Two bidders compete in an all-pay auction to obtain a good. Bidders' values are identically and...
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Two bidders compete in an all-pay auction to obtain a good. Bidders' values are identically and independently distributed according to the uniform distri- bution on [0,1]: v₁, v2 d. U[0,1]. The all-pay auction has the following rules. Each bidder submits a bid, and, as in other standard auctions, the highest bidder wins the object (here the object is winning the election). The unusual aspect of an all-pay auction is that all bidders pay what they bid irrespective of whether they win or not. We want to show that this game has the following Bayesian Nash equilibrium, where each bidder's bidding function is $₁(v) = ß2 (v) = B(v) = 1/v². (a) Consider bidder 1 with value v₁ contemplating a bid of b₁. Suppose that bidder 2 plays according to the proposed strategy B (v₂) = 103. Write the expected payoff of this bidder 1. (b) Show that b₁ =v) maximizes the expected payoff from part (a). (c) Find the expected revenue of the seller in this auction. Hint: the expectation needs to be taken with respect to the uniform distribution of values. Two bidders compete in an all-pay auction to obtain a good. Bidders' values are identically and independently distributed according to the uniform distri- bution on [0,1]: v₁, v2 d. U[0,1]. The all-pay auction has the following rules. Each bidder submits a bid, and, as in other standard auctions, the highest bidder wins the object (here the object is winning the election). The unusual aspect of an all-pay auction is that all bidders pay what they bid irrespective of whether they win or not. We want to show that this game has the following Bayesian Nash equilibrium, where each bidder's bidding function is $₁(v) = ß2 (v) = B(v) = 1/v². (a) Consider bidder 1 with value v₁ contemplating a bid of b₁. Suppose that bidder 2 plays according to the proposed strategy B (v₂) = 103. Write the expected payoff of this bidder 1. (b) Show that b₁ =v) maximizes the expected payoff from part (a). (c) Find the expected revenue of the seller in this auction. Hint: the expectation needs to be taken with respect to the uniform distribution of values.
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Related Book For
Managerial Economics A Problem Solving Approach
ISBN: 978-1133951483
3rd edition
Authors: Luke M. Froeb, Brian T. McCann, Mikhael Shor, Michael R. War
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