decision rule, we have assumed that the expected payoff in monetary terms is the appropriate measure.

In many situations, this is inappropriate.

Accept a 50-50 chance of winning $100,000.

Receive $40,000 with certainty.

Many would pick $40,000, even though the expected payoff on the 50-50 chance of winning $100,000 is $50,000. This is because of risk aversion.

A utility function for money is a way of transforming monetary values to an appropriate scale that reflects a decision maker’s preferences (e.g., aversion to risk).

Utility Theory

The people!!

U(M): utility function for money

Utility Theory

When a utility function for money is incorporated into a decision analysis approach, it must be constructed to fit the current preferences and values of the decision maker.

When the decision maker’s utility function for money is used, Bayes’ decision rule replaces monetary payoffs by the corresponding utilities.

The optimal decision (or series of decisions) is the one that maximizes the expected utility.

Utility Theory

Fundamental Theory:

Under the assumptions of utility theory, the decision maker’s utility function for money has the property that the decision maker is indifferent between two alternatives if the two alternatives have the same expected utility.

Utility Theory

Fundamental Theory:

25% chance of $100,000 =

$10,000 for sure

Both have E(Utility) = 0.25.

50% chance of $100,000 =

$30,000 for sure

Both have E(Utility) = 0.5.

75% chance of $100,000 =

$60,000 for sure

Both have E(Utility) = 0.75.

Utility Theory

Determine the largest potential payoff, M=Maximum.

Assign U(Maximum) = 1.

Determine the smallest potential payoff, M=Minimum.

Assign U(Minimum) = 0.

To determine the utility of another potential payoff M, consider the two aleternatives:

A1: Obtain a payoff of Maximum with probability p. Obtain a payoff of Minimum with probability 1–p.

A2: Definitely obtain a payoff of M.

Question to the decision maker: What value of p makes you indifferent? Then, U(M) = p.

Utility Theory

The possible monetary payoffs in the Goferbroke Co. problem are –130, –100, 0, 60, 90, 670, and 700 (all in $thousands).

Set U(Maximum) = U(700) = 1.

Set U(Minimum) = U(–130) = 0.

To find U(M), use the equivalent lottery method.

For example, for M=90, consider the two alternatives: A1: Obtain a payoff of 700 with probability p Obtain a payoff of –130 with probability 1–p. A2: Definitely obtain a payoff of 90

If Max chooses a point of indifference of p = 1/3, then U(90) = 1/3

Utility Theory

Utility function will be

Utility Theory

Risk Averse

U(ta) = R(1-e-t/R)

Risk Neutral

U(tn) = at + b

Risk Seekers

U(ts) = t2/c

Further study…

Read Chapter 9

Practice problems

9.1, 9.2, 9.3, 9.4, 9.7, 9.19, 9.20, 9.27

The following problems are in Homework 3:

9.16 (modified version)

outcomevalue

xP(X=x)

11/6

21/6

31/6

41/6

51/6

61/6

sum1

outcomevalue

xP(X=x)

even1/2

odd1/2

xP(X=x)x*P(X=x)x*x*P(X=x)

11/60.170.17

21/60.330.67

31/60.501.50

41/60.672.67

51/60.834.17

61/61.006.00

3.52.921.71

E[X]V[X]SD[X]

}

max

{

max

}

min

{

max

]}

min

)[

(

]

max

[

{

max

Bayes' Decision Rule for the Goferbroke Co.

Payoff Table

Expected

Alternative

Oil

Dry

Payoff

Drill

700

-100

100

Sell

Prior Probability

0.25

0.75

State of Nature

Bayes

Bayes' Decision Rule for the Goferbroke Co.
	Payoff Table	State of Nature		Expected	Range Name	Cells
	Alternative	Oil	Dry	Payoff	DrillPayoff	C5:D5
	Drill	700	-100	100	ExpectedPayoff	F5:F6
	Sell	90	90	90	PriorProbability	C8:D8
					SellPayoff	C6:D6
	Prior Probability	0.25	0.75

Sheet1

Payoff

-100

700

Oil (0.25)

Dry (0.75)

Drill

Sell

Payoff Table

Alternative

Oil

Dry

Drill

700

-100

Sell

Maximum Payoff

700

Prior Probability

0.25

0.75

EP (with perfect info)

242.5

State of Nature

Perfect Info

Expected Payoff with Perfect Information
	Payoff Table	State of Nature		Range Name	Cells
	Alternative	Oil	Dry	DryPayoff	D5:D6
	Drill	700	-100	MaximumPayoff	C7:D7
	Sell	90	90	OilPayoff	C5:C6
	Maximum Payoff	700	90	PriorProbability	C9:D9
	Prior Probability	0.25	0.75
		EP (with perfect info)	242.5

Sheet1

Do seismic survey

No seismic survey

Unfavorable

Favorable

Drill

Sell

Drill

Sell

Oil

Dry

Oil

Dry

Oil

Dry

Sell

Drill

Payoff

670

-130

670

-130

700

-100

Do seismic survey

No seismic survey

Unfavorable

Favorable

Drill

-100

Sell

Drill

-100

Sell

Drill

-100

Sell

Oil (0.143)

800

Dry(0.857)

Oil (0.5)

800

Dry (0.5)

Oil (0.25)

800

Dry (0.75)

(0.3)

-30

Payoff

670

-130

670

-130

700

-100

100

270

123

-15.7

270

100

Do seismic survey

No seismic survey

-30

Unfavorable

Favorable (0.3)

Drill

-100

Sell

Drill

-100

Sell

Drill

-100

Sell

Oil (0.143)

800

Dry (0.857)

Oil (0.5)

800

Dry (0.5)

Oil (0.25)

800

Dry (0.75)

(

)

(

)

(

)

(

) Risk averse

(

) Risk seeker

(

) Risk neutral

0.25

0.5

0.75

$10,000

$30,000

$60,000

$100,000

(

)