Auction Theory
Class 4 – Some applications of revenue equivalence
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TodayThe machinery in Myerson’s work is useful in many
settings.
Today, we will see two applications: interesting results that are derived almost “for free” from these tools.
1. Equilibrium in 1st-price auctions.2. “Auctions vs. negotiations” – should we really run
the optimal auction?
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Equilibrium in 1st-price auctionsOld debt:
I promised to prove what is the equilibrium behavior in 1st-price auctions.
This will be an easy conclusion from the results we know.
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Equilibrium in 1st-price auctionsIn a second price auction:
– The usual notation: bidder i with value vi wins with probability Qi(vi).
– i’s expected payment when he wins: the expected value of the highest bid of the other n-1 bidders given that their value is < vi.
– E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ]
ui(vi) = Qi(vi) ( vi - E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ] )
In 1st-price auction: a winning bidder pays her bid. ui(vi) = Qi(vi) ( vi - bi(vi) )
Revenue equivalence: expected utility in 1st and 2nd price must be equal in equilibrium.
Equilibrium bid in 1st-price auctions: bi(vi) = E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ]
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Payment when winning with value vi
Payment when winning with value vi
Equilibrium in 1st-price auctionsExample: the uniform distribution on [0,1].
This is the expected highest order statistic of n-1 draws from the uniform distribution on the interval [0,v]
Conclusion:
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bi(vi) = E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ]
0 v 1
ivn
1ivn
2ivn
n 1ivn
n 2….
iii vn
nvb
1)(
TodayThe machinery in Myerson’s work is useful in many
settings.
Today, we will see two applications: interesting results that are derived almost “for free” from these tools.
1. Equilibrium in 1st-price auctions.2. “Auctions vs. negotiations” – should we really run
the optimal auction?
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Optimal auctions• We saw that Vickrey auctions are efficient, but do
not maximize revenue.
• Myerson auctions maximize revenue, but inefficient.– With i.i.d. distributions, Myerson = Vickrey + reserve price.
• Should one really run Myerson auctions?– We will see: not really…
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Marketing• Two approaches for improving your revenue:
– Optimize your mechanism. Make sure you make all the revenue theoretically possible.
– Increase the market size: invest in marketing.
• Maybe, instead of optimizing a reserve price, we can just expand the market?
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Bulow-Klemperer’s result
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• The efficient auction with one additional bidder earns more revenue than the optimal auction!
• Finding an additional bidder is better than optimizing the reserve price.
Revenue in the optimal auction with n players.
Revenue from the Vickrey auction with n+1 players.
Theorem [Bulow & Klemperer 1996]:
≤
Bulow-Klemperer’s result: discussion
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• Holds for every n:– Even n=1.
• The optimal (Myerson) auction requires knowledge on the distribution, Vickrey does not.
• Auctions with no reserve price may be more popular.
Additional revenue from optimal auctions is minor.
Revenue in the optimal auction with n players.
Revenue from the Vickrey auction with n+1 players.
Theorem [Bulow & Klemperer 1996]:
≤
Bulow & Klemperer: setting
• We consider the basic auction setting:– Values are drawn i.i.d from some distribution F.– Risk Neutrality– F is Myerson-regular (non-decreasing virtual valuation)
• We will define the “must-sell” optimal auction: the auction with the highest expected revenue among all auctions where the item is always sold.
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Bulow & Klemperer: proof
• The proof will follow easily from two simple claims.– Taken from “A short proof of the Bulow-Klemperer auctions vs.
negotiations result” by Rene Kirkegaard (2006)
– Still, a bit tricky.
• (the original proof was not so easy…)
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Bulow & Klemperer: proofClaim 1:
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The revenue in the “must-sell” optimal auction with n+1 bidders
The optimal revenue with n bidders.≥
Bulow & Klemperer: proof
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Proof of claim 1:• The following “must sell” auction with n+1 bidders
achieves the same revenue as the optimal revenue:“Run the optimal auction with n players; if item is unsold, give it to bidder n+1 bidder for free.”
But this “must-sell” auction achieves the same revenue as the optimal auction with n bidders….
• The “must-sell” optimal auction can only do better. The claim follows.
Claim 1: the revenue in the “must-sell” optimal auction with n+1 bidders
The optimal revenue with n bidders.≥
Bulow & Klemperer: proof
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What is the “must-sell” optimal auction?
Claim 2: the “must-sell” optimal auction is the Vickrey auction.
Proof: • Recall: E[revenue] = E[virtual surplus]• When you must sell the item, you would still aim to maximize
expected virtual surplus.• The bidder with the highest value has the highest virtual value.
– When values are distributed i.i.d. from F and F is Myerson regular.
Vickrey auction maximizes the expected virtual surplus when item must be sold.
Bulow & Klemperer: proof
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Conclusion:
the revenue in the “must-sell” optimal auction with n+1 bidders
The revenue in the “must-sell” optimal auction with n+1 bidders
The optimal revenue with n bidders.
The revenue in the Vickrey auction with n bidders.
≥
The optimal revenue with n bidders.
The revenue in the Vickrey auction with n+1 bidders.
≥=
Claim 1Claim 2
SummaryThe tools developed in the literature on optimal
auctions are useful in many environements.
We saw two applications:1. Characterization of the equilibrium behavior in 1st
price auctions.2. Bulow-Klemperer result: running Vickrey with n+1
bidders achieves more revenue than the optimal auction with n bidders.
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