Exclusive dealing through resellers in auctions with stochastic bidder participation.

Deltas, George

1. Introduction

This paper considers a private values auction in which some of an item's potential consumers compete for its possession against professional resellers (or intermediaries). If the resellers win, they market the item to the general public, which includes the consumers who participated in the auction. We show that prohibiting the participation of these potential consumers in the auction can have direct revenue-enhancing effects. In particular, we show that a risk-neutral seller prefers to exclude final consumers from an auction and sell the item exclusively to resellers when these resellers can gain access, at a cost, to a sufficiently bigger market than the seller himself.

This result might at first appear counterintuitive, as increasing the number of bidders increases expected revenue in private values bidding models. Seemingly, a seller could not profit from excluding any set of bidders. The intuition behind our result is that the resellers recoup their expenses for buying the item by reselling it to the final consumers. If some final consumers participate in the first auction and are outbid by the resellers, this is an indication that their values for the item are relatively low. Outbidding part of their customer base is "bad news" for resellers, so their bids are depressed if final consumers compete with them. Indeed, the resellers are less aggressive when competing against a subset of the final consumers even if, by observing the valuation of the participating consumers, they cannot make any inference about the valuations of the nonparticipating consumers, that is, even when consumer valuations are completely independent.

This paper also characterizes the social welfare implications of restricting the participation of consumers. The socially optimal and revenue-maximizing choices of auction format do not necessarily coincide. Even though restricting the participation of consumers may be both socially and privately (for the seller) optimal, it is also possible that restricting participation is socially optimal but privately suboptimal, and vice-versa.

Further, the results of this paper have implications for seller strategy, implications that are particularly relevant in a world where electronic trading is shrinking transactions costs. In Internet auctions, for instance, bids for an item can now be submitted electronically by potential consumers. Sellers who were hitherto unable to sell directly to consumers can now do so through these electronic auctions. However, it appears likely that Internet auctions fail to attract the entire set of possible consumers, as up to 50% of the auctions in some major sites do not result in a sale (see Lucking-Reiley 2000). The results of this paper suggest that if indeed only a small fraction of the potential customers participate directly in the electronic/Internet auction, the seller may find it optimal to exclude them altogether and, instead, sell the item in a dealer auction. Furthermore, such a direct exclusion of final consumers from an auction may not be driven by the desire to reduce transaction costs. In real estat e auctions, for example, it is frequently observed that properties that could be sold separately are sold in large batches. Indeed, in some cases the seller explicitly announces that only wholesalers are welcome to participate. Although such exclusions and sales of items in large batches might economize on transactions costs, they also result in more aggressive bidding from wholesalers. In other words, the results of our paper endogenize the distribution channel in auction markets, without an appeal to transactions costs.

This paper is related to two different strands of the literature that are discussed in the next section. The first strand considers common or affiliated value bidding environments in which preventing the participation of certain bidders can increase seller revenue. The second strand studies auctions with the possibility of postauction resale, and considers both private and common value environments. Our work integrates these two strands by considering the impact that the possibility of resale has on the incentives of the seller to forbid the participation of certain bidders in the auction. Our paper shows that a seller can profitably exclude some bidders from the auction even in a private values environment when the set of bidders is partitioned to those who purchase for their own consumption and those who purchase for the explicit purpose of resale.

Section 2 of the paper summarizes the related literature, whereas section 3 describes the modeling framework and a simple, tractable, benchmark model. The following section solves the seller's problem for the benchmark model, and section 5 analyzes the welfare implications of his choice. Section 6 generalizes the benchmark model to markets of arbitrary size and shows that the key results are robust to different assumptions about bidder entry and seller reserve. Finally, the paper ends with a few concluding remarks. All long proofs are contained in an Appendix.

2. Related Literature

This paper is related to two different lines of research. The first one studies auctions in which resale can arise in equilibrium. The other studies the possibility that the exclusion of some bidders from the auction can actually be revenue increasing. In this section we provide a brief overview of this research and how it relates to our work.

There has been a recent flurry of research on models of auctions that incorporate the possibility of profitable resale. One strand of the literature pursues explanations for resale on the basis of the existence of further gains from trade available after the object is sold via an auction. A potential source of these gains can arise from the nonparticipation of a subset of the buyers in the first auction. The winner in the first auction may then find it profitable to reauction the item (see Haile 1996). Another potential source of gains from trade arises from the existence of uncertainty about the private value of the object, which is resolved after the first auction. A bidder who expects to value the object most highly wins it in the first auction. When uncertainty about private values is resolved, he may find it profitable to sell the object to one of the other bidders (see Haile 2000, 2001, 2002). A third possibility of gains from trade occurs if bidder asymmetry results in an inefficient allocation in the first-price auction. The winner may find it profitable to resell the item to one of the losing bidders (see Gupta and Lebrun 1997, 1999). The results in Gupta and Lebrun (1997) are very interesting in the context of this work because they show that it may be profitable for the seller to forbid resale of the good. Limiting what the buyers can do with the item, which seemingly reduces its value, can actually yield higher expected revenue. In our work, forbidding consumption of the item, that is, limiting the buyers to purchasing for resale, can actually increase expected revenue.

In all of the above models, all participants are potentially "final consumers": They directly value the object that is put up for sale. Another strand of the literature considers participants of two types: The first type consists of participants in the first auction who compete for the object with the intention of reselling it. The second type consists of the final consumers who compete among themselves to purchase the items from the resellers. Bikhchandani and Huang (1989), for instance, analyze a common value auction in which an exogenous number of bidders competes in a multiple object auction to acquire items that will then be resold to consumers. In that paper, the policy question is which auction format to use to award the objects to the resellers rather than to exclude or not to exclude the final consumers from the auction.

This paper adopts the exogenous partition of the players into resellers and final consumers. Unlike the model in Bikhchandani and Huang (1989), ours is a private values model where some or all of the final consumers can be present in the first auction. Furthermore, the policy question we analyze is whether or not the initial seller should allow the participation of the final consumers. Bose and Deltas (1999) is the precursor to this work. In that paper, unlike this one, consumer and reseller participation is deterministic with a single out of N final consumers participating in a seller auction. That paper shows that the seller is never better off by allowing the single consumer to compete for the item. In contrast, in this paper the participation of both types of buyers is stochastic. Under this framework, the optimal seller policy depends on the probabilities of participation and the relative, to the bidder valuations, size of the marketing costs. Furthermore, we are here able to fully analyze the welfare im plications of the exclusion of the final consumers from the seller's auction, derive the comparative statics of the revenue (or welfare)-maximizing auction format, and discuss the robustness of the results to the posting of reserve prices and how stochastic consumer participation can arise endogenously by analyzing the consumer participation decision.

There is a relatively small literature that considers the profitability of exclusionary practices in auctions. Krishna and Morgan (1997) demonstrate that, when consumer valuations are the average of all bidders' signals, excluding one of the bidders at random can potentially result in higher revenue. Bulow and Klemperer (1998) provide a similar example when bidder valuations have a common and a private component. Finally, an example in Haile (1996) shows that when competing buyers know each other's type with probability that becomes arbitrarily close to 1, and all competing buyers have some intrinsic value for the auctioned object, the seller can profitably exclude one of the buyers from the auction. Unlike our study, these three papers rely on the fact that valuations have a common component. Our work shows that, when the set of bidders is partitioned to those who purchase for private consumption and those who purchase for resale, excluding bidders from the auction can be revenue increasing even if consumer valuations conform to the independent private values paradigm. (1)

3. Modeling Framework and the Benchmark Model

In this and the next two sections we describe and solve a benchmark model. This model is parsimonious and simple enough that analytical solutions can be obtained. Nevertheless, the model is sufficiently rich to allow for a range of comparative analyses. In section 6 we consider a series of extensions that demonstrate that the main results of this model are quite general.

Description of Players and Auction Formats

A seller is willing to sell an item to a market that consists of two consumers. The value, [v.sub.i], that consumer i attaches to the item is an independent draw from a nondegenerate, differentiable, distribution F(v) with density function f(v) and is private knowledge. We assume that the probability that each one of these two consumers attends an auction organized by the seller is equal to p. This reflects the possibility that a consumer may not be aware that the auction is taking place. There also exist two intermediaries who specialize in the resale of items of the type being sold. Either one of these intermediaries, upon incurring a marketing cost c > 0, can ensure that both potential customers will be aware that the item is for sale. The intermediaries derive no direct benefit from ownership of the item. Their willingness to pay for it equals the expected revenue they will receive from the resale minus the marketing cost. Each intermediary will participate in the auction organized by the seller with prob ability q. (2) All agents in this model are risk neutral. We assume that the item is perishable in the sense that if it does not sell there are no further opportunities for the seller to receive any revenue from its sale. (3) We further assume that the bidders in any auction can identify whether the competitors are consumers or intermediaries. (4) We also assume that there is no possibility of resale from one consumer to the other, that is, the "marketing cost" for the consumers is too high and, if only one of them attends the auction and wins, it will not pay to locate the other consumer and bargain with him. (5) Finally, all the information given above is common knowledge.

We consider two possible sales mechanisms at the disposal of the seller: A restricted oral (English) auction in which only the intermediaries are allowed to participate and an unrestricted oral auction in which anyone is allowed to participate. The seller must commit to the auction format before he observes the number and type of participants. This reflects the institutional constraint that the rules of an auction must be preannounced. We assume that if the intermediaries win the item they, too, will sell it to the two consumers via an oral auction. (6) Both auctions are assumed to be without reserve. (7) Therefore, the seller and the intermediaries are treated symmetrically with respect to the selling mechanisms they use. (8)

Limiting the model to two consumers allows us to characterize the choice of auction format using only the three parameters defined above and the expected values of the highest and lowest consumer valuations. As we show below, the functional form and any additional properties of f(v) need not be considered. The probability of consumer participation p can be thought of as indicating the size of the market the seller can access without the use of a professional reseller. The marketing cost c indicates the efficiency of these services in gaining access to the remainder of the market. The probability q of reseller participation in the auction indicates the availability of reseller services and the degree of competition between resellers for the provision of these services. A high value of q would result in a higher proportion of the value of these services being appropriated by the seller, as resellers are more likely to compete with each other. Conversely, a low value of q would result in most of the rents accrui ng to the intermediaries. Therefore, this framework is parsimonious but sufficiently rich for us to analyze the factors that drive the seller's choice of auction format and the factors that lead to a conflict between revenue and welfare maximization.

Restricted Auction Revenue

In the restricted auction no consumers are allowed to participate in the seller's auction. Observe that the expected revenue that an intermediary will receive from selling the item to the two consumers, [R.sup.I.sub.r], is equal to

[R.sup.I.sub.r] = E{min([v.sub.i])}.

The reservation value of an intermediary in the restricted auction is equal to [R.sup.I.sub.r] - c. If neither intermediary shows up at the seller auction, the realized price would equal zero. If only one intermediary shows up at the seller auction, the realized price would also equal zero, since there is no reserve. If both intermediaries show up at the seller auction, they will bid up the price until it reaches their reservation value. Therefore, the expected revenue the seller will receive in a restricted auction, [R.sup.S.sub.r], equals:

[R.sup.S.sub.r] = [q.sup.2][[R.sup.I.sub.r] - c] = [q.sup.2][E{min([v.sub.i])} - c].

Note that the intermediaries make positive expected profits, since, if a single intermediary shows up, he will secure the item at a price of zero.

Unrestricted Auction Revenue

If consumers are allowed to participate but none actually appears at the sale the seller's revenue will be the same as in the restricted auction. If both appear, then the intermediaries will not bid, and the item will go to the highest valuation consumer at a price equal to the second highest consumer valuation. To complete the evaluation of the seller's revenue, we must determine the bidding behavior of the intermediaries and the consumer in the case in which a single consumer, say consumer 1 with valuation [v.sub.1], shows up at the seller's auction.

First, let us consider the participating consumer's optimal bidding strategy. If he loses in the seller's auction, he knows that he will be able to compete with the other consumer for the item in the auction organized by the intermediary who won the item. Denote the valuation of the other consumer by [v.sub.2]. Then, in that second auction, the participating consumer's expected rent is

[FORMULA NOT REPRODUCIBLE IN ASCII]

Therefore, if a single consumer with valuation [v.sub.1] participates in the seller's auction, his reservation value, r([v.sub.1]) will be equal to

r([v.sub.1]) = [v.sub.1] - U([v.sub.1]).

Let us now turn to the bidding strategy of the intermediaries when a single consumer participates in the seller's auction. If an intermediary wins the auction, beating a consumer with valuation [v.sub.1], the expected revenue he will reap when he resells the item, [R.sup.I.sub.u]([v.sub.1]), is equal to

[FORMULA NOT REPRODUCIBLE IN ASCII]

The first term is equal to the expected revenue he will receive if the participating consumer has the highest valuation for the item, and the second term is equal to the expected revenue if the other consumer has the highest valuation. We can write the above expression as:

[FORMULA NOT REPRODUCIBLE IN ASCII]

The expected revenue of the intermediaries, if they win the auction, is equal to the reservation value of the bidder they are competing against. Therefore, for any positive marketing cost c the intermediaries will find it unprofitable to outbid the consumer. This implies that if a single consumer shows up in the seller's auction the intermediaries will not bid and the seller's revenue will be zero. Therefore, the expected seller revenue in the unrestricted auction, [R.sup.S.sub.u], is equal to

[R.sup.S.sub.u] = [p.sup.2]E{min([v.sub.i])} + [(1 - p).sup.2][q.sup.2]{min([v.sub.i])} - c].

In the next section we consider the seller's optimal choice of auction format.

4. Seller's Choice of Auction Format

The seller will choose the restricted auction if

[R.sup.S.sub.r] > [R.sup.S.sub.u].

It turns out, as Proposition 1 states below, that the revenue-maximizing auction format is a function of the probability of consumer participation, p, the probability of reseller participation, q, and the ratio of the marketing cost, c, to the revenue obtained in a reseller's auction. We, therefore, find convenient to introduce a definition of the normalized marketing cost:

DEFINITION. The normalized marketing cost, c, is the ratio of the marketing cost, c, to the revenue obtained in a reseller's auction.

c = c/E{min([v.sub.i])}

Suppose that the normalized marketing cost, c, is low enough that the restricted auction yields positive revenue. Then, if the probability of consumer participation, p, is sufficiently low, the seller would prefer to exclude the consumers from the auction. This is formally stated in Proposition 1 below, the proof of which (and of all other results) is to be found in the Appendix.

PROPOSITION 1. For any value of marketing costs c [member of] (0, 1) and probability of intermediary participation q [member of] (0, 1) there exists some critical probability of consumer participation [p.sup.*] [member of] (0, 1) such that for p < [p.sup.*] the seller would prefer to exclude the consumers from the auction. Furthermore, this probability is given by

[p.sup.*] = 2[q.sup.2] 1 - c/1 + [q.sup.2](1 - c).

It follows that [p.sup.*] is an increasing function of the probability of the reseller participation, q, and a decreasing function of the normalized marketing cost, c.

Observe from the expressions for [R.sup.S.sub.r] and [R.sup.2.sub.u] that if the probability of consumer participation is 0, then the seller is indifferent between the two auction formats. If, on the other hand, the probability of consumer participation is 1, then the seller would choose the unrestricted auction. One might presume, on the basis of the above observations, that for any positive probability of consumer participation the seller would prefer the unrestricted auction. Proposition 1 shows that this conjecture is false. The intuition for this is as follows: Since when neither consumer participates, both auction formats yield the same revenue, and the seller weighs the probability that both consumers participate (in which case the unrestricted auction yields the most revenue) with the probability that a single consumer participates (in which case the restricted auction yields the most revenue). For p sufficiently small, the latter event becomes arbitrarily more likely than the former, and as a result the restricted auction becomes the revenue-maximizing format. Further intuition can be obtained by comparing the surplus the seller can extract under the two formats. When both resellers are present in the restricted auction, the seller extracts the full surplus from the market (minus the marketing cost). An auction in which a single consumer participates has the worst possible surplus extraction properties from the point of view of the seller: He obtains zero revenue. Therefore, the restricted auction is preferred by the seller when the marketing cost is small, the probability of reseller participation high, and participation of consumers low, because such an auction has superior surplus extraction properties under such conditions. Furthermore, when the probability of reseller participation is equal to 1, the restricted auction is almost always preferred by the seller, in the sense that it generates higher revenue for all "reasonable" values of the marketing cost.

Note that results analogous to those in Proposition 1 do not exist for the probability of reseller participation, q, and the marketing cost, c. That is, it is not true that no matter what p and q are, there exists some value of c at which the seller would prefer the restricted auction. This is somewhat intuitive: If the probability of consumer participation is near 1 and the probability of reseller participation is near zero, a seller would never want to use a restricted auction even if c is very low. Similarly, it is not true that no matter what p and c are there exists some q so that the seller would prefer to use the restricted auction.

An easy way to demonstrate these results is to observe Figures 1 and 2. (9) Figure 1 plots, for various values of c, the values of p and q for which the restricted auction yields the same revenue as the unrestricted auction. The area below each line corresponds to the region where, for that particular value of c, the restricted auction yields higher expected revenue than does the unrestricted auction. It can be readily seen that for, say, p = 0.8 and = 0.75 there exists no q for which the restricted auction is preferred to the unrestricted one. Figure 2 plots, for various values of q, the values of p and c. for which the restricted auction yields the same revenue as the unrestricted one. The area below each line corresponds to the region for which the restricted auction yields higher expected revenue than the unrestricted one. It can be readily seen that for p = 0.5 and q = 0.5 there exists no for which the restricted auction yields more revenue than the unrestricted one.

5. Welfare Implications

Preventing consumers from bidding in the seller's auction means that, if either reseller participates, then the item will for sure be eventually awarded to the highest valuation consumer. If consumers are allowed to bid then it is possible that the consumer who purchases the item is not the highest valuation consumer. (10) On the other hand, the cost of preventing the consumers from bidding is that the marketing cost, c, will be incurred more often, and that, if neither reseller participates, the item goes unsold. The socially optimal choice of auction format, then, depends on the difference between the valuations of the highest and lowest valuation consumer, the probabilities of consumer and reseller participation, and the value of the marketing cost.

We now turn to the formal analysis of the welfare ranking of the two auction mechanisms. In the restricted auction the presence of one or both intermediaries will ensure that the item will ultimately be bought by the highest valuation consumer. The winning intermediary will incur the marketing cost equal to c. Therefore, the expected social surplus from the restricted auction is equal to:

[W.sub.r] [1 - [(1 - q).sup.2]][E{max([v.sub.i])} - c].

In the unrestricted auction, the expected social surplus is equal to

[W.sub.u] = [(1 - p).sup.2][W.sub.r] + 2p(l - p)E{[v.sub.i]} + [p.sup.2]E{max([v.sub.i])}.

The first term corresponds to the surplus if no consumer shows up in the seller's auction, the second term corresponds to the surplus if a single consumer shows up, and the last term corresponds to the case in which both consumers show up. In the rest of this discussion it will be helpful to define [rho] as the ratio of the expected value of the highest valuation over the expected value of the lowest valuation:

[rho] = E{max([v.sub.i])}/E{min([v.sub.i])}

which can take values in the open interval (l,[infinity]). Proposition 2 below gives the critical probability for which, when it exists, the social surplus is the same for both auction mechanisms.

PROPOSITION 2. Let p be defined as:

p = 1 + [1 - [(1 - q).sup.2]]([rho] - c) - [rho]/[1 - [(1 - q).sup.2]]([rho] - c) - 1

where c is the normalized marketing cost, q is the probability of reseller participation, and [rho] is the ratio of expected value of the highest to the second-highest valuation. Suppose that for some values of the parameters c, q, and [rho], p [member of] (0,1). Then at those parameter values, the social surplus obtained under the restricted format equals that obtained unrestricted format. Furthermore, for consumer participation probabilities p > p the unrestricted format yields higher social surplus than the restricted format, whereas the converse is true for p < p .

Note that p depends on [rho], whereas [p.sup.*] does not. This makes sense since revenues in a second-price auction depend on the value of the second-highest valuation. Welfare, however, depends on the highest valuation. The bigger the difference between the expected values of the highest and the second-highest consumer valuations, the more important it becomes, from the social surplus point of view, that both consumers have a chance to bid for the item. Therefore, for higher values of [rho] the restricted auction will be socially optimal even for relatively high values of p. This, along with other properties of this critical probability, are summarized in Corollary 1 below.

COROLLARY 1. p is (i) increasing in the dispersion of bidder valuations, (ii) decreasing in the value of the (normalized) cost, (iii) increasing in the probability of consumer participation, and (iv) asymptotes, as [rho] goes to infinity, to [p.sub.max], where

[p.sub.max] = 1 - [(1 - q).sup.2]/1 - [(1 - q).sup.2].

Part (i) of Corollary 1 formalizes the intuition discussed above. Parts (ii) and (iii) indicate that p relates to changes in normalized marketing costs and the probability of intermediary participation in qualitatively the same way as [p.sup.*]. Finally, part (iv) indicates that there is a limiting value of the critical probability as [rho] goes to infinity, and this limiting probability depends only on the probability of intermediary participation. This also makes intuitive sense; as [rho] goes to infinity only the probability that the item goes to the highest valuation consumer matters. In fact, it can be shown that, in the limit, the socially optimal auction format is the one that maximizes this probability (and is, therefore, independent of the marketing cost and the probability of consumer participation).

We now turn to the comparison of p and [p.sup.*]. This is best done by plotting the relevant probabilities in the same graph. At first, this seems impossible since p depends on [rho] in addition to q and c. However, Corollary 1 indicates that p is monotonically increasing in [rho] and asymptotes to a particular function as p goes to infinity. One can also observe that when [rho] is low enough, p goes to zero. (11) This suggests the following strategy for constructing a figure: Plot [p.sub.max] against q. Clearly, it is never socially optimal to exclude the consumers when the probability of consumer participation exceeds [p.sub.max]. For p < [p.sub.max] it will be socially optimal to exclude the consumer for sufficiently low values of p, that is, when the variance of consumer valuations is relatively low.

Figure 3 plots [p.sub.max] p, and [p.sup.*] for [rho] = 2, and c = 0.25. It can be readily seen that depending on the value of q and p we might have any of the four possible combinations of socially optimal and revenue-maximizing policies. That is, it is possible that the restricted auction yields (i) higher social surplus and revenue, (ii) lower surplus and revenue, (iii) higher surplus but lower revenue, and (iv) lower surplus but higher revenue. This important result is summarized in the following proposition.

PROPOSITION 3. The revenue ranking of the two auction formats does not necessarily coincide with the socially optimal ranking. In particular, the restricted auction may yield (i) higher social surplus and revenue, (ii) lower surplus and revenue, (iii) higher surplus but lower revenue, and (iv) lower surplus but higher revenue.

Note that higher values of p make the restricted auction more desirable from the social point of view. This is because, as mentioned above, for big relative differences between the highest and second-highest valuation, it is more important that both consumers have a chance to bid on the item. It is worth pointing out that, as Figure 4 shows, even if q = 1, we can get all four possible combinations of optimal policies for different values of [rho].

It is important to contrast the results of this paper with those in Ausubel and Crampton (1998). They show that if the seller cannot restrict supply and prevent resale, the revenue-maximizing auction format is also efficient (it maximizes social welfare) when there is costless consumer-to-consumer resale. [But also see Zheng (2000) for an alternative treatment.] In our framework, the revenue- and welfare-maximizing auction formats would not necessarily coincide even if the participating consumer could costlessly resell the item to the nonparticipating consumer. This is because, unlike in Ausubel and Crampton (1998), our model incorporates (by assumption) an exogenous "supply restriction:" Not all consumers participate in the seller's auction.

6. Extensions to the Benchmark Model

Consumer Market of Arbitrary Size

The stylized model described in the preceding sections limits the number of consumers to a maximum of two. Such a restriction allows us to obtain closed-form solutions for the conditions under which a restricted auction is revenue maximizing. It also allows us to evaluate the welfare consequences of seller's choice of auction format and demonstrate the conflict between revenue and welfare maximization. Finally, restricting the retail market size to only two consumers makes it possible to determine how changes in the market environment affect the revenue-maximizing and welfare-maximizing choices of auction format.

In this section we demonstrate that the main results of the paper remain valid, in a qualitative sense, even when the retail market consists of an arbitrary number, N, of consumers. This is stated formally below.

PROPOSITION 4. Consider a market that consists of (i) N > 2 consumers, each one of which independently participates in the seller's auction with probability p. and (ii) two resellers, each one of which independently participates in the seller's auction with probability q [member of] [[q.sub.crit](N, c), 1], where 0 < [q.sub.crit](N, c), < 1. Let the marketing cost of resellers, c, be positive but sufficiently low so that the seller obtains positive revenue from the restricted auction. Then, there exists some critical probability of consumer participation [p.sup.*] [member of] (0,1) such that for p < [p.sup.*] the seller would prefer to exclude the consumers from the auction.

Notice that when there are more than two consumers in the market, a sufficiently low probability that each of them participates in the seller auction is not enough for the restricted auction to yield a higher revenue than the unrestricted auction. It is also required that the resellers participate with a sufficiently high probability. The details of the proof are tedious and are relegated to the Appendix, in which we also provide the expression for [q.sub.crit]((N, c). However, an outline of the main argument is instructive. Let K denote the number of consumers that participate in the seller auction in a particular realization of this market. We first observe that when no consumer participates (i.e., K = 0), the restricted and unrestricted auctions yield the same revenue. We then show that when only a single consumer participates in the seller's auction (i.e., K = 1) and each of the two resellers participates with probability higher than [q.sub.crit] where 1 > [q.sub.crit] > 0, the seller prefers to forbid that consumer from bidding for all positive values of the marketing cost. The preferences for the restricted auction is weak when it yields zero revenue. We finally show that as the probability of consumer participation, p, decreases, the probability that there are two or more participating consumers goes to zero faster than the probability that there is only one participating consumer. Indeed, the ratio of these two probabilities goes to zero in the limit. Since the difference in the revenue of the two auction formats is finite for all K, it follows that for all values of p below some positive threshold the seller would prefer the restricted auction to the unrestricted auction. Essentially, Proposition 4 works by shrinking p to compensate for an increase in the market size. (12)

We have derived this result without the explicit calculation of optimal bidder strategies for arbitrary K. Such a "brute-force" calculation would present two complications: The first one is conceptual. Suppose there are N > 2 consumers, K > 1 of which participate in the seller's auction. A brute-force calculation of the optimal strategy of the participating consumers requires, as a first step, the calculation of the surplus that the consumer with the highest valuation among them would obtain if he lost the item to a dealer and were to compete for it in the retail market. This surplus depends on whether or not the consumer with the second-highest valuation (among the K participating consumers) would return to compete for the item in the retail market. Formally, he is indifferent between competing and not competing as his payoff is zero in both cases. (13) Therefore, one would need to consider both possibilities. The second complication of a brute-force analysis is that one cannot obtain a closed-form expressio n for the "critical" participation probability of the consumers that would make a seller prefer to prevent their participation in his auction. Our approach sidesteps these difficulties and allows us to demonstrate the existence of such a critical probability, even though its complete characterization is not possible in the N > 2 case.

Other Extensions

In the basic model the probability of participation is exogenous. Consumers participate in the seller's auction with probability less than 1 because the seller was assumed to lack the marketing technology to inform all consumers that there was a sale taking place. The resellers possess a specialized marketing technology and could, at a cost, access both consumers with probability one. This treatment views the consumers as being passive receivers of marketing information. An alternative would be to endogenize the consumer search for information by postulating that the consumers can choose to be informed about the presence of the seller's auction by incurring a search cost. One can show that probabilistic consumer entry into the seller's auction can arise endogenously. Furthermore, when the distribution of bidder values exhibits only moderate dispersion, the symmetric mixed strategy equilibrium is not pareto dominated by any pure strategy equilibrium of the entry game. (14)

This paper purposefully ignores any heterogeneity in reseller marketing costs. As a result the resellers obtain positive profits only when one of them happens not to compete in the seller's auction. The assumption of marketing cost homogeneity is made primarily for simplicity. Introducing cost heterogeneity would have the effect of shifting surplus from the seller to the resellers by lessening reseller competition. However, it is clear that the results will not change qualitatively even if one allows for cost heterogeneity, provided that this heterogeneity is not excessive: The restricted auction revenue would decline by more than the unrestricted auction revenue, but the former would still exceed the latter for sufficiently low probability of consumer participation.

A concern arising from the assumption of cost homogeneity is that it facilitates reseller collusion. The resellers could agree not to bid aggressively so as to earn positive profits even when both are present. Instead, they could agree to bid up to a predetermined price and have the item be awarded to one of them at random. Neither of the two resellers would have an incentive to break such a collusive agreement: In the event that one of them raises his bid above the predetermined price, the other reseller would have a dominant strategy to also raise his bid, resulting in zero surplus for the defecting reseller. Formally, we do not consider the possibility of collusion but rather consider purely noncooperative equilibria. We note, however, that reseller collusion would not change the qualitative nature of the results provided that the seller can post a reserve. Indeed, as we discuss below, the restricted auction can yield higher revenue than the unrestricted auction (in same cases) even when the seller and res ellers can post optimal reserves. The auction with reserve is collusion proof: The seller's reserve is set at the reservation value of the resellers and, therefore, there is no scope for profitable collusion. Less formally, the concern about collusive equilibria (in the absence of a reserve) would be mitigated by the introduction of reseller heterogeneity. As discussed above, such heterogeneity would not alter the qualitative nature of the results and is only omitted for clarity. With the introduction of reseller heterogeneity the restricted auction would be no more susceptible to collusive behavior than any other auction studied in the literature.

In the basic model we did not consider a reserve by either the seller or the resellers. In doing so we follow a large part of the theoretical literature. Ignoring a positive reserve is often meant to reflect the institutional fact that many auctions are lacking of a serious reserve. It is also meant to reflect the commitment of many sellers to sell, a commitment that often arises endogenously as part of an optimal strategy. (15) Nevertheless, one can show that allowing for a positive reserve, though complicating the analysis substantially (in part because the reseller can now use any information he obtains in the seller's auction to set the reserve in his auction), does not alter the main result of the paper: There are still conditions under which the seller prefers to prohibit the participation of consumers in the auction.

Finally, the main result of this paper is likely to also hold if first-price auctions were used. Recall that the main driving force of the results comes from the fact that the resellers bid less aggressively in an auction in which the consumers are present because winning is "bad" news about the valuation of the participating consumers. If anything, we expect the resellers to bid even more cautiously in a first-price environment since in such an environment they cannot observe the consumers' bidding behavior and thus update their estimates of the resale value of the item. However, complete analysis of the game with first-price auctions is too complicated for the following reasons: First, the presence of a consumer in the seller's auction yields an asymmetry between the two consumers in the reseller auction (in the event that the resellers win the seller auctions). Solving for the equilibrium in asymmetric first-price auction is an exceptionally difficult exercise even when one considers particular distributio ns. Second, the nature of the equilibrium strategies depends crucially on features that are not important in an English auction. For example, it would now be important for the consumer who fails to participate in the first auction to know whether another consumer had participated in the seller's auction and failed to win the item.

7. Conclusion

The central result of this paper is to show (in the context of a private values environment) that it can be beneficial for a seller to exclude final consumers from an auction when there is a potential set of intermediaries that, upon securing the item, can gain access to an even bigger market of consumers. This exclusion can sometime also be socially optimal. However, it is possible that excluding consumers from the auction is socially optimal but not revenue maximizing.

Appendix

PROOF OF PROPOSITION 1. The revenue from the restricted auction is greater than the revenue from the unrestricted auction if

[q.sup.2][E{min([v.sub.i])} - c] > [p.sup.2]E{min([v.sub.i])} + [(1 - p).sup.2][q.sup.2][E{min([v.sub.i])} - c].

Canceling out terms, simplifying, and factoring out p, we can rewrite the above inequality as

0 > p[E{min([v.sub.i])} + [q.sup.2](E{min([v.sub.i])} - c)] - 2[q.sup.2][E{min([v.sub.i])} - c].

Given that E{min([v.sub.i])} > c for the intermediaries to participate in any auction, we have:

p < [2q.sup.2](E{min([v.sub.i])} - c)/E{min([v.sub.i])} + [q.sup.2][E{min([v.sub.i])} - c].

The fraction is greater than 0 as both the numerator and denominator are positive. Furthermore, since E{min([v.sub.i])} > [q.sup.2][E{min([v.sub.i])} - c], the fraction is bounded above by 1. Dividing and denominator by E{min([v.sub.i])} yields the expression for [p.sup.*]. Differentiating [p.sup.*] with respect to q we have:

d[p.sup.*]/dq = 4q1 - c/[[1 + [q.sup.2](1 - c)].sup.2] > 0.

Finally, differentiating [p.sup.*] with respect to c we get:

d[p.sup.*]/dc = -2[q.sup.2]/[[1 + [q.sup.2](1 - c)].sup.2] < 0

QED.

PROOF OF PROPOSITION 2. Note that since we have two draws from the distribution we can write

E{[v.sub.i]} = E{max([v.sub.i])} + E{min([v.sub.i])}/2.

Then, the difference, [W.sub.u] - [W.sub.r], in the social surplus between the two formats, can be written as

[[(1 - p).sup.2] - 1][W.sub.r] + 2p(1 - p)E{max([v.sub.i])} + E{min([v.sub.i])}/2 + [p.sup.2]E{max([v.sub.i])}.

Suppose that there exists a value of p, p [member of] (0,1) such that [W.sub.u] - [W.sub.r] = 0. Then, substituting the expression for [W.sub.r], solving for p and simplifying yields

[FORMULA NOT REPRODUCIBLE IN ASCII]

[FORMULA NOT REPRODUCIBLE IN ASCII]

For the critical probability to be in the (0,1) interval, the denominator and the numerator of the fraction in the right-hand side must be of different signs. Because the numerator is smaller than the denominator, this implies that the former must be negative and the latter positive. This also implies, considering the inequality above, that for p > p the unrestricted auction will create higher social surplus than the restricted auction. Dividing both the numerator and denominator by the expected value of the lowest valuation, we finally get

p = 1 + [1 - [(1 - q).sup.2]](p - c) - p/[1 - [(1 - q).sup.2]](p - c) - 1

QED.

PROOF COROLLARY 1. (i) Taking the derivative of p with respect to [rho] yields:

[FORMULA NOT REPRODUCIBLE IN ASCII]

[FORMULA NOT REPRODUCIBLE IN ASCII]

since both the numerator and denominator are positive.

(ii) Dividing through by [rho], the expression for the critical probability can be rewritten as:

[FORMULA NOT REPRODUCIBLE IN ASCII]

The result follows from taking the limit as [rho] goes to infinity and simplifying the numerator.

(iii) Taking the derivative of p with respect to c we have:

[FORMULA NOT REPRODUCIBLE IN ASCII]

This is negative since both terms are negative. [Recall that the denominator of the first fraction is positive and the numerator of the second fraction is negative for p [member of] (0, 1).]

(iv) Taking the derivative with respect to q we get

[FORMULA NOT REPRODUCIBLE IN ASCII]

Factoring out the first term we have

[FORMULA NOT REPRODUCIBLE IN ASCII]

Both terms of the product are positive; therefore, the critical probability is increasing in the probability of reseller participation. [Recall that the denominator of the first fraction is positive and the numerator of the second fraction is negative for p [member of] (0, 1). Also, [rho] > I, whereas c < 1.] QED.

PROOF OF PROPOSITION 4. As a first step we consider a particular realization of the seller's auction in which only one of N consumers participates. In other words, we, for the moment, condition on the event that a single consumer shows up at the seller's auction. Seller's revenue for both the restricted and unrestricted auction formats depends on the size of the market, N, the marketing cost, c, and on the number of resellers that participate, M. Denote by [R.sup.S.sub.r](N, M, c) and [R.sup.S.sub.u](N, M, c) the seller's revenue from the restricted and unrestricted auctions, respectively. Then, the expected revenue from the restricted auction exceeds that of the unrestricted auction if

[q.sup.2][R.sup.S.sub.r](N, w, c) > [q.sup.2][R.sub.S.sub.u[(N, 2, c)] + 2q(1 - q)[R.sup.S.sub.u](N, 1, c)

where the expectation has been taken over the participation of the resellers. Notice that in the restricted auction the seller obtains positive revenue only when both resellers participate (as there are no consumers bidding in the auction). In the unrestricted auction the seller obtains positive revenue if at least one reseller participates (as there is also one consumer bidding in the auction).

The above inequality can be rewritten as

[q.sup.2][[R.sup.S.sub.r](N, 2, c) - [R.sup.S.sub.u](N, 2, c)] > 2q(1 - q)[R.sup.S.sub.u](N, 1, c).

Proposition 6 of Bose and Deltas (1999) shows that the expression in the brackets of the left-hand side of the above expression is non-negative. It is guaranteed to be positive if the revenue from either auction format is positive and c > 0, that is, it is positive for the range of marketing costs that we consider here. Simplifying the above inequality and solving for q we obtain

[FORMULA NOT REPRODUCIBLE IN ASCII]

Notice that the right-hand side of the above inequality, henceforth denoted by [q.sub.crit](n, c), is a non-negative number that is strictly less than 1. Therefore, if a single consumer participates (deterministically) in the seller's auction, the seller would prefer to prohibit this consumer from participating if the probability of reseller participation exceeds [q.sub.crit](N, c).We will next show that when q >[q.sub.crit](N, c), there exists a p* such that for any p < p* the seller prefers to sell by the restricted auction. We generalize the notation: Denote by [R.sup.S.sub.u](N, M, c, K) the revenue of the unrestricted auction in the event that K Out of N consumers participate in the seller's auction. Next, observe that for any finite N the revenue of the restricted and unrestricted auctions is finite. Therefore, the difference in the revenue between the two formats is also finite. Now, define

X = [E.sub.u][[R.sup.S.sub.r](N, M, c)] - [E.sub.M][[R.sup.S.sub.u](N, M, c, 1)]

where the expectation is taken over the number of participating resellers, M. It follows from the first part of the proof that X>0. Next define

Y = [max.sub.K] [E.sub.M][[R.sup.S.sub.r](N, M, c)] - [E.sub.M][[R.sup.S.sub.u](N, M, c, K)].

A sufficient condition for the seller to prefer the restricted auction is

[E.sub.K]{[E.sub.M][[R.sup.S.sub.r](N, M, c)] - [E.sub.M][[R.sup.S.sub.u](N, M, c, K)]} > 0.

Observe that when K = 0 the restricted and unrestricted auctions are equivalent. Therefore, a sufficient condition for the above inequality to be satisfied is

[FORMULA NOT REPRODUCIBLE IN ASCII]

We will show that this inequality is satisfied for sufficiently low values of p, which will demonstrate the result in Proposition 4. In particular, we will demonstrate that the right-hand side of the above inequality has a limit of zero as p goes to zero, which ensures (since the right-hand side is continuous in p) that there exists a p* such that for all p < p* the restricted auction is preferred. First, observe that both the numerator and the denominator of the fraction in the righthand side of the above inequality go to zero as p goes to zero. Then, using L'Hopital's Rule we have

[FORMULA NOT REPRODUCIBLE IN ASCII]

= N - N + 0/N - 0 = 0.

This completes the proof. QED.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Received April 2000; accepted July 2001.

(1.) There is also a strand of literature that partitions bidders on the basis of the information they possess and their ability to process it (e.g., Engelbrecht-Wiggans, Milgrom, and Weber 1983; Deltas and Engelbrecht-Wjggans 2001). These models consider a static pure common values framework. An explicit modeling of resale that constitutes an important aspect of our model is absent from them.

(2.) One might be inclined to believe that the intermediaries, being professionals, will be aware that the auction is taking place and will attend with probability 1. However, intermediaries may be "capacity constrained", that is, be already fully occupied disposing other assets in their possession and be unwilling to purchase other assets. Under this interpretation, the probability that an intermediary participates is equal to the probability that he has an interest in purchasing and reselling this particular item. By allowing for this more general case, we show that our results do not depend on the certainty of the intermediaries' participation.

(3.) Note that the item never goes unsold in the special case when q = 1.

(4.) In Internet auctions, for instance, pasting of a bidder's history and feedback ratings give an indication about whether a bidder is likely to be an occasional consumer or a dealer. Nevertheless, we believe that one could obtain similar results in (more complicated) models in which the bidders do not know with certainty the identity of their competitors: Resellers are likely to bid less aggressively in the unrestricted auction as long as some of their competitors are likely to be final consumers, regardless of whether the resellers know which among their competitors are these final consumers.

(5.) For example, even though the winner of an item in an Internet auction could, in principle, turn around and sell that item in the same site, he is not likely to make a profit by doing so. In fact, the price of an item typically declines in an immediate resale in the same Internet site.

(6.) In practice, resellers often do not sell by auction, but rather use posted prices or negotiations with individual consumers. For example, vehicles purchased by resellers in wholesale auto auctions are resold in used car dealerships, and rare books purchased by resellers in book auctions are resold at antique stores or specialized bookstores. We assume that the resellers use the same mechanism as the original seller to isolate the effects arising from differences in their "marketing technology" rather than any differences in their "selling technology."

(7.) See section 6 for discussion of an extension with optimally set reserves.

(8.) Bose and Deltas (1997) provide an example in which restricting the participation of final consumers in a second-price sealed-bid auction is also increasing the seller's expected revenue. Since second-price sealed-bid and English auctions are not isomorphic in the presence of resellers, this example suggests that the results obtained here are not specific to the use of the English auction mechanism.

(9.) These statements can be verified analytically by solving for the critical c and q instead of solving for the critical p.

(10.) This will happen if a single consumer shows up and this consumer is not the highest valuation consumer.

(11.) From Proposition 2 the expression for p is equal to 1 plus a fraction that has a negative numerator and positive denominator. As [rho] becomes smaller the denominator shrinks toward zero. It therefore follows that the fraction shrinks toward minus one and the critical probability shrinks toward zero.

(12.) For most distributions, holding p constant while increasing the size of the market would eventually make a seller prefer the unrestricted to the restricted auctions.

(13.) One might argue in favor of assuming that he participates since his possibility of winning the item would be positive if the highest valuation consumer "trembles" and does not return to compete for item (say, he drops dead between the two sales). On the other hand, one might argue in favor of assuming that he does not participate if participation entails even infinitesimal costs that are omitted from the formal analysis.

(14.) This approach parallels the work on endogenous entry in auctions by Levin and Smith (1994), Engelbrecht-Wiggans and Nonnenmacher (1999), and Deltas and Engelbrecht-Wiggans (2001). However, even with symmetric entry costs, there also exist asymmetric pure strategy equilibria. In these equilibria, some bidders enter with probability one, whereas others enter with probability zero. See McAfee and MeMillan (1987) and Engelbrecht-Wiggans (1993).

(15.) For instance, Engelbrecht-Wiggans (1993) shows that the optimal reserve in an auction with endogenous bidder participation should be sufficiently close to zero so as not to deter any bidder from participating. Recently, Engelbrecht-Wiggans and Nonnenmacher (1999) have shown that the commitment to sell is particularly important when different venues compete for potential bidders.

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Subir Bose * and George Deltas +

* Department of Economics, Iowa State University, Ames, IA 50011, USA; E-mail bose@iastate.edu.

+ Department of Economics, University of Illinois, Champaign, IL 61820, USA; E-mail deltas@uiuc.edu; corresponding author.

We thank Thomas Jeitschko, Preston McAfee, and Martin Perry for helpful discussion, two anonymous referees for insightful comments, and Janet Fitch for editorial assistance. The usual caveat applies.