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Internet Economics כלכלת האינטרנט

Internet Economics כלכלת האינטרנט. Class 4 – Optimal Auctions. Golden balls. Let’s warm up with some real-game theory: Reality games and game theory… Scene 1 Scene 2. Last week (1/4). How to sell a single item to n bidders? Seller doesn’t know how much bidders are willing to pay

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Internet Economics כלכלת האינטרנט

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  1. Internet Economicsכלכלת האינטרנט Class 4 – Optimal Auctions

  2. Golden balls Let’s warm up with some real-game theory: Reality games and game theory… • Scene 1 • Scene 2

  3. Last week (1/4) • How to sell a single item to n bidders? • Seller doesn’t know how much bidders are willing to pay • vi is the value of bidder i for the item. • Getting this information via an auction. • Game with incomplete information.

  4. Last week (2/4) • The English Auction: • Price starts at 0 • Price increases until only one • bidder is left. • Vickrey (2nd price) auction: • Bidders send bids. • Highest bid wins, pays 2nd highest bid. • Private value model: each person has a privately known value for the item. • We saw: the two auctions are equivalent in the private value model. • Auctions are efficient:dominant strategy for each player: truthfulness.

  5. Last week (3/4) • The Dutch Auction: • Price starts at max-price. • Price drops until a bidder agrees to buy. • 1st-price auction: • Bidders send bids. • Highest bid wins, pays his bid. • Dutch auctions and 1st price auctions are strategically equivalent. (asynchronous vs simple & fast) • No dominant strategies. (tradeoff: chance of winning, payment upon winning.) • Analysis in a Bayesian model: • Values are randomly drawn from a probability distribution. • Strategy: a function. “What is my bid given my value?”

  6. Last week (4/4) • We considered the simplest Bayesian model: • 2 bidders. • Values drawn uniformly from [0,1]. Then: In a 1st-price auction, it is a (Bayesian) Nash equilibrium when all bidders bid • An auction is efficient, if in (Bayesian) Nash equilibrium the bidder with the highest value always wins. • 1st price is efficient!

  7. Remark: Efficiency • We saw that both 2nd –price and 1st –price auctions are efficient. • What is efficiency (social welfare)?The total utility of the participants in the game (including the seller). For each bidder: vi – pi For the seller: (assuming it has 0 value for the item) • Summing:

  8. 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).

  9. Example: Spectrum auctions • One of the main triggers to auction theory. • FCC in the US sells spectrum, mainly for cellular networks. • Improved auctions since the 90’s increased efficiency + revenue considerably. • Complicated (“combinatorial”) auction, in many countries. • (more details further in the course)

  10. New Zealand Spectrum Auctions • A Vickrey (2nd price) auction was run in New Zealand to sale a bunch of auctions. (In 1990) • Winning bid: $100000 Second highest: $6 (!!!!) Essentially zero revenue. • NZ Returned to 1st price method the year after. • After that, went to a more complicated auction (in few weeks). • Was it avoidable?

  11. 1st or 2nd price? • Assume 2 bidders, uniform distribution on [0,1]. • Facts: (1) E[ max(v1,v2) ] = 2/3 (2) E[ min(v1,v2) ] = 1/3 (in general, k’th highest value of n is (n+1-k)/n+1) • Revenue in 2nd price: • Bidders bid truthfully. • Revenue is 2nd highest bid. • Expected revenue = 1/3 • Revenue in 1st price: • bidders bid vi/2. • Revenue is the highest bid. • Expected revenue = E[ max(v1/2,v2/2) ] • = ½ E[ max(v1,v2)] • = ½ × 2/3 = 1/3

  12. Revenue equivalence theorem • No coincidence! • Somewhat unintuitively, revenue depends only on the way the winner is chosen, not on payments. • Auction for a single good. • Values are independently drawn from distribution F (increasing) Theorem (“revenue equivalence”): All auctions where: • the good is allocated to the bidder with the highest value • Bidders can guarantee a utility of 0 by bidding 0. yield the same revenue! (more general: two auction with the same allocation rule yield the same revenue)

  13. Remark: Individual rationality • The following mechanism gains lots of revenue: • Charge all players $10000000 • Bidder will clearly not participate. • We thus have individual-rationality (or participation) constraints on mechanisms:bidders gain positive utility in equilibrium . • This is the reason for condition 2 in the theorem.

  14. All-pay auction (1/3) • Rules: • Sealed bid • Highest bid wins • Everyone pay their bid • Equilibrium with the uniform distribution: b(v)= • Does it achieve more or less revenue? • Note: Bidders shade their bids as the competition increases.

  15. All-pay auction (2/3) • expected payment per each player: herbid. • Each bidder bids • Expected payment for each bidder: • Revenue: from n bidders • Revenue equivalence!

  16. All-pay auction (3/3) • Examples: • crowdsourcing over the internet: • First person to complete a task for me gets a reward. • A group of people invest time in the task. (=payment) • Only the winner gets the reward. • Advertising auction: • Collect suggestion for campaigns, choose a winner. • All advertiser incur cost of preparing the campaign. • Only one wins. • Lobbying • War of attrition • Animals invest (b1,b2) in fighting.

  17. What did we see so far • 2nd-price, 1st-price, all pay: all obtain the same seller revenue. • Revenue equivalence theorem:Auctions with the same allocation decisions earn the same expected seller revenue in equilibrium. • Constraint: individual rationality (participation constraint) • Many assumptions: • statistical independence, • risk neutrality, • no externalities, • private values, • …

  18. 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.

  19. 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 winsPayment: winner pays max{ 2nd highest bid, R} Yes. For bidders, exactly like an extra bidder bidding R. Still Truthful?

  20. 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

  21. 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 Revenue loss here (efficiency loss too) R

  22. 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)

  23. Reservation price • Can increase revenue! • 2 bidders, uniform distribution: optimal reserve price = ½ • Revenue: 5/12=0.412 > 1/3 • n bidders, uniform distribution: optimal reserve price = ½ Theorem: (Myerson ‘81)Vickrey auction with a reserve price maximizes revenue. • For a general family of distributions (uniform, exponential, normal, and many others). • Reserve price is independent of n. (Nobel prize, 2007)

  24. 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 • Surprising? No. We said that the optimal reserve price does not depend on n. Probability that the buyer will accept the price The payment for the seller

  25. 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…

  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.

  27. Next week • Designing dominant-strategy mechanisms for more general environments. • the magic of the VCG mechanism.

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