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auctions.doc

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  • 1. Econ 452 auctions Auctions Real world? Farm/estate auctions Treasury bills Timber Ebay Neo-pets Broadcast spectrum Distinct from other markets? - price-makers - buyers and sellers come together Advantages? - open, objective assignment method - criteria specified in advance - reason for assignment public - determine market prices - promote efficient allocation and investment - assign resource quickly - can incorporate public policy goals Page 1 of 10
  • 2. Econ 452 auctions Types a) By bidders' valuations: Private value Common value b) by bidding/selling rules English / ascending bid auction - oral - high bidder - own bid First price sealed bid - simultaneous - highest bidder - own bid Second price sealed bid - simultaneous - highest bidder - second highest bid Dutch / descending bid auction - oral - first person to stop - last price announced Page 2 of 10
  • 3. Econ 452 auctions Double auction - buyers and sellers submit bids - "auctioneer" finds eq'm price - transfers goods - (gold - London…) Questions? - optimal bid - relation to valuation? - winning bid / seller's revenue? - allocation? 1. Private value auctions - Why care about others' valuations? - analyze sealed bid auctions (simultaneous; derive NE) - do you bid your valuation? Notice: 1 object - only highest, second highest bidders count Page 3 of 10
  • 4. Econ 452 auctions Assume: two bidders, valuations vi , i =1,2 Information? Assume: bid f'n bi = f (vi ) , i =1,2 ; increasing Payoffs? vi − bi if bi > b j ui (b1,b2;v1,v2) = 0.5(vi − bi ) if bi = b j 0 if bi < b j Auction 1: first price sealed bid Claim: Highest expected payoff obtained from bidding amount equal to estimate of second highest valuation, assuming yours is the highest. Why? 1. bid > valuation? 2. Bid = valuation? 3. Bid < valuation? How much? Page 4 of 10
  • 5. Econ 452 auctions How to estimate 2nd highest? Recall: valuations iid, assumed to know distribution With n players, have n draws = n valuations Can arrange these valuations in ascending order, so v1 < v2 <...< vn Note that v(k ) , k=1,…,n is a function of the sample variables, and hence is a statistic, called the k'th order statistic. Simplify for example: assume vi : U [0,1] - then the order statistics are, on average, evenly spaced on the interval - if n = 2, E v(1) =1/3 , and v(2) = 2/3 - if n = 3… Page 5 of 10
  • 6. Econ 452 auctions - in general: uniform dist'n on [0,1] v(n) = n /(n +1) , v(n−1) = (n −1)/(n +1) Uniform distribution on [a,b]? _________________________________ So: optimal bid? Assume your valuation is highest. - remaining (n-1) bids drawn from U [0,vi ] - expected highest is v(n−1) = (n −1)vi / n - bid this value Page 6 of 10
  • 7. Econ 452 auctions Auction 2: second price sealed bid Claim: vi = bi is a weakly dominant strategy Why? - intuition: bidding valuation won't change your price if win (unlike 1st price), but will change probability of winning Notice: - all relevant info about bidders revealed to seller. - equivalent to English auction (expected revenue, identification of highest bidder, valuations) Page 7 of 10
  • 8. Econ 452 auctions 2. Common value auctions - objective value of good(s) being sold - each bidder has private info on this - problem? "winner's curse" - if win, had higher valuation than others - overestimated value - objective: optimal amount to shave bid How? Order statistics, again. - as before, assume everyone knows dist'n + # bidders, as well as own valuation - Use own info to estimate upper bound of possible values - bid = mean of estimate Page 8 of 10
  • 9. Econ 452 auctions - importance of own valuation? - Assume it is largest drawn - ie, 1st order statistic - (why?) Ex: if distribution is uniform on [0,I] If vi is 1st order statistic, and n bidders - then vi = nI /(n +1) - then estimate I as I = (n +1)vi / n This gives optimal bid to avoid winner's curse. Winner's curse in real life? Page 9 of 10
  • 10. Econ 452 auctions Page 10 of 10