Game Theory

                                            Sherif Khalifa, Ph.D.
                                          D...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Introduction


            Game theory is a framework to aid in decision making when your
            payo¤ depends on the...
Simultaneous Move, One Shot Games



            A strategy is a decision rule that describes the actions a player will
  ...
Simultaneous Move, One Shot Games




            A strategy is a dominant strategy if it results in the highest payo¤
   ...
Applications of One-Shot Games
   Pricing Decisions




                                                                  ...
Applications of One-Shot Games
   Advertising and Quality Decisions




                                                  ...
Applications of One-Shot Games
   Coordination Decisions




                                                             ...
Applications of One-Shot Games
   Monitoring Employees Decisions




                                                     ...
Applications of One-Shot Games
   Nash Bargaining




                                                                    ...
In…nitely Repeated Games



    Is a game that is played over and over again forever.
                                    ...
In…nitely Repeated Games




            A trigger strategy is a strategy that is contingent on the past plays of
        ...
In…nitely Repeated Games




            A trigger strategy is a strategy that is contingent on the past plays of
        ...
In…nitely Repeated Games
   Cheat and Cooperate




                                                                     P...
In…nitely Repeated Games
   Cheat and Cooperate



    Example

    Assume i = 0.4. If the …rm does cheat it earns $40 thi...
In…nitely Repeated Games
   Factors A¤ecting Collusion



            Collusion is easier when there are few …rms rather t...
In…nitely Repeated Games
   Factors A¤ecting Collusion



            Collusion is easier when there are few …rms rather t...
In…nitely Repeated Games
   Factors A¤ecting Collusion



            Collusion is easier when there are few …rms rather t...
In…nitely Repeated Games
   Factors A¤ecting Collusion



            Collusion is easier when there are few …rms rather t...
In…nitely Repeated Games
   Factors A¤ecting Collusion



            Collusion is easier when there are few …rms rather t...
Finitely Repeated Games
   Games with an Uncertain Final Period



                                                       ...
Finitely Repeated Games
   Games with an Uncertain Final Period



    Example
    Assume Θ = 0.1. If the …rm does cheat i...
Finitely Repeated Games
   Games with a Known Final Period




                                                           ...
Multistage Games



            An extensive form game summarizes the players, the information
            available to th...
Multistage Games



            An extensive form game summarizes the players, the information
            available to th...
Multistage Games


                                                                                          (10,15)
     ...
Multistage Games


                                                                                          (-1,1)
      ...
Multistage Games


                                                                                      (-5,20)
         ...
Multistage Games


                                                                                   ($99,$1)
           ...
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Game Theory

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Game Theory

  1. 1. Game Theory Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton Fall 2007 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 1 / 22
  2. 2. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  3. 3. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Strategies are planned decisions of the players. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  4. 4. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Strategies are planned decisions of the players. The payo¤s of the players are the pro…ts or losses that result from the strategies, and that depend not only on that player’ strategies but s also on the strategies employed by other players. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  5. 5. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Strategies are planned decisions of the players. The payo¤s of the players are the pro…ts or losses that result from the strategies, and that depend not only on that player’ strategies but s also on the strategies employed by other players. Simultaneous move game is a game in which each player makes decisions without knowledge of the other player’ decisions. s Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  6. 6. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Strategies are planned decisions of the players. The payo¤s of the players are the pro…ts or losses that result from the strategies, and that depend not only on that player’ strategies but s also on the strategies employed by other players. Simultaneous move game is a game in which each player makes decisions without knowledge of the other player’ decisions. s Sequential move game is a game in which one player makes a move after observing the other player’ move. s Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  7. 7. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Strategies are planned decisions of the players. The payo¤s of the players are the pro…ts or losses that result from the strategies, and that depend not only on that player’ strategies but s also on the strategies employed by other players. Simultaneous move game is a game in which each player makes decisions without knowledge of the other player’ decisions. s Sequential move game is a game in which one player makes a move after observing the other player’ move. s One shot game is a game that is played only once. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  8. 8. Introduction Game theory is a framework to aid in decision making when your payo¤ depends on the actions taken by other players. Strategies are planned decisions of the players. The payo¤s of the players are the pro…ts or losses that result from the strategies, and that depend not only on that player’ strategies but s also on the strategies employed by other players. Simultaneous move game is a game in which each player makes decisions without knowledge of the other player’ decisions. s Sequential move game is a game in which one player makes a move after observing the other player’ move. s One shot game is a game that is played only once. Repeated game is a game that is played more than once. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 2 / 22
  9. 9. Simultaneous Move, One Shot Games A strategy is a decision rule that describes the actions a player will take at each decision point. The normal form representation of a game indicates the players in the game, the possible strategies of the players, and the payo¤s to the players that will result from alternative strategies. Player B Strategy Left Right Player A Up 10, 20 15, 8 Down 10, 7 10, 10 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 3 / 22
  10. 10. Simultaneous Move, One Shot Games A strategy is a dominant strategy if it results in the highest payo¤ regardless of the action of the opponent. A secure strategy is one that guarantees the highest payo¤ given the worst possible scenario. Nash equilibrium is a condition describing a set of strategies in which no player can improve her payo¤ by unilaterally changing her own strategy, given the other players’strategies. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 4 / 22
  11. 11. Applications of One-Shot Games Pricing Decisions Player B Strategy Low Price High Price Player A Low Price 0, 0 50, 10 High Price 10, 50 10, 10 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 5 / 22
  12. 12. Applications of One-Shot Games Advertising and Quality Decisions Firm B Strategy Advertise Do not Advertise Firm A Advertise 4, 4 20, 1 Do not Advertise 1, 20 10, 10 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 6 / 22
  13. 13. Applications of One-Shot Games Coordination Decisions Firm B Strategy 120 volt 90 volt Firm A 120 volt 100, 100 0, 0 90 volt 0, 0 100, 100 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 7 / 22
  14. 14. Applications of One-Shot Games Monitoring Employees Decisions Worker Strategy Work Shirk Manager Monitor 1, 1 1, 1 Do not Monitor 1, 1 1, 1 Mixed strategy is a strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her action. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 8 / 22
  15. 15. Applications of One-Shot Games Nash Bargaining Labor Union Strategy 0 50 100 Management 0 0, 0 0, 50 0, 100 50 50, 0 50, 50 1, 1 100 100, 0 1, 1 1, 1 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 9 / 22
  16. 16. In…nitely Repeated Games Is a game that is played over and over again forever. ∞ π1 π2 πt PVFirm = π 0 + + 2 + .......... = ∑ t 1+i (1 + i ) t =0 (1 + i ) If π t = π, then ∞ π π π 1+i PVFirm = π + + 2 + .......... = ∑ t = π 1+i (1 + i ) t =0 ( 1 + i ) i Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 10 / 22
  17. 17. In…nitely Repeated Games A trigger strategy is a strategy that is contingent on the past plays of players in a game, and in which some particular past action triggers a di¤erent action by a player. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 11 / 22
  18. 18. In…nitely Repeated Games A trigger strategy is a strategy that is contingent on the past plays of players in a game, and in which some particular past action triggers a di¤erent action by a player. A player who adopts a trigger strategy continues to choose the same action until some other player takes an action that triggers a di¤erent action by the …rst player. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 11 / 22
  19. 19. In…nitely Repeated Games Cheat and Cooperate Player B Strategy Low Price High Price Player A Low Price 0, 0 50, 40 High Price 40, 50 10, 10 Cheat PVFirm = 50 + 0 + 0 + .......... = 50 Cooperate 10 10 10 (1 + i ) PVFirm = 10 + + 2 + .......... = 1+i (1 + i ) i Cheat 10 (1 + i ) Cooperate PVFirm = 50 = PVFirm i Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 12 / 22
  20. 20. In…nitely Repeated Games Cheat and Cooperate Example Assume i = 0.4. If the …rm does cheat it earns $40 this period and zero afterwards. If it does not cheat, it earns $10 every period. Will the …rm cheat or collude? Cheat PVFirm = 40 + 0 + 0 + .......... = 40 Cooperate 10 10 10 (1 + 0.4) PVFirm = 10 + + 2 + .......... = = 35 1 + 0.4 (1 + 0.4) 0.4 Cheat Cooperate PVFirm = 40 > 35 = PVFirm Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 13 / 22
  21. 21. In…nitely Repeated Games Factors A¤ecting Collusion Collusion is easier when there are few …rms rather than many, as the total number of monitors needed in the market grow as the number of …rms increases. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
  22. 22. In…nitely Repeated Games Factors A¤ecting Collusion Collusion is easier when there are few …rms rather than many, as the total number of monitors needed in the market grow as the number of …rms increases. If the number of …rms is large, the monitoring costs become so high relative to collusive pro…ts. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
  23. 23. In…nitely Repeated Games Factors A¤ecting Collusion Collusion is easier when there are few …rms rather than many, as the total number of monitors needed in the market grow as the number of …rms increases. If the number of …rms is large, the monitoring costs become so high relative to collusive pro…ts. Monitoring costs constitute a greater share of total costs for small than large …rms. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
  24. 24. In…nitely Repeated Games Factors A¤ecting Collusion Collusion is easier when there are few …rms rather than many, as the total number of monitors needed in the market grow as the number of …rms increases. If the number of …rms is large, the monitoring costs become so high relative to collusive pro…ts. Monitoring costs constitute a greater share of total costs for small than large …rms. Tacit collusion occurs when the …rms do not explicitly conspire to collude but accomplish collusion indirectly. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
  25. 25. In…nitely Repeated Games Factors A¤ecting Collusion Collusion is easier when there are few …rms rather than many, as the total number of monitors needed in the market grow as the number of …rms increases. If the number of …rms is large, the monitoring costs become so high relative to collusive pro…ts. Monitoring costs constitute a greater share of total costs for small than large …rms. Tacit collusion occurs when the …rms do not explicitly conspire to collude but accomplish collusion indirectly. In a single price market, the cost of punishing an opponent is higher than in markets in which di¤erent customers are quoted di¤erent prices. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 14 / 22
  26. 26. Finitely Repeated Games Games with an Uncertain Final Period Player B Strategy Low Price High Price Player A Low Price 0, 0 50, 40 High Price 40, 50 10, 10 Assume the interest rate=0. ΠCheat = 50 + 0 + 0 + .......... = 50 FirmA 10 ΠCoop = 10 + (1 Θ) 10 + (1 Θ)2 10 + .......... = FirmA Θ 10 ΠCheat = 50 FirmA = ΠCoop Θ FirmA Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 15 / 22
  27. 27. Finitely Repeated Games Games with an Uncertain Final Period Example Assume Θ = 0.1. If the …rm does cheat it earns $80 this period and zero afterwards. If it does not cheat, it earns $10 every period. Will the …rm cheat or collude? ΠCheat = 80 + 0 + 0 + .......... = 80 FirmA 10 ΠCoop = 10 + (1 FirmA 0.1) 10 + (1 0.1)2 10 + .......... = = 100 0.1 ΠCheat = 80 FirmA 100 = ΠCoop FirmA Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 16 / 22
  28. 28. Finitely Repeated Games Games with a Known Final Period Player B Strategy Low Price High Price Player A Low Price 0, 0 50, 40 High Price 40, 50 10, 10 Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 17 / 22
  29. 29. Multistage Games An extensive form game summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payo¤s resulting from alternative strategies. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 18 / 22
  30. 30. Multistage Games An extensive form game summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payo¤s resulting from alternative strategies. Subgame perfect equilibrium is a condition describing a set of strategies that constitutes a Nash equilibrium and allows no player to improve his own payo¤ at any stage of the game by changing strategies. Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 18 / 22
  31. 31. Multistage Games (10,15) Up B Up Down (5,5) A (0,0) Up Down B Down (6,20) Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 19 / 22
  32. 32. Multistage Games (-1,1) Hard B In Soft (5,5) A Out (0,10) Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 20 / 22
  33. 33. Multistage Games (-5,20) Clone B Introduce Don’t (100,0) Clone A Don’t Introduce (1,1) Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 21 / 22
  34. 34. Multistage Games ($99,$1) Accept U $1 Reject ($0,$0) Accept ($50,$50) M $50 U $99 Reject ($0,$0) Accept ($1,$99) U Reject ($0,$0) Sherif Khalifa, Ph.D. Department of Economics California State University, Fullerton () Fall 2007 22 / 22

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