# Week 91 COS 444 Internet Auctions: Theory and Practice Spring 2009 Ken Steiglitz

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week 91 COS 444 Internet Auctions: Theory and Practice Spring 2009 Ken Steiglitz ken@cs.princeton.edu

Myerson 1981 optimal, not efficient asymmetric bidders the bigger picture, all single item …

week 93 Moving to asymmetric bidders Efficiency: item goes to bidder with highest value Very important in some situations! Second-price auctions remain efficient in asymmetric (IPV) case. Why? First-price auctions do not …

Inefficiency in FP with asymmetric bidders

week 95 New setup: Myerson 81*, also BR 89Myerson 81BR 89 Vector of values v Allocation function Q (v ): Q i (v ) is prob. i wins item Payment function P (v ): P i (v ) is expected payment of i Subsumes A rs easily (check SP, FP) The pair (Q, P ) is called a Direct Mechanism *wins Nobel prize for this and related work, 2007

week 96 New setup: Myerson 81 Definition: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 mechanism can be replaced by an incentive-compatible direct mechanism.

week 97 Revelation Principle Proof: 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. □ This principle is very general and includes any sort of negotiation!

week 99 Asymmetric bidders We can therefore restrict attention to incentive-compatible direct mechanisms! Note: In the asymmetric case, expected surplus is no longer v i F(z) n-1 − P(z) (bidding as if value = z ) Next we write expected surplus in the asymmetric case …

week 910 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

week 911 Asymmetric bidders Similarly for the expected payment of bidder i : Expected surplus is then

week 912 A yet more general RET 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

week 913 yet more general RE Integrate: Or, using S = vQ – P,  In equilibrium, expected payment of every bidder depends only on allocation function Q !

week 914 Optimal allocation Average over v i and proceed as in RS81: where ← no longer a common F

week 915 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

week 916 Optimal allocation, con’t When P i (0 ) ≤ 0 we say bidders are individually rational : They don’t participate in auctions if the expected payment with zero value is positive.

week 917 Optimal allocation The optimal allocation can now be seen by inspection! For each vector of v’s, 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 hold on to the item (seller retains item)

Optimal allocation (inefficient!)

week 919 Payment rule Hint: must reduce to second-price when bidders are symmetric Therefore: Pay the least you can while still maintaining the highest MR This is incentive compatible; that is, bidders bid truthfully! Why?

week 920 Vickrey ’61 yet again

week 921 Wrinkle For this argument to work, MR must be an increasing function. We call F ’s with increasing MR’s regular. (Uniform is regular) 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

week 922 Notice also that this asks a lot of bidders 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. Or, think of MR i (v i ) as i’s bid As usual in game-theoretic settings, distributions are common knowledge---at least the hypothetical auctioneer must know them.

week 923 In the symmetric case…A rs are optimal mechanisms!* By the revelation principle, we can restrict attention to direct mechanisms An optimal direct mechanism in the symmetric case awards item to the highest- value bidder, and so does any auction in A rs All direct mechanisms with the same allocation rule have the same revenue Therefore any auction in A rs has the same allocation rule, and hence revenue, as an optimal (general!) mechanism *Includes any sort of negotiation whatsoever!

week 924 Efficiency Second-price auctions are efficient --- i.e., they allocate the item to the buyer who values it the most. (Even in asymm. case, truthful is dominant.) We’ve seen that optimal (revenue-maximizing) auctions in the asymmetric case are in general inefficient. It turns out that second-price auctions are optimal in the class of efficient auctions. They generalize in the multi-item case to the Vickrey- Clark-Groves (VCG) mechanisms. … More later.

week 925 Laboratory Evidence Generally, there are three kinds of empirical methodologies: Field observations Field experiments Laboratory experiments  Problem: people may not behave the same way in the lab as in the world  Problem: people differ in behavior  Problem: people learn from experience

week 926 Laboratory Evidence Conclusions fall into two general categories: Revenue ranking Point predictions (usually revenue relative to Nash equilibrium) For more detail, see J. H. Kagel, "Auctions: A Survey of Experimental Research", in The Handbook of Experimental Economics, J. Kagel and A. Roth (eds.), Princeton Univ. Press, 1995.

week 927 Best revenue-ranking results for IPV model Second-Price > English Kagel et al. (87) English  truthful=Nash Kagel et al. (87) First-Price ? Second-Price First-Price > Dutch Coppinger et al. (80) First-Price > Nash Dyer et al. (89) Dyer et al. (89) Thus, generally, sealed versions > open versions!

week 928 A violation of theory is the scientist’s best news! Let’s discuss some of the violations… Second-Price > English. These are (weakly) strategically equivalent. But English  truthful = Nash. What hint towards an explanation does the “weakly” give us?

week 929 First-Price > Dutch. These are strongly strategically equivalent. But recall Lucking- Reiley’s pre-eBay internet test with Magic cards, where Dutch > FP by 30%! What’s going on here?

week 930 See also Kagel & Levin 93 for experiments with 3 rd -price auctions that test IPV theoryKagel & Levin 93 More about experimental results for common-value auctions later We next focus for a while on a widely accepted point prediction: One explanation, as we’ve seen, is risk aversion But is here is an alternative explanation… First-price > Nash

week 931 Spite [MSR 03 MS 03]MSR 03MS 03 Suppose bidders care about the surplus of other bidders as well their own. Simple example: Two bidders, second- price, values iid unifom on [0,1]. Suppose bidder 2 bids truthfully, and suppose bidder 1’s utility is not her own surplus, but the difference Δ between hers and her rival’s.

week 932 Spite Now bidder 1 wants to choose her bid b 1 to maximize the expectation of where I is the indicator function, 1 when true, 0 else. Taking expectation over v 2 :

week 933 Spite Maximizing wrt b 1 yields best response to truthful bidding: Intuition?

week 934 Spite Maximizing wrt b 1 yields best response to truthful bidding: Intuition: by overbidding, 1 loses surplus when 2’s bid is between v 1 and her bid. But, this is more than offset by forcing 2 to pay more when he wins. Notice that bidder 2 still cannot increase his absolute surplus. (Why not?) He must take a hit to compete in a pairwise knockout tournament.

week 935 Spite Some results from MSR 03: take the case when bidders want to maximize the difference between their own surplus and that of their rivals. Values distributed as F, n bidders. Then  FP equilibrium is the same as in the risk- averse CRRA case with ρ = ½ (utility is t 1/2 ). Thus there is overbidding.  SP equilibrium is to overbid according to

week 936 Spite  Revenue ranking is SP > FP. (Not a trivial proof. Is there a simpler one?) Thus, this revenue ranking is the opposite of the prediction in the risk-averse case, where there is overbidding in FP but not in SP. (Testable prediction.) This explains overbidding in both first- and second-price auctions, while risk-aversion explains only the first. (Testable prediction.) Raises a question: do you think people bid differently against machines than against people?

week 937 Spiteful behavior in biology This model can also explain spiteful behavior in biological contexts, where individuals fight for survival one-on-one [MS 03]. Example:MS 03 This is a hawk-dove game. Winner type replaces loser type. In a large population where the success of an individual is determined by average individual payoff, there is an evolutionarily stable solution that is 50/50 hawks and doves. If winners are determined by relative payoff in each 1-1 contest, the hawks drive out the doves. Thus, there is an Invasion of the Spiteful Mutants!

week 938 Invasion of the spiteful mutants To see this, suppose in the large population there is a fraction ρ of H’s and (1-ρ ) of D’s. The average payoff to an H in a contest is and to a D The first is greater than the second iff ρ<1/2. A 50/50 mixture is an equilibrium. But if the winner of a contest is determined by who has the greater payoff, an H always replaces a D!

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