A recently discovered painting by Picasso is on auction at Sotheby's. There are twomain bidders Amy and Ben {1,2}. Bidding starts at £10M but the value of the paintingis certainly not more than £20M. Each bidder's valuation v; is independently and uni-formly distributed on the interval [10M, 20M], and this is common knowledge amongthe players: A bidder knows their own valuation but not of their opponent. Consider anauction where an object is allocated to the highest bidder but the price paid by the bidderis determined randomly. With probability 3/4, the bidder pays their own bid, and withprobability 1/4 the bidder pays the losing bid. The person bidding lowest pays nothing. Ifthe bids are equal, each bidder gets the object with probability one-half, and in this case,pays their bid.Suppose that bidder 1 assumes that bidder 2 will bid a constant fraction, 7, of bidder 2'svaluation (and similarly, bidder 2 assumes bidder 1 will bid the same constant propor-tional value of their valuation).(i) Write down the expected payoff to bidder 1, as a function of their own valuation v₁and bid b₁. (Ignore ties, where both bidders bid the same amount.)(ii) Solve for bidder 1's optimal bid, as a function of their valuation and hence solvefory and thereby, find the Bayes-Nash equilibrium of this game, where each bidderbids a constant fraction of their own valuation.(iii) Assume Amy's valuation is set at 17M, how much should she bid?

In the context of bids, the term "payoff" typically refers to the benefits or rewards that a bidder…

(i) Write down the expected payoff to bidder 1, as a function of their own valuation v₁ and bid b₁. (Ignore ties, where both bidders bid the same amount.) (ii) Solve for bidder 1's optimal bid, as a function of their valuation and hence solve for y and thereby, find the Bayes-Nash equilibrium of this game, where each bidder bids a constant fraction y of their own valuation. (iii) Assume Amy's valuation is set at 17M, how much should she bid?

(i) Write down the expected payoff to bidder 1, as a function of their own valuation v₁ and bid b₁. (Ignore ties, where both bidders bid the same amount.) (ii) Solve for bidder 1's optimal bid, as a function of their valuation and hence solve for y and thereby, find the Bayes-Nash equilibrium of this game, where each bidder bids a constant fraction y of their own valuation. (iii) Assume Amy's valuation is set at 17M, how much should she bid?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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