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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. . Moving to asymmetric bidders. Efficiency: item goes to bidder with highest value Very important in some situations!

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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

  3. Moving to asymmetric bidders Efficiency: item goes to bidder with highest value • Very important in some situations! • Second-price auctions remain efficient in asymmetic (IPV) case • First-price auctions do not …

  4. Inefficiency in FP with asymmetric bidders

  5. New setup: Myerson 81 • Vector of values v • Allocation functionQ (v ) : Qi(v ) isprob. i wins item • Payment functionP (v ) : Pi(v ) isexpected payment of i • Subsumes Ars easily (check SP, FP) • The pair (Q , P ) is called a Direct Mechanism

  6. New setup: Myerson 81 • Definition: When agents who participate in a mechanism have no incentive to lie about their values, we say the mechanism is incentive compatible. • The Revelation Principle: In so far as equilibrium behavior is concerned, any auction can be replaced by an incentive-compatible mechanism.

  7. Revelation Principle Proof: Replace the bid-taker with a direct mechanism that computes equilibrium values for the bidders. Then a bidder can bid equilibrium simply by being truthful, and there is never an incentive to lie.

  8. Asymmetric bidders • We can therefore restrict attention to incentive compatible direct mechanisms! • In the asymmetric case, surplus is no longer vi F(z) n-1 − P(z) (bidding as if value = z ) Next we write expected surplus in the asymmetric case …

  9. Asymmetric bidders Notation:v−i =vectorv with the i – th Value omitted. Then the prob. that i wins is Where V-i is the space of all v’s except viand F (v-i )is the corresponding distribution

  10. Asymmetric bidders Similarly for the expected payment of bidder i : Expected surplus is then

  11. Asymmetric bidders: yet more general RE Differentiate wrt z and set to zero when z = vi as usual: But now take the total derivative wrt vi when z = vi : And so

  12. Asymmetric bidders: yet more general RE Integrate: or (S = vQ − P ) Expected payment of every bidder depends only on allocation function Q !

  13. Optimal allocation Average over vi and proceed as in RS81: where

  14. Optimal allocation, con’t The total expected revenue is For participation, Pi (0 ) ≤ 0, and seller chooses Pi (0) = 0 to max surplus. Therefore

  15. Optimal allocation, con’t When Pi (0 ) ≤ 0 we say bidders are individually rational : The don’t participate in auctions if the expected payment with zero value is positive.

  16. Optimal allocation The optimal allocation can now be seen by inspection! Look for the maximum value of MRi (vi). Say it occurs at i = i* , and denote it by MR* . • If MR* > 0, then choose that Q i* to be 1 and all the other Q’s to be 0 (bidder i* gets the item) • If MR* ≤ 0, then hold on to the item (seller retains item)

  17. Optimal allocation (inefficient!)

  18. Payment rule Hint: must reduce to second-price when bidders are symmetric Therefore: Pay the least you can while still maintaining the highest MR Verify: This is incentive compatible; that is, bidders bid truthfully!

  19. Wrinkle • For this argument to work, MR must be an increasing function. We F ’s with increasing MR’s regular. (Uniform OK) • It’s sufficient for the inverse hazard rate (1 – F ) /f to be decreasing. • Can be fixed: See Myerson 81 (“ironing”) • Assume MR is regular in what follows

  20. Notice that this shows that all the auctions in Ars in the symmetric case are optimal auctions. (SP is, and the rest are revenue equivalent.) • Notice also that this asks a lot in the asymmetric case. In the direct mechanism the bidders must understand enough to be truthful, and accept the fact that the highest value doesn’t always win.

  21. Ars are optimal mechanisms! • By the revelation principle, we can restrict attention to direct mechanisms • All direct mechanisms with the same allocation rule have the same revenue • An optimal mechanism in the symmetric case awards item to highest-value bidder, and so does any auction in Ars • Therefore any auction in Ars has the same allocation rule, and hence revenue, as an optimal (general!) mechanism

  22. LaboratoryEvidence Generally, there are three kinds of empirical methodologies: • Field observations • Field experiments • Laboratory experiments Problem: do people behave the same way in the lab as in the world? Problem: people differ in experience; people learn

  23. LaboratoryEvidence Conclusions fall into two general categories: • Revenue ranking • Point predictions (usually revenue relative to Nash equilibrium)

  24. Best results for IPV model • First-Price > Dutch Coppinger et al. (80) • First-Price > Nash Dyer et al. (89) • Second-Price > English Kagel et al. (87) • English  truthful=Nash Kagel et al. (87) • First-Price ? Second-Price Thus, generally, sealed versions > open versions!

  25. See also Kagel & Levin 93 for experiments with 3rd price auctions that test IPV theory More about experimental results for common-value auctions later We next focus a while on First-price > Nash One explanation: risk aversion But is there another explanation…

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