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COS 444 Internet Auctions: Theory and Practice

COS 444 Internet Auctions: Theory and Practice. Spring 2008 Ken Steiglitz ken@cs.princeton.edu. Multi-unit demand auctions (Ausubel & Cramton 98, Morgan 01). Examples: FCC spectrum, Treasury debt securities, Eurosystem: multiple, identical units

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COS 444 Internet Auctions: Theory and Practice

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  1. COS 444 Internet Auctions:Theory and Practice Spring 2008 Ken Steiglitz ken@cs.princeton.edu

  2. Multi-unit demand auctions(Ausubel & Cramton 98, Morgan 01) • Examples: FCC spectrum, Treasury debt securities, Eurosystem: multiple, identical units • Issues: Pay-your-bid (discriminatory) prices v. uniform-price; efficiency; optimality of revenue • The problem: conventional, uniform-price auctions provide incentives for demand-reduction

  3. Multi-unit demand auctions Example 1: (Morgan) 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Suppose bidders bid truthfully; rank bids: $10 bidder 1 10 bidder 1 8 bidder 2  first rejected bid If buyers pay this, surplus (1) = $4 revenue = $16

  4. Multi-unit demand auctions Example 1: But bidder 1 can do better! Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Suppose bidder 1 shades her demand: $10 bidder 1 for her first unit 8 bidder 2 for first unit 0 bidder 1 for her 2nd unit  first rej. bid If buyers pay this, surplus (1) = $10 surplus (2) = $8 inefficient! revenue = $0!

  5. Multi-unit demand auctions Thus, uniform price demand reductioninefficiency The natural generalization of the Vickrey auction (winners pay first rejected bid) is not incentive compatible and not efficient Lots of economists got this wrong!

  6. Multi-unit demand auctions Ausubel & Cramton prove, in a simplified model, that this example is not pathological: Proposition: There is no efficient equilibrium strategy in a uniform-price, multi-unit demand auction. The appropriate generalization of the Vickrey auction is the Vickrey-Clark-Groves (VCG) mechanism…

  7. The VCG auction for multi-unit demand Return to example 1: 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Suppose bidders bid truthfully, and order bids: $10 bidder 1 10 bidder 1 8 bidder 2 Award supply to the highest bidders … How much does each bidder pay?

  8. The VCG auction for multi-unit demand Define: social welfare = W ( v ) = total value received by agents, where v is the vector of values Then the VCG payment of i is W-i( 0, x-i) − W-i ( x ) = welfare to others when i bids 0, minus that when i bids truthfully = sum of kirejected bids (if bidder i gets ki items)

  9. The VCG auction for multi-unit demand Example 1: 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 If bidder 1 bids 0, welfare = $8, and is $0 when 1 bids truthfully…  1 pays $8 for the 2 items

  10. The VCG auction for multi-unit demand Example 2: 3 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Bidder 3: capacity 1, value $6 $10 bidder 1 10 bidder 1  bidder 1 gets 2 items 8 bidder 2  bidder 2 gets 1 item 6 bidder 3 Welfare when 1 bids 0 = $14 Welfare when 1 bids truthfully = $8  1 pays $6 for the 2 items

  11. The VCG auction for multi-unit demand Example 2, con’t 3 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Bidder 3: capacity 1, value $6 $10 bidder 1  bidder 1 gets 2 items 8 bidder 2  bidder 2 gets 1 item 6 bidder 3 Welfare when 2 bids 0 = $26 Welfare when 2 bids truthfully = $20  2 pays $6 for the 1 item (notice that revenue = $12 < $18 =3x$6 in uniform-price case, so not optimal)

  12. VCG mechanisms(Krishna 02) VCG mechanisms are • efficient • incentive-compatible (truthful is weakly dominant) • individually rational • max-revenue among all such mechanisms … but not optimal revenue in general, and prices are discriminatory, “murky”

  13. Bilateral trading mechanisms[Myerson & Satterthwaite 83] An impossibility result: The following desirable characteristics of bilateral trade (not an auction): • efficient • incentive-compatible • individually rational Cannot all be achieved simultaneously!

  14. Bilateral trading mechanisms The setup: • one seller, with private value v1 , distributed with density f1 > 0 on [a1 , b1 ] • one buyer, with private value v2 , distributed with density f2 > 0 on [a2 , b2 ] • risk neutral … Notice: not an auction in Riley & Samuelson’s class!

  15. Bilateral trading mechanisms Outline of proof: We use a direct mechanism (p, x ): where p (v1 , v2 ) = prob. of transfer 12 x (v1 , v2 ) = expected payment 12

  16. Bilateral trading mechanisms Main result: If then no incentive-compatible individually rational trading mechanism can be (ex post) efficient. Furthermore, is the smallest lump-sum subsidy to achieve efficiency.

  17. Bilateral trading mechanisms Examples • f i > 0 is necessary: discrete probs. • Subsidy for efficiency: v1 and v2 both uniform on [0,1]

  18. Auctions vs. Negotiations(Bulow & Klemperer 96) Simple example: IPV, uniform Case 1) Optimal auction = optimal mechanism with one buyer. Optimal entry value v* = 0.5; revenue = 1/4 Case 2) Two buyers, no reserve; revenue = 1/3 > ¼  One more buyer is worth more than setting reserve optimally!

  19. Auctions vs. Negotiations, con’t • Bulow & Klemperer96 generalize to any F, • any number of bidders… • A no-reserve auction with n +1 bidders is more profitable than an optimal auction • (and hence optimal mechanism) with n • bidders

  20. Auctions vs. Negotiations, con’t Optimal reserve, n bidders: No reserve, n+1 bidders

  21. Auctions vs. Negotiations, con’t Facts: … QED

  22. Bidder rings (Graham & Marshall 87) Stylized facts • They exist and are stable • They eliminate competition among ring members; yet ensure ring member with highest value is not undercut • Benefits shared by ring members • Have open membership • Auctioneer responds strategically • Try to hide their existence

  23. Bidder rings Graham & Marshall’s model: Second-price pre-auction knockout (PAKT) • IPV, risk neutral • Value distributions F, common knowledge • Identity of winner & price paid common knowledge • Membership of ring known only to ring members

  24. Bidder rings Pre-auction knock-out (PAKT): • Appoint ring center, who pays P to each ring member, P to be determined below • Each ring member submits a sealed bid to the ring center • Winner is advised to submit her winning bid at main auction; other ring members submit only meaningless bids • If the winner at the sub-auction (sub-winner) also wins main auction, she pays:

  25. Bidder rings If sub-winner wins main auction, she pays: • Main auctioneer P* = SP at main auction • Ring center δ = max{ P̃ − P* , 0 }, where P̃ = SP in PAKT Thus: If the sub-winner wins main auction, she pays SP among all bids

  26. Bidder rings The quantity δ is the amount “stolen” from the main auctioneer, the “booty” The ring center receives and distributes E[δ | sub-winner wins main auction]  so his budget is balanced Each ring member receives P = E[δ | sub-winner wins main auction]/K

  27. Bidder rings Graham and Marshall prove: • Truthful bidding in the PAKT, and following the recommendation of the ring center is SBNE & weakly dominant strategy (incentive compatible) • Voluntary participation is advantageous (individually rational) • Efficient (buyer with highest value gets item) In fact, the whole thing is equivalent to a Vickrey auction

  28. Bidder rings Main auctioneer responds strategically by increasing reserves or shill-bidding Graham& Marshall also prove that • Optimal main reserve is an increasing function of ring size K • Expected surplus of ring member is a decreasing function of reserve prices • Expected surplus of ring member is an increasing function of ring size K So best to be secretive

  29. Term papers due 5pm Tuesday May 13 (Dean’s Date)  Email me for office hours re term papers

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