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Working Some Problems

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Telephone Game How about xexed strategies? Let Winnie call with probability p and wait with probability 1-p. For what values of w is Colleen indifferent? Expected payoff for Colleen from calling is???? Expected payoff for Colleen from waiting is??? C Are there any pure strategy equilibria?

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Listing the N.E Let p be probability that Winnie calls and 1-p the probability that she waits. Let q be the probability that Colleen calls and 1-q the probability that she waits. Nash equilibria are strategy profiles. The mixed strategy equilibria include: a)p=0 and q=1 b) p=1 and q=0 c) p=1/4 and q=1/4

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A Duel FastballCurveball Fastball.35,.65.3,.7 Curveball.2,.8.5,.5 Pitcher throws Batter prepares for Does this game have any pure strategy equilibria? A)Yes B)No

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A Duel FastballCurveball Fastball.35,.65.3,.7 Curveball.2,.8.5,.5 Pitcher throws Batter prepares for In Nash equilibrium if Batter has positive probability of using each strategy, what is the probability that Pitcher throws a fastball? A)1/3 B)2/3 C)½ D)4/7 E)3/5

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Let’s go figure In N.E, Batter will play each strategy with positive probability only if the two strategies have the same expected payoff for him. – Suppose Pitcher throws a fastball with probability p and a curveball with probability 1-p. – Batter’s expected payoff from “Prepare for Fastball” is.35p+.30(1-p) – Batter’s expected payoff from “Prepare for curveball” is.2p+.5(1-p). – These payoffs are equal if.35p+.30(1-p)=.2p+.5(1-p). Solve this equation for p.

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A Duel FastballCurveball Fastball.35,.65.3,.7 Curveball.2,.8.5,.5 Pitcher throws Batter prepares for In a mixed strategy Nash equilibrium what is the probability that Batter prepares for a fastball? A)4/7 B)3/7 C)½ D)6/7 E)4/5

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Does this game have a Nash equilibrium in which Kicker mixes left and right but does not kick to center?

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If there is a Nash equilibrium where kicker never kicks middle but mixes between left and right, Goalie will never play middle but will mix left and right (Why?) If Goalie never plays middle but mixes left and right, Kicker will kick middle. (Why?) So there can’t be a Nash equilibrium where Kicker never kicks Middle. (See why?)

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Problem 4: For what values of x is there a mixed strategy Nash equilibrium in which the victim might resist or not resist and the Mugger assigns zero probability to showing a gun?

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Problem 7.7, Find mixed strategy Nash equilibria

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c dominates a and y dominates z A mixed strategy N.E. strategy does not give positive probability To any strictly dominated strategy Look at reduced game without these strategies

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Problem 7.7 Find mixed strategy Nash equilibia For player 1, Bottom strictly dominates Top. Throw out Top Then for Player 2, Middle weakly dominates Right. Therefore if Player 1 plays bottom with positive probability, player 2 gives zero Probability to Right. There is no N.E. in which Player 1 plays Bottom with zero probability, (Why?) (If he did, what would Player 2 play? Then what would 1 play?)

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A Nash equilibrium is any strategy pair in which the defense defends against the outside run with probability.5 and the offense runs up the middle with probability.75. No matter what the defense does, The offense gets the same payoff from wide left or wide right, So any probabilities pwl and pwr such that pwl+pwr=.25 will be N.E. probabilities for the offense. Problem 8, Chapter 7

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Problem 10. Each of three players is deciding between the pure strategies go and stop. The payoff to go is 120, where m is the number of players that choose go, and the payoff to stop is 55 m (which is received regardless of what the other players do). Find all Nash equilibria in mixed strategies. Let’s find the “easy ones”. Are there any symmetric pure strategy equilibria? How about asymmetric pure strategy equilibria? How about symmetric mixed strategy equilibrium? Solve 40p^2+60*2p(1-p)+120(1-p) 2 =55 40p 2 -120p+65=0

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What about equilibria where one guy is in for sure and other two enter with identical mixed strategies? For mixed strategy guys who both Enter with probability p, expected payoff from entering is (120/3)p+(120/2)(1-p). They are indifferent about entering or not if 40p+60(1-p)=55. This happens when p=1/4. This will be an equilibrium if when the other two guys enter with Probability ¼, the remaining guy is better off entering than not. Payoff to guy who enters for sure is: 40*(1/16)+60*(3/8)+120*(9/16)=92.5>55.

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RockPaperScissors Rock0,0-1,12,-2 Paper1,-10,0-1,1 Scissors-2,21,-10,0 Advanced Rock-Paper-Scissors Are there pure strategy Nash equilibria? Is there a symmetric mixed strategy Nash equilibrium? What is it?

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Remember: Midterm on Thursday Chapters 2-5 and 7 No need to bring bluebooks. Calculators and phones not allowed

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17. (A very brief) Introduction to game theory Varian, Chapters 28, 29.

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