1. Slides by
John
Loucks
St. Edward’s
University

2. Chapter 5 Advanced Linear Programming Applications
Data Envelopment Analysis
Revenue Management
Portfolio Models and Asset Allocation
Game Theory

3. Data Envelopment Analysis
Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives.
DEA creates a fictitious composite unit made up of an optimal weighted average (W1, W2,…) of existing units.
An individual unit, k, can be compared by determining E, the fraction of unit k’s input resources required by the optimal composite unit.
If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient.
If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

4. Data Envelopment Analysis
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. 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. 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.

6. Data Envelopment Analysis
Input
Roosevelt Lincoln Washington
Senior Faculty
Budget ($100,000's)
Senior Enrollments

7. Data Envelopment Analysis
Output
Roosevelt Lincoln Washington
Average SAT Score
High School Graduates
College Admissions

8. Data Envelopment Analysis
Define the Decision Variables
E = Fraction of Roosevelt's input resources required by the composite high school
w1 = Weight applied to Roosevelt's input/output resources by the composite high school
w2 = Weight applied to Lincoln’s input/output resources by the composite high school
w3 = Weight applied to Washington's input/output resources by the composite high school

9. Data Envelopment Analysis
Define the Objective Function
Minimize the fraction of Roosevelt High School's input resources required by the composite high school:
MIN E

10. Data Envelopment Analysis
Define the Constraints
Sum of the Weights is 1:
(1) w1 + w2 + w3 = 1
Output Constraints:
Since w1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt:
(2) 800w w w3 > (SAT Scores)
(3) 450w w w3 > (Graduates)
(4) 140w w w3 > (College Admissions)

11. Data Envelopment Analysis
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:
(5) 37w w w3 < 37E (Faculty)
(6) 6.4w w w3 < 6.4E (Budget)
(7) 850w w w3 < 850E (Seniors)
Nonnegativity of variables:
E, w1, w2, w3 > 0

12. Data Envelopment Analysis
Computer Solution
OBJECTIVE FUNCTION VALUE =
VARIABLE VALUE REDUCED COSTS E W W W

13. Data Envelopment Analysis
Computer Solution (continued)
CONSTRAINT SLACK/SURPLUS DUAL VALUES

14. Data Envelopment Analysis
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.)

15. Revenue Management Another LP application is revenue management.
Revenue management involves managing the short-term demand for a fixed perishable inventory in order to maximize revenue potential.
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.
Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.

16. Revenue Management 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. Revenue Management 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.

18. Revenue Management Fare Class D F ODIF Code IMD IAD ITD IMF IAF ITF
BMD
BAD
BTD
BMF
BAF
BTF
MAD
MTD
MAF
MTF
ODIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Origin
Indianapolis
Baltimore
Memphis
Destination
Memphis
Austin
Tampa
Tampa Austin
Fare
175
275
285
395
425
475
185
315
290
385
525
490
190
180
310
295
Demand 44 25 40 15 10 8 26 50 42 12 16 9 58 48 14 11

19. Revenue Management Define the Decision Variables
There are 16 variables, one for each ODIF:
IMD = number of seats allocated to Indianapolis-Memphis-
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. Revenue Management Define the Decision Variables (continued)
ITF = number of seats allocated to Indianapolis-Tampa-
Full Fare class
BMD = number of seats allocated to Baltimore-Memphis-
Discount class
BAD = number of seats allocated to Baltimore-Austin-
BTD = number of seats allocated to Baltimore-Tampa-
BMF = number of seats allocated to Baltimore-Memphis-
BAF = number of seats allocated to Baltimore-Austin-

21. Revenue Management Define the Decision Variables (continued)
BTF = number of seats allocated to Baltimore-Tampa-
Full Fare class
MAD = number of seats allocated to Memphis-Austin-
Discount class
MTD = number of seats allocated to Memphis-Tampa-
MAF = number of seats allocated to Memphis-Austin-
MTF = number of seats allocated to Memphis-Tampa-

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

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

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

25. Revenue Management Computer Solution
Objective Function Value =
Variable Value Reduced Cost
IMD
IAD
ITD
IMF
IAF
ITF
BMD
BAD

26. Revenue Management Computer Solution (continued)
Variable Value Reduced Cost
BTD
BMF
BAF
BTF
MAD
MTD
MAF
MTF

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

28. Portfolio Model 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 characteristics are listed in the table on the next slide. Note that Unidyde 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.

29. Portfolio Model Exp. Annual After Tax Liquidity Risk
Category Investment Return Factor Factor
Equities Unidyde Corp %
(Stocks) CC’s Restaurants %
First General REIT %
Debt Metropolis Electric %
(Bonds) Unidyde Corp %
Lewisville Transit %
Real Estate Realty Partners 22.0%
First General REIT ( --- See above --- )
Money T-Bill Account %
Money Mkt. Fund %
Saver's Certificate %

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

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

33. Portfolio Model 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

34. Portfolio Model Define the Objective Function
Maximize the total expected after-tax return over the next year:
Max .15X X X X X5
+ .12X X X X X10

35. Portfolio Model Define the Constraints
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:
100X X X3 + 95X4 + 92X5 + 79X6 + 80X X9 >
65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
Weighted average risk factor must be no greater than 55:
60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 <
55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
No more than $60,000 to be invested in Unidyde Corp:
X1 + X5 < 60,000

36. Portfolio Model 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
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. Portfolio Model 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 > j = 1, , 10

38. Portfolio Model Solution Summary
Total Expected After-Tax Return = $64,355
X1 = $ invested in Unidyde Corp. (Equities)
X2 = $80,000 invested in CC’s Restaurants
X3 = $80,000 invested in First General REIT
X4 = $ 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

39. Introduction to Game Theory
In decision analysis, a single decision maker seeks to select an optimal alternative.
In game theory, there are two or more decision makers, called players, who compete as adversaries against each other.
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.
Each player selects a strategy independently without knowing in advance the strategy of the other player(s).
continue

40. Introduction to Game Theory
The combination of the competing strategies provides the value of the game to the players.
Examples of competing players are teams, armies, companies, political candidates, and contract bidders.

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

42. Two-Person Zero-Sum Game Example
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.

43. Two-Person Zero-Sum Game Example
Strategy Choices
Strategy 1: Offer a cash rebate
on a new vehicle.
Strategy 2: Offer free optional
equipment on a
new vehicle.
Strategy 3: Offer a 0% loan

44. Two-Person Zero-Sum Game Example
Payoff Table: Number of Vehicle Sales
Gained Per Week by Dealership A
(or Lost Per Week by Dealership B)
Dealership B
Cash
Rebate b1
Free
Options b2 0%
Loan b3
Dealership A
Cash Rebate a1
Free Options a2
0% Loan a3

45. Two-Person Zero-Sum Game
Step 1: Identify the minimum payoff for each
row (for Player A).
Step 2: For Player A, select the strategy that provides
the maximum of the row minimums (called
the maximin).

46. Two-Person Zero-Sum Game Example
Identifying Maximin and Best Strategy
Dealership B
Cash
Rebate b1
Free
Options b2 0%
Loan b3
Row
Minimum
Dealership A
Cash Rebate a1
Free Options a2
0% Loan a3 1 -3 -2
Best Strategy
For Player A
Maximin
Payoff

47. Two-Person Zero-Sum Game
Step 3: Identify the maximum payoff for each column
(for Player B).
Step 4: For Player B, select the strategy that provides
the minimum of the column maximums
(called the minimax).

48. Two-Person Zero-Sum Game Example
Identifying Minimax and Best Strategy
Dealership B
Best Strategy
For Player B
Cash
Rebate b1
Free
Options b2 0%
Loan b3
Dealership A
Cash Rebate a1
Free Options a2
0% Loan a3
Minimax
Payoff
Column Maximum

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

50. Pure Strategy Example Saddle Point and Value of the Game Dealership B
game is 1
Cash
Rebate b1
Free
Options b2 0%
Loan b3
Row
Minimum
Dealership A
Cash Rebate a1
Free Options a2
0% Loan a3 1 -3 -2
Column Maximum
Saddle
Point

51. Pure Strategy Example Pure Strategy Summary
Player A should choose Strategy a1 (offer a cash rebate).
Player A can expect a gain of at least 1 vehicle sale per week.
Player B should choose Strategy b3 (offer a 0% loan).
Player B can expect a loss of no more than 1 vehicle sale per week.

52. Mixed Strategy
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.
In this case, a mixed strategy is best.
With a mixed strategy, each player employs more than one strategy.
Each player should use one strategy some of the time and other strategies the rest of the time.
The optimal solution is the relative frequencies with which each player should use his possible strategies.

53. Mixed Strategy Example
Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy.
Player B
Row
Minimum b1 b2
Player A
Maximin a1 a2 4 5
Column
Maximum
Minimax

54. Mixed Strategy Example
p = the probability Player A selects strategy a1
(1 - p) = the probability Player A selects strategy a2
If Player B selects b1:
EV = 4p + 11(1 – p)
If Player B selects b2:
EV = 8p + 5(1 – p)

55. Mixed Strategy Example
To solve for the optimal probabilities for Player A
we set the two expected values equal and solve for
the value of p.
4p + 11(1 – p) = 8p + 5(1 – p)
4p + 11 – 11p = 8p + 5 – 5p
11 – 7p = 5 + 3p
Hence,
(1 - p) = .4
-10p = -6
p = .6
Player A should select:
Strategy a1 with a .6 probability and
Strategy a2 with a .4 probability.

56. Mixed Strategy Example
q = the probability Player B selects strategy b1
(1 - q) = the probability Player B selects strategy b2
If Player A selects a1:
EV = 4q + 8(1 – q)
If Player A selects a2:
EV = 11q + 5(1 – q)

57. Mixed Strategy Example
Expected gain
per game
for Player A
Value of the Game
For Player A:
EV = 4p + 11(1 – p) = 4(.6) + 11(.4) = 6.8
For Player B:
Expected loss
per game
for Player B
EV = 4q + 8(1 – q) = 4(.3) + 8(.7) = 6.8

58. Dominated Strategies Example
Suppose that the payoff table for a two-person zero-
sum game is the following. Here there is no optimal
pure strategy.
Player B
Row
Minimum b1 b2 b3
Player A
Maximin a1 a2 a3 -2 -3
Column
Maximum
Minimax

59. Dominated Strategies Example
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.
Player B b1 b2 b3
Player A a1 a2 a3
Player A’s Strategy a3 is dominated by
Strategy a1, so Strategy a3 can be eliminated.

60. Dominated Strategies Example
We continue to look for dominated strategies in order to reduce the size of the game.
Player B b1 b2 b3
Player A a1 a2
Player B’s Strategy b2 is dominated by
Strategy b1, so Strategy b2 can be eliminated.

61. Dominated Strategies Example
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.
Player B b1 b3
Player A a1 a2

62. Two-Person Zero-Sum Game Example #2
Competing for Vehicle Sales
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.)

63. Two-Person Zero-Sum Game Example #2
Payoff Table: Number of Vehicle Sales
Gained Per Week by Dealership A
(or Lost Per Week by Dealership B)
Dealership B
Cash
Rebate b1
Free
Options b2 0%
Loan b3
Dealership A
Cash Rebate a1
Free Options a2
0% Loan a3

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

65. Two-Person Zero-Sum Game Example #2
Let us first consider the game from the point of view of Dealership A.
Dealership A will select one of its three strategies based on the following probabilities:
PA1 = the probability that Dealership A selects strategy a1
PA2 = the probability that Dealership A selects strategy a2
PA3 = the probability that Dealership A selects strategy a3

66. Two-Person Zero-Sum Game Example #2
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
b1 EG(b1) = 2PA1 – 3PA2 + 3PA3
b2 EG(b2) = 2PA1 + 3PA2 – 2PA3
b3 EG(b3) = 1PA1 – 1PA2 + 3PA3

67. Two-Person Zero-Sum Game Example #2
Define GAINA to be the optimal expected gain in vehicle sales for Dealership A, which we want to maximize.
Thus, the individual expected gains, EG(b1), EG(b2) and EG(b3) must all be greater than or equal to GAINA.
For example,
2PA1 – 3PA2 + 3PA3 > GAINA
Also, the sum of Dealership A’s mixed strategy probabilities must equal 1.
This results in the LP formulation on the next slide …..

68. Two-Person Zero-Sum Game Example #2
Dealership A’s Linear Programming Formulation
Max GAINA
s.t.
2PA1 – 3PA2 + 3PA3 – GAINA > (Strategy b1)
2PA1 + 3PA2 – 2PA3 – GAINA > 0 (Strategy b2)
1PA1 – 1PA2 + 0PA3 – GAINA > 0 (Strategy b3)
PA1 + PA2 + PA3 = (Prob’s sum to 1)
PA1, PA2, PA3, GAINA > (Non-negativity)

69. Two-Person Zero-Sum Game Example #2
Computer Solution: Dealership A
OBJECTIVE FUNCTION VALUE =
VARIABLE VALUE REDUCED COSTS PA PA PA
GAINA

70. Two-Person Zero-Sum Game Example #2
Computer Solution: Dealership A
CONSTRAINT SLACK/SURPLUS DUAL VALUES

71. Two-Person Zero-Sum Game Example #2
Dealership A’s Optimal Mixed Strategy
Offer a cash rebate (a1) with a probability of 0.833
Do not offer free optional equipment (a2)
Offer a 0% loan (a3) 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.

72. Two-Person Zero-Sum Game Example #2
Let us now consider the game from the point of view of Dealership B.
Dealership B will select one of its three strategies based on the following probabilities:
PB1 = the probability that Dealership B selects strategy b1
PB2 = the probability that Dealership B selects strategy b2
PB3 = the probability that Dealership B selects strategy b3

73. Two-Person Zero-Sum Game Example #2
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
a1 EL(a1) = 2PB1 + 2PB2 + 1PB3
a2 EL(a2) = -3PB1 + 3PB2 – 1PB3
a3 EL(a3) = 3PB1 – 2PB2 + 3PB3

74. Two-Person Zero-Sum Game Example #2
Define LOSSB to be the optimal expected loss in vehicle sales for Dealership B, which we want to minimize.
Thus, the individual expected losses, EL(a1), EL(a2) and EL(a3) must all be less than or equal to LOSSB.
For example,
2PA1 + 2PA2 + 1PA3 < LOSSB
Also, the sum of Dealership B’s mixed strategy probabilities must equal 1.
This results in the LP formulation on the next slide …..

75. Two-Person Zero-Sum Game Example #2
Dealership B’s Linear Programming Formulation
Min LOSSB
s.t.
2PB1 + 2PB2 + 1PB3 – LOSSB < (Strategy a1)
-3PB1 + 3PB2 – 1PB3 – LOSSB < (Strategy a2)
3PB1 – 2PB2 + 3PB3 – LOSSB < (Strategy a3)
PB1 + PB2 + PB3 = (Prob’s sum to 1)
PB1, PB2, PB3, LOSSB > (Non-negativity)

76. Two-Person Zero-Sum Game Example #2
Computer Solution: Dealership B
OBJECTIVE FUNCTION VALUE =
VARIABLE VALUE REDUCED COSTS PB PB PB
LOSSB

77. Two-Person Zero-Sum Game Example #2
Computer Solution: Dealership B
CONSTRAINT SLACK/SURPLUS DUAL VALUES

78. Two-Person Zero-Sum Game Example #2
Dealership B’s Optimal Mixed Strategy
Do not offer a cash rebate (b1)
Offer free optional equipment (b2) with a probability of 0.333
Offer a 0% loan (b3) with a probability of 0.667
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
the expected gain for Dealership A. (There is a zero-
sum for the expected payoffs.)

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

80. End of Chapter 5