# 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.

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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 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

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 werent 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.

Dominant strategies

Clicker Question 1, 3 5, 3 2, 4 7, 2 Player 2 Strategy A Strategy B Player 1 Strategy A 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

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.

Clicker Question 2,2 0, 3 3, 0 1,1 Player 2 Strategy A Strategy B Player 1 Strategy A 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

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

Dominant strategies? 1 0, 10 0, 11 1 1, 0 1, 1 Strategy A Strategy B Player 1 Player 2 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?

How about this one? 1 0, 10 0, 10 1 0, 0 1, 1 Strategy A Strategy B Player 1 Player 2 Does either strategy strictly dominate the other for Player 1? Does either strategy weakly dominate the other for Player 1? How about player 2?

Clicker Question 1 0, 10 0, 10 1 0, 0 1, 1 Strategy A Strategy B Player 1 Player 2 If I were playing this game just once with a stranger whom I would never meet again, I would: A)Play Strategy A B)Play Strategy B

Rousseaus Stag Hunt 2, 2 0, 1 1, 0 1, 1 Stag Hare Player 1 Player 2 Are any strategies weakly dominated? Are any strategies strictly dominated? How would you play?

Clicker Question 2, 2 0, 1 1, 0 1, 1 Stag Hare Player 1 Player 2 If you were playing Rosseaus stag hunt with a stranger, whom you will never meet again, which strategy would you play? A) Stag B) Hare

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

What went on in the pigpen

The Entry Game Challenger Stay out 0 Challengers payoff 1 Incumbents payoff Challenge Incumbent Give in Fight 1010 Challengers payoff Incumbents payoff

Strategic Form of Entry Game 0,1 1,0-1,-1 Give inFight Stay out Enter Challenger Incumbent

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.

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.

Kidnapping with imperfect information

Strategic Form

Dominated strategies? Guy doesnt have any dominated strategies But for Vivica, Dont Pay dominates Pay. What does iterated elimination of dominated strategies tell us? If Guy knows that Vivica is rational, he knows she wont pay ransom. If Guy knows that Vivica wont pay ransom, he is better off not kidnapping.

Kidnapping with Perfect Information

Kidnapping with complete information Pay RansomDont Pay Ransom Kidnap-- Kill if R, Kill if NR4,12,2 KidnapRelease if R, Kill if NR5,32,2 Kidnap Kill if R, Release if NR4,11,4 KidnapRelease if R, Release if NR5,31,4 Dont Kidnap– Kill, Kill3,5 Dont KidnapRelease, Kill3,5 Dont kidnap--Kill, Release3,5 Dont kidnapRelease, release3,5 Vivica Guy Are any strategies strictly dominated for either player?

Dominated strategies? Neither strategy dominates for Vivica For Guy, KidnapRelease 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.

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

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

If each knows the other wont play a dominated strategy, we have a smaller game. The game After first round of Eliminaton

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?

Reduced Game after 2 rounds of iterated elimination of strictly dominated strategies. (Note that x couldnt have been eliminated in the first round.)

Reduced Game after 3 rounds of iterated elimination. a is eliminated. This couldnt 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, Were going to need more tools.

Iterated elimination and Common Knowledge Strategy a dominates c for Player 1. Strategy y dominates w and x for Player 2. Rational players wont use these strategies. If each knows other is rational, then Player 2 know s that 1 wont play c and 1 knows that 2 wont play w or x. If both are rational and believe other is rational, Player 1 knows that 2 wont play x or y, so Player 1 can eliminate b. Player 2 knows that Player 1 wont 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?

…And Steer Clear of Dominated Strategies See you on Thursday

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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