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The Random Dresser

The Random Dresser. Wilbur has 3 left shoes, all of different colors 5 right shoes, all of different colors 4 right gloves, all of different colors 2 left gloves of different colors. How many different ways can Wilbur dress himself?. Iraq War Game. 1. 3. 2.

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The Random Dresser

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  1. The Random Dresser • Wilbur has • 3 left shoes, all of different colors • 5 right shoes, all of different colors • 4 right gloves, all of different colors • 2 left gloves of different colors. • How many different ways can Wilbur dress himself?

  2. Iraq War Game 1 3 2

  3. What are the strategies for Saddam? • For Saddam we have to say what he would do at each node where it is his turn. There are 3 nodes. We list actions at each node to specify a strategy: • No WMD, Deny, Deny • No WMD, Deny, Allow • No WMD, Deny, Allow and Hide • No WMD, Allow, Deny • No WMD, Allow, Allow • No WMD, Allow, Allow and Hide • WMD, Deny, Deny • WMD, Deny, Allow • WMD, Deny, Allow and Hide • WMD, Allow, Deny • WMD, Allow, Allow • WMD, Allow, Allow and Hide

  4. What does that mean? • How do we interpret a strategy like, No WMD, Allow, Deny? This means 1) he chooses to have no wmd, 2) if the UN asks to inspect and he has no wmd, he allows them to inspect 3) if the UN asks to inspect and he has wmd, he denies inspection. • So, you ask, if he chooses no wmd, why do we need to specify what he will do at decision node 3? • Imagine that Saddam commanded No WMD, but discovers that his orders weren’t followed and he realizes that he has WMD and the UN wants to inspect. What will he do? • Game theory needs to deal with such possibilities.

  5. Dominant strategies

  6. Clicker Question Player 2 Strategy A Strategy B Strategy A Player 1 Strategy B A ) Strategy A strictly dominates Strategy B for both Players. B) Strategy B strictly dominates A for Player 1. Strategy A weakly dominates B for Player 2. C) Strategy B strictly dominates A for Player 1. Strategy A strictly dominates B for Player 2. D) Strategy B strictly dominates Strategy A for both players. E) No strategy in this game is strictly dominated

  7. Strict and Weak Dominance • Strategy A strictly dominates strategy B for a player if that player gets a higher payoff from doing A than from doing B no matter what the other player(s) do. • Strategy A weakly dominates strategy B for a player gives at least as high a payoff no matter what the other player(s) do and for some actions of the others gives a higher payoff.

  8. Clicker Question Player 2 Strategy A Strategy B Strategy A Player 1 Strategy B A ) Strategy A strictly dominates Strategy B for both Players. B) Strategy B strictly dominates A for Player 1. Strategy A weakly dominates B for Player 2. C) Strategy B strictly dominates A for Player 1. Strategy A strictly dominates B for Player 2. D) Strategy B strictly dominates Strategy A for both players. E) No strategy in this game is strictly dominated

  9. Game Theory Doctrine (A tautology) • A rational player who understands the payoffs of a game and who tries to maximize his own payoff will • never use a strictly dominated strategy. • will always use a strictly dominant strategy if one exists.

  10. Dominant strategies? Player 2 Strategy B Strategy A Strategy A Player 1 Strategy B Does either strategy strictly dominate the other for Player 1? Does either strategy strictly dominate the other for Player 2? What is the predicted outcome? What are games like this called?

  11. How about this one? Player 2 Strategy B Strategy A Strategy A Player 1 Strategy B Does either strategy strictly dominate the other for Player 1? Does either strategy weakly dominate the other for Player 1? How about player 2?

  12. Clicker Question Player 2 Strategy B Strategy A Strategy A Player 1 Strategy B If I were playing this game just once with a stranger whom I would never meet again, I would: Play Strategy A Play Strategy B

  13. Rousseau’s Stag Hunt Player 2 Hare Stag Stag Player 1 Hare Are any strategies weakly dominated? Are any strategies strictly dominated? How would you play?

  14. Clicker Question Player 2 Hare Stag Stag Player 1 Hare If you were playing Rosseau’s stag hunt with a stranger, whom you will never meet again, which strategy would you play? A) Stag B) Hare

  15. Gaming Pigs(Iterated dominance) Are there dominated strategies for Big Pig? How about Little Pig? How would you “solve” this game?

  16. What went on in the pigpen

  17. The Entry Game Challenger Challenge Stay out 0 Challenger’s payoff 1 Incumbent’s payoff Incumbent Give in Fight 1 0 -1 -1 Challenger’s payoff Incumbent’s payoff

  18. Strategic Form of Entry Game Incumbent Give in Fight Stay out Challenger Enter

  19. Dominance in Entry Game? • No dominant strategy for Challenger. Which is better depends on what incumbent will do. • Give-in is weakly dominant for Incumbent. • If Challenger believes that Incumbent is rational, Challenger believes that Incumbent will give in. • In this case, predicted outcome is Challenger enters and incumbent gives in.

  20. What if incumbent could precommit? • Could the incumbent make a credible threat to fight if challenger enters. If he could, he could get challenger to stay out. • On blackboard we will draw a game that allows incumbent the choice of committing to to be badly punished if he gives in. • Lets do this so that the “solution” is that the incumbent chooses to make this commitment and the challenger stays out. • Tools for understanding this “solution” will arrive later in this course.

  21. Kidnapping with imperfect information

  22. Strategic Form

  23. Dominated strategies? • Guy doesn’t have any dominated strategies • But for Vivica, Don’t Pay dominates Pay. • What does iterated elimination of dominated strategies tell us? • If Guy knows that Vivica is rational, he knows she won’t pay ransom. • If Guy knows that Vivica won’t pay ransom, he is better off not kidnapping.

  24. Kidnapping with Perfect Information

  25. Kidnapping with complete information Vivica Guy Are any strategies strictly dominated for either player?

  26. Dominated strategies? • Neither strategy dominates for Vivica • For Guy, Kidnap—Release if Ransom, Kill if No ransom weakly dominates all other strategies that start with Kidnap. • So if Vivica believe that Guy is rational, then she believes that if Guy Kidnaps, he will kill if no ransom and release if ransom. • So Vivica would pay ransom • So Guy would Kidnap and release after receiving ransom.

  27. Does Player 1 have a dominated strategy? Hint: Compare b and d.

  28. Iterated Elimination of Dominated Strategies-Stage 1 Does Player 2 have a dominated strategy? Hint: Compare y and z.

  29. The game After first round of Eliminaton If each knows the other won’t play a dominated strategy, we have a smaller game.

  30. Reduced Game after one iteration. This is the game if each knows that the other is rational and each knows that the other knows that the other is rational. Are there any dominated strategies?

  31. Reduced Game after 2 rounds of iterated elimination of strictly dominated strategies. (Note that x couldn’t have been eliminated in the first round.)

  32. Reduced Game after 3 rounds of iterated elimination. a is eliminated. This couldn’t have been done in earlier rounds. Are there any strictly dominated strategies in this game? We have eliminated 12 of 16 strategies, but to get any further, We’re going to need more tools.

  33. Iterated elimination and Common Knowledge Strategy a dominates c for Player 1. Strategy y dominates w and x for Player 2. Rational players won’t use these strategies. If each knows other is rational, then Player 2 know s that 1 won’t play c and 1 knows that 2 won’t play w or x. If both are rational and believe other is rational, Player 1 knows that 2 won’t play x or y, so Player 1 can eliminate b. Player 2 knows that Player 1 won’t play c, so Player 2 can eliminate y. If Player 1 knows that Player 2 knows that Player 1 is rational, then Player 1 knows Player 2 will Play z. What will Player 1 do?

  34. See you on Thursday …And Steer Clear of Dominated Strategies

  35. Clicker Question A) No strategies are strictly dominated for Player 1. Strategy w is dominated for Player 2 B) No strategies are strictly dominated for either player. C) Strategy c is strictly dominated for Player 1. Strategies w and x are strictly dominated for Player 2. D) Strategy d is strictly dominated for Player 1. Strategies x and y are Strictly dominated for Player 2. E) No strategies are strictly dominated for Player 1. Strategies x and y are Strictly dominated for Player 2.

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