Auction Theory

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

2. 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?

3. 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.

4. 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

5. 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 0 1 ….

6. 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?

7. 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…

8. 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?

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

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

11. 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.

12. 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…)

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

14. Bulow & Klemperer: proof ≥ • 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.

15. 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.

16. 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:

17. 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.