# Class 4 – Some applications of revenue equivalence

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Class 4 – Some applications of revenue equivalence
Auction Theory Class 4 – Some applications of revenue equivalence

Today The 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. Equilibrium in 1st-price auctions. “Auctions vs. negotiations” – should we really run the optimal auction?

Equilibrium in 1st-price auctions
Old 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.

Equilibrium in 1st-price auctions
In 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 ] Payment when winning with value vi Payment when winning with value vi

Equilibrium in 1st-price auctions
Example: 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: bi(vi) = E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ] v 1 ….

Today The 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. Equilibrium in 1st-price auctions. “Auctions vs. negotiations” – should we really run the optimal auction?

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…

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?

Bulow-Klemperer’s result
Theorem [Bulow & Klemperer 1996]: Revenue in the optimal auction with n players. Revenue from the Vickrey auction with n+1 players. 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.

Bulow-Klemperer’s result: discussion
Revenue in the optimal auction with n players. Revenue from the Vickrey auction with n+1 players. Theorem [Bulow & Klemperer 1996]: 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.

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.

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…)

Bulow & Klemperer: proof
Claim 1: The revenue in the “must-sell” optimal auction with n+1 bidders The optimal revenue with n bidders.

Bulow & Klemperer: proof
Claim 1: the revenue in the “must-sell” optimal auction with n+1 bidders The optimal revenue with n bidders. 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.

Bulow & Klemperer: proof
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
The revenue in the Vickrey auction with n bidders. The revenue in the Vickrey auction with n+1 bidders. 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 optimal revenue with n bidders. = Claim 1 Claim 2 Conclusion:

Summary The tools developed in the literature on optimal auctions are useful in many environements. We saw two applications: Characterization of the equilibrium behavior in 1st price auctions. Bulow-Klemperer result: running Vickrey with n+1 bidders achieves more revenue than the optimal auction with n bidders.

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