Systematic Price Differences Between Successive Auctionsare no Anomaly

SYSTEMATIC PRICE DIFFERENCES BETWEEN SUCCESSIVE AUCTIONS ARE NO ANOMALY

JANE BLACK AND DAVID DE MEZA University of Exeter

England

ldentical cases of wine are often auctioned one immediately after another. Ashenfelter (2989) reports that on average, the later lots fetch less. Such a systematic price difference seems anomalous, the more so because it is shown here that rational expectations imply not equal, but rising, prices. Risk aversion is an obvious way of reconciling the evidence with rational behavior. There is an alternative explanation. The auctions observed by Ashenfelter involved a buyer's option, whereby the first-round winner could purchase further cases at the same price. I t is shown that this feature may both account for the observed price trajectory and raise seller revenue.

1. INTRODUCTION

Two identical cases of wine are listed for auction, one to be sold immediately after the other. All bidders are rational, risk neutral, and are buying for personal consumption. Everyone is fully informed of the quality of the wine, but bidders do not know each others' tastes. As the owner of one of the cases, would you prefer it to be auctioned first or second? At first sight the order of sale should make no difference to the expected price, for, were it otherwise, rational bidders would surely raise their bids in the auction with the lower expected price. It is shown later that this is not so; the equilibrium expected price in the second auction exceeds that in the first. This result not only challenges intuition but also appears to contradict the finding of Weber (1983) that under the specified circumstances, the sequence of prices is a martingale. But Weber assumes that bidders only want one item, whereas here additional purchases have a positive value.

Rather than later lots selling for more, Ashenfelter (1989) reports evidence from wine auctions that price falls are typical. He attributes this violation of the law of one price to bidder risk aversion. Here it is shown that the price decline may instead be explained by the pres-

We are grateful to John Black, Martin Cripps, Paul Milgrom, and seminar participants at Warwick for their comments and to Gareth Myles for generous help with the computing.

0 1993 The Massachusetts Institute of Technology. Journal of Economics & Management Strategy, Volume 1, Number 4, Winter 1992

608 Journal of Economics & Management Strategy

ence of a ”buyer’s option,” whereby the winner of the first auction has the opportunity to buy further items at the same price.

Even without a buyer’s option, at first sight there would appear to be several possible reasons for why the intuitively appealing arbitrage argument should be wrong. Buyers might bid up in the first round in order to try to acquire both items so that on average, the first item would sell for more than the second. On the other hand, buyers might bid less than the price they expect the second item to fetch, on the grounds that this expected price is an average that includes some prices in excess of their own valuation and they are interested only in the expected price conditional on their making a purchase. Third, when one decides how much to bid in the first auction, it is rational to evaluate what would have happened in the second round conditional on the bid succeeding. In fact the price of the first item is determined by a bid that fails, which therefore underestimates the strength of the competition.

It is only this last argument that has an element of truth. To see this more precisely, consider a two-bidder English auction, initially with no buyer’s option. Assume that the value to a bidder of acquiring both lots is less than twice the value of a single lot. Suppose the price of the first unit has reached P* and that, given her valuation pair, it is borderline whether it is worth a buyer’s while to raise the bidding fractionally. Such a person must be indifferent between losing at P* or winning at an epsilon more. To determine P*, the consequences of letting her rival win when he too has a price threshold of P* must be evaluated. Because the rival is also at the margin, his valuation of a second unit must be less than her valuation of a single unit, so, if he wins the first item, she is sure to win the second. Therefore, she can only be at the margin if she expects to pay P* in the second auction. If both bidders choose the same drop-out price, the price of the first unit is thus an unbiased predictor of the second price. However, there is virtually no chance that the preferences of the two bidders are such that they would both wish to set the same maximum bid. With probability one the winner would have been prepared to bid more if forced to. Come the second round, the winner of the first item will bid up to his or her second valuation. Therefore, the price in the second auction will be at least equal to this value. Under plausible assumptions, there is a positive correlation between the maximum bid a person will make in the first round and his or her second valuation. Hence, because P* is the expected price of the second unit conditional on the winner of the first unit being willing to bid at most P* for it, if the winner would, if necessary, have bid even more, P* is an underestimate of the expected price of the second unit.

Suppose now that a buyer’s option is introduced. A bidder can no

Successive Auctions 609

longer assume that she will win at least one item if her first valuation is higher than her rival's second valuation, because he may win the first round and exercise the option. If she is at the margin of whether to raise the bidding, she is concerned not only with whether doing so would result in her winning at the current price but also with whether this would be enough to block the exercise of the option by a rival with a higher bid threshold than her own. It will never be advantageous for a bidder to bid in excess of her own valuation, and so the winner will always exercise the option if the selling price is below his second valuation. But in the two-bidder case, if the option is not exercised, the second auction price is the first-round winner's second valuation. Hence, the second item must necessarily sell for less than the first.

The standard result that English and second-price auctions have identical outcomes when valuations are private extends to sequential auctions if there are two items and just two bidders. With three or more bidders, the exact correspondence breaks down because in the first English auction, the course of the bidding may reveal information relevant to predicting the price that the second item will fetch. The formal analysis that follows is for second-price, sealed-bid auctions, but it is argued later that the relationship between the expected prices will also apply to an English auction with several bidders.

The focus of the paper is on positive economics, and issues of optimal multiunit auction design are avoided. It is nevertheless of relevance whether a buyer's option increases expected revenue from the sale. In our numerical examples, introducing a buyer's option is in the seller's interest. If one assumes away transactions costs and so forth, even better schemes can be devised (Maskin and Riley, 1989). It is implicitly assumed that it is too costly to administer the exotic auction forms proposed. It would certainly be interesting to have a theory of why these forms are not typically observed. On this and other issues, see the excellent survey of the theory and practice of auctions provided by McAfee and McMillan (1987).

The succeeding sections of the paper develop the analysis more generally, providing intuitive accounts of the steps in the argument but relegating formal proofs to the appendix. Numerical results are also provided. Comparisons with other models are drawn in the conclusions, and key empirical implications are identified.

2. T H E M O D E L

Two items are to be sold by sequential second-price, sealed-bid auctions. Each of the n bidders has a valuation pair (b ,P) . Agents can only observe their own b and 0, but they know the form of the distribu-

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tions from which their rivals’ valuations are drawn. The b values are independent drawings from a distribution f(b). Two different assumptions about the process generating the j3 values are made. The simpler is that j3 = kb, where k is the same for all bidders. The other is that j3 is drawn from a distribution g(P!b) that is conditioned on the particular b value drawn.

If an agent secures just one item, his utility is given by b less whatever he has had to pay. It is assumed that income effects are absent, so that the benefit of winning a second unit is /3 less its cost, and this is independent of the price paid for the first unit. j3 is assumed to be less than or equal to b, which rules out situations where items are more valuable in pairs than singly. For the case where /3 is random, it is, however, assumed that the higher is b the more likely an agent is to have a high p value, that is, 5 0. This stochastic dominance property is a more stringent condition than simply assuming that the expected value of /3 is increasing in b. One justification for assuming stochastic dominance is that if an individual has characteris- tics such as a high income that lead him to value a single item highly, then it is likely that these same factors will also lead to a second unit being highly valued. Alternatively, suppose that the two valuations are drawn independently from some distribution h(.). If only one item is acquired, then this is worth whichever is the larger of the two values, while the smaller drawing gives the worth of an additional item. The conditional distribution function for j3 is H(P)IH(b). This is decreasing in b and so satisfies the required condition.

After the first auction, the winner is informed. The release of more comprehensive bid information would not affect the outcomes. This is because in the second auction, the standard result applies that for a second-price, sealed-bid auction, there is a dominant strategy. The agent will bid his true valuation: /3 if he was successful in the first auction and b if he was not. The question is what will an agent with valuation pair (b,P) bid in the first auction?

Consider first the case where /3 = kb. If one assumes that the bidder believes his rivals’ bids are increasing in b, he is concerned with the highest and the second highest of the n - 1 rival b values. Label these bH and b,.

PROPOSITION I: For the case where p = kb, the unique symmetric- equilibrium bid function for the first auction is

z(b) = E[max(kb, b,) I b, = b1.

This proposition states that the equilibrium strategy is to bid what you would expect to pay in the second auction if the highest b value of all


your rivals is the same as your own, and this rival wins the first auction.

It can be readily checked that it does not pay to deviate if others are following this strategy, The function z(b), given earlier, is increasing in b, so a bidder knows that the highest rival bid will be made by the bidder with valuation bH. Suppose the deviant bids just below z and by doing so, he just fails to win the first auction. The highest rival b value must therefore be the same as his own, and so he will acquire the second item at either kb, = kb or b,, whichever is greater. But the expected value of the maximum of these is equal to z(b). If he deviates by bidding slightly more than z and thereby wins the first auction, he will lose the second auction. A risk-neutral bidder will be indifferent between just winning and just losing the first auction at bid z. Unique- ness is addressed in the appendix.

Now consider the case where p is stochastic with cumulative distribution function G(Plb) where dG(plb)lab I 0. Define PH as the p value of the rival with b value bH.

PROPOSITION 2: If dG(plb)ldb I 0, a symmetric equilibrium bid function, z(b,p), is

z(b,p) = E[max@,,b,)lb, = bl.

For n 2 3 bidders, this is the unique symmetric equilibrium bid function.

As for the simpler case previously, in the first round, a bidder with first valuation b bids what he expects to pay in the second auction if he loses the first auction, and the highest of his rivals’ b valuations is the same as his own.

Again it can be checked that this is an equilibrium strategy by considering the consequences of deviating. Note that the bid function, z(b,P), earlier, while it depends upon expectations about the p value of a rival with the same b as the bidder, does not depend upon the bidder’s own p value. Given the assumption that aG(plb)/ab I 0, z(b,P) is increasing in b, and so, if others are following this strategy, the highest rival bid will be made by whichever of the other bidders has the highest b value. If a deviant bids marginally below z(b,p) and by doing so just loses the auction, he loses to a rival with the same b value as himself, and, hence, he will win the second auction at an expected price equal to z(b$). If he deviates by bidding a little more than z and so wins the first auction, then he will pay price z. He cannot hope to acquire both items by bidding up, because, if by so doing he displaces a rival with a b value equal to his own, this rival bidder will certainly win the second auction. The uniqueness of the equilibrium for n z- 3 bidders is demonstrated in the appendix.


PROPOSITION 3: If aG(Plb)/db 5 0 and n = 2, the following also consti- tutes a symmetric equilibrium bid function

z(b,P) = P. If there are only two bidders, then the loser of the first auction will win the second auction provided his b value is greater than his rival’s P. Suppose a bidder believes his rival is following the strategy of bidding her own P value. If by bidding slightly less than z , she loses the first auction, when by bidding slightly more, she could have won it, then she will win the next auction by paying her rival’s p value, but this must equal z, and so she is indifferent between winning and losing the first auction at price z .

PROPOSITION 4: A sufficient condition for the expected price at which the second item is sold to be above that at which the first item is sold is that either P = kb or that, for stochastic P, t?G(P/b)iab I 0.

The first item is sold at a price equal to the second highest bid. For the case where P = kb and for the case where p is stochastic and n L 3, this will be the bid of the person with the second highest b value. He bids what he expects to pay in the second auction if he is just defeated in the first auction by a rival with the same b value as his own, that is, the expected value of the maximum of the winner’s P value and the highest remaining b value conditional on this being less than his own. This is a biased estimate of the price that an item will on average fetch in the second auction, because with probability one the winner of the first auction will have a higher b value than the runner-up. If the winner of the first auction has a p value in excess of the runner-up’s b value, then he dso acquires the second item paying the b value of the first-round loser, which must be greater than the runner-up’s first auction bid. Even if the runner-up does win the second auction, he will pay more on average than his bid of z , because while the runner- up’s bid is correctly conditioned on the highest remaining b value being less than his own, he makes his calculations about the likely P value of the winner on the assumption that the winner‘s b value is the same as his own. In fact, it will be higher and so, given the assumptions about how P is related to b, he will underestimate the average value of his rival’s P.

This is proved more formally in the appendix.

3. THE BUYER’S OPTION

With a buyer’s option, the successful bidder in the first auction can purchase as many items as he likes (or are available) at the first auc-


tion price. Introducing such an option can reverse the conclusions about the price trajectory.

With the buyer's option, there is no guarantee that there will be a second auction. A bidder must now take into account that even if raising his bid would not enable him to win the first auction, its level matters. If a higher bid exceeds the winner's p, this will block the exercise of the option and so give the bidder the chance of acquiring the second item. Unlike the no-option case, the equilibrium bid function may depend upon p as well as b. Without the option, marginally increasing one's bid above the equilibrium level does not increase the probability of securing both items, but with the option it does. Con- sider first the case where j3 = kb. Assuming the equilibrium bid function, z(b), is increasing in b, with inverse function b-'(z), a bidder must consider the joint distribution of the two highest rival b values, b, and b,. Define rn(b,), as the marginal distribution of b,.

PROPOSITION 5: If p = kb and there is a buyer's option, the symmetric, pure strategy, equilibrium bid function that is nondecreasing in b must satisfy

db (z - E[max(kb, b,)lb, = b])m(b) - dz

= E b - maX(b,,Z)lkb, = z,b, < b P[bL < blkb, = z]rn(z/k). [ 1 Suppose a bidder considers bidding slightly below the equilibrium bid function, z . At bid z his chance of winning the item is P[b, < b-'(z)]. A marginal reduction in z lowers this probability by rn(b-'(z))db-'(z)/dz. Evaluated at z = z(b), this reduction is m(b)db/dz. If he does just lose, it is to the rival with b, = b, so rather than acquiring the item at price z, he will win the second auction at an expected price of E[max(kb,b,)(b, = b]. He must also take into account the blocking potential of his bid. This is only relevant if his is the second highest bid, that is, if b, < b. If his is the second highest bid, then the chance that his rival will exercise the option is P[b, > z/k]. By marginally lowering his bid, he increases this probability by rn(z/k). Should this marginal deviation result in the winner exercising the option, then he loses the opportunity to acquire the second item at an expected price equal to the expected maximum of bL and kb, = z . For an equilibrium, it must not pay to deviate, so combin- ing these consequences gives the condition of Proposition 5. Note that z(b) is greater than kb, and, hence, the motivation to increase z in order to try to secure both items does not apply here because the bidder would not wish to exercise at price t. A more formal derivation of the equilibrium condition is given in the appendix.

From Proposition 5 it follows:


PROPOSITION 6: If p = kb and mtb,) > 0 for all values of bH within the support of the distribution, a bidder will bid more in the first auction under the buyer's option rules than without a buyer's option, unless the bid in the absence of an option would be sufficiently high to block the exercise of the option by all other bidders.

If m(z/k) = 0, given the assumption that m(.) > 0 within the support of the distribution, no rival could have a b value as large as zik. If this is the case, the right-hand side of the equation given in Proposition 5 is zero, and, thus, z(b) = E[max(kb.b$, = b], but this is the equilibrium bid function for the no-option case. If m(z/k) > 0, then, because the right-hand of the equation must be positive, the equilibrium bid exceeds the no-option bid.

PROPOSITION 7: If p = kb a buyer's option will reverse the price trajectory so that the expected price at which the second item sells is less than that fetched by the first item if n = 2 and may reverse it for n > 2 .

The option increases the first round bids for all or for some bidders compared to the no-option case. Define b, as the highest b value of the n bidders and p, as the p value of this bidder. Define bn-, and b,-2 as the next two highest b values. If p,, > z the winner will exercise the option and both items sell for z, but if p, < z , then there will be a second auction, and the item will sell for p, or bn-*, whichever is larger. If p, is the larger value, then, because this is less than z , the second item necessarily commands a lower price. For n = 2, p, is the only relevant value, and so, if there is a second auction, the second item must sell for less than the first. If n 2 3 the second item may sell for bn-2. If this exceeds z , then the second item will sell for more on these occasions. We need to consider the expected price fetched by the second item. The bidder with value b,-, determines the selling price in the first auction. His bid depends on the two components-what he would pay in the second auction if he just lost to a bidder with the same b value as his own, and what his expected benefit would be from just blocking the exercise of the option by someone with a higher b than his own. He underestimates the average price he will pay in the second auction if he is the runner-up in the first because he loses to someone, not with the same b value as his own, but with a higher b value. This was the reason for the upward price trajectory in the no-option case. The question is whether the effect of the blocking motive is sufficient to offset this bias. It can be shown that the trajectory may indeed be reversed. The equilibrium bid function can be found for particular assumptions about the distribution f(b) from which the b values are drawn by solving the equation given in Proposition 5. In Table I the expected prices fetched


TABLE 1.

Expected selling prices of the two items

with the option without the option

n = 3,k = 5

Pl P2

0.246 0.228

Mean 0.237 PI

P2

0.375 0.361

Mean 0.368

0.167 0.250 0.208 0.313 0.375 0.344

by the two items are given for the case where b is uniformly distributed over the interval (O,l), for n = 2 and n = 3. The second item sells for less on average. Figure 1 shows the equilibrium bid function for n = 3. Details of the computations are given in the appendix.

PROPOSITION 8: For p = kb, introducing a buyer's option may increase the total expected receipts from selling the two items.

The option increases the price for which the first item is sold, but it depresses the price for the second because whenever the option is exercised, the second item would have fetched a higher price without the option. The net effect on revenue is not clear. However, for the examples given in Table I, the option does increase the combined expected revenue. The option may result in an inefficient allocation because sometimes the second item is not acquired by the bidder who places the highest valuation on it. Nevertheless, from the seller's viewpoint, the option may increase expected receipts.

Finding the equilibrium conditions for the case where p is stochastic is more difficult as any equilibrium bid function will depend on both b and p. It will no longer be the case that the highest bid is necessarily made by the bidder with the highest b value or that the runner-up will have the second highest b value.

PROPOSITION 9: For stochastic p , with a buyer's option the expected price for which the second item is sold is less than that commanded by the first if there are just two bidders. With three bidders, a sufficient condition for expecting price declines is that the conditional distribution of p given b is such that g(pIp < z) does not depend on b V z < b.l

1. One assumption that would imply this condition on the distribution of p is that the two valuations are the larger and the smaller of two random drawings from the same distribution.


FIGURE I . BID FUNCTION FOR N = 3

For n = 2 the argument used earlier must apply. If the option is not exercised, then the first-round winner's p value must be less than z , and, because his p value will set the second price, the trajectory must be downward. The case where N > 2 is more difficult. For stochastic /3 any symmetric equilibrium bid function will depend upon the bidder's p value as well as his b because if he has a high p value, he will want to bid up to try to secure both items. There is no guarantee either that the first item will be won by the bidder with the highest b value or that the second highest bid will be made by the bidder with the second highest b value.

Consider the bidder who makes the second highest bid in the first auction. She must weigh up the gains and losses frommarginallyreduc- ing her bid from the equilibrium level, z. Let P, be the probability that if she loses to a bid of z , this bidder then exercises the option. Conditional on the winner having a p value less than z , let P2 be the probability that she loses the second auction. If she does lose, it must be to someone who bid less than z in the first auction. Let P, = 1 - P, be the probability that she wins the second auction if there is one, and let V be what she expects to pay in this event. If by not deviating she would have won the first auction, her net losses from a downward deviation are

( b - Z ) - (1 - PI)P3(b - V) = P,b + (1 - P1)P2b + (1 - P,)P,V - Z.


She may also be concerned to bid sufficiently high to reduce the likelihood of the exercise of the option by a rival bidding more than Z.

Her bid, therefore, will be at least equal to:

bP, + (1 - P,)(bP, + P3TI).

Now suppose the second highest bidder makes a bid of z , and the option is not exercised. The second item will be sold for either the p value of the highest bidder, which must be less than z , or at the second highest remaining b value. If the distribution of p, given /3 < z, does not depend upon a bidder's b value, the second highest bidder correctly estimates what she will pay if she wins the second auction, that is, she forms an unbiased estimate of V. Given she has rational expectations, she also has correct beliefs about the probability that she will win the auction in the event that the option is not exercised and that other bidders have all bid less than z , that is, she knows P,. She also correctly estimates P,, the chance that if the option is not exercised and all other bidders have bid less than z , she will nonetheless lose the second auction. If this occurs and n = 3, the second item will sell for her own b value because she must be the runner-up in the second auction, but in forming her bid of z , she weights this probability by her own b value. Conditional on the option not being exercised, if she is the second highest bidder and bids z , she correctly estimates the price at which the second item will be sold and bids an amount at least equal to this. If n = 3 the second item on average sells for an amount no greater than the second highest bid, and will on average sell for less if the second highest bid incorporates a bloclung element. This is shown somewhat more formally in the appendix.

4. ENGLISH AUCTIONS

The price trajectory result extends to English auctions. Explicit model- ing requires an assumption concerning what happens when more than one of the potential buyers are willing to raise the bidding. This matters because, in the first auction, agents use the bids made so far to update their estimates of the distribution of their rivals' valuations and, hence, the price that they expect the second item to fetch. So in any particular auction, the price of the first item may be different according to whether bids are made in strict sequence, or it is random which of those willing to raise the bidding actually does so. Under either assumption, in the absence of a buyer's option, the expected price of the second item exceeds that of the first.

To establish the price trajectory results, the reasoning is essen- tially the same as in a second-price auction. Consider the penultimate


bid in the first English auction. For sufficiently small bid-increments, this bid will be made by the person with the second highest b value. As in the sealed-bid auction, he estimates what the second item would fetch if a rival with the same b as his own were to win the first unit given that no one has an even higher b value. In comparison with the second-price, sealed-bid auction, being able to observe the sequence of bids will sharpen the bidder’s estimates of the price in the second auction, because the bidding will reveal information about the third highest b value. But just as in the second-price auction, it is rational for the person with the second highest b to bid as if no one else has a higher b, so it is in the English auction. Outbidding a rival in the first auction does not increase a bidder’s chances of winning both items because if he does beat someone with a higher b than his own, he will certainly not win the second auction. First-auction bids therefore depend only on the bidder’s b value. A bidder has to decide whether to raise the bidding if it stands against him.

Postulate the following strategy: Assume that the opponent who has made the current highest bid has the same b value as one’s own and that other bidders will not raise the price. (This assumption may well be false, but then the auction continues and expectations are recomputed.) On these assumptions the bidder can work out what he expects to pay in the second auction if he lets his rival win. If the bidding is currently less than this amount, he should raise the bidding. This will be an equilibrium strategy if it does not pay any bidder to deviate from it. Suppose all other players do follow this strategy. If a deviant outbids a rival with a higher b than his own, then he will pay marginally more than the average price that an item will fetch in the second auction. Thus, there is no point in trying to outbid a rival with a higher b. If the deviant drops out of the bidding at a point below that implied by the proposed strategy, then the expected price in the second auction is higher than his final bid. Hence, he has an incentive to raise the bidding. Therefore, the strategy is an equilibrium one.

Just as in the sealed-bid auction, on average the second item will sell for more than the first. Each bidder acts on the ”as if” assumption that his rival has the same b value as his own, and the first item will sell on average for the amount that the person with the second highest b estimates the second item would sell for, making this assumption. However, sometimes the second auction will be won by the person with the highest b value at price bn-l, a possibility that is rationally ignored by the losing bidder, and sometimes it will be sold at price P,, which will on average be greater than the expected value of P that the loser used in computing his bid, because he has made his bid assuming his rival’s b to be the same as his own. These two factors bias the


first price downward compared to the second auction outcome, just as they did in the sealed-bid case. Also by analogy with the sealed-bid case, introducing a buyer’s option will induce bidders to increase their first auction bids to try to block the exercise of the option.

5. EXTENSIONS, PREDICTIONS, AND CONCLUSIONS

There are a variety of alternative sequential auction settings in which to examine price trajectories. In the common-value model, the items have an objective worth the same for all participants. At the time of sale, it is not known for sure what this value is. Bidders have access to different private information, and so their estimates of value differ. Ortega- Reichert (1968) and Hausch (1986), among others, have shown that under these circumstances, because agents infer from their rivals what signals they have observed, there is a strategic incentive to put in a low bid for the first item. The aim is to fool rivals into believing that there is evidence that the items are of low value. If the pretence succeeds, a second-round bargain is obtained. Of course, rational rivals recognize the incentive and see through the attempted deception. First-round bidding behavior is nevertheless affected. However, in the second and last round of the auction, there is no strategic cause for underbidding. It follows that on average, the first item sells for less than its true worth, while the second unit fetches its expected value, resulting in an upward price trajectory. This strategic reason for a rising price path is not relevant to the private value model analyzed in this paper. When there are just two items to be sold, second-round bids are based solely on the valuation a bidder places on acquiring the item that is not influenced by how others have bid in the first round.

Krishna (1990) considers multiunit private value auctions in which bidders are fully informed of each other’s preferences. Here it is possible that price declines may be observed. The reason is that a bidder who could profitably outbid a rival in all the auctions may nonetheless increase her surplus by letting the rival win the earlier items if this means that in the subsequent auctions, the rival drops out of the bidding at a lower price. This can be illustrated for a two- bidder, two-item example. Suppose bidder 1 has valuations @,@, and bidder 2 has valuations @,a) where b > p > a > a.2 If bidder 1 wins

2. A primary focus of Krishna’s paper is on situations in which it is worth more than twice as much to acquire two items as one, for example, because doing so aug- ments market power over resales. Her general conclusions discussed in the text also apply to such cases.

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the first auction at price z, she will have to pay a for the second item. Her total surplus is b + /3 - z - a. If she lets her rival win the first round, her surplus is b - a. She therefore will pay up to z+ = p - (a - a). If a > z*, that is, a > (p + a)/2, her opponent will outbid her. The first item then sells for z*, and the second for a, but a is less than z*, thus giving a declining price trajectory.3 For an English auction, this result requires the assumption that a bidder will enter an auction even if she knows that she is not going to win. For the previous example, the bidder with the highest valuation pair has no positive incentive to bid up to z+ because her rival will outbid her. If she stays out of the first auction and her rival stays out of the second, then the price trajectory is level-both items selling for zero. Finally, Von der Fehr (1991) argues a case for declining price trajectories based on the argument that if there are participation costs, bidders will not enter auctions when they know that they are bound to lose.

Notice that in Krishna’s setting, price rises cannot occur, and so a buyer’s option would never be exercised. Hence, her model cannot be directly applied to explain Ashenfelter’s (1989) observations, in which options were frequently taken up. Ashenfelter’s own preferred explanation, developed by McAfee and Vincent (1991)) is that the pattern of price declines is due to buyer’s risk aversion. Because both risk aversion and a buyer’s option give rise to the same prediction, it will not be easy to use field data to distinguish between them. Ideally we would like observations on price paths before and after a buyer’s option has been introduced to discover whether it imparts an extra downward impetus to price trajectories. Experiments may be the best way to proceed.4

Even in the absence of a buyer’s option, some testable predictions do emerge from our analysis. One implication is that price rises are more likely to be found if the goods being sold are cheap, because risk aversion can then be neglected, and if there are only a few bidders. The upward bias in the price trajectory arises because the winner of the first auction may win or be runner-up in the second auction. With many bidders, there is lower chance of this occurring, and the expected price

3. If the number of bidders exceeds the number of items, a necessary condition for a price decline is that in an efficient allocation, all items would be acquired by a single agent. Notice that in this model, full information precludes a price rise. In the model we develop, equal ignorance rules out declining price paths.

4. Engelbrecht-Wiggans (1992) and Gale and Hausch (1992) consider the case of sequential private auctions with heterogeneous objects that are nevertheless “stochastically equivalent” in the sense that valuations are drawn from the same distribution. If one assumes bidders want at most one item and under various assumptions about when valuations become known, declining price are shown to be possible.


in the second auction approaches that in the first. A rather strong prediction, because it is implied even under risk aversion, is that, in the absence of a buyer’s option, if the same bidder wins both items, then the second item must fetch a higher price than the first.

Extending the analysis to three or more items is not so trivial if all the bids are revealed at the end of each round. It is then in each bidder‘s interest to bid low in the first round to attempt to deceive his rivals into believing that his valuations are low, although this will be self- defeating in equilibrium. This is analogous to the feigning motive in the two-item common value model. It is conjectured that increasing the number of items and mixing private and common value assumptions reinforces the upward bias in the price trajectory for the no-option case.

REFERENCES

Ashenfelter, Orley, 1989, “How Auctions Work for Wine and Art,” Journal of Economic

Engelbrecht-Wiggans, Richard, 1992, “Sequential Auctions of Stochastically Equivalent

Gale, Ian and Donald Hausch, 1992, “Bottom-Fishing and Declining Prices in Sequen-

Hausch, Donald B., 1986, “Multi-Object Auctions: Sequential vs Simultaneous Sales,”

Krishna, Kala, 1990, ”Auctions with Edogeneous Valuations; the Snowball Effect, and

Maskin, Eric and John Riley, 1989, in F. Hahn, ed., The Economics of Missing Markets:

McAfee, R. Preston, and John McMillan, 1987, “Auctions and Bidding,” Journal of

~~ , and Daniel Vincent, 1991, ”The Afternoon Effect,” CMSEMS discussion

Ortega-Reichert, A., 1968, “Models for Competitive Bidding under Uncertainty,” Depart-

Weber, Robert J., 1983, “Multiple-Object Auctions” in R. Engelbrecht-Wiggans et al.,

Von Der Fehr, Nils-Henrik, 1991, “Predatory Bidding in Sequential Auctions,” Nuffield

Perspectives, Summer, 23-36.

Objects,” mimeo, University of Illinois.

tial Auctions,” mimeo, University of Wisconsin.

Management Science, 32, 1599-1612.

Other Applications,” NEBR working paper 3483.

Information and Games, Oxford: Clarendon Press.

Economic Literature, June, 699-738

paper no 961, Northwestern University.

ment of Operations Research Technical Report No. 8, Stanford University

eds., Auctions, Bidding and Contracting, New York NYU Press, 165-691.

College discussion paper in Economics 61.

APPENDIX

UNIQUENESS OF S Y M M E T R I C EQUILIBRIUM BIDS GIVEN IN

PROPOSITIONS 1 AND 3

For the case where p = kb bids will depend only on b. For the case where p is stochastic, it is first necessary to show that if n 2 3, the equilibrium bid function depends only on b and not on p.


In a symmetric equilibrium bidders have identical beliefs about the distribution of the highest rival bid, y , of what the highest rival bid, w, in the second auction will be if they win the first, given y , and of what will be the highest rival bid, v, in the second auction conditional on y if they lose the first auction. Let the three relevant pdfs be m(y) , q(wly), and h(v1y). The expected gain from bidding z given the valuation pair (b,@ is

P

G(z,b,P) = 11 i’(b - y)m(y)dy + i’ 1 ( P - w)q(wly)m(y)dwdy

+ I, 1 (b - vP(vly)m(y)dvdy

dG/dz = {(b - Z ) + (p - w)q(wl~)dw - (b - u ) ~ ( u ~ z ) ~ v } I T z ( z ) . i p I‘ This must equal zero if z is at the equilibrium level.

dzGldzdb = P[v > b/y = z ] dZG/dzdp = P[w < ply = 21.

Unless these probabilities are both zero, should two bidders make the same equilibrium bid of z , it cannot be the case that both the b and the /3 values of one bidder lie below those of the other. w is at least as great as the b value of the first auction runner-up, so if he too has bid z, w must be at least as great as the p value of the winner. The first integral in the expression for dG/dz must therefore be zero when evaluated at the equilibrium bid. Thus, the equilibrium bid must satisfy

z = bP[v > bly = z ] + E[vlv < b, y = z]P[v < bly = 21.

Unless P[v > b/y = z ] = 0, two bidders can only make the same bid of z if they have the same b value, and, hence, z must be a function of b only. Now suppose this probability is zero. Consider the case where n 2 3 . v is the maximum of the p value of the first auction winner and of the highest remaining b value, b,, but P[b, > b/y = z ] implies that the by, the b value of the first auction winner, is no greater than b. Because he too has bid z , b can be no greater than by, and so both bidders must have the same b value. Thus, the bid function depends only on b if n z 3.

The next step is to establish that for n 2 3, the bid function must be increasing in b. The equilibrium symmetric bid function cannot be everywhere decreasing in b, because in equilibrium a bidder must be indifferent between just outbidding and just failing to outbid the highest rival bidder. Suppose the bid function is decreasing in b. If a bidder just fails to outbid another bidder, also bidding z(b), the other bidders must have higher b valuations, and so the loser will be outbid in the second auction. By deviating, he would suffer a finite loss of b - I.


h. h, h, b - I - d -a

FIGURE 2 . AN lNCONSISTEhrT BID FUNCTION

Discontinuities in otherwise increasing bid functions can also be ruled out. Consider a possible bid function as shown in Figure 2.

For n 2 3 a bidder with valuation b, from the first-order condition for z must bid at least b,, so z(bJ > b,, but z(b,) > z(bJ > b,. It cannot be rational for a bidder to bid more than his b value, and SO this condition cannot be satisfied. For n 2 3 the bid function depends only on b and is increasing in b. Given this result the highest rival bidder must be the bidder with the highest b value and, hence,

z = E[vlb, = b] = E[max(P,,b,)lb, = b],

where if p = kb, PH = kb. Notice that the fact, already established, that if two bidders have

different b's, they cannot be willing to make the same bid (for the second-round payoffs from just losing cannot then be the same) rules out mixed strategy first-round equilibria.

Now consider the case where n = 2. For the case where /3 = kb in a symmetric equilibrium, the bidder can infer his rival's b value from his bid, 6 = b-'(z). The equilibrium condition must be that

b - z = b - kb-l(z).

Thus

z = kb.


If P is stochastic the argument used above in the case where n 2 3, that P[v > bly = z] = 0 implies that the highest bidder must be the rival with the highest b value does not apply as v = Py where P,, is the P value of the first-round winner. This only needs to be less than b for the probability to be zero. Thus, z(b,P) = P or z = EIPlbH = b] or any linear combination of these are possible symmetric equilibrium bid functions.

Proof of Proposition 4. For n = 2 the proof is trivial. The proof below is for n 2 3 and where P is stochastic. It is easily adapted for P = kb.

The first auction price is determined by the bid of the person with the second highest value of b, bn-,. Let b, and b, be the two largest drawings from a sample size of n - l, and let PH be the /3 value associated with b,. The expected price is therefore found by taking the expecta- tion over bn-l of ~ ( b ~ - ~ , p ~ - , ) :

E[pll = Ebn-1EbL,3H max(bL,PH)lbH=bn-l . [ 1 The joint distribution of b , , P H , and bn-, given b, = bn-, is identical to be joint distribution of bn-2, Pn-l, and bn-,.

Thus,

E[pll = Eb,-1Ebn-Z,3n-l maX(bn-ptPn -1) I bn- 1 ] 1

and hence,

If one writes I , for the expression in the square brackets,

up11 = E b, - b, -z [I1 1 . If one takes expectations over b,,

In the second auction, the price is given by

Thus,


If one writes I, for the expression in the square brackets,

QP21 = Ebnb,-lb,-2[121

E [ p 2 1 - ELPI] = Eb,b,_lb,_z['2 - '11'

If one integrates by parts,

12 - 1, = Jbn-l [ G(Plb,-l) - G(Plb.)] "n-2

Thus, if the distribution of ,!3, given b = b,, stochastically dominates that of p given b = bn-l, it follows that ( I , - 11) is positive and, hence,

Proof of Proposition 5. Consider an individual with valuation b. He faces n - 1 rivals. The joint distribution of bH and b,, the two highest rival valuations, is

0 2 1 ' E[Pll.

h ( b L l b H ) m ( b H ) = (n - l)(n - 2)(F(bL))"-3~f(bL)f(bH).

If he bids an amount z*, and his rivals bid according to the bid function z(b), his expected gain is

where 6 = 0 if z* > kb, 6 = 1 if z* < kb. For a Symmetric Bayesian Nash equilibrium to exist, it must not

pay an individual to deviate, and so the derivative of this expression evaluated at 2% = z(b) must equal zero. Thus, if a symmetric bid

t must satisfy the condition function exists,

1 (b - bL)h(bL/b)dbL - ( b - kb)h(bLlb)dbL

+ ( b - z)h(b,/z/k)db, m(z/k) = 0. (1) (b - b,)h(b,lz/k)db, + [ 1 Now

1 bh(b,lb)db, = b.

626 Journal of Economics & Management SLrategy

If one rewrites eq. (l),

( l]h (b - b,)h(b,)z/k)db, + (b - z)h(b,lz/k)db,

but

Because the right-hand side of eq. (2) is nonnegative, this implies z 2 kb, and so 6 = 0. Therefore, eq. (2) gives

( z - E [ max(bL,kb)lb, = b m(b) 1 L ) (b - b,)h(bL/dk)db, + (b - Z)h(b,lz/k)db, m(zlk)dz/db. (3)

If there is a finite upper bound to b, 6, then m(z/k) will be zero if z/k > 6. For z > k6, eq. (2) gives

=

z = E [ max(b,,kb)Ib, = b].

This is the equilibrium bid function if there is an option, that is, once z is sufficiently large that there is a zero chance that a rival will exercise the option, there is no strategic reason to bid up in order to block the exercise of the option. However, if E[max(b,,kb)lb, = b] < kh, then if one inspects eq. (2), it can be seen that the equilibrium z , provided it exists, must exceed the bid appropriate to the no-option case. 0

SOLVING FOR THE U N I F O R M DISTRIBUTION

For the particular case where b has a uniform distribution over the interval [0,1] eq. (3) reduces to

(4) dz p->{(n - 1)z - b(n - 2 + p-1)) = 1 (p-1- n-1 z k

where A = 0 if z 2 k, A = 1 otherwise. This is a nonlinear homogeneous equation that can be solved

using standard methods. Numerical solutions have been found for the cases k = 0.5, for n = 2, and for n = 3.


For k = 0.5, n = 2, eq. (4) becomes

(5) dz z - 0.5b = 2(b - z)h 3 A = 0 if z L 0.5.

If z = 0.5, the LHS of the equation is zero when b = 1. If one defines v = z/b to satisfy eq. (4) and picks the constant of

integration to satisfy the condition v = 0.5 when b = 1, z, must satisfy

1.2368 z, + (5°.5 - 1)/4 ) 3+505b2005= ( 5°.5 + 1)/4 - v ) 3-505.

This can be solved numerically, and the resulting expected prices at which the first and second items are sold can be derived. The expected first price is E[v(b,)b,], where b, is the lower of the two valuations. The second item sells for v(b,) if 0.5b2 > v(b,), that is, if the option is exercised and for 0.5b2 if the option is not exercised. The second item must necessarily sell for less than the first for this two- bidder case.

(

For n = 3, k = 0.5, eq. (4) becomes

dz db b{z - 0.625b) = 4(b2 - z2)h - h = 0 if b > 0.8, h = 1 otherwise. (6)

In the absence of an option, the equilibrium bid is z = 0.625b. For b > 0.8 the no-option bid is sufficiently high that any exercise of the option is blocked. For b < 0.8 the solution to eq. (6) gives z in excess of 0.625b. Graph 1 gives a plot of z(b).5 With three bidders on those occasions when the option is not exercised, the second item may sell for more than the first. This will be true if b, > z(bJ and 0.5b3 < z(b2). The numerical calculations, however, show that for this case, the average price commanded by the second item is less than that of the first. (See Table I of the main text.) The option again raises the expected total revenue, by about 7.4% for this case. As the sample size increases, the expected price of both the first and the second item will tend toward the expected value of the third highest b both with and without an option, so it seems plausible that the beneficial effects on expected revenue of introducing an option will tend to zero.

Proof of Proposition 9. Define y as the highest rival bid and py as the value of this bidder. Let x be the second highest rival bid, and let b, be

5. Defining v = zib the solution to eq. (6) satisfies

2v + a WInb = 2(1 - aZ)ln(a - v ) - (1 + 2a2)1n(v2 + av + a ~ ) - 2 ~ 3 arctan( __ a*

where a = (a)+, and c is set to satisfy v = 0.625 when b = 0.8, v = 0.625 for b > 0.8.


the highest b value of other bidders apart from the rival making bid y. If by lowering her bid, a bidder just loses an auction she would otherwise have won, she will win the second auction provided &, < z and b, < b. Her net losses are therefore

b[ l - f"P, < zh, < bly = 211 + E[max(P,,b,>lP, < z,b, < b,y = z]P[P, < z,b, < bly = z] - z + S(p - z ) , where 6 = 1 if p > z.

In addition, by lowering her bid, she may fail to block the exercise of the auction when y > z . Her bid, therefore, must be at least as great as

b[l - J"P, < zh, < bly = 211 + E[max(P,,,b,)lP,, < z,b, < b,y = z]P[P, < z,b, < b(y = z ]

- -

bP(& > z(y = z ] + bP[P, < z,b, > bly = 21 + E[max(P,,b,)JP, < z,b, < b,y = ZIP[/$ < z,b, < b(y = z]

bP[py > zIy = z ] + {bP[b, > b(x < z] + E[max(pY,b,)l/3, < z,b, < b,y = z]P[b, < blx < z]}P[/3, < zIy = 21.

< z does not depend

rn = bP[b, > blx < 21 + E[max(p,,b,)lP, < z,b, < b,x < zlP[b, < blx < 21.

- -

Assume that the distribution of p, given that upon b, her bid is at least as great as

The price in the first auction is equal to the second highest bid. As- sume this is z and that this bidder has a first valuation of b.. If the option is exercised, the second item also sells for 2. If it is not, then p, < 2. Conditional on this event, either the second item is lost by the person making bid z , in which case the item is sold for b, or it is won by the z bidder, in which case it is sold for the maximum of p, and b,. The expected price is therefore equal to the previous expression m. Butt 2 rn, hence, the second item sells for no more than the first and typically will sell for less. 0

Systematic Price Differences Between Successive Auctionsare no Anomaly

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