1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 5 Advanced Linear Programming Applications n Data Envelopment Analysis n Revenue Management n Portfolio Models and Asset Allocation n Game Theory

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Data Envelopment Analysis n Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives. n DEA creates a fictitious composite unit made up of an optimal weighted average ( W 1, W 2,…) of existing units. n An individual unit, k, can be compared by determining E, the fraction of unit k ’s input resources required by the optimal composite unit. n If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient. n If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Data Envelopment Analysis n The DEA Model MIN E s.t.Weighted outputs > Unit k ’s output (for each measured output) Weighted inputs < E [Unit k ’s input] (for each measured input) Sum of weights = 1 E, weights > 0

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved The Langley County School District is trying to determine the relative efficiency of its three high schools. In particular, it wants to evaluate Roosevelt High. The district is evaluating performances on SAT scores, the number of seniors finishing high school, and the number of students who enter college as a function of the number of teachers teaching senior classes, the prorated budget for senior instruction, and the number of students in the senior class. Data Envelopment Analysis

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved n n Input Roosevelt Lincoln Washington Roosevelt Lincoln Washington Senior Faculty 37 25 23 Budget ($100,000's) 6.4 5.0 4.7 Senior Enrollments 850 700 600 Data Envelopment Analysis

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved n n Output Roosevelt Lincoln Washington Roosevelt Lincoln Washington Average SAT Score 800 830 900 High School Graduates 450 500 400 High School Graduates 450 500 400 College Admissions 140 250 370 College Admissions 140 250 370 Data Envelopment Analysis

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Decision Variables E = Fraction of Roosevelt's input resources required by the composite high school E = Fraction of Roosevelt's input resources required by the composite high school w 1 = Weight applied to Roosevelt's input/output resources by the composite high school w 2 = Weight applied to Lincoln’s input/output resources by the composite high school w 3 = Weight applied to Washington's input/output resources by the composite high school Data Envelopment Analysis

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Objective Function Minimize the fraction of Roosevelt High School's input resources required by the composite high school: MIN E Data Envelopment Analysis

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Constraints Sum of the Weights is 1: (1) w 1 + w 2 + w 3 = 1 (1) w 1 + w 2 + w 3 = 1 Output Constraints: Output Constraints: Since w 1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: Since w 1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: (2) 800 w 1 + 830 w 2 + 900 w 3 > 800 (SAT Scores) (3) 450 w 1 + 500 w 2 + 400 w 3 > 450 (Graduates) (3) 450 w 1 + 500 w 2 + 400 w 3 > 450 (Graduates) (4) 140 w 1 + 250 w 2 + 370 w 3 > 140 (College Admissions) (4) 140 w 1 + 250 w 2 + 370 w 3 > 140 (College Admissions) Data Envelopment Analysis

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Constraints (continued) Input Constraints: The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are: The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are: (5) 37 w 1 + 25 w 2 + 23 w 3 < 37 E (Faculty) (6) 6.4 w 1 + 5.0 w 2 + 4.7 w 3 < 6.4 E (Budget) (6) 6.4 w 1 + 5.0 w 2 + 4.7 w 3 < 6.4 E (Budget) (7) 850 w 1 + 700 w 2 + 600 w 3 < 850 E (Seniors) (7) 850 w 1 + 700 w 2 + 600 w 3 < 850 E (Seniors) Nonnegativity of variables: Nonnegativity of variables: E, w 1, w 2, w 3 > 0 E, w 1, w 2, w 3 > 0 Data Envelopment Analysis

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution OBJECTIVE FUNCTION VALUE = 0.765 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS E 0.765 0.000 E 0.765 0.000 W1 0.000 0.235 W1 0.000 0.235 W2 0.500 0.000 W2 0.500 0.000 W3 0.500 0.000 W3 0.500 0.000 Data Envelopment Analysis

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution (continued) CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.000 -0.235 1 0.000 -0.235 2 65.000 0.000 2 65.000 0.000 3 0.000 -0.001 3 0.000 -0.001 4 170.000 0.000 4 170.000 0.000 5 4.294 0.000 5 4.294 0.000 6 0.044 0.000 6 0.044 0.000 7 0.000 0.001 7 0.000 0.001 Data Envelopment Analysis

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved n Conclusion The output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint (#4)). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.) Data Envelopment Analysis

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Another LP application is revenue management. n Revenue management involves managing the short- term demand for a fixed perishable inventory in order to maximize revenue potential. n The methodology was first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare. n Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management LeapFrog Airways provides passenger service for LeapFrog Airways provides passenger service for Indianapolis, Baltimore, Memphis, Austin, and Tampa. LeapFrog has two WB828 airplanes, one based in Indianapolis and the other in Baltimore. Each morning the Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies to Tampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved LeapFrog uses two fare classes: a discount fare D LeapFrog uses two fare classes: a discount fare D class and a full fare F class. Leapfrog’s products, each referred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares and forecasted demand. LeapFrog wants to determine LeapFrog wants to determine how many seats it should allocate to each ODIF. Revenue Management

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Decision Variables There are 16 variables, one for each ODIF: IMD = number of seats allocated to Indianapolis-Memphis- Discount class Discount class IAD = number of seats allocated to Indianapolis-Austin- Discount class ITD = number of seats allocated to Indianapolis-Tampa- Discount class IMF = number of seats allocated to Indianapolis-Memphis- Full Fare class IAF = number of seats allocated to Indianapolis-Austin-Full Fare class

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Decision Variables (continued) ITF = number of seats allocated to Indianapolis-Tampa- Full Fare class Full Fare class BMD = number of seats allocated to Baltimore-Memphis- Discount class Discount class BAD = number of seats allocated to Baltimore-Austin- Discount class Discount class BTD = number of seats allocated to Baltimore-Tampa- Discount class Discount class BMF = number of seats allocated to Baltimore-Memphis- Full Fare class Full Fare class BAF = number of seats allocated to Baltimore-Austin- Full Fare class Full Fare class

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Decision Variables (continued) BTF = number of seats allocated to Baltimore-Tampa- Full Fare class Full Fare class MAD = number of seats allocated to Memphis-Austin- Discount class Discount class MTD = number of seats allocated to Memphis-Tampa- Discount class Discount class MAF = number of seats allocated to Memphis-Austin- Full Fare class Full Fare class MTF = number of seats allocated to Memphis-Tampa- Full Fare class Full Fare class

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Objective Function Maximize total revenue: Max (fare per seat for each ODIF) Max (fare per seat for each ODIF) x (number of seats allocated to the ODIF) x (number of seats allocated to the ODIF) Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF + 490BTF + 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF + 190MAD + 180MTD + 310MAF + 295MTF

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Constraints There are 4 capacity constraints, one for each flight leg: Indianapolis-Memphis leg Indianapolis-Memphis leg (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 Baltimore-Memphis leg Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 Memphis-Austin leg Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 Memphis-Tampa leg Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF < 120 (4) ITD + ITF + BTD + BTF + MTD + MTF < 120

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Constraints (continued) There are 16 demand constraints, one for each ODIF: (5) IMD < 44(11) BMD < 26(17) MAD < 5 (5) IMD < 44(11) BMD < 26(17) MAD < 5 (6) IAD < 25(12) BAD < 50(18) MTD < 48 (6) IAD < 25(12) BAD < 50(18) MTD < 48 (7) ITD < 40(13) BTD < 42(19) MAF < 14 (7) ITD < 40(13) BTD < 42(19) MAF < 14 (8) IMF < 15(14) BMF < 12(20) MTF < 11 (8) IMF < 15(14) BMF < 12(20) MTF < 11 (9) IAF < 10(15) BAF < 16 (9) IAF < 10(15) BAF < 16 (10) ITF < 8(16) BTF < 9 (10) ITF < 8(16) BTF < 9

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n The Management Scientist Solution Objective Function Value = 94735.000 Variable Value Reduced Cost IMD44.000 0.000 IMD44.000 0.000 IAD 3.000 0.000 IAD 3.000 0.000 ITD 40.000 0.000 ITD 40.000 0.000 IMF 15.000 0.000 IMF 15.000 0.000 IAF 10.000 0.000 IAF 10.000 0.000 ITF 8.000 0.000 ITF 8.000 0.000 BMD 26.000 0.000 BMD 26.000 0.000 BAD 50.000 0.000 BAD 50.000 0.000

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n The Management Scientist Solution (continued) Variable Value Reduced Cost BTD 7.000 0.000 BTD 7.000 0.000 BMF 12.000 0.000 BMF 12.000 0.000 BAF 16.000 0.000 BAF 16.000 0.000 BTF 9.000 0.000 BTF 9.000 0.000 MAD 27.000 0.000 MAD 27.000 0.000 MTD 45.000 0.000 MTD 45.000 0.000 MAF 14.000 0.000 MAF 14.000 0.000 MTF 11.000 0.000 MTF 11.000 0.000

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Models and Asset Management n Asset allocation involves determining how to allocate investment funds across a variety of asset classes such as stocks, bonds, mutual funds, real estate. n Portfolio models are used to determine percentage of funds that should be made in each asset class. n The goal is to create a portfolio that provides the best balance between risk and return.

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved John Sweeney is an investment advisor who is John Sweeney is an investment advisor who is attempting to construct an "optimal portfolio" for a client who has $400,000 cash to invest. There are ten different investments, falling into four broad categories that John and his client have identified as potential candidate for this portfolio. The investments and their important The investments and their important characteristics are listed in the table on the next slide. Note that Unidyde Corp. Corp. under Equities and Unidyde Corp. under Debt are two separate investments, whereas First General REIT is a single investment that is considered both an equities and a real estate investment. Portfolio Model

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Exp. Annual Exp. Annual After Tax Liquidity Risk After Tax Liquidity Risk Category Investment Return Factor Factor Equities Unidyde Corp. 15.0% 100 60 (Stocks)CC’s Restaurants 17.0% 100 70 First General REIT 17.5% 100 75 First General REIT 17.5% 100 75 Debt Metropolis Electric 11.8% 95 20 (Bonds) Unidyde Corp. 12.2% 92 30 Lewisville Transit 12.0% 79 22 Lewisville Transit 12.0% 79 22 Real Estate Realty Partners 22.0% 0 50 First General REIT ( --- See above --- ) First General REIT ( --- See above --- ) Money T-Bill Account 9.6% 80 0 Money Mkt. Fund 10.5% 100 10 Money Mkt. Fund 10.5% 100 10 Saver's Certificate 12.6% 0 0 Saver's Certificate 12.6% 0 0

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Formulate a linear programming problem to Formulate a linear programming problem to accomplish John's objective as an investment advisor which is to construct a portfolio that maximizes his client's total expected after-tax return over the next year, subject to the limitations placed upon him by the client for the portfolio. (Limitations listed on next two slides.)

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Portfolio Limitations Portfolio Limitations 1. The weighted average liquidity factor for the portfolio 1. The weighted average liquidity factor for the portfolio must to be at least 65. must to be at least 65. 2. The weighted average risk factor for the portfolio must 2. The weighted average risk factor for the portfolio must be no greater than 55. be no greater than 55. 3. No more than $60,000 is to be invested in Unidyde 3. No more than $60,000 is to be invested in Unidyde stocks or bonds. stocks or bonds. 4. No more than 40% of the investment can be in any one 4. No more than 40% of the investment can be in any one category except the money category. category except the money category. 5. No more than 20% of the total investment can be in 5. No more than 20% of the total investment can be in any one investment except the money market fund. any one investment except the money market fund.continued

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Portfolio Limitations (continued) Portfolio Limitations (continued) 6. At least $1,000 must be invested in the Money Market 6. At least $1,000 must be invested in the Money Market fund. fund. 7. The maximum investment in Saver's Certificates is 7. The maximum investment in Saver's Certificates is $15,000. $15,000. 8. The minimum investment desired for debt is $90,000. 8. The minimum investment desired for debt is $90,000. 9. At least $10,000 must be placed in a T-Bill account. 9. At least $10,000 must be placed in a T-Bill account.

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n Define the Decision Variables X1 = $ amount invested in Unidyde Corp. (Equities) X2 = $ amount invested in CC’s Restaurants X3 = $ amount invested in First General REIT X4 = $ amount invested in Metropolis Electric X5 = $ amount invested in Unidyde Corp. (Debt) X6 = $ amount invested in Lewisville Transit X7 = $ amount invested in Realty Partners X8 = $ amount invested in T-Bill Account X9 = $ amount invested in Money Mkt. Fund X10 = $ amount invested in Saver's Certificate X10 = $ amount invested in Saver's Certificate

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n Define the Objective Function Maximize the total expected after-tax return over the next year: Max.15X1 +.17X2 +.175X3 +.118X4 +.122X5 +.12X6 +.22X7 +.096X8 +.105X9 +.126X10 +.12X6 +.22X7 +.096X8 +.105X9 +.126X10

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Total funds invested must not exceed $400,000: (1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 = 400,000 Weighted average liquidity factor must to be at least 65: (2)100X1 + 100X2 + 100X3 + 95X4 + 92X5 + 79X6 + 80X8 + 100X9 > 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) Weighted average risk factor must be no greater than 55: (3)60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) No more than $60,000 to be invested in Unidyde Corp: (4)X1 + X5 < 60,000 n Define the Constraints

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n Define the Constraints (continued) No more than 40% of the $400,000 investment can be in any one category except the money category: (5) X1 + X2 + X3 < 160,000 (6) X4 + X5 + X6 < 160,000 (7)X3 + X7 < 160,000 No more than 20% of the $400,000 investment can be in any one investment except the money market fund: (8) X2 < 80,000(12) X7 < 80,000 (9) X3 < 80,000(13) X8 < 80,000 (10) X4 < 80,000(14) X10 < 80,000 (11) X6 < 80,000

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n Define the Constraints (continued) At least $1,000 must be invested in the Money Market fund: (15) X9 > 1,000 The maximum investment in Saver's Certificates is $15,000: (16) X10 < 15,000 The minimum investment the Debt category is $90,000: (17) X4 + X5 + X6 > 90,000 At least $10,000 must be placed in a T-Bill account: (18) X8 > 10,000 Non-negativity of variables: Xj > 0 j = 1,..., 10 Xj > 0 j = 1,..., 10

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n Solution Summary Total Expected After-Tax Return = $64,355 X1 = $0 invested in Unidyde Corp. (Equities) X2 = $80,000 invested in CC’s Restaurants X3 = $80,000 invested in First General REIT X4 = $0 invested in Metropolis Electric X5 = $60,000 invested in Unidyde Corp. (Debt) X6 = $74,000 invested in Lewisville Transit X7 = $80,000 invested in Realty Partners X8 = $10,000 invested in T-Bill Account X9 = $1,000 invested in Money Mkt. Fund X10 = $15,000 invested in Saver's Certificate X10 = $15,000 invested in Saver's Certificate

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved Introduction to Game Theory n In decision analysis, a single decision maker seeks to select an optimal alternative. n In game theory, there are two or more decision makers, called players, who compete as adversaries against each other. n It is assumed that each player has the same information and will select the strategy that provides the best possible outcome from his point of view. n Each player selects a strategy independently without knowing in advance the strategy of the other player(s). continue

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved Introduction to Game Theory n The combination of the competing strategies provides the value of the game to the players. n Examples of competing players are teams, armies, companies, political candidates, and contract bidders.

41. 41 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-person means there are two competing players in the game. n Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player. n The gain and loss balance out so that there is a zero- sum for the game. n What one player wins, the other player loses. Two-Person Zero-Sum Game

42. 42 Slide © 2008 Thomson South-Western. All Rights Reserved n Competing for Vehicle Sales Suppose that there are only two vehicle dealer- ships in a small city. Each dealership is considering three strategies that are designed to take sales of new vehicles from the other dealership over a four-month period. The strategies, assumed to be the same for both dealerships, are on the next slide. Two-Person Zero-Sum Game Example

43. 43 Slide © 2008 Thomson South-Western. All Rights Reserved n Strategy Choices Strategy 1: Offer a cash rebate Strategy 1: Offer a cash rebate on a new vehicle. on a new vehicle. Strategy 2: Offer free optional Strategy 2: Offer free optional equipment on a equipment on a new vehicle. new vehicle. Strategy 3: Offer a 0% loan Strategy 3: Offer a 0% loan on a new vehicle. on a new vehicle. Two-Person Zero-Sum Game Example

44. 44 Slide © 2008 Thomson South-Western. All Rights Reserved 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B n Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A Gained Per Week by Dealership A (or Lost Per Week by Dealership B) (or Lost Per Week by Dealership B) -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Two-Person Zero-Sum Game Example

45. 45 Slide © 2008 Thomson South-Western. All Rights Reserved n Step 1: Identify the minimum payoff for each row (for Player A). row (for Player A). n Step 2: For Player A, select the strategy that provides the maximum of the row minimums (called the maximum of the row minimums (called the maximin). the maximin). Two-Person Zero-Sum Game

46. 46 Slide © 2008 Thomson South-Western. All Rights Reserved n Identifying Maximin and Best Strategy RowMinimum 1-3-2 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Best Strategy For Player A MaximinPayoff Two-Person Zero-Sum Game Example

47. 47 Slide © 2008 Thomson South-Western. All Rights Reserved n Step 3: Identify the maximum payoff for each column (for Player B). (for Player B). n Step 4: For Player B, select the strategy that provides the minimum of the column maximums the minimum of the column maximums (called the minimax). (called the minimax). Two-Person Zero-Sum Game

48. 48 Slide © 2008 Thomson South-Western. All Rights Reserved n Identifying Minimax and Best Strategy 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Column Maximum 3 3 1 3 3 1 Best Strategy For Player B MinimaxPayoff Two-Person Zero-Sum Game Example

49. 49 Slide © 2008 Thomson South-Western. All Rights Reserved Pure Strategy n Whenever an optimal pure strategy exists: n the maximum of the row minimums equals the minimum of the column maximums (Player A’s maximin equals Player B’s minimax) n the game is said to have a saddle point (the intersection of the optimal strategies) n the value of the saddle point is the value of the game n neither player can improve his/her outcome by changing strategies even if he/she learns in advance the opponent’s strategy

50. 50 Slide © 2008 Thomson South-Western. All Rights Reserved RowMinimum 1-3-2 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Column Maximum 3 3 1 3 3 1 Pure Strategy Example n Saddle Point and Value of the Game 2 2 1 2 2 1 SaddlePoint Value of the game is 1

51. 51 Slide © 2008 Thomson South-Western. All Rights Reserved Pure Strategy Example n Pure Strategy Summary n Player A should choose Strategy a 1 (offer a cash rebate). n Player A can expect a gain of at least 1 vehicle sale per week. n Player B should choose Strategy b 3 (offer a 0% loan). n Player B can expect a loss of no more than 1 vehicle sale per week.

52. 52 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy n If the maximin value for Player A does not equal the minimax value for Player B, then a pure strategy is not optimal for the game. n In this case, a mixed strategy is best. n With a mixed strategy, each player employs more than one strategy. n Each player should use one strategy some of the time and other strategies the rest of the time. n The optimal solution is the relative frequencies with which each player should use his possible strategies.

53. 53 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example b1b1b1b1 b2b2b2b2 Player B 11 5 a1a1a2a2a1a1a2a2 Player A 4 8 4 8 n Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy. ColumnMaximum 11 8 11 8 RowMinimum 4 5 Maximin Minimax

54. 54 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example p = the probability Player A selects strategy a 1 (1  p ) = the probability Player A selects strategy a 2 If Player B selects b 1 : EV = 4 p + 11(1 – p ) If Player B selects b 2 : EV = 8 p + 5(1 – p )

55. 55 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example 4 p + 11(1 – p ) = 8 p + 5(1 – p ) To solve for the optimal probabilities for Player A we set the two expected values equal and solve for the value of p. 4 p + 11 – 11 p = 8 p + 5 – 5 p 11 – 7 p = 5 + 3 p -10 p = -6 p =.6 Player A should select: Strategy a 1 with a.6 probability and Strategy a 1 with a.6 probability and Strategy a 2 with a.4 probability. Strategy a 2 with a.4 probability. Hence, (1  p ) =.4

56. 56 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example q = the probability Player B selects strategy b 1 (1  q ) = the probability Player B selects strategy b 2 If Player A selects a 1 : EV = 4 q + 8(1 – q ) If Player A selects a 2 : EV = 11 q + 5(1 – q )

57. 57 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example n Value of the Game For Player A: EV = 4 p + 11(1 – p ) = 4(.6) + 11(.4) = 6.8 For Player B: EV = 4 q + 8(1 – q ) = 4(.3) + 8(.7) = 6.8 Expected gain per game for Player A Expected loss per game for Player B

58. 58 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example RowMinimum -2 0-3 b1b1b1b1 b3b3b3b3 b2b2b2b2 Player B 1 0 3 1 0 3 3 4 -3 3 4 -3 a1a1a2a2a3a3a1a1a2a2a3a3 Player A ColumnMaximum 6 5 3 6 5 3 6 5 -2 6 5 -2 Suppose that the payoff table for a two-person zero- sum game is the following. Here there is no optimal pure strategy. Maximin Minimax

59. 59 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example b1b1b1b1 b3b3b3b3 b2b2b2b2 Player B 1 0 3 1 0 3 Player A 6 5 -2 6 5 -2 If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategies in order to reduce the size of the game. If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategies in order to reduce the size of the game. 3 4 -3 3 4 -3 a1a1a2a2a3a3a1a1a2a2a3a3 Player A’s Strategy a 3 is dominated by Player A’s Strategy a 3 is dominated by Strategy a 1, so Strategy a 3 can be eliminated.

60. 60 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example b1b1b1b1 b3b3b3b3 Player B Player A a1a1a2a2a1a1a2a2 Player B’s Strategy b 2 is dominated by Player B’s Strategy b 2 is dominated by Strategy b 1, so Strategy b 2 can be eliminated. b2b2b2b2 1 0 3 1 0 3 6 5 -2 6 5 -2 We continue to look for dominated strategies in order to reduce the size of the game. We continue to look for dominated strategies in order to reduce the size of the game.

61. 61 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example b1b1b1b1 b3b3b3b3 Player B Player A a1a1a2a2a1a1a2a2 1 3 1 3 6 -2 6 -2 The 3 x 3 game has been reduced to a 2 x 2. It is now possible to solve algebraically for the optimal mixed-strategy probabilities. The 3 x 3 game has been reduced to a 2 x 2. It is now possible to solve algebraically for the optimal mixed-strategy probabilities.

62. 62 Slide © 2008 Thomson South-Western. All Rights Reserved n Competing for Vehicle Sales Let us continue with the two-dealership game Let us continue with the two-dealership game presented earlier, but with a change to one payoff. If both Dealership A and Dealership B choose to offer a 0% loan, the payoff to Dealership A is now an increase of 3 vehicle Sales per week. (The revised payoff table appears on the next slide.) Two-Person Zero-Sum Game Example #2

63. 63 Slide © 2008 Thomson South-Western. All Rights Reserved 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B n Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A Gained Per Week by Dealership A (or Lost Per Week by Dealership B) (or Lost Per Week by Dealership B) -3 3 -1 3 -2 3 3 -2 3 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Two-Person Zero-Sum Game Example #2

64. 64 Slide © 2008 Thomson South-Western. All Rights Reserved n The maximin (1) does not equal the minimax (3), so a pure strategy solution does not exist for this problem. n The optimal solution is for both dealerships to adopt a mixed strategy. n There are no dominated strategies, so the problem cannot be reduced to a 2x2 and solved algebraically. n However, the game can be formulated and solved as a linear program. Two-Person Zero-Sum Game Example #2

65. 65 Slide © 2008 Thomson South-Western. All Rights Reserved n Let us first consider the game from the point of view of Dealership A. n Dealership A will select one of its three strategies based on the following probabilities: PA 1 = the probability that Dealership A selects strategy a 1 PA 2 = the probability that Dealership A selects strategy a 2 PA 3 = the probability that Dealership A selects strategy a 3 Two-Person Zero-Sum Game Example #2

66. 66 Slide © 2008 Thomson South-Western. All Rights Reserved Two-Person Zero-Sum Game Example #2 n Weighting each payoff by its probability and summing provides the expected value of the increase in vehicle sales per week for Dealership A. Dealership B Strategy Expected Gain for Dealership A Dealership B Strategy Expected Gain for Dealership A b 1 EG ( b 1 ) = 2 PA 1 – 3 PA 2 + 3 PA 3 b 1 EG ( b 1 ) = 2 PA 1 – 3 PA 2 + 3 PA 3 b 2 EG ( b 2 ) = 2 PA 1 + 3 PA 2 – 2 PA 3 b 3 EG ( b 3 ) = 1 PA 1 – 1 PA 2 + 3 PA 3 b 3 EG ( b 3 ) = 1 PA 1 – 1 PA 2 + 3 PA 3

67. 67 Slide © 2008 Thomson South-Western. All Rights Reserved Two-Person Zero-Sum Game Example #2 n Define GAINA to be the optimal expected gain in vehicle sales for Dealership A, which we want to maximize. n Thus, the individual expected gains, EG ( b 1 ), EG ( b 2 ) and EG ( b 3 ) must all be greater than or equal to GAINA. n For example, 2 PA 1 – 3 PA 2 + 3 PA 3 > GAINA n Also, the sum of Dealership A’s mixed strategy probabilities must equal 1. n This results in the LP formulation on the next slide …..

68. 68 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership A’s Linear Programming Formulation Max GAINA s.t. 2 PA 1 – 3 PA 2 + 3 PA 3 – GAINA > 0 (Strategy b 1 ) 2 PA 1 + 3 PA 2 – 2 PA 3 – GAINA > 0 (Strategy b 2 ) 1 PA 1 – 1 PA 2 + 0 PA 3 – GAINA > 0 (Strategy b 3 ) 1 PA 1 – 1 PA 2 + 0 PA 3 – GAINA > 0 (Strategy b 3 ) PA 1 + PA 2 + PA 3 = 1 (Prob’s sum to 1) PA 1, PA 2, PA 3, GAINA > 0 (Non-negativity) Two-Person Zero-Sum Game Example #2

69. 69 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership A OBJECTIVE FUNCTION VALUE = 1.333 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS PA1 0.833 0.000 PA1 0.833 0.000 PA2 0.000 1.000 PA2 0.000 1.000 PA3 0.167 0.000 PA3 0.167 0.000 GAINA 1.333 0.000 GAINA 1.333 0.000 Two-Person Zero-Sum Game Example #2

70. 70 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership A CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.833 0.000 1 0.833 0.000 2 0.000 -0.333 2 0.000 -0.333 3 0.000 -0.667 3 0.000 -0.667 4 0.000 1.333 4 0.000 1.333 Two-Person Zero-Sum Game Example #2

71. 71 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership A’s Optimal Mixed Strategy Offer a cash rebate ( a 1 ) with a probability of 0.833Offer a cash rebate ( a 1 ) with a probability of 0.833 Do not offer free optional equipment ( a 2 )Do not offer free optional equipment ( a 2 ) Offer a 0% loan ( a 3 ) with a probability of 0.167Offer a 0% loan ( a 3 ) with a probability of 0.167 The expected value of this mixed strategy is a gain of 1.333 vehicle sales per week for Dealership A. Two-Person Zero-Sum Game Example #2

72. 72 Slide © 2008 Thomson South-Western. All Rights Reserved n Let us now consider the game from the point of view of Dealership B. n Dealership B will select one of its three strategies based on the following probabilities: PB 1 = the probability that Dealership B selects strategy b 1 PB 2 = the probability that Dealership B selects strategy b 2 PB 3 = the probability that Dealership B selects strategy b 3 Two-Person Zero-Sum Game Example #2

73. 73 Slide © 2008 Thomson South-Western. All Rights Reserved n Weighting each payoff by its probability and summing provides the expected value of the decrease in vehicle sales per week for Dealership B. Dealership A Strategy Expected Loss for Dealership B a 1 EL ( a 1 ) = 2 PB 1 + 2 PB 2 + 1 PB 3 a 1 EL ( a 1 ) = 2 PB 1 + 2 PB 2 + 1 PB 3 a 2 EL ( a 2 ) = -3 PB 1 + 3 PB 2 – 1 PB 3 a 3 EL ( a 3 ) = 3 PB 1 – 2 PB 2 + 3 PB 3 a 3 EL ( a 3 ) = 3 PB 1 – 2 PB 2 + 3 PB 3 Two-Person Zero-Sum Game Example #2

74. 74 Slide © 2008 Thomson South-Western. All Rights Reserved Two-Person Zero-Sum Game Example #2 n Define LOSSB to be the optimal expected loss in vehicle sales for Dealership B, which we want to minimize. n Thus, the individual expected losses, EL ( a 1 ), EL ( a 2 ) and EL ( a 3 ) must all be less than or equal to LOSSB. n For example, 2 PA 1 + 2 PA 2 + 1 PA 3 < LOSSB n Also, the sum of Dealership B’s mixed strategy probabilities must equal 1. n This results in the LP formulation on the next slide …..

75. 75 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership B’s Linear Programming Formulation Min LOSSB s.t. 2 PB 1 + 2 PB 2 + 1 PB 3 – LOSSB < 0 (Strategy a 1 ) 2 PB 1 + 2 PB 2 + 1 PB 3 – LOSSB < 0 (Strategy a 1 ) -3 PB 1 + 3 PB 2 – 1 PB 3 – LOSSB < 0 (Strategy a 2 ) 3 PB 1 – 2 PB 2 + 3 PB 3 – LOSSB < 0 (Strategy a 3 ) 3 PB 1 – 2 PB 2 + 3 PB 3 – LOSSB < 0 (Strategy a 3 ) PB 1 + PB 2 + PB 3 = 1 (Prob’s sum to 1) PB 1 + PB 2 + PB 3 = 1 (Prob’s sum to 1) PB 1, PB 2, PB 3, LOSSB > 0 (Non-negativity) PB 1, PB 2, PB 3, LOSSB > 0 (Non-negativity) Two-Person Zero-Sum Game Example #2

76. 76 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership B OBJECTIVE FUNCTION VALUE = 1.333 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS PB1 0.000 0.833 PB1 0.000 0.833 PB2 0.333 0.000 PB2 0.333 0.000 PB3 0.667 0.000 PB3 0.667 0.000 LOSSB 1.333 0.000 LOSSB 1.333 0.000 Two-Person Zero-Sum Game Example #2

77. 77 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership B CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.000 0.833 1 0.000 0.833 2 1.000 0.000 2 1.000 0.000 3 0.000 0.167 3 0.000 0.167 4 0.000 -1.333 4 0.000 -1.333 Two-Person Zero-Sum Game Example #2

78. 78 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership B’s Optimal Mixed Strategy Do not offer a cash rebate ( b 1 )Do not offer a cash rebate ( b 1 ) Offer free optional equipment ( b 2 ) with a probability of 0.333Offer free optional equipment ( b 2 ) with a probability of 0.333 Offer a 0% loan ( b 3 ) with a probability of 0.667Offer a 0% loan ( b 3 ) with a probability of 0.667 The expected payoff of this mixed strategy is a loss of The expected payoff of this mixed strategy is a loss of 1.333 vehicle sales per week for Dealership B. Note that expected loss for Dealership B is the same as Note that expected loss for Dealership B is the same as the expected gain for Dealership A. (There is a zero- sum for the expected payoffs.) Two-Person Zero-Sum Game Example #2

79. 79 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-Person, Constant-Sum Games (The sum of the payoffs is a constant other than zero.) (The sum of the payoffs is a constant other than zero.) n Variable-Sum Games (The sum of the payoffs is variable.) (The sum of the payoffs is variable.) n n -Person Games (A game involves more than two players.) (A game involves more than two players.) n Cooperative Games (Players are allowed pre-play communications.) (Players are allowed pre-play communications.) n Infinite-Strategies Games (An infinite number of strategies are available for the players.) (An infinite number of strategies are available for the players.) Other Game Theory Models

80. 80 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 5