1. 1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2. 2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Agenda n Some Review from Last Class n Data Envelopment Analysis n Revenue Management n Game Theory Concepts

3. 3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 5 Advanced Linear Programming Applications n Data Envelopment Analysis Compares one unit to similar others Compares one unit to similar others Ie branch of a bank, franchise of a chain Ie branch of a bank, franchise of a chain n Revenue Management Maximize revenue with a fixed inventory Maximize revenue with a fixed inventory n Portfolio Models and Asset Allocation Determine best portfolio composition Determine best portfolio composition n Game Theory Competition with a zero sum Competition with a zero sum

4. 4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Data envelopment analysis (DEA): used to determine the relative operating efficiency of units with the same goals and objectives. n DEA creates a hypothetical composite optimal weighted average ( W 1, W 2,…) of existing units. optimal weighted average ( W 1, W 2,…) of existing units. n E – Efficiency Index Allows comparison between composite and unit Allows comparison between composite and unit “what the outputs of the composite would be, given the units inputs” “what the outputs of the composite would be, given the units inputs” If E < 1, unit is less efficient than the composite unit If E = 1, there is no evidence that unit k is inefficient. If E < 1, unit is less efficient than the composite unit If E = 1, there is no evidence that unit k is inefficient.

5. 5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n The DEA Model MIN E s.t.OUTPUTS INPUTS Sum of weights = 1 E, weights > 0

6. 6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis 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.Outputs: performances on SAT scores, the number of seniors finishing high school the number of students who enter college Inputs number of teachers teaching senior classes the prorated budget for senior instruction number of students in the senior class.

7. 7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n n Input Roosevelt1 Lincoln2 Washington3 Roosevelt1 Lincoln2 Washington3 Senior Faculty 37 25 23 Budget ($100,000's) 6.4 5.0 4.7 Senior Enrollments 850 700 600

8. 8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n n Output Roosevelt1 Lincoln2 Washington3 Roosevelt1 Lincoln2 Washington3 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

9. 9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis 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

10. 10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Define the Objective Function minimize the fraction of Roosevelt High School's input resources required by the composite high school: Since our objective is to DETECT INEFFICIENCIES, we want to minimize the fraction of Roosevelt High School's input resources required by the composite high school: MIN E

11. 11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis 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 n Output Constraints General form for each output: General form for each output: output for composite >= output for Rooseveltoutput for composite >= output for Roosevelt Output for composite = Output for composite = (Output for Roosevelt * weight for Roosevelt ) + (output for Lincoln * weight for Lincoln ) + (output for Washington * weight for Washington ) +(Output for Roosevelt * weight for Roosevelt ) + (output for Lincoln * weight for Lincoln ) + (output for Washington * weight for Washington ) +

12. 12 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis 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)

13. 13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Input Constraints General Form General Form Input for composite <= input for Roosevelt * EInput for composite <= input for Roosevelt * E Input for composite = Input for composite = (Input for Roosevelt * Input for Roosevelt ) + (Input for Lincoln * Input for Lincoln ) + (Input for Washington * Input for Washington )(Input for Roosevelt * Input for Roosevelt ) + (Input for Lincoln * Input for Lincoln ) + (Input for Washington * Input for Washington ) (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 : E, w 1, w 2, w 3 > 0 Nonnegativity : E, w 1, w 2, w 3 > 0

14. 14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n MIN E n ST n (1) w 1 + w 2 + w 3 = 1 (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) (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) (8) E, w 1, w 2, w 3 > 0 (8) E, w 1, w 2, w 3 > 0

15. 15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Computer 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 (R) 0.000 0.235 W1 (R) 0.000 0.235 W2 (L) 0.500 0.000 W2 (L) 0.500 0.000 W3 (W) 0.500 0.000 W3 (W) 0.500 0.000 *Composite is 50% Lincoln, 50% Washington *Roosevelt is no more than 76.5% efficient as composite

16. Data Envelopment Analysis n Computer Solution (continued) CONSTRAINT SLACK/SURPLUS DUAL VALUES 1 0.000 -0.235 1 0.000 -0.235 2 (SAT) 65.000 0.000 2 (SAT) 65.000 0.000 3 (grads) 0.000 -0.001 3 (grads) 0.000 -0.001 4 (college) 170.000 0.000 4 (college) 170.000 0.000 5 (fac) 4.294 0.000 5 (fac) 4.294 0.000 6 (budget) 0.044 0.000 6 (budget) 0.044 0.000 7 (seniors) 0.000 0.001 7 (seniors) 0.000 0.001 Zero Slack – Roosevelt is 76.5% efficient in this area (ie grads) Positive slack – Roosevelt is LESS THAN 76.5% efficient (ie SAT) ie SAT scores are 65 points higher in the composite school

17. 17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Another LP application is revenue management. n Revenue management managing the short-term demand for a fixed perishable inventory in order to maximize revenue potential. n first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare.

18. 18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n General Form n MAX (revenue per unit * units allocated) n ST CAPACITY CAPACITY DEMAND DEMAND NONNEGATIVE NONNEGATIVE

19. 19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

20. 20 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 how many seats it should allocate to each ODIF. LeapFrog wants to determine how many seats it should allocate to each ODIF. Revenue Management

21. 21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. INDBAL MEM AUSTAM Each day a plane Leaves both IND And BAL for AUS and TAM Respectively. Both flights lay over In MEM No return flights (for simplicity) Each plane holds 120 Leg 1 Leg 2 Leg 3 Leg 4

22. 22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. OrigDest INDMEM INDAUS INDTAM BALMEM BALAUS BALTAM MEMAUS MEMTAM 8 different origin-destination combinations Plus two different fare classes: Discount and Full Fare 8 Orig-Desination combinations * 2 fare classes = 16 combinations

23. 23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. ODIF12345678910111213141516OriginIndianapolisIndianapolisIndianapolisIndianapolisIndianapolisIndianapolisBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreMemphisMemphisMemphisMemphisDestinationMemphisAustinTampaMemphisAustinTampaMemphisAustinTampaMemphisAustinTampaAustin Tampa Austin Tampa FareClassDDDFFFDDDFFFDDFFODIFCodeIMDIADITDIMFIAFITFBMDBADBTDBMFBAFBTFMADMTDMAFMTF Fare175275285395425475185315290385525490190180310295Demand442540151082650421216958481411 Revenue Management

24. 24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

25. 25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

26. 26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

27. 27 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

28. 28 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

29. 29 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Constraints (continued) Demand Constraints Limit the amount of seats for each ODIF There are 16 demand constraints, one for each ODIF: (5) IMD < 44(11) BMD < 26(17) MAD < 58 (5) IMD < 44(11) BMD < 26(17) MAD < 58 (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

30. Revenue Management 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 ST: IMD + IAD + ITD + IMF + IAF + ITF < 120 BMD + BAD + BTD + BMF + BAF + BTF < 120 IAD + IAF + BAD + BAF + MAD + MAF < 120 ITD + ITF + BTD + BTF + MTD + MTF < 120 IMD < 44, BMD < 26, MAD < 58, IAD < 25, BAD < 50 MTD < 48, ITD < 40, BTD < 42, MAF < 14, IMF < 15 BMF < 12, MTF < 11, IAF < 10, BAF < 16, ITF < 8 BTF < 9 IMD, IAD, ITD, IMF, IAF, ITF, BMD, BAD, BTD, BMF, BAF, BTF, MAD, MTD, MAF, MTF > 0

31. 31 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Computer Solution n Revenue Contribution is $96265

32. 32 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Computer Solution (continued) -IMD dual value is 90 -IMF dual value is 310

33. 33 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

34. 34 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

35. 35 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

36. 36 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

37. 37 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

38. 38 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

39. 39 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

40. 40 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

41. 41 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

42. 42 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

43. 43 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

44. 44 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

45. 45 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

46. 46 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

47. 47 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

48. 48 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 )

49. 49 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

50. 50 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 )

51. 51 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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