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Business System Analysis & Decision Making- Lecture 5

Zhangxi Lin

ISQS 5340

July 2006

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Outline of Modeling Preferences

Probability Basics – Conditional probability Risk Attitude and Expected Utility Game Model with Complete Information

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Conditional Probability P(A|B) = P(A and B) / P(B) Example, there are 40 female students in a class of 100. 10 of

them are from some foreign countries. 20 male students are also foreign students. Even A: student from a foreign country Even B: a female student

If randomly picking up one of students to give a talk in the class. The probability the student is a female: P(B) = 0.4 The probability the student is from a foreign country: P(A) = (10

+ 20) / 100 = 0.3 The student is female and from a foreign country: P(A and B) =

10 / 100 = 0.1 If randomly choosing a female student to present in the class,

the probability she is a foreign student: P(A|B) = 10 / 40 = 0.25, or P(A|B) = P (A and B) / P (B) = 0.1 / 0.4 = 0.25

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Venn Diagrams

FemaleForeignstudent

Female foreign student

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Questions

What is the probability of female students who are not foreign students regarding the whole class?

What is the probability of male students who are foreign students regarding the whole class?

What is the probability of male students who are not foreign students regarding the whole class?

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Confusion Matrix

Model M1 PREDICTED CLASS

ACTUALCLASS

Bad Good

Bad 50 50

Good 150 250

100

400

200 300

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Utility Functions

Many of the examples and problems that we have considered so far have been analyzed in terms of expected monetary value (EMV). EMV, however, does not capture risk attitudes.

For example, consider the Texaco-Pennzoil example. If Pennzoil were afraid of the prospect that Pennzoil could end up with nothing at the end of the court case, the company might be willing to take the $2 billion that Texaco offered.

When discussing risk attitudes, we need to think of a utility function.

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Utility Function Curve

x: Payoff

U(x): Utility

x1 x3x2

U(x) = f(x) Utility function

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Utility Function

Concave utility functions U(x) = log(x) U(x) = 1 – ex/R

U(x) = +x0.5

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Risk Premium

Payoff

Utility

lottery

10P=0.6

100P=0.4

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Concave utility function

Convex utility function

Positive premium

Negative premium

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Risk Attitudes

Risk averse – positive risk premium Risk seeking – negative risk premium Risk neutral –risk premium = 0, i.e. EMV can

be used as the utility function

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Quiz 1 Question

Suppose you are looking for an apartment. There are two choices: 2br, $700/month, built 2004, cross street to

the university 2br, $400/month, built 1988, about 5 miles

from the university Draw a table as follow and input weights for

each criterion to conduct your decision making. You can set any scalar for the different criteria and use the weight value between 1-100 or in percentage.

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Factor Rating (Scale: 1-10)

Criteria Weight Apartment A Apartment B

Rent price 1 4 10

Year built 1 8 3

Close to the University 1 10 5

Total 22 18

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Decision Tree for Apartment Decision

Apartment B

Apartment A

You

Price

w2

w1

Distance

Year built

w3

w2

w1

w3

Price

Distance

Year built

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“Pennzoil vs. Texaco” Revisit

2

5

10.3

5

0

10.3

5

0

4

Counteroffer$5 billion

$Billion

TexacoCounteroffer$4 billion(0.33)

Texaco refuse

(0.50)

Texaco accepts $5 billion

(0.17)

Accepts $2 billion

Accept $4 billion

Court decision

Court decision

(0.2)

(0.5)

(0.3)

(0.2)

(0.5)

(0.3)

4.56

4.564.63

4.56

Making decisions Weighting payoffs

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Utility Function Assessment

Take into account of utility function:

U(x) = x0.5

Calculations: U(10.3) = 10.30.5 = 3.21 U(5) = 50.5 = 2.24 U(0) = 0 Total = 3.21 * 0.2 + 2.24 * 0.5 + 0 = 1.76 U(4) = 2

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“Pennzoil vs. Texaco” Revisit

4

5

10.3

5

0

10.3

5

0

4

Counteroffer$5 billion

$Billion

TexacoCounteroffer$4 billion(0.33)

Texaco refuse

(0.50)

Texaco accepts $5 billion

(0.17)

Accepts $4 billion

Accept $4 billion

Court decision

Court decision

(0.2)

(0.5)

(0.3)

(0.2)

(0.5)

(0.3)

U = 2

U = 1.76U=1.92

U = 2.24

U=1.41

Utility function: U(x) = x0.5

U = 1.76

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“Pennzoil vs. Texaco” Revisit

4

5

10.3

5

0

10.3

5

0

4

Counteroffer$5 billion

$Billion

TexacoCounteroffer$4 billion(0.33)

Texaco refuse

(0.50)

Texaco accepts $5 billion

(0.17)

Accepts $4 billion

Accept $4 billion

Court decision

Court decision

(0.2)

(0.5)

(0.3)

(0.2)

(0.5)

(0.3)U = 1.39

U = 1.27U=1.36

U = 1.61

U=1.39

Utility function: U(x) = ln(x)

U = 1.39

U = 2.33

U = 1.61

U = 0

U = 1.27

U = 2.33

U = 1.61

U = 0

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Exercise 2.1

Draw the decision tree of “Pennzoil vs. Texaco”

Recalculate the problem by assuming U(x) = log(x).

Put the outcomes in the tree and check the final decision

Try to draw the utility function curve to explain the final outcome.

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Exercise 2.2

Let U(x)=+x0.5, calculate Question 4 in Chapter 3. Explain the outcome with the story told in Chapter 3.

This is the assignment of homework 2 in addition to the one online

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Prisoner’s Dilemma Problem

In this game, each player has two strategies available: confess and not confess.1) If prisoner 1 chooses not confess and another confesses, the prisoner 1 will be sentenced to stay in jail for 9 month and prisoner 2 will be released. 2) If both confess, they will stay in jail for 6 months. 3) If both do not confess, they will only stay in the jail for one month.

-1, -1 -9, 0

-6, -6 0, -9

Not Confess

Confess

Not Confess confess

Prisoner 1

Prisoner 2

Nash Equilibrium

(-6, -6) is a Nash equilibrium for the two prisoners

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The Concepts in a Game Model

Information set: Complete or incomplete Strategies: (Confess, not confess) Payoff: How much the players will be

benefited/punished with regard to different outcomes of the game.

Nash equilibrium: A set of strategies composed of the ones adopted by each player is called Nash equilibrium if, for any player, his responding strategy is the best one to others. Implying that any deviation of a player from the

strategy of Nash equilibrium will cause the player worse off.

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Not confess

Confess

Prisoner 1

Not confess

Confess-6, -6

Confess

Not confess

Game tree – Extended Form of the Game

Prisoner 2

0, -9

-9, 0

-1, -1

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The Battle of the Sex Problem

Pat and Bob must choose to attend either the opera or a prize fight. Both players would rather spend the evening together than apart, but Pat would rather they be together at the prize fight while Bob would rather they be together at the opera.

2, 1 0, 0

1, 2 0, 0

Opera

Fight

Opera Fight

Bob

Pat

Nash Equilibrium

There are two Nash equilibria

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Matching Pennies

Assume Pat and Bob decide to play a game to determine whether they will go to the opera or the prize fight. They are flipping two pennies. If both are heads up or tails up Bob win. If the outcomes are different, Pat win.

-1, 1 1, -1

-1, 1 1, -1

Heads

Tails

Heads Tails

Bob

Pat

There is no Nash equilibrium in this game

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Two-stage dynamic game of complete but imperfect information: Bank runs Two investors have each deposited D with a bank. The bank has

invested these deposits in a long-term project. If the bank is forced to liquidate its investment before the project matures, a total of 2r can be recovered, where D > r > D / 2. If the bank allows the investment to reach maturity, the project will pay out a total of 2R, where R > D.

There are two dates at which the investors can make withdrawals from the bank: date 1 is before the bank’s investment matures; date 2 is after. If both investors make withdrawals at date 1 then each receives r.

If only one investor makes a withdrawal at date 1 then that investor receives D, the other receives 2r – D

If neither investor makes withdrawal decisions at date 1 then the project matures and the investors make withdrawals at date 2. They will receive R.

If only one investor makes a withdrawal at date 2 then he receives 2R – D, and the other receives D.

If neither makes a withdrawal at date 2 then the bank returns R to them.

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Subgame perfect equilibrium

r, r D, 2r - D

2r - D, D Next stage

Withdraw don’t

Withdraw

don’t

Date 1

R, R 2R - D, D

D, 2R - D R , R

Withdraw don’t

Withdraw

don’t

Date 2

r, r D, 2r - D

2r - D, D R, R

Withdraw don’t

Withdraw

don’t=

R > D > r > D / 2

Two pure strategy subgame perfect equilibrium: (r, r) and (R, R)

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Discussion

There are two pure strategy subgame perfect equilibria in this game

It is different from classical Prisoner’s Dilemma game. The latter only has one unique equilibrium that is inefficient, while here the model has one extra equilibrium that is efficient.