1 Business System Analysis & Decision Making - Lecture 5 Zhangxi Lin ISQS 5340 July 2006

2 Outline of Modeling Preferences Probability Basics – Conditional probability Risk Attitude and Expected Utility Game Model with Complete Information

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

4 Venn Diagrams Female Foreign student Female foreign student

5 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?

6 Confusion Matrix Model M 1 PREDICTED CLASS ACTUAL CLASS BadGood Bad50 Good

7 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.

8 Utility Function Curve x: Payoff U(x): Utility x1x3x2 U(x) = f(x) Utility function

9 Utility Function Concave utility functions U(x) = log(x) U(x) = 1 – e x/R U(x) = +x 0.5

10 Risk Premium Payoff Utility lottery 10 P= P= Concave utility function Convex utility function Positive premium Negative premium

11 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

12 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 or in percentage.

13 Factor Rating (Scale: 1-10) CriteriaWeightApartment AApartment B Rent price 1410 Year built 183 Close to the University 1105 Total 2218

14 Decision Tree for Apartment Decision Apartment B Apartment A You Price w2 w1 Distance Year built w3 w2 w1 w3 Price Distance Year built

15 “Pennzoil vs. Texaco” Revisit Counteroffer $5 billion $Billion Texaco Counteroffer $4 billion (0.33) Texaco refuse (0.50) Texaco accepts $5 billion (0.17) Accepts $2 billion Accept $4 billion Court decision (0.2) (0.5) (0.3) (0.2) (0.5) (0.3) Making decisions Weighting payoffs

16 Utility Function Assessment Take into account of utility function: U(x) = x 0.5 Calculations: U(10.3) = = 3.21 U(5) = = 2.24 U(0) = 0 Total = 3.21 * * = 1.76 U(4) = 2

17 “Pennzoil vs. Texaco” Revisit Counteroffer $5 billion $Billion Texaco Counteroffer $4 billion (0.33) Texaco refuse (0.50) Texaco accepts $5 billion (0.17) Accepts $4 billion Accept $4 billion Court decision (0.2) (0.5) (0.3) (0.2) (0.5) (0.3) U = 2 U = 1.76 U=1.92 U = 2.24 U=1.41 Utility function: U(x) = x 0.5 U = 1.76

18 “Pennzoil vs. Texaco” Revisit Counteroffer $5 billion $Billion Texaco Counteroffer $4 billion (0.33) Texaco refuse (0.50) Texaco accepts $5 billion (0.17) Accepts $4 billion Accept $4 billion Court decision (0.2) (0.5) (0.3) (0.2) (0.5) (0.3) U = 1.39 U = 1.27 U=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

19 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.

20 Exercise 2.2 Let U(x)=+x 0.5, 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

21 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, -60, -9 Not Confess Not Confess confess Prisoner 1 Prisoner 2 Nash Equilibrium (-6, -6) is a Nash equilibrium for the two prisoners

22 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.

23 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

24 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, 10, 0 1, 20, 0 Opera Fight Opera Fight Bob Pat Nash Equilibrium There are two Nash equilibria

25 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, 11, -1 -1, 11, -1 Heads Tails Heads Tails Bob Pat There is no Nash equilibrium in this game

26 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.

27 Subgame perfect equilibrium r, rD, 2r - D 2r - D, DNext stage Withdraw don’t Withdraw don’t Date 1 R, R2R - D, D D, 2R - DR, R Withdraw don’t Withdraw don’t Date 2 r, rD, 2r - D 2r - D, DR, R Withdraw don’t Withdraw don’t = R > D > r > D / 2 Two pure strategy subgame perfect equilibrium: (r, r) and (R, R)

28 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.