Robert Wilson (1977), A Bidding Model of Perfect Competition, Review of Economic

Transcription

1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 10 Oct No lecture on Thursday New HW, and solutions to last one, online Today s papers: Robert Wilson (1977), A Bidding Model of Perfect Competition, Review of Economic Studies 44 Paul Milgrom (1981), Rational Expectations, Information Acquisition, and Competitive Bidding, Econometrica 49 Wolfgang Pesendorfer and Jeroen Swinkels (1997), The Loser s Curse and Information Aggregation in Common Value Auctions, Econometrica 65 Ilan Kremer (2002), Information Aggregation in Common Value Auctions, Econometrica 70 Ronald Harstad, Aleksandar Pekec, Ilia Tsetlin (2008), Information Aggregation in Auctions with an Unknown Number of Bidders, Games and Economic Behavior 62 Stepping back a little bit... The standard models of general equilibrium and rational expectations are appealing for a number of reasons, but don t explicitly model price formation that is, prices appear out of nowhere and individual players best-response to them. A feature of many of these models is that whatever information is available is somehow incorporated into equilibrium prices. Thus, markets are assumed to aggregate information even when information is dispersed and held in small amounts by large numbers of individuals, all of it is assumed to filter into equilibrium prices, making them fully revealing. (In a sense, this is one of the central tenets of the notions of efficient markets and rational expectations) Ilan Kremer puts it this way: A key question in information economics and finance is whether prices aggregate information in a competitive environment. That is, when agents are endowed with private information, does competition lead in the limit to prices that would occur if all information were public? 1

2 While general equilibrium models do not explicitly model price formation, auctions do prices in an auction come directly from the equilibrium behavior of individuals trying to maximize their profit. So auctions are a natural setting in which to try to answer this question. To rephrase the question, consider a common value auction, where the true value of the good for sale is unknown, but where each bidder gets some signal related to this underlying value. Consider what happens as the number of bidders grows, so that a decent econometrician with access to all the bidders signals would have a very good estimate of the true value of the good. Does the price of the good that is, the price paid by the winner of the auction under equilibrium behavior naturally converge toward this same value? This problem was first considered by Bob Wilson and then Paul Milgrom, in the late 1970s and early 1980s. Milgrom ( Rational Expectations, Information Acquisition, and Competitive Bidding, Econometrica 1981) considered a world with k identical common-value objects up for sale using a k+1 st -price auction; thus, as in the standard rational expectation equilibrium models, bidders were price takers, since they could not directly influence the price at which they transacted. (In a k+1 st -price auction, the highest losing bid sets the price, so no bidder can simultaneously affect the price and transact.) k + 1 st -price auctions are also relatively easy to analyze, since equilibrium bidding behavior is relatively clean. Milgrom s Model Milgrom allows for a fairly general model private- and common-value components, risk-aversion but we ll use the simple case for ease of notation Bidders are symmetric Conditional on an unobserved state of the world (true value of the object) z, bidders signals x i are conditionally independent The value of winning the object is z b Signals are related to the true value via the monotone likelihood ratio property: for f(y z) the distribution of bidders signals conditional on the true value of the object, and z > z, the ratio is increasing in y Of course, this just means that for y > y, f(y z) f(y z ) f(y z) f(y z ) f(y z) f(y z ) 2

3 or f(y z)f(y z ) f(y z )f(y z) If we rewrite the conditional distribution as f(y z) = f(y, z)/f z (z), the denominators drop out from both sides, and this is equivalent to f(y, z)f(y, z ) f(y, z )f(y, z) so the monotone likelihood ratio property is exactly the same as our familiar condition that the joint distribution of y and z is log-supermodular, or y and z are affiliated. (Note that MLRP generally means the ratio is weakly increasing; strict MLRP means the ratio is strictly increasing, which is the same as the affiliation inequality holding strictly.) 3

4 Equilibrium in Milgrom s Model Let X i be bidder i s signal, and Y 1,..., Y k be the order statistics of the other bidders signals. Then it won t come as that big a surprise that a symmetric equilibrium of the k-object auction is for each bidder to bid b i (x) = E(Z X i = x, Y k = x) That is, bidder i bids the expected value of the object conditional on his being pivotal just on the fence between winning the object and losing it. (When k = 1, this is the equilibrium we already looked at I compute the expected value of the object, conditional on tying for the highest signal. When k > 1, the intuition is the exact same I condition on the k th highest of my opponents having the same signal as me. The proof that this is a best response even if I knew Y k is exactly the same as we ve already seen.) Note that the joint density of the underlying value Z = s, my own signal X i = x, and the k th -highest of my opponents signals Y k = y is ( ) N 1 g(s) f(x s) Nf(y s) (1 F (y s)) k 1 F N k 1 (y s) k 1 so the conditional density of Z, conditional on particular realizations of x and y, is this expression divided by the unconditional joint density of (X i, Y k ) at (x, y). We can write E(Z X i = x, Y k = y) = sg(s x, y)ds g(s x, y)ds Canceling a bunch of constant terms that appear in both numerator and denominator and then setting y = x gives us b i (x) = sf(x s)f(x s)(1 F (x s)) k 1 F n k 1 (x s)dg(s) f(x s)f(x s)(1 F (x s)) k 1 F n k 1 (x s)dg(s) 4

5 Information Aggregation in Milgrom s Model Fix k, and let W n be a random variable comprising the winning bid in a k-object auction with n bidders. The main information aggregation result of the Milgrom paper is: Theorem 1. W n Z converges to zero in probability as n if and only if for each two possible values z and z of Z with z > z, inf x f(x z) f(x z ) = 0 Recall that convergence to 0 in probability means that for any ɛ > 0 and δ < 1, there is an N such that Pr( W n Z < ɛ) < δ for all n > N. Thus, W n Z converging to 0 in probability means that as n grows, the price paid by the winning bidders converges on the true value of the object with probability 1 - that is, information aggregates. Milgrom spins the theorem as a positive result the theorem indicates that prices may give a good approximation of value when n is large but the necessary-andsufficient condition is very strong, so some authors read this as a negative result. The condition is easiest to understand with discrete values and a finite number of signals: then for any value z, there is some signal that occurs with positive probability at z but is impossible at all lower values z < z (To see the equivalence, take z i to be each signal below z, and let y i be the signal at which f(y i z i ) f(y i z ) = 0. (The inf must be attained for each z i, since there are finite signals to take it over.) The MLRP says that the ratio f(y z i) f(y z ) is decreasing in y, so it must be 0 at all y y i. So if we take max{y i }, the ratio must be 0 for all z i.) So for each possible value z, there is some signal that rules out all lower values z < z, so a bidder with that signal must bid at least the z that s their lower bound on value As n grows, the probability goes to 1 that at least k + 1 of the bidders get the signal that reveals the actual value, and by assumption, any higher signals are impossible (This condition was found to be sufficient in Wilson; Milgrom shows that it is also necessary.) With continuous types and signals, the condition is harder to interpret, but it s very strong 5

6 Pesendorfer and Swinkels Pesendorfer and Swinkels, ( The Loser s Curse and Information Aggregation in Common Value Auctions, Econometrica 1997, recently added to the syllabus) explain Milgrom s result this way: when k is fixed but n grows unboundedly large, the winner s curse becomes overwhelming, since I am now conditioning on basically an infinite number of competitors all (or nearly all) having signals lower than mine. Thus, unless individual high signals are extremely powerful, high bids are never optimal; but since the true value will sometimes be high, information aggregation is likely to fail. (Picture a model with types and signals both drawn from [0, 1]. Information aggregation would require someone to bid 1 given the highest signal. But this would require E(z x i = 1, max j i x j < 1) = 1 If the strength of a single bidder s information is limited, this would not happen.) Pesendorfer and Swinkels, however, point out that if we want to use auctions as an analogy for markets, we may want to consider large numbers of sellers as well as buyers That is, if we re letting n go to, why not let k get large as well? So Pesendorfer and Swinkels consider a series of auctions {A r } r=1, where auction S r has n r bidders and k r objects for sale (using a k r + 1 st -price auction) Pesendorfer and Swinkels point out that in addition to the usual winner s curse, there is a corresponding loser s curse when I lose at a signal x i, I learn that at least k r of my opponents had signals greater than that And when kr n r is away from both 0 and 1, these two curses are of the same magnitude, so when I conditional on being pivotal, I am neither massively optimistic nor massively pessimistic P and S specifically assume Milgrom s condition does not hold, that is, any one signal cannot cause you to update your prior on a particular true value by an unbounded amount; so when k is fixed, information aggregation would fail 6

7 Pesendorfer and Swinkels model Values are drawn from a distribution F on [0, 1], with a density f which is bounded above and below away from 0: for some η 1 > 0, for all v [0, 1]. 1 η 1 > f(v) > η 1 Conditional on a value v, signals are drawn from a distribution G(x v) with a density g(x v) such that... g(x v) continuously differentiable in v g and g(x v) v is continuous in x g(x v) bounded above and below away from 0: there is some η 2 > 0 such that 1 η 2 > g(x v) > η 2 for all (x, v). (This is in contrast to Milgrom.) monotone likelihood ratio holds for some η 3, g(1 v) v g(0 v) > η 3 for all v, so higher signals are strictly better news (The last assumption ensures that, while some signals may be the same, not all signals are the same; that is, at the very least, the best possible signal is better news than the worst possible signal. While Pesendorfer and Swinkels generally work in a continuous world, they also give an example with continuous values but just two signals, a good signal and a bad signal. Without this last assumption, information could not aggregate, because all the bidders signals would not be enough to predict v.) They do allow for some range of signals which are equivalent as is the case when there are just good and bad but they point out that in the symmetric equilibrium, bidders still make different bids at different signals, even when the signals convey the same information. (This is the same as saying that if there is a particular signal received with positive probability, bidders play a mixed strategy at that signal.) 7

8 Results in P and S Pesendorfer and Swinkels also offer a proof that the equilibrium considered by Milgrom and in the other papers is the only symmetric equilibrium, which is nice. The formula for equilibrium bids is the same as above. P and S define double largeness along the series of auctions {A r } as both the number of winners and the number of losers in the auction going to infinity, that is, the sequence satisfies double largeness if both k r and n r k r go to infinity as r. Double largeness turns out to be both necessary and sufficient for information aggregation: Theorem 2. The sequence of auctions {A r } satisfies full information aggregation (that is, p r v converges to 0 in probability) if and only if it satisfies double-largeness. The fact that double-largeness is necessary follows from Milgrom. If k r does not go to infinity, it might as well be constant; but since Milgrom s condition on informativeness of signals is violated, information aggregation does not hold. In the case of n r k r being finite, the exact opposite intuition holds. Now, equilibrium bids are conditioned on an infinite number of signals higher than your own, with nothing to balance them; so low bids are no longer optimal unless Milgrom s condition holds. One way to think about the intuition for this. First, suppose that instead of the actual equilibrium bid function, each bidder were to bid E(Z) conditional on him being the k + 1 st -highest signal, that is, having exactly k above him and n k 1 below him. In that case, the winning bid is the expected value of the object, conditional on the k + 1 st -highest signal being equal to what the k + 1 st -highest signal actually is. But as k and n get large, the k + 1 st highest of n becomes less and less stochastic, and more and more closely converges to the (k + 1)/n point in the distribution of signals conditional on the true underlying value. (That is, the k+1 st signal out of n converges in probability to G 1 (k+1/n v).) But since this distribution shifts with v, this reveals v exactly; since the k + 1 st bidder assumes he is k + 1 st, his bid is conditioned on the correct value of v, and his bid sets the price. Of course, the winning bidder actually bids as if he s tied for k th highest signal, rather than being below the k th highest signal; but as n gets large, this difference in E(Z) due to moving one bidder s signal vanishes. What s interesting about this result is that it does not require k r /n r to be an interior point. That is, although k r and n r must be unbounded, they do not have to approach infinity at the same rate; so the ratio k r /n r could go to 0 or 1 without any problem. (For instance, it could be that n r = r and k r = r.) Pesendorfer and Swinkels explain it this way: when k r grows slower than n r, so k r /n r goes to 0, a large fraction of the 8

9 bidders are inactive, that is, submit bids very close to 0; so the ratio of k r to the number of serious bidders ends up bounded away from 0, and the loser s curse is not overpowering. Degrading Information In Wilson/Milgrom and Pesendorfer and Swinkels, it was assumed that the informativeness of each bidder s signal did not degrade as n increased That is, the relationship between signals and value was constant as the number of bidders grew P and S also give an example where, as k r and n r grow, a smaller and smaller fraction of the population actually has useful information They consider a model where there are three possible signals: a good signal, a bad signal, and an uninformative signal, that is, a signal that does not cause its recipient to update his prior on v. They show that information can still aggregate if the fraction of bidders who get informative signals shrinks as r grows, just so long as it does not shrink too quickly. If a fraction 2γ r of bidders get an informative signal in auction r, information aggregates if, in addition to double-largeness, γ r nr. So γ r must go to 0 slower than 1 nr. Ilan Kremer ( Information Aggregation in Common Value Auctions, Econometrica 2002) greatly simplifies the proofs of many of the existing results, and recasts things in terms of some new definitions. Kremer also gives an example where signals get less informative as the number of bidders grows and information fails to aggregate. case, but it s still an interesting example. He does it in a bit of an edge Suppose that in auction r, there are r sellers and 2r buyers, that is, k r n r = 2r. Suppose that signals are uniformly U[0, 1], and that { 1 1 if V = n sn > otherwise = r and The pivotal bidder will have signal very close to 1 2, and will bid as if half the signals are below his and half above; but because of the discontinuity, this could be either 0 or 1 with positive probability, so information aggregation fails. 9

10 Uncertain Bidders Note that in the models above, in each auction A r, there are k r sellers and n r bidders, and both these facts are common knowledge That is, in each auction, even as they grow large, the number of prizes, and the number of bidders, is known A recent working paper by Harstad, Pekec, and Tsetlin ( Information Aggregation in Auctions with an Unknown Number of Bidders ) shows that this assumption is very nearly required for information aggregation Suppose that each auction A r is actually stochastic: for i from 1 to something finite, with probability π i r, auction A r has k i r objects for sale and n i r bidders kr Harstad, Pekec and Tsetlin show that as r goes to infinity, if max i k i min n i r i i r n i r not go to 0, information does not aggregate does That is, if uncertainty persists as to the fraction of bidders who will win as the auctions get big, the price paid will not always match the value of the object (They basically show that the unique well-behaved symmetric equilibrium involves bidders bidding a weighted average of what they would bid in the auction with k i r winners and n i r bidders if they knew k i r and n i r; and that as long as there remains uncertainty about what they call the success ratio, k i r/n i r, the winning bid cannot match the value in every i state.) The assumption that bidders know how many opponents they re up against is a standard one we make for tractability, and probably makes sense in small settings; but as n and auctions begin to resemble large markets, it s reasonable to question whether this is the case (Harstad, Pekec and Tsetlin further suggest proportional selling if information aggregation is a goal; that is, if the seller can commit to selling to a fixed fraction of the bidders who show up, then k/n is fixed and information aggregates. However, I m not sure why this would be in the seller s interest, either ex ante or ex post.) Up next... No lecture Thursday. Next Tuesday: auctions with endogenous participation (entry). Next Thursday: auctions with endogenous information. 10