1. week 10 1 COS 444 Internet Auctions: Theory and Practice Spring 2008 Ken Steiglitz ken@cs.princeton.edu

2. 

3. week 103 Moving to asymmetric bidders Efficiency: item goes to bidder with highest value Very important in some situations! Very important in some situations! Second-price auctions remain efficient in asymmetic (IPV) case Second-price auctions remain efficient in asymmetic (IPV) case First-price auctions do not … First-price auctions do not …

4. Inefficiency in FP with asymmetric bidders

5. week 105 New setup: Myerson 81 Vector of values v Vector of values v Allocation function Q (v ) : Allocation function Q (v ) : Q i (v ) is prob. i wins item Q i (v ) is prob. i wins item Payment function P (v ) : Payment function P (v ) : P i (v ) is expected payment of i P i (v ) is expected payment of i Subsumes A rs easily (check SP, FP) Subsumes A rs easily (check SP, FP) The pair (Q, P ) is called a The pair (Q, P ) is called a Direct Mechanism Direct Mechanism

6. week 106 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. 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. The Revelation Principle: In so far as equilibrium behavior is concerned, any auction can be replaced by an incentive- compatible mechanism.

7. week 107 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. 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.

9. week 109 Asymmetric bidders We can therefore restrict attention to incentive compatible direct mechanisms! We can therefore restrict attention to incentive compatible direct mechanisms! In the asymmetric case, surplus is no longer v i F(z) n-1 − P(z) In the asymmetric case, surplus is no longer v i F(z) n-1 − P(z) (bidding as if value = z ) (bidding as if value = z ) Next we write expected surplus in the asymmetric case … Next we write expected surplus in the asymmetric case …

10. week 1010 Asymmetric bidders Notation: v −i = vector v with the i – th Value omitted. Then the prob. that i wins is Where V -i is the space of all v’s except v i and F (v -i ) is the corresponding distribution F (v -i ) is the corresponding distribution

11. week 1011 Asymmetric bidders Similarly for the expected payment of bidder i : Expected surplus is then

12. week 1012 Asymmetric bidders: yet more general RE Differentiate wrt z and set to zero when z = v i as usual: But now take the total derivative wrt v i when z = v i : And so

13. week 1013 Asymmetric bidders: yet more general RE Integrate:or (S = vQ − P )  Expected payment of every bidder depends only on allocation function Q !

14. week 1014 Optimal allocation Average over v i and proceed as in RS81: where

15. week 1015 Optimal allocation, con’t The total expected revenue is For participation, P i (0 ) ≤ 0, and seller chooses P i (0) = 0 to max surplus. Therefore

16. week 1016 Optimal allocation, con’t P i (0 ) ≤ 0 we say bidders are individually rational : The don’t participate in auctions if the expected payment with zero value is positive. When P i (0 ) ≤ 0 we say bidders are individually rational : The don’t participate in auctions if the expected payment with zero value is positive.

17. week 1017 Optimal allocation The optimal allocation can now be seen by inspection! Look for the maximum value of MR i (v i ). 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 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) If MR* ≤ 0, then hold on to the item (seller retains item)

18. Optimal allocation (inefficient!)

19. week 1019 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!

20. week 1020 Wrinkle For this argument to work, MR must be an increasing function. We F ’s with increasing MR’s regular. (Uniform OK) 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. It’s sufficient for the inverse hazard rate (1 – F ) /f to be decreasing. Can be fixed: See Myerson 81 (“ironing”) Can be fixed: See Myerson 81 (“ironing”) Assume MR is regular in what follows Assume MR is regular in what follows

21. week 1021 Notice that this shows that all the auctions in A rs in the symmetric case are optimal auctions. (SP is, and the rest are revenue equivalent.) Notice that this shows that all the auctions in A rs 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. 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.

22. week 1022 A rs are optimal mechanisms! By the revelation principle, we can restrict attention to direct mechanisms By the revelation principle, we can restrict attention to direct mechanisms All direct mechanisms with the same allocation rule have the same revenue 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 A rs An optimal mechanism in the symmetric case awards item to highest-value bidder, and so does any auction in A rs Therefore any auction in A rs has the same allocation rule, and hence revenue, as an optimal (general!) mechanism Therefore any auction in A rs has the same allocation rule, and hence revenue, as an optimal (general!) mechanism

23. week 1023 Laboratory Evidence Generally, there are three kinds of empirical methodologies: Field observations Field observations Field experiments Field experiments Laboratory experiments Laboratory experiments  Problem: do people behave the same way in the lab as in the world?  Problem: people differ in experience; people learn

24. week 1024 Laboratory Evidence Conclusions fall into two general categories: Revenue ranking Revenue ranking Point predictions (usually revenue relative to Nash equilibrium) Point predictions (usually revenue relative to Nash equilibrium)

25. week 1025 Best results for IPV model First-Price > Dutch Coppinger et al. (80) First-Price > Dutch Coppinger et al. (80) First-Price > Nash Dyer et al. (89) First-Price > Nash Dyer et al. (89) Second-Price > English Kagel et al. (87) Second-Price > English Kagel et al. (87) English  truthful=Nash Kagel et al. (87) English  truthful=Nash Kagel et al. (87) First-Price ? Second-Price First-Price ? Second-Price Thus, generally, sealed versions > open versions! sealed versions > open versions!

26. week 1026 See also Kagel & Levin 93 for experiments with 3 rd price auctions that test IPV theory More about experimental results for common- value auctions later We next focus a while on First-price > Nash First-price > Nash One explanation: risk aversion But is there another explanation…