Learning in sequential auctions.

Jeitschko, Thomas D.

1. Introduction

A fundamental question of economics concerns the impact of market structure and institutions on the allocation of resources. A class of trading mechanisms that has received much attention in this respect over the past several years is the transfer of resources through auctions. While tremendous advances have been made in understanding single-unit auctions (see the surveys by Milgrom 1985; McAfee and McMillan 1987; Wilson 1992; and Wolfstetter 1996), only recently have auction theorists begun to embed this analysis in broader and more realistic institutional settings.(1)

Auctions are often part of an institutionalized market in which trades are recurring events, for example, markets for government treasury securities, used cars, fresh fish, and many wholesale agricultural commodities (flowers, cheese, tobacco, and so on). The focus of this study is the potential impact of information transmission and learning when goods in these markets are auctioned sequentially.

The transmission of information and the resolution of uncertainty during a sequential auction can have important implications for bidding behavior and auction outcomes. First, each participant learns about the valuations of other bidders before it is "too late" to act on the information. Specifically, a trader is able to use the behavior of rivals in earlier rounds to update beliefs about rivals' valuations for the item currently on the trading block. This leads to the second informational effect. When deciding on bids in earlier auctions, bidders must take into account how these bids affect the flow of information. Specifically, bidders must account for how their own bids affect the information they obtain about their rivals in later auctions. Hence, sequential auctions exhibit two learning effects not present in static auctions: a direct effect in which information gleaned from earlier rounds is used to formulate current bids and an anticipation effect in which bidders incorporate the impact of their earlier bids on the direct effect in future rounds.

Despite the potential significance of information transmission in sequential auctions, most of the existing literature in this area has focused on other aspects of these markets. Gale and Hausch (1994), for example, present a model in which two "stochastically equivalent" objects are auctioned in sequence to two bidders with unit demands. However, because there is only one bidder left in the second auction, the learning effects described above are not present in this setting. In similar models, Engelbrecht-Wiggans (1994) and Bernhardt and Scoones (1994) consider scenarios in which values are assumed to be independent across both bidders and auctions and the bidders do not know their value of a particular object until it is up for sale. Together, these assumptions eliminate any learning during the sequence of sales. Finally, several papers on sequential auctions (e.g., Black and de Meza 1992) examine auction formats in which bidders have dominant strategies that do not depend on information.

Most research studying informational issues in sequential auctions has focused on the common value paradigm (see, e.g., Engelbrecht-Wiggans and Weber 1983; Hausch 1986; and Bikhchandani 1988). The focus of this study, however, is how bidding behavior is affected when bidders have independent private values, so that learning is not about the value of the objects for auction but about opponents' types. Since many sequential auctions are wholesale markets in which the goods purchased are retailed in downstream markets, the independent private values assumption is relevant if demand conditions in the downstream markets are independent of each other. Moreover, even if there is a common value component to bidders' values, the private values paradigm is relevant if all bidders are equally (not necessarily fully) informed about the common value elements of the good, so that the opponents' strategies reveal no additional information about one's own value of the good.

The papers most closely related to this investigation are Milgrom and Weber (1982), Weber (1983), and McAfee and Vincent (1993). In these studies, identical goods are sold sequentially through first price sealed bid auctions to bidders who have independent private values and unit demands. Rather than information transmission, however, the focus of these papers is the equilibrium price path. In fact, information transmission is of little importance in these models. In equilibrium, bidders assess their option value of continuing in further auctions if they lose the current auction. This option value depends solely on the probability of winning or losing the first auction, not, however, on estimations of how high the winning bid actually turns out to be. Given this option value, bidders bid their estimation of the highest valuation of their opponents, conditioning on their valuation being the highest. Because of this conditioning, bids are independent of the prior history of the game. In sum, bidders' strategies depend only on which auction they are currently in but not on any past price realizations or on estimations of future price realizations. Hence, bidders do not update their beliefs about their opponents in the course of the auction and therefore need not anticipate learning either.

The reason for the absence of learning and the anticipation of information is that valuations are modeled to come from a continuous distribution, so there is zero probability that any two bidders have the same valuations. If, however, a bidder thinks there is a positive probability that another bidder has the same valuation for an object for auction, this bidder can no longer condition his bids on him having the highest valuation. In this instance the bidder uses a mixed strategy to avoid tying bidders of the same type. This strategy depends critically on the probability of another bidder having the same valuation. Bidders' beliefs about this probability are updated in the course of the auction, and thus bidders' expected payoffs are affected by their learning about rivals' types. This illustrates the direct effect of information transmission mentioned above. To appreciate the indirect effect, note that in the first auction bidders must trade off the benefits associated with winning the auction and retiring against the benefits of losing the first round and learning more about their rivals' valuation in the second auction. In sum, learning and the anticipation of information being generated affect bidders' optimal strategies and hence expected payoffs throughout the sequence of sales.

The remainder of this paper contains three sections. In section 2 the model is formalized and the equilibrium discussed. In section 3 properties of the equilibrium are studied. It is shown that bidders who learn from the outcome of the first auction have higher expected payoffs. Moreover, the anticipation of information leads to greater payoffs when compared to "myopic" bidding in the first auction because myopic bidders are unaware of the trade-off between winning the first auction and being better informed in the second auction.

In equilibrium, prices may fluctuate. The probability that prices increase or decrease depends on the information generated in the first auction. However, regardless of this information, and thus independent of the probability of increasing or decreasing prices, the price in the second auction is, on average, the same as the price in the first auction. This result is also obtained in the case for a continuum of bidders' types (see, e.g., Weber 1983). Thus, although bidders' strategies are affected differently when types are distributed discretely as opposed to continuously, the equilibrium price path forms a martingale in either case. Another analogy is found when considering the institution chosen for the sale of the objects, namely, contrasting the sequential auction with a simultaneous (static) auction. Despite the fact that learning has no role, the static auction yields the same expected final allocation of goods and bids, as does the sequential auction. Thus, expected revenue equivalence studied in Weber (1983) and Engelbrecht-Wiggans (1988) carries over to the case of a discrete distribution of types.

Some brief concluding remarks appear in section 4, and the appendix contains the proofs to the propositions and theorems.

2. The Model and the Equilibrium

Two identical objects are auctioned in sequence to three bidders. Each bidder can be one of two possible types: have either a high valuation for a unit or a low valuation for a unit. Without loss of generality, valuations are normalized to 1 and 0. Bidders have an ex ante probability of [Rho] [Epsilon] (0,1) of having a high valuation for a unit of the goods, and 1 - [Rho] that they have a low valuation. They know their own valuation for a unit of the good and have beliefs [Rho] that any given opponent has a high valuation for a unit of the good. Bidders have unit demands; that is, a bidder's value for a second unit is nonpositive. Bidders have a reservation utility of zero, which they obtain if they do not win one of the goods; otherwise, their utility is given by the difference between their valuation of a unit and the price they pay. In both auctions the strategy space is the real line; that is, negative bids are permitted. Ties are broken by the roll of a die. The auction format is a first price sealed bid auction in which only the winning bid is announced.(2)

Notice that the auction format does not maximize expected revenue of the sellers. However, in most markets in which sequential auctions are used, the institutional framework is agreed on by both sellers and buyers, as these markets are often arranged by trade associations or the like. Thus, one would expect these markets to be efficient yet not necessarily producer surplus maximizing.(3) The model under consideration, which yields efficient allocations, serves as a formalization to highlight the informational issues that bidders are confronted with in these markets.

An equilibrium to this model is a strategy and set of beliefs, in each auction for each bidder, such that given the opponents' strategies the expected payoff of every bidder is maximized. The equilibrium presented and studied is the symmetric noncooperative perfect Bayesian equilibrium. The model illustrates how the observation of the outcome of the first auction allows bidders to learn and thus affects behavior in the second auction. It also shows how the anticipation of information available in the second auction impacts bidders' strategies in the first auction.

In equilibrium, bidders with a high valuation use behavioral strategies to avoid tying. This is particularly intuitive in the types of markets under consideration as bidders often encounter the same set of opponents in consecutive sequential auctions and thus, in mixing their bids, remain unpredictable to these opponents. Bidders with a low valuation for the good bid their own value and receive a payoff of zero. This is a reflection of the fact that there are more bidders than goods to be auctioned, and hence there is never excess supply.

The equilibrium strategies in the two auctions are summarized in the following proposition. It is understood that in the second auction the winner of the first auction does not participate. The derivation of these strategies and the proof that these strategies constitute an equilibrium are found in the appendix. The bidders' updated beliefs after resolution of the first auction are denoted by [[Rho].sub.L]; that is, [[Rho].sub.L] denotes the subjective belief that a bidder's opponent in the second auction has a high valuation for the good.(4)

PROPOSITION 1. If the type is high, the strategy in the first auction is [F.sub.1](b) = [(b + [b.sup.1/2])/(1 - b)][(1 - [Rho])/[Rho]], on [0, [[Rho].sup.2]], and the expected payoff is 1 - [[Rho].sub.2]. If the type is high, the strategy in the second auction is [F.sub.2](b) = [b/(1 - b)][(1 - [[Rho].sub.L])/[[Rho].sub.L]], on [0, [[Rho].sub.L]], and the expected payoff is 1 - [[Rho].sub.L]. If the type is low, the strategy in both auctions is zero, and the expected payoff is zero.

All proofs are in the appendix.

Notice that in the limit as (1 - [Rho])[Rho] goes to 0, [[Rho].sub.L] must tend to [Rho]; that is, in the full information environment there is no updating. Hence, as the probability of all bidders having a valuation of 1 goes to one, the bids in both auctions go to 1, and as the probability of all bidders having a valuation of 0 goes to one, the bids in both auctions go to 0.

Since the equilibrium is found through backward induction, it is best to focus on the second auction before studying the first auction.

Learning and the Equilibrium Strategies in the Second Auction

For bidders to determine their optimal strategies in the second auction, they must form beliefs regarding their opponents' valuations and strategies. After the conclusion of the first auction, the winning bid is revealed. Bidders use this information to update their beliefs regarding the types of their opponents. In the first auction a low type bids 0 with probability one. The probability that a high type bids 0 or below that is zero. It follows that by observing the winning bid, bidders infer the type of the winner.

Thus, after completing the first auction, all bidders know their own type as well as the type of the winner of the first auction. However, since the winner of the first auction does not compete in the second auction, any information regarding the winner's type is unimportant to the bidders in the second auction. In other words, bidders are concerned only with the valuations of their opponents in the second auction (i.e., the other losers of the first auction). To solve the bidders' problem in the second auction, it is necessary to establish the information that bidders have regarding their opponents who lost the first auction. The fact that the losers did not win the first auction implies that their bids are below the winning bid, which allows bidders to learn about the type of their opponents. Let (1 - [[Rho].sub.L]) denote the posterior probability that any loser of the first auction is a low type. Let Pl denote the winning bid in the first auction.

PROPOSITION 2. In equilibrium, the posterior probability that a loser of the first auction is a low type is

(1 - [[Rho].sub.L]) = (1 - [Rho])/[(1 - [Rho]) + [Rho][F.sub.1]([P.sub.1])].

The proof is an application of Bayes' rule.

Since [F.sub.1]([p.sub.1]) [element of] [0,1], (1 - [[Rho].sub.L]) [greater than or equal to] (1 - [Rho]). That is, any bidder who loses the first auction is more likely to be a low type, so the average valuation is nonincreasing in the course of the auction. This is so because if a high type is present in the first auction, another high type has a chance of winning the first auction; a low type, however, does not.

Proposition 2 applies to equilibrium actions in the first auction. However, beliefs must also be specified if in the first auction a bidder uses a strategy other than the equilibrium strategy. Since only the winning bid is revealed, the only way that bidders can become aware that an out-of-equilibrium strategy was used is if the winning bid is above the support of the equilibrium mixed strategy used by high types. All other winning bids would be mistaken for bids resulting from an equilibrium strategy. Should the winning bid be above the support, bidders do not know which type placed the bid. However, assuming that the winner's unit demand is fulfilled, beliefs about this bidder's type are irrelevant. The remaining bidders participate in the second auction with their posterior beliefs coinciding with their prior beliefs, as no information is revealed about the losers of the first auction. Hence, Proposition 2 describes bidders' learning in the sequential auction with the provision that the bidders' beliefs are subjective beliefs.(5)

The Anticipation of Information in the First Auction

In determining equilibrium strategies in the first auction, bidders account for the possibility of participating in the second auction. For low-type bidders this is irrelevant since their future payoff is zero. For the high-type bidder the future payoff (i.e., the payoff in the second auction) depends on the history of the game. That is, the high type's payoff in the second auction depends on the outcome of the first auction. Specifically, high types who win the first auction have their unit demands satisfied and do not participate in the second auction, receiving a payoff of zero in the second auction. However, a high type who loses the first auction participates in the second auction and expects to obtain the second good with some probability. In other words, there is an opportunity cost of winning the first auction, as this necessarily implies the loss of the expected payoff in the second auction that accrues only to those who lose the first auction.

To consider the significance of this opportunity cost, one must determine the expected payoff that the high types obtain when participating in the second auction (given that they lose the first auction). Applying Proposition 2 to the payoff given in Proposition 1, one obtains (1 - [[Rho].sub.L]) = (1 - [Rho])/[(1 - [Rho]) + [Rho][F.sub.1]([p.sub.1])]. Hence, the high type's payoff in the second auction depends on the winning bid in the first auction, [p.sub.1], and the first auction mixed strategy of the high types, [F.sub.1]. However, prior to the first auction the winning bid is not known. It is at this point that information impacts bidders' strategies. Since the winning bid is unknown prior to the first auction, bidders take expectations regarding the winning bid to determine their expected future payoff. The distribution of the winning bid, however, depends on the strategies used in the first auction. Moreover, since the expectation is taken conditional on the bidder losing the first auction, the conditional distribution of the winning bid is different for any bid under consideration. In sum, a bidder's future expected payoff depends on the information generated in the first auction; the expectation of this information, in turn, depends on the distribution of the winning bid; and, finally, the distribution of the winning bid, in turn, depends on which bid the high type who loses places in the first auction.

In particular, in assessing the implications of winning the first auction in terms of forgone second auction payoffs, high types trade off winning the first auction and retiring by placing higher bids against learning and increased second auction payoffs by placing lower bids. That is, the higher the bid the high type places is, the greater the probability of winning the first auction is, yet, at the same time, the lower the second auction payoff is expected to be if he loses the first auction. This is illustrated in the following proposition, in which [E.sub.1] denotes the expectations operator prior to the first auction.

PROPOSITION 3. The expectation of the payoff in the second auction of a high type conditioned on losing the first auction when placing the bid b is decreasing in the high type's first auction bid; that is,

(d/db)[E.sub.1][(1 - [[Rho].sub.L])[where]b [less than] [p.sub.1]] [less than] 0.

The trade-off between winning the first auction and having higher second auction expected payoffs becomes particularly clear when considering a high type who in the first auction bids the lower end of the support. If this is the winning bid, then clearly the opponents must be low types. But in this case the winning bid in the second auction is 0 with certainty, and the bidder is indifferent between winning the first or the second auction since both are concluded at the same price.

The implications for the strategies of the high types in both auctions of the anticipation of information in the first auction and learning in the second auction are the subject of the next section.

3. The Impact of Information and Learning

As indicated in the previous section, information and learning impact bidders' strategies and payoffs in both auctions. Thus, information and learning create a two-way link between the two auctions. Consider first the second auction. After observing the outcome of the first auction, bidders update their beliefs regarding their opponents' types. In doing so, the strategy of high-type bidders in the second auction is influenced by the outcome of the first auction. That is, the second auction equilibrium strategy is a function of the information generated. Since the second auction equilibrium strategies affect the expected payoff in the second auction, the second auction expected payoff is also a function of the information generated.

Regarding the first auction, high-type bidders consider the opportunity cost of winning the first auction. Specifically, the expected opportunity cost of obtaining the good from the first auction is the expected second auction payoff, which the losers of the first auction obtain. In taking the expectation of the second auction payoff, bidders take into account the effect that learning has on the second auction payoff. High-type bidders anticipate the information generated by the outcome of the first auction for each bid on the support of their equilibrium strategy. In sum, the second auction equilibrium is affected by the outcome of the first auction through changes in beliefs; and the first auction equilibrium is affected through bid-dependent opportunity costs.

To understand the importance of information on the two auctions, a basis of comparison is needed. The equilibrium is contrasted to auction outcomes when bidders are unaware of informational effects. That is, when bidders determine their strategies in the second auction, they do not take into account the information generated in the first auction. Consequently, in the first auction, bidders are myopic - they do not anticipate the information being generated when determining their strategies. Therefore, in the comparison bidding behavior, the prior and posterior beliefs of all bidders are the same; that is, their posterior beliefs are given by [Rho], and the expectation regarding the posterior beliefs are also [Rho]. The resulting bidding behavior in this comparison auction allows one to study the impact of information at each stage of the two-auction equilibrium. The benchmark strategies are derived as in the previous section with [Rho] replacing [[Rho].sub.L] and [E.sub.1]([[Rho].sub.L]), respectively, and are given below with superscript us.

If the type is high, the strategy in the first auction is [Mathematical Expression Omitted] and the strategy in the second auction is [Mathematical Expression Omitted], on [0, [Rho]]. If the type is low, the strategy is zero in both auctions.

Compared to the benchmark scenario, the information generated in the first auction has a positive value to high-type equilibrium bidders. This is independent of whether in the first auction bidders are myopic or anticipate information and use the equilibrium strategies. The value of information is manifested in higher expected payoffs as, on average, bidders place lower bids in the equilibrium case than in the benchmark case. Specifically, the value of information to a high-type loser of the first auction is the difference in the expected payoffs in the second auction. This difference, denoted by V([p.sub.1]), where [p.sub.1] is the winning bid in the first auction, is zero for low-type bidders. For high-type bidders, V([p.sub.1]) is given in the following theorem.

THEOREM 1. The value of information to the high-type loser is

V([p.sub.1]) = b[(1 - [[Rho].sub.L]) - [[Rho].sub.L](1 - [Rho])/[Rho]].

Moreover, the value of information to the high-type loser is positive and strictly decreasing in the first auction winning bid; that is,

[Mathematical Expression Omitted]

and

dV([p.sub.1])/[dp.sub.1] [less than] 0.

To understand the intuition for dV([p.sub.1])/[dp.sub.1] [less than] 0, recall that, in equilibrium, bidders update their beliefs regarding losers of the first auction. For high types to lose the first auction, it is necessary that their bid be below the winning bid, which is more likely if the winning bid is high. The lower the winning bid, the less likely it is that a loser drew a bid from the distribution [F.sub.1]([center dot]). In other words, it is more likely that the bidder is a low type and placed the bid 0. This means that the informational content of the winning bid is decreasing in the winning bid. This can be seen clearly in the limit cases. First, let [Mathematical Expression Omitted] then [[Rho].sub.L] = [Rho] and the winning bid does not allow any inferences about the type of the loser of the first auction (i.e., the winning bid contains no additional information). Next, let [p.sub.1] = 0; then [[Rho].sub.L] = 0 (i.e., there is full information).

The next theorem shows how the information generated in the first auction affects the bidding behavior in the second auction. Regardless of the outcome of the first auction, the probability of high bids is lower in the second auction of the equilibrium model than in the benchmark model. This is a reflection of the fact that information has positive value to high types. Furthermore, the difference in bids increases as the first auction winning bid declines; that is, the lower the first auction winning bid, the lower, on average, are the bids placed in the second auction. This is a reflection of the fact that the value of information is decreasing in the first auction winning bid.

THEOREM 2. The second auction benchmark distribution of high types first degree stochastically dominates the equilibrium mixed strategy distribution; that is,

[Mathematical Expression Omitted], [Mathematical Expression Omitted].

Furthermore, given two first auction prices [p[prime].sub.1] and [p[double prime].sub.1] with [p[prime].sub.1] [greater than] [p[double prime].sub.1], the second auction mixed strategy resulting from [p[prime].sub.1] first degree stochastically dominates the second auction mixed strategy resulting from [p[double prime].sub.1]; that is,

[Mathematical Expression Omitted].

The second part of Theorem 2 follows from the fact that the lower the first auction winning bid, the more informative the winning bid is to the losers of the first auction.

These theorems demonstrate the impact of learning on the second auction equilibrium strategies of the high-type bidders. Regarding the first auction, recall that bidders anticipate the generation of information in the first auction when determining their optimal bidding strategies. Specifically, high types face an opportunity cost of winning the first auction, namely, not having an expected payoff in the second auction after having won the first auction. This opportunity cost incurs regardless of whether bidders are sophisticated or myopic. However, only the sophisticated bidders take into account that the expected payoff in the second auction depends on the bid placed in the first auction and hence on the expected information generated by placing any particular bid. That is, sophisticated bidders are aware that there is a trade-off between increasing the probability of winning the good in the first auction by placing higher bids and increasing the amount of information by placing lower bids and thus increasing the expected payoff in the second auction. Since the myopic bidders overlook the latter effect, on average, in the first auction they place higher bids than the sophisticated bidders. This is formalized in the following theorem.

THEOREM 3. The strategy used by myopic bidders in the first auction first degree stochastically dominates the strategy used by sophisticated bidders in the first auction; that is,

[Mathematical Expression Omitted], [Mathematical Expression Omitted].

The Equilibrium Price Path

Thus far it has been shown how information and learning impact the strategies and payoffs at each stage of the auction. In this section properties of the equilibrium price path are studied. It is shown that, in equilibrium, prices may decrease from the first to the second auction. However, there is also a probability that the second price is higher than the first price. The probability that the second price is above or below the first price depends on the first price. Specifically, the higher the first auction price, the more likely it is that the second auction price is greater than the first auction price. The reason for this is twofold. First, the higher the first price, the more likely it is that many high-type bidders participated in the first auction. This suggests a high probability of high types competing in the second auction. Second, the higher the first price, the less (additional) information is generated. By Theorem 2, this leads high types to place higher bids in the second auction, as their mixed strategies stochastically dominate any equilibrium mixed strategy that would have been used had a lower first auction price been observed.(6) Let [p.sub.i] denote the price in the ith auction and let [W.sub.2]([p.sub.2][where][p.sub.1]) denote the distribution of the second auction price given [p.sub.1].

THEOREM 4. For all first auction prices greater than 0, the second price is different, with probability one, than the first price; that is,

Pr{[p.sub.2] = [p.sub.1][where][p.sub.1] [greater than] 0} = 0.

Moreover, the higher the first auction price, the more likely it is that the price sequence is increasing. In other words, the lower the first auction price, the more likely it is that prices decrease; that is,

d[W.sub.2]([p.sub.1][where][p.sub.1])/[dp.sub.1] [less than] 0.

Despite the properties of the equilibrium price path, prior to the first auction the expected price sequence is constant; that is, on average, both prices are the same. Even more striking is the fact that after resolution of the first auction (i.e., after [p.sub.1] is known), the expected second auction price, [p.sub.2], is equal to the first price, [p.sub.1]. In other words, no matter how likely it is that the second price will be below the first price, given the first price there will always be occasional second auction prices so high that, on average, the second auction price is equal to the first price.

Weber (1983) proves that the expectation of the second price is equal to the first price for the case of a continuum of bidder types. Unlike the case of a continuum of types, however, in this model with a discrete distribution, the result critically depends on bidders drawing the correct inferences when observing the winning bid of the first auction.(7) In the case of a discrete distribution, the result is explained by observing that there are three differences between the distributions of the winning bid in the first and second auctions.

First, in the second auction there is one less bidder than in the first auction since the winner of the first auction drops from competition in the second auction because the unit demand of the winner is met. Since average bids (and thus the winning bid) are increasing in the number of bidders, this difference in the distributions of the winning bids leads to lower bids in the second auction.

Second, since in equilibrium a low type does not win the first auction when there are high types present, it is less likely that the bidders participating in the second auction are high types (i.e., [[Rho].sub.L] [less than] [Rho]). This leads to lower bids, on average, because the distribution of the winning bid puts more weight on bids placed by low types. Moreover, high types place lower bids, on average, as it is more likely that their opponents are low types. Both of these differences, which lead to lower prices in the second auction, are in contrast to the third difference in the distributions of the winning bids in the two auctions.

The third difference in the distributions of the winning bids is that bidders in the second auction value winning more highly, as there is no opportunity cost to winning the second auction. That is, bidders who lose the first auction participate in the second auction and then have a chance of obtaining a good. However, if bidders also lose the second auction, the auction ends without them having obtained a good. Due to the worse implications of losing the second auction when compared to losing the first auction, high types place, on average, higher bids than they otherwise would. The third difference in the distributions of the winning bids leads to higher prices in the second auction. In equilibrium, this increased bidding in the second auction exactly offsets the first two effects, so that the expected second price is equal to the first price.

These properties of the equilibrium price path are formalized in Theorem 5, where [E.sub.i] denotes the expectations operator prior to the ith auction.

THEOREM 5. Ex ante, the difference in expected prices is zero; that is,

[E.sub.1] ([p.sub.1] - [p.sub.2]) = 0.

Moreover, the expected second price, given the first price, is equal to the first price. That is, prices form a Martingale:

[E.sub.2]([p.sub.2][where][p.sub.1]) = [p.sub.1], [Mathematical Expression Omitted].

Finally, another analogy to the case of a continuum of bidder types is worth pointing out. Despite the importance of information and learning throughout the auctions, a surprising result is that a static auction format in which the goods are auctioned simultaneously rather than sequentially yields the same expected final allocation.(8)

THEOREM 6. The sequential auction and the simultaneous auction yield the same expected final allocations.

The equilibrium of the static auction has an interesting property: Although there are only two possible values for the goods, they may be sold at different prices. Different prices occur with certainty if one or more of the bidders has a high valuation for the good. This is because there are more bidders than there are goods, so bidders avoid tying by using mixed strategies.

The reason that both auctions yield the same expected final allocation is that the economic problem in both auctions is the same: There are two goods and three bidders, whose only costs are winning bids. Thus, this particular change in the institutional arrangements of the market does not affect the final allocation. However, the importance of the auction format is significant in determining equilibrium strategies. As illustrated, in the sequential auction bidders must account for informational effects when determining equilibrium strategies. However, the information is public (i.e., observable by all bidders). Thus, by symmetry of the strategies, informational gains offset each other in the course of the sequential auction. That is, in the second auction all bidders have the same beliefs regarding the probability of facing a specific type of bidder.

4. Conclusion

In this paper, the impact on bidders' strategies and payoffs is studied when in the course of a sequential auction winning bids are disclosed. In the model, bidders are one of two types (high or low value), so bidders face the possibility that rivals are of the same type as themselves. It is shown that the revelation of the winning bid leads bidders to update their beliefs regarding their opponents' types at the auction. This updating affects the bidders' strategies in such a way that, on average, they place lower bids than bidders unaware of the information would. Consequently, in equilibrium, bidders have increased expected payoffs because of their learning after information is revealed.

Bidders are aware of this impact of information on the later strategies and hence anticipate this impact in earlier auctions. In participating in the early auction, however, a bidder's action directly influences the flow of information. Hence, when determining strategies in the early auction, bidders take into consideration how current actions influence their expectations regarding the revelation of information and its impact on subsequent behavior. Specifically, bidders trade off increasing the probability of winning the early auction with being better informed and hence increased expected payoffs in the later auction. Since the value of information in the later auction is decreasing in the winning bid, this trade-off leads bidders to place lower bids, on average, in the early auction when compared to bidders who are unaware of the impact of information and learning in the course of the sequential auction.

Thus, this paper helps to further the understanding of the importance of information in determining bidders' optimal strategies in sequential auctions. However, there are still many unresolved issues regarding the impact of information on bidders' optimal strategies in markets in which sequential auctions are utilized. For instance, it is worth exploring how other auction formats and policies regarding the revelation of information impact bidders' strategies in sequential auctions. Moreover, these questions should be studied for other assumptions regarding the distribution of bidders' types, such as affiliated values or a richer type space.

Finally, just as the anticipation of information and learning increase the bidders' expected payoffs, the anticipation of information and learning decrease the sellers' expected revenues. Thus, it is worth examining how sellers can manipulate the flow of information to increase expected revenue.(9) For instance, the sellers could improve expected revenue if they had the option to charge a fixed price in the first auction. This could lead to increased expected revenue in the first auction and, because it blocks learning in the second auction, also increase revenue in the second auction.

This paper is based on the first chapter of my Ph.D. dissertation written under the guidance of Leonard J. Mirman, whose help and encouragement I gratefully acknowledge. I thank Curtis Taylor, Rajiv Sarin, Ray Battalio, Jonathan Hamilton, and two anonymous reviewers for helpful comments and the Private Enterprise Research Center for financial support.

1 See, e.g., Rothkopf and Harstad (1994) and the literature cited therein.

2 This is the informational equivalent to the descending bid (Dutch) auction. In a Dutch auction, a bid clock is successively lowered until one of the bidders stops the clock. Thus, in the Dutch auction, only the winner and the winning bid are revealed.

3 In some markets revenue-maximizing schemes are particularly unlikely. For instance, in the sequential auctions for tobacco in the United States, there are fewer than ten tobacco curing companies bidding for units from many small tobacco farmers (there are in excess of 350,000 tobacco quota owners), so that one might expect the bidders to have more discretion over the auction rules than the sellers (cf. Raper, Love, and Shumway 1997).

4 Notice that the equilibrium is isomorphic to a common-value sequential auction in which the value of the goods is known to be 1, yet there are an unknown number of bidders, each of three potential bidders having a probability of [Rho] of participating in the auctions.

5 These subjective beliefs coincide with objective probabilities only in equilibrium or when only one bidder deviates from the equilibrium and this bidder wins the first auction. However, bidders are necessarily unaware of any possible discrepancy between subjective beliefs and objective probabilities.

6 Ashenfelter (1989) and Ashenfelter and Genesove (1992) show that in sequential auctions of wine and condominiums, prices on average declined. This price decline is termed the afternoon effect. The ex ante probability of observing the afternoon effect in this model depends on the parameters of the model.

7 Recall that in the case of a continuum of types, the equilibrium strategies are independent of past price realizations, so bidders need not draw any inferences about the other bidders.

8 Because of revenue equivalence of the discriminatory format and other static auctions for multiple objects when bidders are risk neutral, this result does not depend on the particular format of the static auction (cf. Vickrey 1962). In Jeitschko (1995) it is shown that even risk-averse bidders obtain the same expected payoff in the sequential and simultaneous auctions.

9 This issue was brought to my attention by a reviewer. Taylor (1998) examines the issue in the case of the sale of a single item of which the quality is uncertain.

10 Notice that in the proof of Proposition 3 only the fact that high types use a mixed strategy is used. In particular, the distribution of the high types' bids is not needed in the proof.

Appendix

Proof of Proposition 1. Consider first the second auction strategies. Given the bidders' posterior beliefs, [[Rho].sub.L], it must be shown that no bidder type can improve on the payoffs resulting from the proposed strategies. Low types bid their own value and receive a payoff of zero on which, given the other bidders' strategies, they cannot improve. The distribution of the high types' bids must be continuous, as otherwise (by symmetry) they have positive probability of tying their opponents at the discontinuity. This is dominated by bidding infinitesimally above any atom in the distribution. It is clear that the lower end of the support of the strategy must be the bid placed by low types (i.e., 0). Finally, high types must be indifferent between any particular bid placed with positive probability. This yields the following equation, where [Mathematical Expression Omitted] denotes the upper end of the support of the strategy,

[Mathematical Expression Omitted]. (A1)

Equation A1 is used to determine the equilibrium expected payoff of the high type who participates in the second auction. Letting [Phi] denote the expected payoff,

[Mathematical Expression Omitted] (A2)

Setting Equations A1 and A2 equal to each other yields the equilibrium mixed strategy [F.sub.2]. Specifically,

(1 - b)[(1 - [[Rho].sub.L] + [[Rho].sub.L])[F.sub.2](b)] = (1 - [[Rho].sub.L]) [tautomer] [F.sub.2](b) = [b/(1 - b)][(1 - [[Rho].sub.L])/[[Rho].sub.L]].

Clearly, a bidder cannot improve by placing a bid off the support, as this either implies losing the auction or winning with an unnecessarily high bid.

Consider now the first auction. The argument is the same for low types since their future expected payoff is zero. Hence, low types bid their value and obtain an expected payoff of zero in the first auction. For high types, the expected payoff of the entire game must be constant on the support of their mixed strategy. Letting [Pi] denote the high types' expected payoff from the entire game, the analogous equation to Equation A1 is given by

[Mathematical Expression Omitted]. (A1[prime])

By Proposition 3 below,(10) a high type's expected future payoff is a function of the bid placed in equilibrium. Applying Equation A3 from the proof of Proposition 3, Equation A A1[prime] becomes

[Mathematical Expression Omitted] (A1[double prime])

This yields the expected payoff of the entire two-auction game to the high type. Analogous to Equation A2, one obtains

[Mathematical Expression Omitted]. (A2[prime])

Setting Equations A1[double prime] and A2[prime] equal to each other gives [F.sub.1](b) as a quadratic equation of which only the positive root yields a distribution function. That is,

[Mathematical Expression Omitted].

Given that the high type's second auction payoff depends on the opponents' beliefs, it must be verified that a high type does not benefit by manipulating the others' beliefs by deviating from the proposed equilibrium. As noted above, however, only a bid placed above the support would be detected as an out-of-equilibrium bid, and this bid clearly decreases the bidder's expected payoff. Finally, consider a high type bidding below the support of the proposed strategy. In this case the bidder loses the first auction and expects to obtain a payoff of (1 - [[Rho].sub.L]) (cf. Equation A2) in the second auction. However, the value of [[Rho].sub.L] depends on the winning bid in the first auction, but, as in the proof of Proposition 3, it is possible to calculate the expected value of [[Rho].sub.L] The distribution of the winning bid, [W.sub.1]([p.sub.1]), conditioned on a bid below the support of the mixed strategy is [[(1 - [Rho]) + [Rho][F.sub.1]([p.sub.1])].sup.2]. Hence, the expectation of the second auction payoff of a high type who places a bid below the lower end of the support is

[Mathematical Expression Omitted],

which is exactly the payoff that the bidder receives when playing the equilibrium mixed strategy. Hence, there is no increase in the expected payoff by bidding below the support of the mixed strategy, and the proposed strategies constitute a Nash equilibrium.

Proof of Proposition 3. Since the high type who is taking the expectation conditions on one of the opponents winning the first auction, the distribution of the winning bid comes from the bids that the opponents place. Since there are two opponents, the distribution of the highest of the opponents' bids is [W.sub.1]([p.sub.1]) = [[(1 - [Rho]) + [Rho][F.sub.1]([p.sub.1])].sup.2]. However, since the high type places a bid in the first auction, the distribution must be conditioned on the winning bid, [p.sub.1], being greater than the bid, b, that the high type places when taking the expectation. Thus, the conditional distribution of the winning bid is [W.sub.1]([p.sub.1][where][p.sub.1] [greater than] b) = [[W.sub.1]([p.sub.1]) - [W.sub.1](b)]/[1 - [W.sub.1](b)], with density [w.sub.1]([p.sub.1][where][p.sub.1] [greater than] b) = 2[(1 - [Rho]) + [Rho][F.sub.1]([p.sub.1])][Rho][f.sub.1]([p.sub.1])/{1 - [[(1 - [Rho]) + [Rho][F.sub.1](b)].sup.2]}. Given (1 - [[Rho].sub.L]) in Proposition 2, the expected posterior beliefs that a high type who loses the first auction has regarding the losing opponent of the first auction, conditional on the bid that the type whose beliefs are in question places, is

[Mathematical Expression Omitted]. (A3)

The derivative of this with respect to the bid being placed, b, has the same sign as does

-[f.sub.1](b){1 - [[(1 - [Rho]) + [Rho][F.sub.1](b)].sup.2]} + [1 - [F.sub.1](b)]2[(1 - [Rho]) + [Rho][F.sub.1](b)][Rho][f.sub.1](b),

which simplifies to -[[Rho].sup.2][f.sub.1](b)[[1 - [F.sub.1](b)].sup.2]. Hence, (d/db)[E.sub.1][(1 - [[Rho].sub.L])[where]b [less than] [p.sub.1]] [less than] 0.

Proof of Theorem 1. The benchmark expected payoff in the second auction to high-type losers of the first auction is given by the payoff they receive, given the bid they place, (1 - b), multiplied by the probability of winning given that bid. The probability of winning, given that bid, depends on the objective posterior probability of the type of their opponent. Because the bidder does not learn and has subjective beliefs [Rho], the objective probability of facing a low-type opponent differs from the subjective beliefs. The objective posterior probability is given by (1 - [[Rho].sub.L]); that is, it is the same as the posterior beliefs of the informed bidders. Therefore, the probability of winning, given a bid, b, is [Mathematical Expression Omitted] Hence, the expected payoff is

[Mathematical Expression Omitted].

Thus,

[Mathematical Expression Omitted].

Furthermore, leaving open which strategy bidders use in the first auction and letting G denote either [F.sub.1] or [F[double prime].sub.1], d(1 - [[Rho].sub.L])/d[p.sub.1] = -(1 - [Rho])[Rho]g([p.sub.1])/[[(1 - [Rho]) + [Rho]G([p.sub.1])].sup.2] [less than] 0. That is, the lower [p.sub.1], the greater (1 - [[Rho].sub.L]) and the greater the value of information; that is, dV([p.sub.1])/[dp.sub.1] [less than] 0.

Proof of Theorem 2. Regarding the first statement in the theorem, first consider [F.sub.2](b) and [Mathematical Expression Omitted]. Then

[Mathematical Expression Omitted].

From Proposition 2, (1 - [[Rho].sub.L]) = (1 - [Rho])/[(1 - [Rho]) + [Rho][F.sub.1]([p.sub.1])]. So, (1 - [[Rho].sub.L])/[[Rho].sub.L] = (1 - [Rho])/[Rho][F.sub.1]([p.sub.1], so the inequality becomes [F.sub.1]([p.sub.1]) [less than] 1. Next, consider the distributions for [Mathematical Expression Omitted]. Then, [Mathematical Expression Omitted] for all [Mathematical Expression Omitted]. Regarding the second part of the theorem, the inequality similarly reduces to [F.sub.1]([p[prime].sub.1]) [less than] [F.sub.1]([p[double prime].sub.1]).

Proof of Theorem 3. The statement is true for [Mathematical Expression Omitted] since [Mathematical Expression Omitted]. It remains to be shown that

[Mathematical Expression Omitted].

The function h is continuous on its domain [Mathematical Expression Omitted] or, equivalently, {(b,p)10 [less than] b [less than] 1 [and] [b.sup.1/2] [less than] p [less than] 1}. Notice that [Delta]h/[Delta][Rho] = b/[([Rho] - b).sup.2] [greater than] 0, [for every] b [greater than] 0. Hence, h must obtain its infimum, where p is smallest (i.e., at p = [b.sup.1/2]) or, equivalently, where b = [[Rho].sup.2]. In other words, h reaches its infimum where the function g([Rho]) [equivalent to] h(b = [[Rho].sup.2], [Rho]) has its infimum. However,

g([Rho]) = 1/[(1 - [Rho]).sup.2] - 1(1 - [Rho]) [greater than] 0, [for every] 0 [less than] [Rho] [less than] 1.

Hence, h(b,[Rho]) is positive on its domain.

Proof of Theorem 4. The distribution of the winning bid in the second auction, [W.sub.2]([p.sub.2][where][p.sub.1] [greater than] 0) = [[(1 - [[Rho].sub.L]) + [[Rho].sub.L][F.sub.2]([p.sub.2][where][p.sub.1] [greater than] 0)].sup.2], is continuous on [Mathematical Expression Omitted], so the probability that any particular price [p.sub.2] = [p.sub.1] [greater than] 0 is realized is zero.

Given the equilibrium mixed strategy in the second auction, [F.sub.2], [W.sub.2]([p.sub.2][where][p.sub.1]) = [[(1 - [[Rho].sub.L])/(1 - [p.sub.2])].sup.2]. Inserting [F.sub.1] from Proposition 1 into Proposition 2 yields [Mathematical Expression Omitted]. The probability that the second auction price is lower than the first auction price is [Mathematical Expression Omitted]. Hence,

[Mathematical Expression Omitted].

Proof of Theorem 5. Since the latter statement implies the former, only the latter is proved: [E.sub.2]([p.sub.2][where][p.sub.1]) = [integral of] [p.sub.2] d[W.sub.2] between limits [[Rho].sub.L] and 0 = [[Rho].sub.L] - [integral of] [W.sub.2] d[p.sub.2] between limits [[Rho].sub.L] and 0. Given [W.sub.2]([p.sub.2]) = [[(1 - [[Rho].sub.L])/(1 - [p.sub.2])].sup.2], as in the proof to Theorem 4, [Mathematical Expression Omitted]. Now notice that, as in the proof of Theorem 4, [Mathematical Expression Omitted].

Proof of Theorem 6. Because of the same reasoning as in the sequential auction, low types bid their own value in the simultaneous auction and have expected payoffs of zero in either auction. Analogous to the sequential auction, the high types' expected payoffs in the simultaneous auction are given by (1 - b)Pr{winning[where]b}. High types win whenever they submit the first- or second-highest bid. Whenever they face either one or two low types, they win for sure. The probability of this occurring is [(1 - [Rho]).sup.2] + 2(1 - [Rho])[Rho]. In all other cases they face two high types using some mixed strategy [F.sub.1]([center dot]) with [F.sub.s](0) = 0. Hence, their payoff is [(1 - [Rho]).sup.2] + 2(1 - [Rho])[Rho], which is the same as for the sequential auction as given in Equation A2[prime].

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