Report

COS 444 Internet Auctions: Theory and Practice Spring 2009 Ken Steiglitz [email protected] week 9 1 the bigger picture, all single item … Myerson 1981 optimal, not efficient asymmetric bidders 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 … week 9 3 Inefficiency in FP with asymmetric bidders New setup: Myerson 81*, also BR 89 *wins Nobel prize for this and related work, 2007 • Vector of values v • Allocation function Q (v ): Qi (v ) is prob. i wins item • Payment function P (v ): Pi (v ) is expected payment of i • Subsumes Ars easily (check SP, FP) • The pair (Q , P ) is called a Direct Mechanism week 9 5 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 mechanism can be replaced by an incentive-compatible direct mechanism. week 9 6 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. □ This principle is very general and includes any sort of negotiation! week 9 7 Asymmetric bidders • We can therefore restrict attention to incentive-compatible direct mechanisms! • Note: In the asymmetric case, expected 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 … week 9 9 Asymmetric bidders Notation: v−i = vector v with the i – th Value omitted. Then the prob. that i wins is Qi ( z ) Qi ( z, vi ) d F (v -i ) Vi Where V-i is the space of all v’s except vi and F (v-i ) is the corresponding distribution week 9 10 Asymmetric bidders Similarly for the expected payment of bidder i : Pi ( z ) Pi ( z, vi ) dF(vi ) Vi Expected surplus is then Si ( z) viQi ( z) Pi ( z) week 9 11 A yet more general RET Differentiate wrt z and set to zero when z = vi as usual: viQ (vi ) Pi (vi ) 0 ' i ' But now take the total derivative wrt vi when z = vi : Si' (vi ) viQi' (vi ) Qi (vi ) Pi ' (vi ) And so week 9 S (vi ) Qi (vi ) ' i 12 yet more general RE Integrate: vi S (vi ) Si (0) Qi ( x) dx 0 Or, using S = vQ – P , vi Pi (vi ) Pi (0) viQi (vi ) Qi ( x) dx 0 In equilibrium, expected payment of every bidder depends only on allocation function Q ! week 9 13 Optimal allocation Average over vi and proceed as in RS81: E[ Pi (v)] Pi (0) MR i (vi ) Qi (v) dF (v) V where 1 Fi (vi ) ←no longer a common F MRi (vi ) vi f i (vi ) week 9 14 Optimal allocation, con’t The total expected revenue is R Pi (0) MR i (vi ) Qi (v) dF (v) i V i For participation, Pi (0 ) ≤ 0, and seller chooses Pi (0) = 0 to max surplus. Therefore R MR i (vi ) Qi (v) dF (v) V week 9 i 15 Optimal allocation, con’t When Pi (0 ) ≤ 0 we say bidders are individually rational : They don’t participate in auctions if the expected payment with zero value is positive. week 9 16 Optimal allocation The optimal allocation can now be seen by inspection! R MR i (vi ) Qi (v) dF (v) V i For each vector of v’s, 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 Qi* 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) week 9 17 Optimal allocation (inefficient!) 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 9 19 Vickrey ’61 yet again week 9 20 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 9 21 • 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 MRi(vi) as i’s bid • As usual in game-theoretic settings, distributions are common knowledge---at least the hypothetical auctioneer must know them. week 9 22 In the symmetric case…Ars 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 highestvalue bidder, and so does any auction in Ars • All direct mechanisms with the same allocation rule have the same revenue • Therefore any auction in Ars has the same allocation rule, and hence revenue, as an optimal (general!) mechanism *Includes any sort of negotiation whatsoever! week 9 23 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 VickreyClark-Groves (VCG) mechanisms. … More later. week 9 24 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 9 25 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 9 26 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) Thus, generally, sealed versions > open versions! week 9 27 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 9 28 • First-Price > Dutch. These are strongly strategically equivalent. But recall LuckingReiley’s pre-eBay internet test with Magic cards, where Dutch > FP by 30%! What’s going on here? week 9 29 • 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 for a while on a widely accepted point prediction: First-price > Nash • One explanation, as we’ve seen, is risk aversion • But is here is an alternative explanation… week 9 30 Spite [MSR 03 MS 03] • Suppose bidders care about the surplus of other bidders as well their own. Simple example: Two bidders, secondprice, 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 9 31 Spite • Now bidder 1 wants to choose her bid b1 to maximize the expectation of (v1, v2 ) (v1 v2 ) Ib 1 v 2 (v2 b1) Ib 1 v 2 where I is the indicator function, 1 when true, 0 else. • Taking expectation over v2 : b1 0 1 (v1 v2 ) dv2 (v2 b1 ) dv2 b1 b1v1 1 / 2 b1 b12 week 9 32 Spite • Maximizing wrt b1 yields best response to truthful bidding: v1 1 b1 2 • Intuition? week 9 33 Spite • Maximizing wrt b1 yields best response to truthful bidding: v1 1 b1 2 • Intuition: by overbidding, 1 loses surplus when 2’s bid is between v1 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 9 34 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 riskaverse CRRA case with ρ = ½ (utility is t1/2 ). Thus there is overbidding. SP equilibrium is to overbid according to b(v ) v 1 v week 9 (1 F ( y ))2 dy (1 F (v))2 35 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 9 36 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: H D • This is a hawk-dove game. H 1/ 2 1 D 0 1/ 2 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 9 37 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 (1 / 2) (1 )(1) and to a D (0) (1 )(1 / 2) • 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! week 9 38