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Auction Theory Class 3 – optimal auctions 1

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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). 2

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

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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{ 2 nd highest bid, R} Still Truthful? Yes. For bidders, exactly like an extra bidder bidding R. 6

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Can we get better revenue? Let’s have another look at 2 nd price auctions: 0 1 0 1 1 wins 2 wins x 1 wins and pays x (his lowest winning bid) x v1v1 v2v2 7

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R Can we get better revenue? I will show that some reserve price improve revenue. v1v1 0 1 0 1 v2v2 1 wins 2 wins Revenue increased When comparing to the 2 nd -price auction with no reserve price: Revenue loss here (efficiency loss too) R 8

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Can we get better revenue? Gain is at least 2R(1-R) R/2 = R 2 -R 3 Loss is at most R 2 R = R 3 0 1 0 1 1 wins 2 wins We will be here with probability R(1-R) Average loss is R/2 When R 2 -2R 3 >0, reserve price of R is beneficial. (for example, R=1/4) We will be here with probability R 2 Loss is always at most R 9 v1v1 v2v2

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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 10

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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. – 1 st -price may earn same revenue, but no explanation needed… 11

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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? 12

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Optimal auctions So auctions with the same allocation has the same revenue. But what is the mechanism that obtains the highest expected revenue? 13

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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 14

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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. 15 A key insight (Myerson 81’): In equilibrium, E[ revenue ] = E[ virtual valuation ]

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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. 16

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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 distribution is Myerson-regular. 17

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Optimal auctions: proof where the virtual valuations is: (Note: this theorem does not require that the virtual valuation is Myerson-monotone.) 18 A key insight (Myerson 81’): In equilibrium, E[revenue] = E[virtual valuation]

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Calculus reminder: Integration by parts 19 Integrating: And for definite integral (אינטגרל מסויים):

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Optimal auctions: proof We saw: consider a truthful mechanism where the probability of a player that bids v’ to win is Q i (v). Then, bidder i’s expected payment must be: The expected payment of bidder i is the average over all his possible values: 20

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Optimal auctions: proof 21 Let’s simplify this term….

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Recall that: Optimal auctions: proof Formula of integration by parts: 22 where

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Optimal auctions: proof 23 Let’s simplify this term….Taking out a factor of Q i (x)f(x)

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Optimal auctions: proof 24 Expected payment of bidder I Expected virtual valuation of player i Expected revenueExpected virtual valuation

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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 25

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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. 26

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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) 27

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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. 28

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