Distinct from other markets?
- buyers and sellers come together
- open, objective assignment method
- criteria specified in advance
- reason for assignment public
- determine market prices
- promote efficient allocation and
- assign resource quickly
- can incorporate public policy goals
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a) By bidders' valuations:
b) by bidding/selling rules
English / ascending bid auction
- high bidder
- own bid
First price sealed bid
- highest bidder
- own bid
Second price sealed bid
- highest bidder
- second highest bid
Dutch / descending bid auction
- first person to stop
- last price announced
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- buyers and sellers submit bids
- "auctioneer" finds eq'm price
- transfers goods
- (gold - London…)
- optimal bid - relation to valuation?
- winning bid / seller's revenue?
1. Private value auctions
- Why care about others' valuations?
- analyze sealed bid auctions
(simultaneous; derive NE)
- do you bid your valuation?
1 object - only highest, second highest bidders
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Assume: two bidders, valuations vi , i =1,2
Assume: bid f'n bi = f (vi ) , i =1,2 ; increasing
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
Highest expected payoff obtained from bidding
amount equal to estimate of second highest
valuation, assuming yours is the highest.
1. bid > valuation?
2. Bid = valuation?
3. Bid < valuation? How much?
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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…
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- 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
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Auction 2: second price sealed bid
Claim: vi = bi is a weakly dominant strategy
- intuition: bidding valuation won't change
your price if win (unlike 1st price), but will
change probability of winning
- all relevant info about bidders revealed to
- equivalent to English auction
(expected revenue, identification of
highest bidder, valuations)
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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
- 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
- bid = mean of estimate
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- importance of own valuation?
- Assume it is largest drawn - ie, 1st order
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?
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