# Section c solve two of the following three problems 6

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Section C. Solvetwoof the followingthreeproblems6. Find all pure and mixed Nash equilibria in the following bimatrix game:DEFA3;41;31;0B2;73;60;3C0;22;15;67. Two players play a version of Rock-Paper-Scissor game. Paper beats Rock, Rock beats Scissors, Scissorsbeats Paper. The two players simultaneously make their choice. The first player can choose any object.The second player can choose Rock or Paper. The winner receives 1 rouble from the loser. In case of adraw the wealth of a player does not change.(a) Construct the payoff matrix of the game.(b) Find all pure and mixed Nash equilibria8. Two players are trying to bribe the judge. The possible amount of bribe is any real number between 0and 1 million roubles. The player who gives the biggest bribe is announced as the winner of the affairby the judge. The winner receives 1 million roubles. The loser gets nothing. Obviously bribes are notreturned by the judge. In the case of equal bribes each player gets nothing.(a) Are there any pure Nash equilibria in this game?(b) Find at least one mixed Nash equilibrium.Short tips:(a) No(b) If the second player choses his move according continuous distribution functionFthen the ex-pected payoff of the first player for the bribebis equal to1·P(b2b)b= 1·F(b)bIf a rational player uses mixed strategies he is indifferend between the corresponding pure strate-gies. That means thatF(b)b=constfor pure strategies that are mixed. For pure strategiesthat are mixed the density functionf(b) =F(b) = 1. Do I know such a random variable? Yes, Iknow! A uniform on[0; 1].4.7MFE, retake exam, 19.09.2012Time allowed 120 minutes.Students should answer all of the following eight questions. Calculators are not permitted in the exam. Markswill be deducted for insufficient explanations within your answers.Section A: 10 points each question.1. It is known that the functionsf1(x)andf2(x)are concave up. Is it possible that the functionh(x) =max{f1(x), f2(x)}is concave down?2. The implicit functionz(x, y)is given by the equationxz=f(yz)wherefis some unknowndifferentiable function. Find∂z∂x+∂z∂y.

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Term
Spring
Professor
Christopher Dougherty
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