Optimal Design Of English Auctions With Discrete Bid Levels*

Transcription

1 Optial Design Of English Auctions With Discrete Bid Levels* E. David, A. Rogers and N. R. Jennings Electronics and Coputer Science, University of Southapton, Southapton, SO7 BJ, UK. J. Schiff Departent of Matheatics, Bar-Ilan University, Raat-Gan 52900, Israel. S. Kraus Departent of Coputer Science, Bar-Ilan University, Raat-Gan 52900, Israel. M. H. Rothkopf Rutgers Business School and RUTCOR: the Rutgers Center for Operations Research, Piscataway, NJ , USA. This paper considers a for of ascending price English auction widely used in both live and online auctions. This discrete bid auction requires that the bidders subit bids at predeterined discrete bid levels, and thus, there exists a inial increent by which the bid price ay be raised. In contrast, the acadeic literature of optial auction design deals alost solely with continuous bid auctions. As a result, there is little practical guidance as to how an auctioneer, seeking to axiize its revenue, should deterine the nuber and value of these discrete bid levels, and it is this oission that is addressed here. To this end, a odel of a discrete bid auction fro the literature is considered, and an expression for the expected revenue of this auction is derived. This expression is used to deterine both nuerical and analytical solutions for the optial bid levels, and unifor and exponential bidder s valuation distributions are copared. Finally, the liiting case where the nuber of discrete bid levels is large is considered. An analytical expression for the distribution of the optial discrete bid levels is derived, and an intuitive understanding of how this distribution axiizes the revenue of the auction is developed. Key words: discrete bids, English auction, optial auction design History: This paper was first subitted on 0 Noveber Introduction Online internet auctions continue to attract any custoers and are currently estiated to sell goods worth over $30 billion annually. While any fors of auction protocols have been developed for both live and online auctions, at this tie over 80% of these auctions ipleent a single protocol: the open ascending price or English auction (Lucking-Reiley 2000). Under this protocol, the auctioneer announces that an ite * A preliinary version of this work appeared in David et al. (2005).

2 2 is for sale, fixes the opening bid and then allows bidders to increase the bid by a fixed discrete aount. The auction proceeds until no bidder is willing to further increase the bid, and the ite is then awarded to the current highest bidder in exchange for payent equal to its bid. Now, in contrast to these actual ipleentations, ost of the acadeic literature on auction theory assues that the bid increent is continuous and thus bidders ay subit extreely sall increents in order to outbid the current highest bidder. As such, the literature iplicitly akes two assuptions: (i) that bidders have no tie constraints, and (ii) that bidding is not a costly process. However, the prevalence of the discrete bid protocol within actual online auctions challenges both these assuptions. Specifically, the use of discrete bid levels radically reduces the nuber of bids subitted during the course of the auction (because the price increases to the expected closing price of the auction through uch larger bid increents) and thus reduces both the tie that the auction takes and the counication costs required to infor all of the participants of the current state of the auction. The ipleentation of discrete bid levels within these English auctions also causes any of the well known results fro the continuous bid auction literature to fail. For exaple, the bidders within the auction no longer have a doinant bidding strategy as they ust decide whether or not to bid at each bid level (Yu 999). In addition, as the ite is no longer guaranteed to be allocated to the bidder with the highest valuation, the revenue equivalence theore no longer applies, and thus the revenue that the auction generates will be dependent on the specific ipleentation details, such as the nuber and distribution of the discrete bid levels (Chwe 989). Thus, despite the wide-spread use of English auctions with discrete bid levels, the standard acadeic auction literature provides little insight or guidance for an auctioneer attepting to axiize its revenue. Moreover, what little work that has been done in this area has addressed the question for very liited cases (see section 2 for ore details). For exaple, Rothkopf and Harstad considered several cases where the nuber of bidders or the nuber of discrete bid levels was restricted to two (Rothkopf and Harstad 994). In the case of two bidders with valuations that are independently drawn fro a unifor distribution, they showed that it was optial to use a fixed bid increent with evenly spaced bid levels. However, it proved difficult to generalize these results to instances with a larger nubers of bidders, whose valuations were drawn fro arbitrary distributions. Thus, without any further guidance, ost online auctions ipleent discrete bid levels with a fixed bid increent, despite the liited applicability of this result. Thus, against this background, it is our ai to address this lack of guidance. Specifically, we seek to deterine the optial auction design for English auctions with discrete bid levels. In particular, we ai to deterine both the reserve price of the auction and the nuber and distribution of the discrete bid levels that The revenue equivalence theore states that all efficient auction protocols will yield the sae revenue at equilibriu.

3 3 yield the axiu auction revenue in the general case of an arbitrary nuber of bidders, whose valuations were drawn fro arbitrary distributions. In so doing, we extend the state of the art in this area in four key ways: We consider the sae odel of an ascending price auction with a bounded nuber of discrete bid levels that was proposed by Rothkopf and Harstad (994). But, rather than considering particular instances with liited nubers of bidders or bid levels, we derive, for the first tie, a general expression for the expected revenue of the auction. This expression relates the expected auction revenue to the specific discrete bid levels used in that auction and is valid for any nuber of bidders and any distribution of bidders private valuations. 2 We deonstrate how this expression is used to deterine the optial bid levels analytically, and, in addition, we present an algorith to calculate the nuerically. In order to copare our results with the earlier work of Rothkopf and Harstad, we consider two exaple cases where the bidders valuations are drawn independently fro a unifor and an exponential distribution. In the case of the unifor distribution, we prove that when there are ore than two bidders participating within the auction, a decreasing bid increent is optial and thus the interval between bid levels decreases with each bid level. For the first tie, we are able to calculate both analytically and nuerically, how this decrease should proceed for any nuber of bid levels and for any nuber of bidders. In addition, in the case of the exponential distribution, we are able to calculate nuerically the optial distribution of discrete bid levels. 3 Building on this analysis, we extend the initial auction odel to consider two additional cases that extend its scope and realis. First, we consider the ore realistic case that the nuber of bidders within the auction is not a known fixed value, but is described by a Poisson distribution whose ean the auctioneer knows (or can estiate). Second, we explicitly include within the odel an expression that describes the auctioneer s costs (e.g. an increental cost for each bid level that the auction progresses through). As before, we derive expressions for the expected revenue of the auction in both cases and nuerically solve for the optial discrete bid levels in the case of unifor and exponential bidders valuation distributions. 4 Finally, in order to develop an intuition into the optial distribution of the discrete bid levels, we consider the distribution of these discrete bid levels when their nuber approaches infinity (and assuing that in this liit the bid levels get closer and closer together). In this case, we are able to derive an analytic expression that describes the density of the discrete bid levels. We show that this expression is siilar (but not identical) to the distribution of the expected closing price of the auction. However, we

4 4 show that the later distribution (being easier to estiate fro historical auction data) can serve as an estiate for the forer distribution. The reainder of the paper is organized as follows: section 2 presents related work, section 3 introduces the initial auction odel that we consider, section 4 derives a general expression for the expected revenue of the auction, and in section 5 this is used to show how the optial bid levels can be derived analytically and deterined nuerically. Section 6 extends the initial odel to cover the two new cases discussed above, and we use our nuerical algorith to calculate the optial discrete bid levels in these cases. Section 7 considers the liiting behavior of these discrete bid levels and derives an expression for the density of the discrete bid levels, and finally, section 8 concludes and discusses future work. 2. Related Work The proble of optial auction design has been studied extensively for the case of auctions with continuous bid increents and independent private valuations (Riley and Sauelson 98, Myerson 98). In such auctions, the revenue equivalence theore states that all feasible efficient auctions generate the sae revenue, thus the interesting design question concerns the reserve price of the auction (i.e. in continuous English auctions, the price at which the bidding coences). In general, setting a reserve price increases the revenue of the auction and, thus, optial auction design has typically been concerned with finding the reserve price that axiizes the expected revenue of the auctioneer. In contrast to the literature of continuous bid auctions, the case of discrete bid levels has received little attention, although soe preliinary works exists. Much of this work is based on the assuption that there is a fixed bid increent and thus the price of the auction ascends in fixed size steps (Yaey 972, Chwe 989, Yu 999, Bapna et al. 2002, 2003). In ore detail, Yaey first considered this scenario and coented that such bidding rules appear to have the effect of speeding up the auction proceedings and hence reduce the costs of both the auctioneer and the bidders (Yaey 972). He concluded that if the fixed bid increent is sall, the expected revenue of the auction will approxiate the second highest price. Chwe also assued fixed bid increents, but considered a first-price sealed bid auction where bidders independent valuations were uniforly distributed (Chwe 989). He showed that a syetric unique Nash equilibriu bidding strategy exists and that this equilibriu converges to the equilibriu of the continuous bid auction, as the bid increent reduces to zero. In addition, he showed that the expected revenue of the discrete bid auction is always less than that of the equivalent continuous bid auction. Thus, the auctioneer has an incentive to ake the bid increents as sall as possible, assuing that the tie and counication costs of the bidding can be ignored. Yu also considered auctions with fixed bid increents, but studied each of the four coon auction protocols: the first-price sealed-bid, second-price sealed-bid, English and Dutch auctions (Yu 999). Extending

5 5 Chwe s result, she showed that in each of the auction protocols a syetric pure strategy equilibriu exists. Specifically, no doinant strategy was identified for the English protocol. However, for each of the protocols, she proved that as the nuber of bid levels becoe very large (i.e. the bid increent becoes sall), the equilibriu bids converge to the equilibriu bids of the corresponding continuous bid auction. In contrast to this work, Rothkopf and Harstad considered the ore general question of deterining the optial nuber and value of the bid levels (Rothkopf and Harstad 994). They provided a full discussion of how the discrete bid levels affect the expected revenue of the auction and they considered two different distributions for the bidders private valuations: a unifor and an exponential distribution. In the case of the unifor distribution, they considered two specific instances: (i) two bidders with any nuber of allowable bid levels, and (ii) two allowable bid levels and any nuber of bidders. In the first instance, even spacing of bid levels (i.e. a fixed bid increent) was found to be the optial. In the second instance, the optial bid increent was shown to decrease as the auction progressed (this decrease was described analytically). In the case of the exponential distribution of bidders valuations, the instance of just two bidders was again considered and the optial bid increent was shown to increase as the auction progressed. Now, in this paper, we extend the work of Rothkopf and Harstad. We initially consider the sae odel of the ascending price auction, but derive the optial bid levels in the general case with any distribution of bidders valuations, any nuber of bid levels, and any nuber of bidders. Moreover, we then extend this odel to incorporate the ore realistic case that there is uncertainty in the nuber of bidders who ay enter the auction. In addition, we explicitly consider the costs of the auctioneer and, in both cases, we are able to deterine optial bid levels. 3. Auction Model Initially, we consider an auction odel where n risk neutral bidders are attepting to buy a single ite fro a risk neutral auctioneer. The bidders have independent private valuations, v i, drawn fro a coon continuous probability density function, f(v), within the range [v, v]. This probability density function has a cuulative distribution function, F(v), and with no loss of generality, we can state that F(v) = 0 and F(v) =. These bidders participate in an ascending price auction, whereby the bids are restricted to discrete levels which are deterined by the auctioneer. We assue there are + discrete bid levels, l 0 < l <... < l and we note that the value of explicitly bounds the tie and costs of the auctioneer (in section 6 we extend this odel to relax this constraint and explicitly consider the costs of the auctioneer). At this point, we ake no constraints on the actual nuber of these bid levels, nor on the intervals between the. In the work of Rothkopf and Harstad, a standard oral English auction was considered. That is, the auctioneer proposes each bid level, and the first bidder to indicate to the auctioneer his willingness to bid

6 6 this aount, becoes the current highest bidder. In traditional English auction houses, this indication is norally accoplished by raising a paddle or by a prearranged signal to the auctioneer. However, as discussed earlier, there is no doinant bidding strategy within this protocol, and bidders ust strategize over whether or not to bid at each bid level. To siplify their analysis, Rothkopf and Harstad assued that the bidders did not attept to strategize, but instead adopted the siple policy of pedestrian bidding. That is, bidders sequentially raised the bid price through the discrete bid levels, until their own private valuation was exceeded. Indeed, they showed that in the case of two bidders whose valuations are drawn fro any non-increasing distribution, such as the unifor and exponential distributions considered here, this policy is an equilibriu. However, in our work, we odify this standard auction protocol to iprove its applicability within online auctions. Thus, under our protocol, the auction coences with the auctioneer announcing the first discrete bid level, and all the bidders have a fixed predeterined tie interval in which to indicate their willingness to pay this bid level. Having received indications fro all willing bidders, the auctioneer then randoly selects one of these bidders, noinates this bidder as the current highest bidder, and announces this noination to all participants. The auction proceeds, with the price ascending through the discrete bid levels proposed by the auctioneer, until there are no bidders willing to pay the offered bid price. At this tie, the auction closes and the ite is allocated to the current highest bidder. Now, to ensure that bidders do not need to strategize over whether or not to bid at any level, we introduce an additional clearing rule. Should the eventual winner of the auction also have been noinated as the current highest bidder at the previous bid level, then the price he pays is that of the lower bid level (i.e. bidders do not pay ore when they out-bid theselves). Our odified auction protocol has three key properties which ake it particularly attractive within coputational settings where the bidders are likely to be autoated trading agents with liited functionality. First, unlike the standard oral auction, bidders within our protocol have a siple doinant bidding strategy; they should continue to participate in the auction and thus bid at each bid level, until the current bid level exceeds their private valuation. Second, as the rounds of the auction have a predeterined and fixed duration, there is no advantage in attepting to subit a bid earlier than an opponent, and thus, bidders with greater coputational or counication resources cannot gain an unfair advantage. Third, bidders ay enter and leave the auction at any tie and need not be present at the coenceent of the auction 2 (an iportant consideration where online auctions are subject to counication drop-outs). Crucially, however, the analysis of how an auction using our odified protocol closes at a particular discrete bid level (presented in the next section) is identical to the analysis of the standard oral auction perfored by Rothkopf and Harstad (994). Thus, the results that we show are not liited to our particular auction protocol, but also apply in this ore general setting. 2 Although clearly they ay iss the opportunity to buy the ite at a low price should there be few other bidders present.

7 7 Case Case 2 l i l i l i+ Two or ore bidders have valuations between [l i,l i+ ) and no bidders have valuations v l i+. One bidder has a valuation v l i+, one or ore bidders have valuations in the range [l i,l i+ ) and the bidder with the highest valuation was the current highest bidder at l i. Case 3 One bidder has a valuation v l i, one or ore bidders have valuations in the range [l i,l i ), and the bidder with the highest valuation was not the current highest bidder at l i. Figure Diagra showing the three cases whereby the auction closes at the bid level l i. In each case, the circles indicate a bidder s private valuation and the arrow indicates the bid level at which that bidder was selected as the current highest bidder. 4. The Auction Revenue In order to calculate the optial bid levels, we ust first find an expression for the expected revenue of the auctioneer, given the specific discrete bid levels used in that auction. Following the work of Rothkopf and Harstad, we can describe the probability of the auction closing at any particular bid level by considering three exhaustive and utually exclusive cases (Rothkopf and Harstad 994). These three cases are shown in figure and they describe all the possible configurations of bidders valuations that lead to the auction closing at a bid level of l i. In the diagra, the valuations of the bidders are shown as circles and the arrows indicate which bidder was noinated as the current highest bidder at each bid level. We can describe each case as follows: Case Two or ore bidders have valuations greater than bid level l i, but none of these bidders have valuations greater than l i+. Thus, once the bid price has reached l i, no bidder is willing to increase the bid any further, and the ite is allocated to the current highest bidder. In this case, the revenue earned by the auctioneer is less than that which would have been earned in a continuous auction (i.e. the second highest valuation) and the outcoe ay be inefficient as the ite is not necessarily allocated to the bidder with the highest valuation. Case 2 Two or ore bidders have valuations between l i and l i+ and a single bidder has a valuation

8 8 greater than l i+. As this single bidder was also noinated when the bid level reached l i, none of the other bidders have valuations sufficient to raise the bid to l i+. Thus, the auction closes at the price l i, and the ite is allocated to the bidder with the highest valuation. Again, the revenue earned by the auctioneer is less than that which would have been earned in a continuous auction, but the outcoe is allocatively efficient. Case 3 Two or ore bidders have valuations between l i and l i, a single bidder has a valuation greater than l i, but, unlike in case 2, this bidder was not noinated when the bid level reached l i. Thus, this bidder is forced to raise the bid level, and the auction closes at l i rather than at l i. Again this case is allocatively efficient, however the revenue earned by the auctioneer is actually greater than that earned in a continuous auction. The expected revenue of the auction is dependent on the probability of each of these three cases occurring. Each of these probabilities can be described in ters of the cuulative distribution function of the bidders valuations, F(v). We write P(case,l i ) for the probability that case one occurs and that the auction closes at bid level l i. This probability can be coputed by considering the probability of having k bidders with valuations between bid levels l i and l i+ (this happens with probability [F(l i+ ) F(l i )] k ) while the other n k bidders have valuations below l i (this happens with probability F(l i ) n k ). Suing over all possible values of k gives P(case,l i ) = n k=2 ( ) n F(l i ) n k [F(l i+ ) F(l i )] k. () k We can perfor a siilar calculation for case two, where we have k bidders with valuations between l i and l i+, one bidder with a valuation greater than l i+ and n k bidders with valuations below l i. In this case, we ust also consider the probability that the bidder with the highest valuation is the current highest bidder. Under our assuption that this selection is rando, this probability is siply given by, and thus the k+ whole expression is P(case2,l i ) = n k= ( n k ) n k+ F(l i) n k [F(l i+ ) F(l i )] k [ F(l i+ )]. (2) Finally, we consider case three, which is identical in for to case two, with the exception that the bidder with the highest valuation was not noinated as the current highest bidder at bid level l i and ust thus raise the price to l i. The probability of this occurring is k, rather than k+ k+ as in case two. Note that this description iplies that there exists a bid level below l i and thus the expression that we derive is only valid for bid levels l,...,l. In order to include the instance in which the auction closes at the bid level l 0, we note that this requires all but one bidder to have valuations below l 0. Thus, the final expression is

9 nf(l 0 ) [ F(l 0 )] i = 0 P(case3,l i ) = ( ) n n kn k k+ F(l i ) n k [F(l i ) F(l i )] k [ F(l i )] i > 0. k= 9 (3) Now, as these three expressions copletely describe all the possible ways in which the auction ay close at any particular bid level, we can find the expected revenue of the auctioneer by siply suing over all possible bid levels and weighting each by the revenue that it generates, l i. Thus the expected revenue of the auction is given by E = i=0 l i [P(case,l i )+P(case2,l i )+P(case3,l i )]. (4) The resulting expression at this stage is extreely coplex due to the cobinatorial sus in equations, 2 and 3. However, as detailed in appendix A, it is possible to siplify this expression significantly (noting, in so doing, that with no loss of generality we can define F(l + ) = ), to give the final result E = i=0 F(l i+ ) n F(l i ) n [ ] l i ( F(l i )) l i+ ( F(l i+ ). (5) F(l i+ ) F(l i ) This expression is a key result, and any of the results that we present in this paper ste fro the fact that we have been able to express the revenue of the auction in a relatively copact for. Unlike previous work that has considered siple instances of the auction, for exaple, those with just two bidders or two bid levels, this expression is for the general case. It relates the revenue of the auction to the actual bid levels used, and is valid for any nuber of bid levels, any nuber of bidders, and for any valuation distribution function which is described by F(v). Also, unlike the earlier work, we ake no assuptions about the positions of the first and last bid levels. Whereas Rothkopf and Harstad fixed these at the extrees of the bidders valuation distribution (i.e. l 0 = v and l = v), we ake the free paraeters and allow the to take any value. Since l 0 is equivalent to the reserve price of the auction, we thus deterine the optial reserve price and the optial bid levels by the sae process. 5. Optial Auction Design The expression derived in the last section describes the expected revenue of the auction when discrete bid levels l 0,...,l are used. Now, our goal is to attept to deterine the discrete bid levels that axiize the revenue of the auctioneer. Initially we present analytical results applying this ethodology to a unifor bidders valuation distribution. However, since it is not always possible to derive analytical results, we also present a nuerical algorith that is applicable in the general case.

10 0 5.. Analytical Solutions Now, in order to solve for the discrete bid levels that generate the axiu expected revenue for the auctioneer, we ust find the partial derivatives of the revenue expression given in equation 5, with respect to each individual bid level l i. We can then solve the equations E/ l i = 0 to find the values of l i that axiize the revenue. To perfor this differentiation, we ust note that each l i occurs in the suation of equation 5 twice. For exaple, the bid level l 5 occurs in the suand when i = 5, as F(l i ), and also in the preceding ter when i = 4, as F(l i+ ). Thus, for a unifor bidders valuation distribution, we substitute the analytical expression F(l i ) = l i v into these two ters and differentiate to give v v E l i = (l i+ v) n (l i v) n (v v) n + nl i (l i v) n nl i+ (l i v) n (v v) n. (6) In order to find the value of l i that axiizes the revenue, we can then siply ake this partial derivative equal to zero (i.e. E/ l i = 0) and solve the resulting expression. Doing so gives l i = v+ n (l i+ v) n (l i v) n. (7) n(l i+ l i ) This expression relates any individual optial bid level to the bid levels on either side of it. Thus, if we consider the specific case where n = 2, we can siplify this expression to l i = l i + l i+. (8) 2 Thus, the value of l i is idway between l i and l i+, and as this is true for all l i, the optial discrete bid levels are evenly spaced with a fixed bid increent. This results confirs the analysis of Rothkopf and Harstad (994) who considered exactly this two bidder case. However, given our general odel, we can also consider the case of ore bidders (i.e. when n > 2), and, in this case, we can show that: l i > l i + l i+. (9) 2 Again this is true for all l i, so the optial distribution of bid levels consists of a decreasing bid increent, whereby the bid levels becoe closer together as the auction progresses (see Appendix B for a proof of this result). Thus, despite the widespread use of a fixed bid increent in real auctions, it is only optial for the auctioneer in the very liited case where there are just two bidders.

11 for i=0: a+i (v a)/ where a = ax(v,v/2) // unifor l i { /α+i (2/α) // exponential d while d > stopping condition, l 0 argax E (l 0,...,l ) where v l 0 < l l 0 for i=:- l i argax E (l 0,...,l ) where l i < l i < l i+ l i l argax E (l 0,...,l ) where l < l v l d 0 for i=0:, d ax(d,abs(l i l i)) l i l i Figure 2 Pseudo-code for a nuerical algorith based on Jacobi iteration to calculate solutions for the optial bid levels with arbitrary bidders valuation distributions Nuerical Solutions When we apply the analytical ethod presented above to arbitrary bidders valuation distributions, we often find that solving equation 6 is intractable. Thus, since we would like to copare the optial discrete bid levels in the ore general case of arbitrary bidders valuation distributions, we ust adopt a nuerical approach to axiizing the expected auction revenue. There are any nuerical optiization algoriths available (see Nuerical Recipes by Press et al. (992) for exaples), but two key features of this proble guide our choice. First, since each ter in the suation in equation 5 contains only pairs of bid levels (i.e. l i and l i+ ), we note that axiizing this expression, or solving E/ l i = 0, is equivalent to solving a tri-diagonal set of + siultaneous equations, that, by denoting E/ l i asg i, we can write as G 0 (l 0,l ) = 0 G i (l i,l i,l i+ ) = 0 for i = to (0) G (l,l ) = 0. Second, the solutions to these equations are constrained by requiring their ordering to be fixed (i.e. l i < l i < l i+ ). Typically, a general purpose optiization package will fail to exploit the first feature and will be heavily constrained by the second. However, we can produce a siple and efficient nuerical algorith by ipleenting a version of the Jacobi iteration for solving iteratively a syste of + siultaneous equations (Hagean and Young 98). That is, while fixing all other bid levels, we find the value of l i that axiizes equation 5, allowing l i to vary in the range l i < l i < l i+. Between these liits, the expression is well behaved and has a single axiu that can be found using hill clibing or any well established one-diensional gradient based ethod. We sequentially update all l i and then iterate the process until the bid levels converge to the necessary accuracy.

12 2 We present this nuerical algorith in pseudo-code in figure 2 and note that the expression E(l 0,...,l ) represents the revenue expression shown in equation 5. While our purpose here is not to prove the convergence properties of this iterative algorith, in our experients it was found to converge reliably and rapidly, given that two starting conditions for l i were satisfied. Specifically, at the initial iteration, no bid level ay be outside the upper liit of the bidders valuation distribution and l 0 ust be greater or equal to the reserve price predicted for the equivalent continuous bid auction. In the first two lines of the algorith, we provide suitable starting conditions for the two valuation distributions that we consider in the next section Coparison Of Bidders Valuation Distributions The nuerical solution described in the previous section allows us to calculate the optial discrete bid levels for any value of n (i.e. the nuber of bidders present in any auction) and any bidders valuation distribution. In this section, we copare the optial bid levels over a range of values of n for two different bidders valuation distributions; the exponential distribution, proposed by Rothkopf and Harstad, and the unifor distribution. To allow us to copare these two directly, we chose their paraeters so that the expected closing prices of the auctions are siilar. Thus, in the case of a unifor distribution, we consider a range of [0,] eaning f(v) = v v and F(v) = where v = 0 and v =. For the exponential distribution, v v v v we take f(v) = αe αv and F(v) = e αv where α = 4. The resulting optial discrete bid levels are shown in figure 3, for three different nubers of bidders (n = 2, 0 and 20) and over a continuous range fro 2 to 20. In both cases, we use 0 bid levels (i.e. = 0), as this akes clear the differences between the two cases 3. Rothkopf and Harstad, and our preceding analytical analysis, proved that when there are two bidders whose valuations are drawn fro a unifor distribution, the optial discrete bid levels are evenly spaced with a fixed bid increent. This sae result is observed in our nuerical results here. In addition, when there are ore than two bidders, we observe that the optial bid levels becoe closer together and the bid increent decreases as the bid price increases, as proved in the previous section. The case of the exponential valuation distribution is ore coplex. When there are two bidders, we see an increasing bid increent, as was shown by Rothkopf and Harstad. However, as the nuber of bidders increases, we observe that, rather than increasing, the bid increent initially decreases, reaches a iniu size and then increases again. We also observe that in both cases, as the nuber of bidders increases, the value of the first bid level, l 0, increases. Rothkopf and Harstad fixed the values of the first and last bid level at the extrees of the valuation distribution (i.e. for the unifor case, l 0 = v and l = v). However, we ake no such restriction, 3 Note that while changing the nuber of bid levels does affect their value, it does not affect the general for of the distribution seen in the plot.

13 3 Optial Bid Levels (Unifor) f(v) v = 0 v = 0.9 l 0 (a) n = 2 n = 0 l 0 l 0 l 0 l n = 20 l 0 l Nuber of Bidders (n) l 0 v = 0.5 Optial Bid Levels (Exponential) l 0 f(v) v =.25 (b) n = 2 n = 0 n = 20 l 0 l 0 l 0 l 0 l 0 l l Nuber of Bidders (n) Figure 3 Optial bid levels for (a) unifor and (b) exponential bidders valuation distributions for the initial auction odel considered here. and thus the values of l 0 and l are optiized at the sae tie as the other bid levels 4. Since l 0 is equivalent to the reserve price of the auction (i.e. the ite will not sell if there are no bidders willing to pay at least l 0 ), the results indicate that, in contrast to the literature of optial continuous bid auctions, the optial reserve price of an auction with discrete bid levels is dependent on the nuber of bidders. In general, we see that when the nuber of bid levels is large, or the nuber of bidders is sall, the value of l 0 tends toward the optial reserve price of the equivalent continuous bid auction (i.e. given by Riley and Sauelson (98): v = [ F(v )]/F (v ). For the unifor valuation distribution, v = ax(v,v/2), and for the exponential valuation distribution v = /α). Intuitively we can understand these effects by the fact that given a fixed nuber of bid levels, we should position the closer together in areas where they are ost likely to differentiate the bidders with the highest valuations. Thus, in the case of the unifor distribution, the bid levels becoe closer together nearer to the upper liit of the distribution. While in the exponential distribution, they becoe closer together in the area where we expect to find the bidder with the second highest valuation. In section 7 we expand upon 4 This is particularly iportant in the case of the exponential distribution, where the upper liit of the probability distribution is infinity. We do not force l = and thus ake ore efficient use of the constrained nuber of bid levels.

14 4 this intuition and analytically calculate the density of bid levels by considering the liiting case where the nuber of bid levels becoes large. 6. Extensions to the Initial Auction Model Having derived these results for our initial auction odel, we consider two increental extensions to it that increase its realis and extend its applicability. We first consider the ore general setting in which the nuber of bidders participating in the auction is not fixed, but is described by a probability distribution. We then explicitly incorporate a odel of the costs of the auctioneer into the revenue calculation. In both cases, we are able to derive an expression for the expected revenue of the auctioneer and thus use the nuerical algorith, described in the previous section, to calculate representative results. 6.. Uncertainty in the Nuber of Bidders The initial auction odel that we considered assued that the nuber of bidders in the auction, n, is fixed and known to the auctioneer. In soe settings this ay be the case, and thus, the auction can be designed using this specific knowledge of the nuber of bidders who will participate. However, in general this it is not so. It is ore likely that while the auctioneer ay have an estiate of the nuber of bidders who will participate, it will be described by a probability distribution 5. There are a nuber of candidates for this probability distribution. Levin and Sith considered an auction odel in which the nuber of bidders participating was endogenously deterined (Levin and Sith 994). They odeled a pool of potential bidders, and showed that, at equilibriu, each potential bidder has a fixed probability of actually participating in (or entering) the auction. The nuber of bidders participating in any auction was thus described by a binoial distribution. More recently, Bajari and Hortacsu (2003) considered a siilar odel and copared their odel to data collected fro ebay auctions selling collectible U.S. coins. They note that in such online auctions, the pool of potential bidders is extreely large. However, the fact that, in general, only a sall nuber of bids are observed, suggests that the probability that a potential bidder participates in any individual auction is very low. Thus they deduce that, in such cases, a Poisson distribution is an appropriate approxiation for the binoial proposed by Levin and Sith. This observation, was confired by Jiang and Leyton-Brown (2005), who copared paraeterized odels to real ebay auction data and found that the nuber of bidders within the auctions was well described by such a Poisson distribution 6. 5 Indeed, in the case of the standard oral auction, it is not possible to deterine the nuber of bidders who are participating, even once the auction has coenced; since, in general, the nuber of received bids can be less than the nuber of participants. 6 Note that Jiang and Leyton-Brown also attepted to infer the presence of bidders who participated but did not have valuations sufficient to allow the to bid

15 5 Optial Bid Levels (Unifor) f(v) v = 0 v = 0.9 l 0 (a) λ = 2 λ = 0 λ = 20 l 0 l 0 l 0 l 0 l 0 l l Mean Nuber of Bidders (λ) v = 0.5 Optial Bid Levels (Exponential) l 0 f(v) v =.25 (b) λ = 2 λ = 0 λ = 20 l 0 l 0 l 0 l 0 l 0 l l Mean Nuber of Bidders (λ) Figure 4 Optial bid levels for (a) unifor and (b) exponential valuation distributions where the nuber of bidders participating within the auction is described by a Poisson distribution who expected value is λ. In light of this work, we describe the nuber of bidders participating in any auction by a Poisson distribution where the probability that n bidders participate is P(n) = λn e λ. () n! Here, the paraeter λ describes the ean of this distribution and thus represents the expected nuber of participants in any individual auction. Given this distribution, we can extend the expression derived in section 4 and thus describe the expected revenue of the auction in ters of the paraeter λ, rather than n. To do so, we siply su the product of the expected revenue of the auctioneer, given a fixed nuber of bidders, E n, and the probability of that nuber of bidders actually occurring: E λ = n=0 P(n)E n. (2) Substituting equations 5 and into this expression and aking use of the identity n=0 to derive E λ = i=0 λ n n! = eλ allows us e λ[f(l i+) ] e λ[f(l i) ] [ [ l i F(li ) ] [ l i+ F(li+ ) ]]. (3) F(l i+ ) F(l i ) As before, we can use this expression within the nuerical algorith presented in section 5.2 to calculate the optial discrete bid levels. Again, we copare unifor and exponential valuation distributions, and

16 6 in figure 4 we show these results for a range of values of λ fro 2 to 20. In each case, λ represents the expected nuber of bidders, but the actual nuber of bidders who participate in an auction is described by the Poisson distribution. The figure shows that when λ is large, there is little difference between this case and the case of a fixed nuber of bidders considered in the previous sections. This is siply due to that fact that a large value of λ results in a Poisson distribution with a peak close to λ and a relatively sall standard deviation λ. Thus the results are little different fro those with an equivalent value of n. However, when λ is sall, the standard deviation of the Poisson distribution is larger relative to the ean value, and thus there is a significant probability that auctions occur in which the nuber of bidders differs fro the expected value by a large percentage. It is the effect of having fewer bidders that doinates, and thus, we observe that these optial discrete bid levels are ore evenly spaced, and hence closer to a fixed bid increent. In addition, the value of l 0 is lower and thus is closer to the optial reserve price of the equivalent continuous bid auction Explicit Auctioneer Costs Now, in the previous analysis, we assued that a fixed nuber of discrete bid levels have been ipleented by the auctioneer. This fixed nuber of discrete bid levels places a strict bound on the tie and costs of the auction, and our goal has been to calculate the value of these bid levels, in order to axiize the expected revenue of the auctioneer. However, we have not explicitly included the costs that each bid level incurs when calculating this revenue. Here we do so by assuing that each bid level that the auction proceeds through costs the auctioneer a fixed aount that represents both the tie and counication costs of the auctioneer 7. This cost is denoted by c, and it is now siple to extend the expression for the expected revenue derived in the previous section to deduct this cost, giving E = i=0 e λ[f(l i+) ] e λ[f(l i) ] [ ] [l i c(i+)]( F(l i )) [l i+ c(i+2)]( F(l i+ ). (4) F(l i+ ) F(l i ) We can use this expression within our nuerical algorith to calculate the optial discrete bid levels. In figure 5 we copare unifor and exponential valuation distributions, but this tie, we fix the expected nuber of bidders and the auctioneer s cost per bid level (taking λ = 0 and c = 0.005), and we vary the nuber of discrete bid levels (taking = 2, 0 and 20). Including this explicit cost has ore effect than the previous extension to the odel. First, it results in a sall increase in the value of all the discrete bid levels, since the costs of the auction progressing through these bid level ust be recovered. More significantly, while in the previous exaples increasing the nuber of bid levels would have resulted in levels that were closer together and an increase in the expected revenue of the auctioneer, this is no longer true. Now, 7 Note that each auction could also incur a fixed overhead cost, but since this will not affect the optial bid levels, we do not incorporate it here.

17 7 Optial Bid Levels (Unifor) f(v) v = 0 v = 0.9 l 2 l (a) = 2 = 0 l 0 l 2 l 0 l l l 0 = 20 l 0 l Nuber of Bid Levels () f(v) v = 0 v = Optial Bid Levels (Exponential) l (b) = 2 = 0 = 20 l 0 l 2 l 0 l 0 l 0 l l 2 l l Nuber of Bid Levels () Figure 5 Optial bid levels for (a) unifor and (b) exponential valuation distributions for an auction where we explicitly odel the costs of the auctioneer. In this case, λ = 0 and the cost to the auctioneer of the auction progressing through each district bid level, c = increasing the nuber of bid levels could potentially result in a net loss, since each level that the auction progresses through incurs an additional cost. Thus, as we increase the nuber of bid levels, these extra bid levels are positioned so that they are unlikely to affect the outcoe of the auction. With the unifor valuation distribution, these bid levels appear, close together, in the extree of the distribution (i.e. close to v). With the exponential distribution, they again appear in the extree of the distribution, but since this distribution is unbounded (i.e. v = ), they do not appear close together, but have increentally increasing values. Most notable, however, is that due to this effect, as we increase the nuber of bid levels that we calculate (i.e. increase ), the values of the first few bid levels rapidly reach static values that do not change. It is the position of these bid levels that deterine the expected revenue of the auctioneer (it is extreely unlikely that the bidders will have high enough valuations to continue bidding past these bid levels), and thus, we can calculate these significant bid levels for an arbitrarily large total nuber of levels, and in so doing we reove the paraeter fro the design of the auction. In the results we have presented, we have assued a relatively sall value for this cost per bid level

18 8 (i.e. c = 0.005). If we decrease the cost, the results approach those presented earlier (i.e. the case in which the auction costs are not calculated directly and the expected revenue is deterined by the strict bound of the nuber of discrete bid levels, ). At the other extree, if we increase this cost further, it becoes increasingly difficult to recover the costs of the auction process through the appropriate distribution of discrete bid levels, and thus the expected bid level at which the auction closes approaches l 0 (i.e. the auction closes at the first announced price). At this point, there is effectively no longer an auction as such, and thus, when we calculate the value of l 0, we are effectively calculating the optial single take-it-or-leave-it price, that the auctioneer should announce to the bidders. 7. Discrete Bid Level Density Function In section 5, we deonstrated that in our basic auction odel we can calculate the optial discrete bid levels, using a nuerical algorith, for any nuber of bidders and any bidders valuation distribution. In addition, we then showed that the sae nuerical algorith can be applied to various extensions of the odel. In this section, in order to develop an intuitive understanding of these nuerical results, we go back to consider what happens in our basic odel when the nuber of discrete bid levels becoes very large. Now in the previous section we have seen that when we do this in an extended odel with a cost ter, then the optial arrangeent is for ost of the bid levels to be clustered toward the upper end of the valuation range adding low bid levels is likely just to increase the cost of the auction without increasing the sale price. However, this is not the case when there is no cost ter. For typical valuation distributions, including the unifor and exponential distributions we have considered here, discrete bid auctions yield a lower revenue than continuous bid auctions (since, in the language of section 4, the losses due to cases and 2 exceed the gains due to case 3). Thus we expect that as the nuber of peritted bid levels is increased, the optial bid levels should get closer together. In calculating the details of this effect, we can derive an analytic expression for the density of the discrete bid levels, and thus, gain an intuitive understanding of how they are distributed that can then be applied in ore general cases. Now, suppose then that for soe large value of the bid levels are closely spaced. Then we can find a sooth function l(t), defined over the range 0 t, such that the actual bid levels l i are equally spaced saples of this function ( ) i l i = l for i = 0,..., (5) and the end points of the range are given by l 0 = l(0) and l = l(). By substituting this ter into the revenue expression, given in equation 5, and expanding in inverse powers of (see appendix C for ore details on this calculation) we obtain an approxiation to the auction revenue

19 E = n l() l(0) F(v) n [vf (v)+f(v) ]dv+l()( F(l()) n ) n(n ) F(l(t)) n 2[ 2F (l(t)) 2 +( F(l(t)))F (l(t)) ] l (t) 3 dt + O( 3 ). (6) Now, the ters in the first line of this expression give the revenue in a continuous price auction with n bidders, a starting price l(0) and a axiu possible price l() (see for exaple Riley and Sauelson (98) and note that since it is typically assued that F(l()) =, the second ter vanishes). The ters in the second line give the loss in revenue due to discrete bid levels (note that for coon distributions, including the unifor and exponential distributions used above, the quantity 2F 2 +( F)F is positive). As noted by Rothkopf and Harstad, the loss due to discrete bid levels is of order, and thus, as the nuber 2 of bid levels increases, the revenue of the auction rapidly approaches that of the equivalent continuous bid auction. Thus, in order to axiize the revenue of the auction, it is necessary to iniize the integral on the second line in the above expression. This is a standard proble in the calculus of variations (and details are again provided in appendix C), with the result that l(t) ust satisfy 9 l (t) = [F(l(t))n 2 (2F (l(t)) 2 +( F(l(t)))F (l(t)))] /3 C (7) where C is a constant 8. Now since l(t) describes the values of the bid levels in ters of the continuous variable t, a plot of /l (t) against l(t) represents the optial density of the discrete bid levels as a function of the bid level (all this, we reiterate, is valid when is large). Thus [ Discrete Bid Level Density F(v) n 2( 2F (v) 2 +( F(v))F (v) )] /3. (8) In order to copute the proportionality constant we need to know the value of the iniu and axiu bid levels, l(0) and l(). To a first approxiation, these can be taken to be the values required to give optial revenue in a continuous price auction. Thus, as the nuber of discrete bid levels becoes larger, the 8 Note that all the considerations of this section can be extended to the odel of section 6., in which the nuber of bidders in the auction is not fixed, but is a Poisson process with ean λ. The large approxiation to the revenue is l() E = λe λ(f(v) )[ vf (v)+f(v) ] dv+l()( e λ(f(l()) ) ) l(0) λ2 2 2 e λ(f(l(t)) )[ ] 2F (l(t)) 2 +( F(l(t)))F (l(t)) l (t) 3 dt + O( 3 ). 0 The optial bid level distribution satisfies [e λ(f(l(t)) )( 2F (l(t)) 2 +( F(l(t)))F (l(t)) )] /3 l (t) =. C