[SOLVED] CS ISE 562; Dr. Criteria Decision Theory

ISE 562; Dr. Criteria Decision Theory
ISE 562; Dr.
Decision making criteria
Decision making under certainty

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Decision making under uncertainty
Decision analysis and the value of information
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ISE 562; Dr. of decision making
Decision alternatives-choices
Decision attributes (profit, cost, etc.)
Uncertain (future) outcomes (states of nature) that affect the decision
Payoffs (profits or costs) for each alternative under each state of nature
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ISE 562; Dr. of these components can be displayed in a payoff table. Consider problem of farmer considering planting one of 3 crops next year. The return will be determined by whether a trade bill with China passes the Senate. The profits under the two outcomes are displayed in the following payoff table.
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ISE 562; Dr.
Payoff Table
States of nature
Trade Bill
ISE 562; Dr. making under certainty
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ISE 562; Dr. making criteria: rules for selecting the best decision
Each rule has assumptions
Uncertainty not considered because:
Not available
Not important
Not enough time
ISE 562; Dr. making criteria (no uncertainty):
Maximax criterion
Maximin criterion
Minimax regret criterion
Hurwicz criterion
Equal likelihood criterion
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ISE 562; Dr. making criteria: Maximax criterion
Maximum of the maximum payoffs
Overly optimistic and ignores large negative consequences
Decision: plant corn
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Trade Bill
ISE 562; Dr. making criteria: Maximin criterion
Maximum of the minimum payoffs
Pessimistic; (Note if table contained costs, would use Minimax)
Decision: plant soybeans
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Trade Bill
ISE 562; Dr. making criteria: Minimax regret criterion (opportunity loss)
Difference between actual choice and best choice Example.1 Example.2
Trying to minimize the maximum regret; subtract each state of nature from the maximum payoff state of nature and redraw table
Then select the alternative with minimum regret.
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ISE 562; Dr. each state of nature from the maximum payoff state of nature and select minimum regret value
Decision: plant corn
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Trade Bill
35,000- 35,000=0
20,000-8,000 =12,000
35,000-18,000 =17,000
20,000-12,000 =8,000
35,000-22,000 =13,000
20,000- 20,000=0

ISE 562; Dr. making criteria: Hurwicz criterion
Compromise between maximax/maximin by allowing the degree of optimism to be specified.
Coefficient of optimism, 01, if =0 decision maker pessimistic; =1 optimistic. (1-) is coefficient of pessimism.
For each alternative, calculate
(max payoff) + (1-) (min payoff)
Select largest (if cost, smallest)
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ISE 562; Dr. making criteria: Hurwicz criterion with =0.30
Trade Bill
H(corn)=.3(35000)+(1-.3)(8000)=16,100 H(pnut)=.3(18000)+(1-.3)(12000)=13,800 H(soy)=.3(22000)+(1-.3)(20000)=20,600
Decision: plant soybeans
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Trade Bill
ISE 562; Dr.
Maximax criterion corn
Maximin criterion soybean
Minimax regret criterion corn
Hurwicz criterion soybeans
Equal likelihood criterion corn
Which one should decision maker use?
Depends on attitude toward outcomes and payoffs.
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ISE 562; Dr. making criteria: Equal likelihood criterion
Assumes states of nature equally likely Multiply by 1/(no states of nature) and sum; Pick maximum
=.5(35000)+(.5)(8000)=21,500 =.5(18000)+(.5)(12000)=15,000 =.5(22000)+(.5)(20000)=21,000
Decision: plant corn
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ISE 562; Dr. example. A real estate firm is considering the following development projects. The financial success of these projects depends on interest rate movements in the next 5 years. The financial returns of each project for different interest rate conditions is shown in the payoff table.
Determine the best investment using
a. Maximax
b. Maximin
c. Equal likelihood
d. Hurwicz criterion, =.30
e. Minimax regret
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ISE 562; Dr. returns ($M):
Interest Rates
Office park
Office Bldg
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ISE 562; Dr. criterion
Decision: Office park
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Interest Rates
Office park
Office Bldg
ISE 562; Dr. criterion
Decision: Office bldg.
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Interest Rates
Office park
Office Bldg
ISE 562; Dr. Smith .33(.5)+.33(1.7)+.33(4.5)=2.21
Equal likelihood criterion
Decision: Office park or mall
1.35 2.21 1.75
Interest Rates
Office park
Office Bldg
ISE 562; Dr. Smith .30(4.5)+.70(0.5)=1.7
Hurwicz criterion, =0.30
Decision: Office bldg
1.21 1.57 1.38
Interest Rates
Office park
Office Bldg
ISE 562; Dr. regret criterion: subtract payoff-max payoff for each state of nature
Interest Rates
Office park
3.2-0.5 =2.7
2.4-1.7 = 0.7
4.5-4.5 =0.0
Office Bldg
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ISE 562; Dr. regret criterion: choose min of max regrets for each alternative
Decision: Office bldg
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Interest Rates
Office park
Office Bldg

ISE 562; Dr. making under uncertainty
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ISE 562; Dr. making under uncertainty
Maximum Likelihood Criterion
Select highest outcome for most probable state
Expected Monetary Value Criterion: Payoffs x probability distribution of outcomes
More frequent states of nature receive more weight
Less likely states of nature receive less weight Expected Opportunity Loss (EOL) Criterion:
Losses x (probability distribution of outcomes)
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ISE 562; Dr.:
Maximum Likelihood:
ML Maxprobability state , payoff Expected value:
E[x] x P(x )
Expected opportunity loss:
EOL[x]n L(x)P(x),L(x)x*x
ISE 562; Dr. likelihood criterion: choose max of most likely state of nature
Decision: Office Park
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Interest Rates
Decline P=0.15
Stable P=0.25
Increase P=0.60
Office park
Office Bldg
Trade Bill
P(pass)=.60
P(fails)=0.40
Trade Bill
P(pass)=.60
P(fails)=0.40
ISE 562; Dr. Likelihood Criterion
=.6(35000)+(.4)(8000)=24,200 =.6(18000)+(.4)(12000)=15,600
=.6(22000)+(.4)(20000)=21,200
Maximum likelihood decision = plant corn
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ISE 562; Dr. Monetary Value
Expected value decision = plant corn
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=.6(35000)+(.4)(8000)=24,200 =.6(18000)+(.4)(12000)=15,600
=.6(22000)+(.4)(20000)=21,200

Trade Bill
P(pass)=.60
P(fails)=0.40
ISE 562; Dr.. Suppose the chance of interest rates declining is .15, .25, and .60. Determine the decision with highest expected value.
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Interest Rates
Office park
Office Bldg
ISE 562; Dr. Opportunity Loss (choose option with minimum EOL)
=.6(0)+(.4)(12000)=4800 =.6(17000)+(.4)(8000)=13,400
=.6(13000)+(.4)(0)=7,800
EOL decision = plant corn
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ISE 562; Dr. Value
E[Office park]=.15(.5)+.25(1.7)+.60(4.5)=3.20
Interest Rates
Office park
Office Bldg
Decision: Office park
3.20 2.20 1.21 2.87 1.22
ISE 562; Dr.. Determine the decision with lowest EOL. Take expectations of regrets (opportunity losses)
Interest Rates
Office park
Office Bldg
Decision: Office park
ISE 562; Dr. Tree Diagrams
Used for multi-stage decisions
Organizes decisions, states of nature (outcomes), and payoffs
Maximize expected value of decision through roll-back of tree from branches to root.
Time progresses from left to right 9/13/2022 35
Decisions represented by choice
ISE 562; Dr. Smith
Alternative 1
Alternative n
States of nature represented by
chance nodes

ISE 562; Dr. Part Decision
Shoulder Decision
Americo Case
ISE 562; Dr. Smith
Utility Concepts Decision Theory
ISE 562; Dr. expected monetary value (EMV) criterion:
Multiply each payoff by its probability
of achieving that payoff and sum.
Intuitive
Many practical applications
Can also be used for cost (minimize expected cost, EC)
Can also be used for opportunity cost (EOL)
ISE 562; Dr.:
Expected value:
L(x ) x* x ii
EMV[x] Expected opportunity loss:
x P(x ) ii
L(x )P(x ) ii
ISE 562; Dr.
Top 15 Things Money Cant Buy
Time. Happiness. Inner Peace. Integrity. Love. Character. Manners. Health. Respect. Morals. Trust. Patience. Class. Common sense. Dignity.
. Bennett, The Light in the Heart
If you want to know what God thinks of money, just look at the people he gave it to.
This planet has or rather had a problem, which was this: most of the people living on it were unhappy for pretty much of the time. Many solutions were suggested for this problem, but most of these were largely concerned with the movement of small green pieces of paper, which was odd because on the whole it wasnt the small green pieces of paper that were unhappy.
, The Hitchhikers Guide to the Galaxy
Dont think money does everything or you are going to end up doing everything for money.
Voltaire
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ISE 562; Dr.
Problems with Expected monetary value (EMV) and Expected Loss (EL)
Some decisions do not involve monetary value (safety, number of side effects, time saved, down-time, time- between failure)
Attitude toward risk
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ISE 562; Dr.
Consider the following bet (fair coin):
Win $1 Pw=1/2
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EMV= 12(1) + 12(-.75) = $0.125
if you take the bet, zero if you dont
Lose -$0.75
Rational choice is to take the bet.
ISE 562; Dr.
Now consider the following bet (fair coin): Win $100,000
Pw=1/2 PL=1/2
EMV= 12(100k) + 12(-75k) = $12500 if you take the bet, zero if you dont
Lose -$75,000
Rational choice is to take the bet. But, would you?
ISE 562; Dr.
The choice under the EMV criterion is to take both bets. If you dont, its a violation of the EMV criterion. The monetary payoffs are clear so why the hesitation?
It is the risk of the big loss of $75,000. 9/13/2022 45
ISE 562; Dr. Smith
was the son of. He was born in Groningen while his father held the chair of mathematics there. His older brother was Nicolaus(II) Bernoulli and his uncle was so he was born into a family of leading mathematicians but also into a family where there was unfortunate rivalry, jealousy and bitterness. Johann was determined that Daniel should become a merchant and he tried to place him in an apprenticeship. However Daniel was as strongly opposed to this as his own father had been and soon Johann relented but certainly not as far as to let Daniel study mathematics. Johann declared that there was no money in mathematics and so he sent Daniel back to to study medicine. This Daniel did spending time studying medicine at Heidelberg in 1718 and Strasbourg in 1719. He returned to Basel in 1720 to complete his doctorate in medicine.
An important work which Daniel produced while in St Petersburg was one on probability and political economy. Daniel makes the assumption that the moral value of the increase in a persons wealth is inversely proportional to the amount of that wealth. He then assigns probabilities to the various means that a person has to make money and deduces an expectation of increase in moral expectation. Daniel applied some of his deductions to insurance.
did produce other excellent scientific work during these many years back in Basel. In total he won the Grand Prize of the Paris Academy 10 times, for topics in astronomy and nautical topics. He won in 1740 (jointly with Euler) for work on Newtons theory of the tides; in 1743 and 1746 for essays on magnetism; in 1747 for a method to determine time at sea; in 1751 for an essay on ocean currents; in 1753 for the effects of forces on ships; and in 1757 for proposals to reduce the pitching and tossing of a ship in high seas. Another important aspect ofs work that proved important in the development of mathematical physics was his acceptance of many of Newtons theories and his use of these together with the tolls coming from the more powerful calculus of Leibniz. Daniel worked on mechanics and again used the principle of conservation of energy which gave an integral of Newtons basic equations. He also studied the movement of bodies in a resisting medium using Newtons methods.
Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html
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ISE 562; Dr.
The classic violation of EMV principle: St. Petersburg paradox
Toss a fair coin until a head appears When head appears on kth toss, you win
2k-1 dollars.
How much would you be willing to pay to
play this game?
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ISE 562; Dr.
If heads occurs on kth toss, then it is preceded by (k-1) tails.
So probability of k-1 tails followed by a headis(12)k-1 (12)=(12)k
To get EMV of the gamble:
k11k 1 EMV[gamble]2 2 2
k1 k1 So expected payoff is infinite
But how much would you be willing to
pay to play?
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ISE 562; Dr.
Suppose you were willing to pay $127. To break even you would need to win 127 + $1.
To determine when heads occurs find k that yields $128. 2k-1 128, so k=8. and P(heads on k=8)=( 12 )8=0.004
So chance of winning $128 only 0.004 9/13/2022 49
ISE 562; Dr.
What if we lower the entry cost to $31?
To break even you would need to win
To determine when heads occurs find k
that yields $32. 2k-1 32, so k=6. and
P(heads on k=6)=( 12 )6=0.02
So chance of winning $32 only 0.02;
would you pay $31 to enter this bet?
At $3 probability is .125
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ISE 562; Dr.
Which bet would you take?
Pw=1/2 PL=1/2
Win $10,000,000
EMV= 12(10M) + 12(10M) = $10M
Win $10,000,000
Win $100,000,000
EMV= 12(100M) + 12(0M) = $50M
Most People
ISE 562; Dr.
Basic message
Value of a dollar varies from person to
person (decision maker to decision maker). Deal or No Deal Examples: Example 1; Example 2
How to address this?
Theory of expected utility; also called the
EU criterion.
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ISE 562; Dr.
Utility allows us to measure the relative value of different levels of consequences to the decision maker.
Consequences can be dollars, hours, reliability, number of fatalities, or other attributes.
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ISE 562; Dr.
The utility function, U(X) is a preference relationship with the following axioms:
1. If payoff X1 is preferred to payoff X2, then U(X1)>U(X2); if X2 preferred to X1, then U(X2)>U(X1); if neither preferred (indifferent), then U(X1)=U(X2)
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ISE 562; Dr.
2. If you are indifferent between receiving
a) payoff X1 for certain and
b) a gamble where you receive payoff
X2 with probability p and payoff X3 with probability (1-p), then
U(X1)=pU(X2)+(1-p)U(X3)
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ISE 562; Dr.
Assessment of utility functions
Suppose we are buying a car and
selecting from among 10 different cars
with various mileage ratings (mpg).
Let Xo be the worst mileage = 10 mpg
Let X* be the best mileage = 40 mpg
Define U(Xo)= 0.0; U(X*)= 1.0
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ISE 562; Dr.
If we stop here with 2 points we have the following utility function (risk neutral)
ISE 562; Dr.
To get more precision we offer the decision maker the following choice:
Which do you prefer?
Receive car with 40 mpg for certain
IF RATIONAL, SHOULD CHOOSE THIS OPTION
Receive car with 40 mpg Receive car with 10 mpg
ISE 562; Dr.
Now bump the sure thing up to 14 the
interval from worst value (10+ 14 (40-
Receive car with 40 mpg Receive car with 10 mpg
Receive car with 17.5 mpg for certain
THE CHOICE WILL BE BASED ON RISK ATTITUDE SUPPOSE THEY CHOOSE B
ISE 562; Dr.
Now split the interval from 17.5 to 40 or (17.5+40)/2 = 28.8
Receive car with 28.8 mpg for certain
SUPPOSE THEY CHOOSE A
Receive car with 40 mpg Receive car with 10 mpg

ISE 562; Dr.
Now split the interval from 17.5 to 28.8 or (17.5+28.8)/2 = 23.2
NOW SUPPOSE THEY CANNOT DECIDETHEY ARE INDIFFERENT BETWEEN A AND B
Receive car with 23.2 mpg for certain
Receive car with 40 mpg Receive car with 10 mpg
ISE 562; Dr.
The value of 23.2 is called the certainty equivalent for the gamble.
It is the amount the decision maker is willing to accept to avoid the risk of getting a car with only 10 mpg.
If the attribute were $ this could also be called a risk premiumwhat the DM should be willing to pay to avoid
the chance of getting the worst case. 9/13/2022 62
ISE 562; Dr. indifferent at 23.2, then U(23.2)=.5U(10) + .5U(40)
= .5(0) + .5(1) = .50
Graphically we have the midpoint of the
0.0 23.2 9/13/2022 10
X, mpg 40 63
ISE 562; Dr. Smith
To obtain the utility at 0.25 we work between 10 and 23.2 and repeat the procedure
Receive car with 23.2 mpg for certain
IF RATIONAL, SHOULD CHOOSE THIS OPTION
Receive car with 23.2 mpg Receive car with 10 mpg
ISE 562; Dr.
Now bump the sure thing up to 14

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