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# Auction Theory

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1. Auction Theory Class 3 – optimal auctions

2. Optimal auctions • Usually the term optimal auctions stands for revenue maximization. • What is maximal revenue? • We can always charge the winner his value. • Maximal revenue: optimal expected revenue in equilibrium. • Assuming a probability distribution on the values. • Over all the possible mechanisms. • Under individual-rationality constraints (later).

3. Next: Can we get better revenue? • Can we achieve better revenue than the 2nd-price/1st price? • If so, we must sacrifice efficiency. • All efficient auction have the same revenue…. • How? • Think about the New-Zealand case.

4. Vickrey with Reserve Price • Seller publishes a minimum (“reserve”) price R. • Each bidder writes his bid in a sealed envelope. • The seller: • Collects bids • Open envelopes. • Winner: Bidder with the highest bid, if bid is above R. Otherwise, no one wins.Payment: winner pays max{ 2nd highest bid, R} Yes. For bidders, exactly like an extra bidder bidding R. Still Truthful?

5. Can we get better revenue? 1 • Let’s have another look at 2nd price auctions: 2 wins v2 1 wins x 1 wins and pays x (his lowest winning bid) 0 x 0 v1 1

6. Can we get better revenue? 1 • I will show that some reserve price improve revenue. Revenue increased 2 wins v2 1 wins R Revenue increased 0 0 v1 1 When comparing to the 2nd-price auction with no reserve price: Revenue loss here (efficiency loss too) R

7. Can we get better revenue? 1 We will be here with probability R(1-R) v2 • Gain is at least 2R(1-R) R/2 = R2-R3 • Loss is at most R2 R = R3 2 wins We will be here with probability R2 1 wins Average loss is R/2 Loss is always at mostR 0 v1 0 1 • When R2-2R3>0, reserve price of R is beneficial.(for example, R=1/4)

8. Reservation price Let’s see another example:How do you sell one item to one bidder? • Assume his value is drawn uniformly from [0,1]. • Optimal way: reserve price. • Take-it-or-leave-it-offer. • Let’s find the optimal reserve price:E[revenue] = ( 1-F(R) ) × R = (1-R) ×R  R=1/2 Probability that the buyer will accept the price The payment for the seller

9. Back to New Zealand • Recall: Vickrey auction.Highest bid: \$100000. Revenue: \$6. • Two things to learn: • Seller can never get the whole pie. • “information rent” for the buyers. • Reserve price can help. • But what if R=\$50000 and highest bid was \$45000? • Of the unattractive properties of Vickrey Auctions: • Low revenue despite high bids. • 1st-price may earn same revenue, but no explanation needed…

10. Optimal auctions: questions. • Is indeed Vickrey auctions with reserve price achieve the highest possible revenue? • If so, what is the optimal reserve price? • How the reserve price depends on the number of bidders? • Recall: for the uniform distribution with 1 bidder the optimal reserve price is ½. What is the optimal reserve price for 10 players?

11. Optimal auctions • So auctions with the same allocation has the same revenue. • But what is the mechanism that obtains the highest expected revenue?

12. Virtual valuations • Consider the following transformation on the value of each bidder: • This is called the virtual valuation. • Like bidders’ values: The virtual valuation is when a player wins and zero otherwise. • Example: the uniform distribution on [0,1] • Recall: f(v)=1, F(v)=v for every v

13. Optimal auctions • Why are we interested in virtual valuations? • Meaning: for maximizing revenue we will need to maximize virtual values. • Allocate the item to the bidder with the highest virtual value. • Like maximizing efficiency, just when considering virtual values. A key insight (Myerson 81’): In equilibrium, E[ revenue ] = E[ virtual valuation ]

14. Optimal auctions • An optimal auction allocates the item to the bidder with the highest virtual value. • Can we do this in equilibrium? • Is the bidder with the highest value is the bidder with the highest virtual value? • Yes, when the virtual valuation is monotone non-decreasing. • And when values are distributed according to the same F • Therefore, Vickrey with a reserve price is optimal. • Will see soon what is the optimal reserve price.

15. Optimal auctions • Bottom line:The optimal auction allocates the item to the bidder with the highest virtual value, and this is a truthful mechanism when is non-decreasing. • Vickrey auction with a reserve price. • Remark: distribution for which the virtual valuation is non-decreasing are called Myerson-regular. • Example: for the uniform distributionis Myerson-regular.

16. Optimal auctions: proof A key insight (Myerson 81’): In equilibrium, E[revenue] = E[virtual valuation] where the virtual valuations is: (Note: this theorem does not require that the virtual valuation is Myerson-monotone.)

17. Calculus reminder: Integration by parts Integrating: And for definite integral (אינטגרל מסויים):

18. Optimal auctions:proof • We saw:consider a truthful mechanism where the probability of a player that bids v’ to win is Qi(v). Then, bidder i’s expected payment must be: • The expected payment of bidder i is the average over all his possible values:

19. Optimal auctions: proof Let’s simplify this term….

20. Optimal auctions: proof Formula of integration by parts: where Recall that:

21. Optimal auctions: proof Let’s simplify this term…. Taking out a factor of Qi(x)f(x)

22. Optimal auctions: proof Expected payment of bidder I Expected virtual valuation of player i Expected revenue Expected virtual valuation

23. Optimal auctions • Bottom line:The optimal auction allocates the item to the bidder with the highest virtual value, and this is a truthful mechanism when is non-decreasing. • The auction will not sell the item if the maximal virtual valuation is negative. • No allocation  0 virtual valuation. • The optimal auction is Vickrey with reserve price p such that

24. Optimal auctions: uniform dist. • The virtual valuation: • The optimal reserve price is ½: • The optimal auction is the Vickrey auction with a reserve price of ½.

25. Remarks • Reservation price is independent of the number of bidders • With uniform distribution, R=1/2 for every n. • With non-identical distributions (but still statistically independent), the same analysis works • Optimal auction still allocate the item to the bidder with the highest virtual valuation. • However, Vickrey+reserve-price is not necessarily the optimal auction in this case. • (it is not true anymore that the bidder with the highest value is the bidder with the highest virtual value)

26. Summary: Efficiency vs. revenue Positive or negative correlation ? • Always: Revenue ≤ efficiency • Due to Individual rationality. • More efficiency makes the pie larger! • However, for optimal revenue one needs to sacrifice some efficiency. • Consider two competing sellers: one optimizing revenue the other optimizing efficiency. • Who will have a higher market share? • In the longer terms, two objectives are combined.