advmicroex2_09 - U high quality low quality buy 6,10 u 2,4...

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Advanced Microeconomics - Game Theory Problem Set 2 1 . Consider a first-price sealed-bid auction with two bidders. Each bidder i ’s valuation of the object is v i , which is known to both bidders. Each bidder submits a bid in a sealed envelope, with the bidder who has submitted the highest bid winning the object and paying the auctioneer the amount of his bid. If the bidders submit the same bid, each gets the object with probability 1/2. Bids must be in euro multiples (assume that valuations are as well). a . Are any strategies strictly dominated? If yes, which ones? b . Are any strategies weakly dominated? If yes, which ones? c . Is there a Nash equilibrium? If yes, what is it and is it unique? 2 . Consider a game between a Buyer and a Seller, in which the two players choose their strategies simultaneously. The Buyer must choose between Buy and Don’t Buy, and the Seller may choose High or Low quality for his product. There game comes in two variants, Game A and Game B. Game A: Buyer u2193 Seller u2192 high quality
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Unformatted text preview: U high quality low quality buy 6,10 u 2,4 dont buy 0,2 0,0 (buyers payoff, sellers payoff) Game B: Buyer u Seller U high quality low quality buy 6,4 u 2,10 dont buy 0,0 0,2 (buyers payoff, sellers payoff) a . Are there any dominant strategies for the seller and/or the buyer in game A? Are there any in game B? If yes, identify them. What is the Nash equilibrium of game A, what is the Nash equilibrium of game B? b . Now suppose that the Buyer does not know whether she is playing Game A or Game B. Her beliefs are that she plays Game A with probability and Game B with probability . Draw the game tree. c . Derive the Bayesian Nash equilibrium of the incomplete-information game. 3 . Derive the (mixed-strategy) Nash equilibrium of the following extensive-form game between player 1 and player 2: Player 1 u Player 2 U A B C A 0, 1 1, 0 1, 0 B 2, 0 0, 2 2, 0 C 3, 0 3, 0 0, 3 4 . Jehle/Reny, exercises 7.16 and 7.17....
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