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Lec 24 Game Applications

Lec 24 Game Applications. Chapter 29. Nash Equilibrium. In any Nash equilibrium (NE) each player chooses a “best” response to the choices made by all of the other players. A game may have more than one NE. How can we locate every one of a game’s Nash equilibria?

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Lec 24 Game Applications

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  1. Lec 24 Game Applications Chapter 29

  2. Nash Equilibrium In any Nash equilibrium (NE) each player chooses a “best” response to the choices made by all of the other players. A game may have more than one NE. How can we locate every one of a game’s Nash equilibria? If there is more than one NE, can we argue that one is more likely to occur than another?

  3. Best Responses Think of a 2×2 game; i.e., a game with two players, A and B, each with two actions. A can choose between actions aA1 and aA2. B can choose between actions aB1 and aB2. There are 4 possible action pairs;(aA1, aB1), (aA1, aB2), (aA2, aB1), (aA2, aB2). Each action pair will usually cause different payoffs for the players.

  4. Best Responses Suppose that A’s and B’s payoffs when the chosen actions are aA1 and aB1 areUA(aA1, aB1) = 6 and UB(aA1, aB1) = 4. Similarly, suppose thatUA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7.

  5. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7.

  6. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB1 then A’s best response is ??

  7. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB1 then A’s best response is action aA1 (because 6 > 4).

  8. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB1 then A’s best response is action aA1 (because 6 > 4). If B chooses action aB2 then A’s best response is ??

  9. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB1 then A’s best response is action aA1 (because 6 > 4). If B chooses action aB2 then A’s best response is action aA2 (because 5 > 3).

  10. Best Responses If B chooses aB1 then A chooses aA1. If B chooses aB2 then A chooses aA2. A’s best-response “curve” is therefore + aA2 A’s bestresponse + aA1 B’s action aB1 aB2

  11. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7.

  12. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA1 then B’s best response is ??

  13. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA1 then B’s best response is action aB2 (because 5 > 4).

  14. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA1 then B’s best response is action aB2 (because 5 > 4). If A chooses action aA2 then B’s best response is ??.

  15. Best Responses UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA1 then B’s best response is action aB2 (because 5 > 4). If A chooses action aA2 then B’s best response is action aB2 (because 7 > 3).

  16. Best Responses If A chooses aA1 then B chooses aB2. If A chooses aA2 then B chooses aB2. B’s best-response “curve” is therefore aA2 A’s action aA1 B’s best response aB1 aB2

  17. Best Responses If A chooses aA1 then B chooses aB2. If A chooses aA2 then B chooses aB2. B’s best-response “curve” is therefore Notice that aB2 is astrictly dominantaction for B. aA2 A’s action aA1 B’s best response aB1 aB2

  18. Best Responses & Nash Equilibria + aA2 + aA1 aB1 aB2 How can the players’ best-response curves beused to locate the game’s Nash equilibria? A’s response A’s choice B A aA2 aA1 aB1 aB2 B’s choice B’s response

  19. Best Responses & Nash Equilibria + aA2 + aA1 aB1 aB2 How can the players’ best-response curves beused to locate the game’s Nash equilibria? Put one curve on top of the other. A’s response A’s choice B A aA2 aA1 aB1 aB2 B’s choice B’s response

  20. Best Responses & Nash Equilibria + aA2 + aA1 aB1 aB2 How can the players’ best-response curves beused to locate the game’s Nash equilibria? Put one curve on top of the other. A’s response A’s choice B A aA2 aA1 aB1 aB2 B’s choice B’s response

  21. Best Responses & Nash Equilibria How can the players’ best-response curves beused to locate the game’s Nash equilibria? Put one curve on top of the other. A’s response + aA2 Is there a Nash equilibrium? + aA1 aB1 aB2 B’s response

  22. Best Responses & Nash Equilibria How can the players’ best-response curves beused to locate the game’s Nash equilibria? Put one curve on top of the other. A’s response + aA2 Is there a Nash equilibrium?Yes, (aA2, aB2). Why? + aA1 aB1 aB2 B’s response

  23. Best Responses & Nash Equilibria How can the players’ best-response curves beused to locate the game’s Nash equilibria? Put one curve on top of the other. A’s response + aA2 Is there a Nash equilibrium?Yes, (aA2, aB2). Why? aA2 is a best response to aB2.aB2 is a best response to aA2. + aA1 aB1 aB2 B’s response

  24. Best Responses & Nash Equilibria Player B Here is the strategicform of the game. aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 aA2 is the only best response to aB2.aB2 is the only best response to aA2.

  25. Best Responses & Nash Equilibria Player B Here is the strategicform of the game. aB1 aB2 Is there a 2nd Nasheqm.? 6,4 3,5 aA1 Player A 4,3 5,7 aA2 aA2 is the only best response to aB2.aB2 is the only best response to aA2.

  26. Best Responses & Nash Equilibria Player B Here is the strategicform of the game. aB1 aB2 Is there a 2nd Nasheqm.? No, becauseaB2 is a strictlydominant actionfor Player B. 6,4 3,5 aA1 Player A 4,3 5,7 aA2 aA2 is the only best response to aB2. aB2 is the only best response to aA2.

  27. Best Responses & Nash Equilibria Player B aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 Now allow both players to randomize (i.e., mix)over their actions.

  28. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 Now allow both players to randomize (i.e., mix)over their actions.

  29. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. Given B1, whatvalue of A1 is bestfor A? aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2

  30. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. Given B1, whatvalue of A1 is bestfor A? aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1.

  31. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. Given B1, whatvalue of A1 is bestfor A? aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1.EVA(aA2) = 4B1 + 5(1 - B1) = 5 - B1.

  32. Best Responses & Nash Equilibria EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 - B1.3 + 3B1 5 - B1 as B1 ?? >=< >=< A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A?

  33. Best Responses & Nash Equilibria EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 - B1.3 + 3B1 5 - B1 as B1 ½. >=< >=< A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A?

  34. Best Responses & Nash Equilibria EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 - B1.3 + 3B1 5 - B1 as B1 ½. A’s best response is: aA1 if B1 > ½ aA2 if B1 < ½ aA1 or aA2 if B1 = ½ >=< >=< A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A?

  35. Best Responses & Nash Equilibria EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 - B1.3 + 3B1 5 - B1 as B1 ½. A’s best response is: aA1 (i.e. A1 = 1) if B1 > ½ aA2 (i.e. A1 = 0) if B1 < ½ aA1 or aA2 (i.e. 0  A1 1) if B1 = ½ >=< >=< A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A?

  36. Best Responses & Nash Equilibria A’s best response is: aA1 (i.e. A1 = 1) if B1 > ½ aA2 (i.e. A1 = 0) if B1 < ½ aA1 or aA2 (i.e. 0  A1 1) if B1 = ½ A’s best response A1 1 0 B1 0 ½ 1

  37. Best Responses & Nash Equilibria A’s best response is: aA1 (i.e. A1 = 1) if B1 > ½ aA2 (i.e. A1 = 0) if B1 < ½ aA1 or aA2 (i.e. 0  A1 1) if B1 = ½ A’s best response A1 1 0 B1 0 ½ 1

  38. Best Responses & Nash Equilibria A’s best response is: aA1 (i.e. A1 = 1) if B1 > ½ aA2 (i.e. A1 = 0) if B1 < ½ aA1 or aA2 (i.e. 0  A1 1) if B1 = ½ A’s best response A1 1 0 B1 0 ½ 1

  39. Best Responses & Nash Equilibria A’s best response is: aA1 (i.e. A1 = 1) if B1 > ½ aA2 (i.e. A1 = 0) if B1 < ½ aA1 or aA2 (i.e. 0  A1  1) if B1 = ½ A’s best response A1 1 This is A’s best responsecurve when players areallowed to mix over theiractions. 0 B1 0 ½ 1

  40. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. Given A1, whatvalue of B1 is bestfor B? aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2

  41. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. Given A1, whatvalue of B1 is bestfor B? aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 EVB(aB1) = 4A1 + 3(1 - A1) = 3 + A1.

  42. Best Responses & Nash Equilibria Player B A1 is the prob. Achooses action aA1. B1 is the prob. Bchooses action aB1. Given A1, whatvalue of B1 is bestfor B? aB1 aB2 6,4 3,5 aA1 Player A 4,3 5,7 aA2 EVB(aB1) = 4A1 + 3(1 - A1) = 3 + A1.EVB(aB2) = 5A1 + 7(1 - A1) = 7 - 2A1.

  43. Best Responses & Nash Equilibria EVB(aB1) = 3 + A1.EVB(aB2) = 7 - 2A1.3 + A1 7 - 2A1 as A1 ?? >=< A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given A1, what value of B1 is best for B? >=<

  44. Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given A1, what value of B1 is best for B? EVB(aB1) = 3 + A1.EVB(aB2) = 7 - 2A1.3 + A1 < 7 - 2A1 for all 0  A1 1.

  45. Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? EVB(aB1) = 3 + A1.EVB(aB2) = 7 - 2A1.3 + A1 < 7 - 2A1 for all 0  A1 1.B’s best response is: aB2 always (i.e. B1 = 0 always).

  46. Best Responses & Nash Equilibria B’s best response is aB2 always (i.e. B1 = 0 always). A1 1 This is B’s best responsecurve when players areallowed to mix over theiractions. 0 B1 0 ½ 1 B’s best response

  47. Best Responses & Nash Equilibria B A A’s best response A1 A1 1 1 0 0 B1 B1 0 ½ 1 0 ½ 1 B’s best response

  48. Best Responses & Nash Equilibria Is there a Nash equilibrium? B A A’s best response A1 A1 1 1 0 0 B1 B1 0 ½ 1 0 ½ 1 B’s best response

  49. Best Responses & Nash Equilibria A’s best response A1 A1 1 1 0 0 B1 B1 0 ½ 1 0 ½ 1 B’s best response Is there a Nash equilibrium? B A

  50. Best Responses & Nash Equilibria A1 1 0 B1 0 ½ 1 B’s best response Is there a Nash equilibrium? A’s best response

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