slides_Allpay

# slides_Allpay - less than some amount x between 0 and 100....

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All-Pay Auctions • In an all-pay auction, every bidder pays what they bid regardless of whether or not they win. • Examples: –E lec t ions – Almost any kind of contest or sports event – Research and Development –Wa rs

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All-Pay Auctions • A simple all-pay auction: – The object for sale is worth 100 to all of n identical bidders and all of them know this valuation exactly. – Clearly no one will bid more than 100. – Each bidder’s bid is an amount x in the interval [0,100]. – The winner is the highest bidder and gets a payoff of 100 – x – The payoff of all other bidders is – x • There is no pure strategy equilibrium of this game. • A mixed strategy of a player is a probability distribution over the possible bids between 0 and 100. • We represent a mixed strategy by a probability distribution function P, where P ( x ) which gives the probability that a player bids

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Unformatted text preview: less than some amount x between 0 and 100. All-Pay Auctions We look for a symmetric equilibrium. Suppose bidders 2,,n used the mixed strategy P and consider bidder 1. Suppose bidder 1 bids x. Her expected payoff is given by P(x) n-1 (100-x) + (1- P(x) n-1 )(-x) = 100P(x) n-1 - x In order for bidder 1 to be willing to randomize we require 100P(x) n-1 - x = c. All-Pay Auctions Moreover, since P is a probability distribution function we require P(0)=0 Hence, c=0 Hence, 1 1 100 ) ( = n x x P All-Pay Auctions Equilibrium mixed strategies in this example: 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x P(x) n = 2 n = 5 n = 10 Cummulative Distribution Function of Bids for All Pay Auction Spring 2011 0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00% 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Bid 2 bidders 5 bidders 10 bidders...
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## This note was uploaded on 12/26/2011 for the course ECON 177 taught by Professor Garratt during the Fall '09 term at UCSB.

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slides_Allpay - less than some amount x between 0 and 100....

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