- Game theory

2,170 views

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
2,170
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
64
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.
  • This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues.
  • This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues.
  • - Game theory

    1. 1. To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Game Theory Prepared by Lee Revere and John Large
    2. 2. Learning Objectives <ul><li>Students will be able to: </li></ul><ul><ul><li>Understand the principles of zero-sum, two-person games. </li></ul></ul><ul><ul><li>Analyze pure strategy games and use dominance to reduce the size of the game. </li></ul></ul><ul><ul><li>Solve mixed strategy games when there is no saddle point. </li></ul></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
    3. 3. To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 <ul><li>Game theory is the study of how </li></ul><ul><li>optimal strategies are formulated </li></ul><ul><li>in conflict. Game theory has been </li></ul><ul><li>effectively used for: </li></ul><ul><ul><li>War strategies </li></ul></ul><ul><ul><li>Union negotiators </li></ul></ul><ul><ul><li>Competitive business strategies </li></ul></ul>Introduction
    4. 4. To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 <ul><li>Game models are classified by the number of players, the sum of all payoffs, and the number of strategies employed. </li></ul><ul><li>A zero sum game implies that what is gained by one player is lost for the other. </li></ul>Introduction (continued)
    5. 5. Language of Games To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Consider a duopoly competitive business market in which one company is considering advertising in hopes of luring customers away from its competitor. The company is considering radio and/or newspaper advertisements. Let’s use game theory to determine the best strategy.
    6. 6. Language of Games (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Below is the payoff matrix (as a percent of change in market share) for Store X. A positive number means that X wins and Y loses, while a negative number implies Y wins and X loses. Note : Although X is considering the advertisements (therefore the results favor X), Y must play the game. STORE X’s PAYOFFs Y’s strategy 1 (use radio) Y’s strategy 2 (use newspaper) X’s strategy 1 (use radio) 3 5 X’s strategy 2 (use newspaper) 1 -2
    7. 7. Language of Games (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Note : Although X is considering the advertisements (therefore the results favor X), Y must play the game. Store X’s Strategy Stores Y’s Strategy Outcome (% change in market share) X1: Radio Y1: Radio X wins 3 Y loses 3 X1: Radio Y2: Newspaper X wins 5 Y loses 5 X2: Newspaper Y1: Radio X wins 1 Y loses 1 X2: Newspaper Y2: Newspaper X loses 2 Y wins 2
    8. 8. The Minimax Criterion To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The minimax criterion is used in a two-person zero-sum game. Each person should choose the strategy that minimizes the maximum loss. Note: This is identical to maximizing one’s minimum gains.
    9. 9. The Minimax Criterion (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The upper value of the game is equal to the minimum of the maximum values in the columns. The lower value of the game is equal to the maximum of the minimum values in the rows.
    10. 10. The Minimax Criterion (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Lower Value of the Game: Maximum of the minimums Upper Value of the Game: Minimum of the maximums STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 (radio) 3 5 3 X2 (newspaper) 1 -2 2 Maximum 3 5
    11. 11. The Minimax Criterion (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Saddle point: Both upper and lower values are 3. A saddle point condition exists if the upper and lower values are equal. This is called a pure strategy because both players will follow the same strategy. STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 (radio) 3 5 3 X2 (newspaper) 1 -2 2 Maximum 3 5
    12. 12. The Minimax Criterion (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Saddle point Let’s look at a second example of a pure strategy game. Lower value Upper value STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 (radio) 10 6 6 X2 (newspaper) -12 2 -12 Maximum 10 6
    13. 13. Mixed Strategy Game To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy. Note: The expected gain is determined using an approach very similar to the expected monetary value approach.
    14. 14. Mixed Strategy Games (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Each player seeks to maximize his/her expected gain by altering the percent of time (P or Q) that he/she use each strategy. Set these two equations equal to each other and solve for Q Set these two equations equal to each other and solve for P Y1 (P) Y2 (1-P) Expected Gain X1 (Q) 4 2 4P + 2(1-P) X2 (1-Q) 1 10 1p + 10(1-P) Expected Gain 4Q + 1(1-Q) 2Q + 10(1-Q)
    15. 15. Mixed Strategy Games (continued) <ul><li>4P + 2(1-P) = 1P + 10(1-P) 4P – 2P – 1P + 10P = 10 – 2 P = 8/11 and 1-P = 3/11 </li></ul><ul><li>Expected payoff: 1P + 10(1-P) = 1(8/11) + 10(3/11) = 3.46 </li></ul><ul><li>4Q + 1(1-Q) = 2Q + 10(1-Q) 4Q – 1Q – 2Q + 10Q = 10 – 1 Q = 9/11 and 1-Q = 2/11 </li></ul><ul><li>Expected payoff: 2Q + 10(1-Q) </li></ul><ul><li>= 2(9/11) + 10(2/11) = 3.46 </li></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
    16. 16. Exercise <ul><li>Player A has a $1 bill and $20 bill, and player B has a $5 bill and $10 bill. Each player will select a bill from the other player without knowing what bill the other player selected. If the total of the bills selected is odd player A gets both bills, but if the total is even, player B gets both bills. </li></ul><ul><li>Develop the payoff table for this problem. </li></ul><ul><li>Determine the value of the game. </li></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
    17. 17. Dominance To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dominance is a principle that can be used to reduce the size of games by eliminating strategies that would never be played. Note: A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding outcomes of another strategy.
    18. 18. Dominance (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Initial game X3 is a dominated strategy Game after removal of dominated strategy Y1 Y2 X1 4 3 X2 2 20 X3 1 1 Y1 Y2 X1 4 3 X2 2 20
    19. 19. Dominance (continued) To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Initial game Game after removal of dominated strategies Y1 Y2 Y3 Y4 X1 -5 4 6 -3 X2 -2 6 2 -20 Y1 Y4 X1 -5 -3 X2 -2 -20
    20. 20. Question1 <ul><li>What is the value of the following game and the strategies for A and B? </li></ul><ul><li>B1 B2 </li></ul><ul><li>A1 19 20 </li></ul><ul><li>A2 5 -4 </li></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
    21. 21. Questions2 <ul><li>Shoe town and fancy foot are both vying for more share of the market. If Shoe town does no ad, it will not lose any share of the market if Fancy Foot does nothing. It will lose 2% of market if Fancy Foot invests $10,000 in ad, and it will lose 5% of the market if Fancy Foot invests $20,000 in ad. On the other hand, if Shoe town invests $15,000 in ad, it will gain 3% of the market if Fancy Foot does nothing; it will gain 1% of the market if Fancy Foot invests $10,000 in ad; and it will lose 1% if Fancy Foot invests $20,000 in ad. </li></ul><ul><li>Q: Develop a payoff table for this problem. </li></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
    22. 22. Question3 <ul><li>For the following 2-person, zero-sum game, are there any dominated strategies? If so, eliminate any dominated strategy and find the value of the game. </li></ul><ul><li>Y1 Y2 Y3 </li></ul><ul><li>X1 4 5 10 </li></ul><ul><li>X2 3 4 2 </li></ul><ul><li>X3 8 6 9 </li></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
    23. 23. Question4 <ul><li>Solve the following the following game: </li></ul><ul><li>Y1 Y2 </li></ul><ul><li>X1 -5 -10 </li></ul><ul><li>X2 12 8 </li></ul><ul><li>X3 4 12 </li></ul><ul><li>X4 -40 -5 </li></ul>To accompany Quantitative Analysis for Management,9 e by Render/Stair/Hanna M4- © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

    ×