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Vivek Bhattacharya

1 , James W. Roberts

2 , and Andrew Sweeting

2,3

1 Department of Economics, Massachusetts Institute of Technology

2 Department of Economics, Duke University and NBER

3 Department of Economics, University of Maryland and NBER

January 22, 2014

Abstract.

In standard models of procurement auctions with endogenous entry, potential bidders simultaneously decide whether to enter based on their own private information about their costs. The resulting entry decisions may be inefficient because, when firms base their decisions on only their own private information and cannot coordinate, either too many or too few bidders may enter. We compare the relative performance of a standard first-price (i.e., lowest bid) auction with free entry and an alternative ‘entry rights’ mechanism where the auctioneer regulates entry by first auctioning off a fixed number of rights to participate in a final auction before running the auction for the contract itself. We show that the relative performance of these schemes from an efficiency perspective is ambiguous and depends non-monotonically on the precision of the information that potential bidders have about their costs of completing the project. As an empirical application, we estimate parameters from first-price auctions with free entry for bridge-building contracts that were let by the

Departments of Transportation in Oklahoma and Texas. In this case, we predict that an entry rights auction procedure would increase efficiency and reduce procurement costs significantly.

Keywords.

Auctions, entry, information, procurement, selection.

JEL Codes.

C72, D44, L20, L92.

Contact.

bhattacv@mit.edu

, j.roberts@duke.edu

, atsweet@duke.edu

. We are very grateful to Dakshina De Silva,

Hubbard, Jon Levin, Rob Porter, Paulo Somaini, Steve Tadelis and John Asker (the co-editor) and three referees provided very useful feedback that improved the paper. Bhattacharya would like to thank the Davies Fellowship from the Duke

University Department of Economics for financial support. Any errors are our own.

Bhattacharya, Roberts, and Sweeting

1. INTRODUCTION

2

Given the importance of public procurement for many economies around the world (e.g., 12% of GDP for OECD countries (OECD (2011))), there is natural interest in improving the way that contracts are awarded as even small improvements for individual contracts could lead to a valuable cumulative reduction in procurement costs or increases in efficiency.

1 . Under the assumption, which we maintain of independent private values (or more accuratley costs), standard auction formats, such as first-price / low bid auctions with suitably chosen reserve prices, can minimize procurement costs and maximize efficiency when the number of bidders is fixed. However, they may not be optimal or efficient when entry into auctions is endogenous as it will be, for example, when bidders have to spend considerable time and effort in ‘due diligence’ to figure out how much it will cost them to complete the contract. The existence of entry costs, and the implications of endogenous entry for auction outcomes, has recently been one of the focuses of the theoretical and empirical auction literatures (for instance, Li and Zheng (2009) and Krasnokutskaya and Seim (2011)). In this paper we consider the relative performance of a standard first-price auction with unregulated, or “free”, entry with an alternative mechanism where the procurer regulates entry by holding an initial auction for a fixed number of rights (an “entry rights auction”) to compete in the final auction for the contract.

Currently, first-price auctions are the predominant method for letting public contracts (Bajari, McMillan, and Tadelis (2009)). When there are non-trivial costs of participating in the auction, these procedures are usually modelled as a two-stage game. In the first-stage, a set of potential entrants (for example, a set of local contractors who purchase or are issued with specifications for the contract) simultaneously and non-cooperatively decide whether to pay the cost of entering the auction based on any private information, or ‘signals’, they have about their costs of completing the contract. Having paid the entry costs, the subset of entering firms find out their costs of completing the contract and in the second stage they submit bids, with the contract being allocated to the firm with the lowest bid, as long as this is less than any reserve price, at a price equal to its bid. We will call this format a“first price auction with free entry”

(FPAFE). In the equilibrium of an FPAFE, the marginal entrant in the first stage will be indifferent between entering the auction, which involves paying the entry cost but allows the bidder to potentially win the contract, and staying out.

1 See Bajari and Tadelis (2001) for extensive references to previous papers studinyg procurement contracts.

Regulating bidder participation in auctions

3

A natural question is whether these entry decisions are efficient, in the sense of being consistent with the minimization of the sum of the expected costs of completing the contract and total expenditures on entry costs. As shown theoretically by Levin and Smith (1994) and Gentry and Li (2012a) under different assumptions about the private information that bidders have about their values prior to entry, entry decisions will be efficient in the sense that a social planner who could only choose whether a potential bidder should enter based on that firm’s private information alone would choose the same entry rules as the potential bidders themselves choose in the symmetric equilibrium.

2

However, inefficiencies in entry can still arise from the fact that the entry decisions of different potential bidders are not coordinated, so that the auctioneer could end up with outcomes where there are too many or too few entrants (or possibly none at all, in which case the auction might have to re-run or the contract let by an alternative procedure).

For example, when potential bidders have no private information about their values when taking entry decisions, as assumed by Levin and Smith (1994) (LS), the symmetric equilibrium involves the potential bidders mixing over entry. As a result, the number of entrants will be stochastic, and, because potential bidders are identical when they decide whether to enter, the expected social surplus will be concave in the realized number of entrants.

3

Given this logic, Milgrom (2004) shows that efficiency would be increased if the auctioneer just chose a fixed number of potential bidders to enter the auction rather than allowing free entry.

In general the assumption that potential bidders have no private information about their costs seems implausible for many procurement settings, as before performing meaningful due diligence a firm is likely to at least know something about its free capacity and how much it will cost for it to secure some of the necessary inputs even if it is only imperfectly informed about other elements of its costs. If so, free entry will tend to lead to the most efficient firms entering, i.e. the free entry process will tend to select the bidders with the lowest costs, and an arrangement where the auctioneer randomly selected entrants without using this private information will have a natural disadvantage. However, in this case there could still be scope for the auctioneer to regulate the number of entrants by using an “entry rights auction”

(ERA) format where the potential bidders bid for a fixed number of rights to perform due diligence and

2

The typical Mankiw and Whinston (1986) result that free entry results in excess entry in a homogenous goods market does not hold because an entrant only takes business from other firms when it is socially optimal for it to do so. The Levin and Smith and Gentry and Li results are conditional on the auctioneer setting a reserve price that is equal to the cost of completing the project outside of the auction if there is no winner. This is the auctioneer-optimal reserve in the Levin and

Smith model, but not the Gentry and Li model although it is the natural ‘non-strategic’ reserve price in that setting as well.

3 This reflects the fact that the expected lowest cost amongst n ex-ante identical entrants will be convex in n .

Bhattacharya, Roberts, and Sweeting

4 enter the auction for the contract based on their private information. As long as first-stage bids are monotonic in the bidders’ private signals then the ERA will also be selective, and the firms likely to have the lowest costs will be chosen, but the ERA will have the advantage of controlling how many firms enter.

We show, using an example, that whether an FPAFE or an ERA is more efficient can depend in a non-monotonic way on the precision of the private information have about their costs. When potential bidders have very imprecise signals about their costs, the ERA dominates as one would expect given the informational assumptions made by LS and Milgrom in deriving the result mentioned above. When potential bidders are almost certain about their costs prior to entering, in which case the model tends towards the Samuelson (1985) (S) model where potential entrants know their costs exactly, the ERA also dominates because it is efficient to have only the firm with the best (lowest cost) signal paying the entry cost and this is the outcome that the ERA ensures will happen whereas no firm or more than one firm may enter under the FPAFE. However, when potential bidders’ signals are moderately informative about their costs an FPAFE can be more efficient than an ERA. The reason is that with free entry the number of entering firms will tend to depend on the private information that bidders have, and auctioneer does not, and this can be desirable. For example, while on average it might be desirable for two firms to enter because signals are only imperfectly correlated with costs, it may be best if only one firm enters if there is only one firm with a signal that indicates that its costs are likely to be low.

Given this ambiguity in which mechanism is more efficient, we estimate our model using data on a sample of contracts for work on bridges let by the Departments of Transportation (DoTs) in Oklahoma and Texas using FPAFEs in order to understand which mechanism might be perform best in a real-world context. To do so, we develop methodologies for solving and estimating a parametric model of FPAFEs where potential bidders have imperfect information about their costs when deciding whether to enter. We solve the model using the Mathematical Programming with Equilibrium Constraints (MPEC) approach proposed by Su and Judd (2012).

4 We estimate our model using a simulated method of moments estimator where the moments are computed using importance sampling, following Ackerberg (2009). This approach enables us to allow for observed and unobserved heterogeneity in the parameters across auctions, unlike the one previous attempt to estimate this type of model of which we are aware (Marmer, Shneyerov, and

4

Hubbard and Paarsch (2009) use this approach to solve first-price auction models under the S model assumption that potential bidders know their values when deciding whether to enter. We extend their solution method to allow for noisy signals about values (in our case, costs), creating imperfect selection.

Regulating bidder participation in auctions

5

Xu (2011)).

5

We estimate that entry is moderately selective for these contracts. However, given the other parameters of our model, our model predicts that overall efficiency would have been increased by using the ERA format. For example, for the representative (average) auction in our data we predict that the sum of the expected cost of the winning bidder and total entry costs would be 2.41% lower in the ERA than the FPAFE and that the total cost to the DoT of letting the contract would fall by a similar amount.

This fall in DoT procurement costs would is much greater than we predict would be achieved by using a more standard design adjustment to an FPAFE, such as adding an optimal reserve price, which for our representative case, would only reduce costs by 0.2%.

We are not aware of any previous attempts in the literature to compare ERAs and auctions with free entry using real-world parameters. A related theoretical literature has considered so-called “indicative bidding” schemes where potential bidders submit non-binding indications of what they are likely to be willing to bid which the seller may use to select a subset of firms to participate in the auction. In contrast in an ERA first-round bids are binding and can result in payments by firms that do not win the contract.

Indicative schemes are often used by banks and as part of the complicated processes by which defense equipment is purchased (Hendricks and Quint (2013), Foley (2003), Welch and Fremond (1998)). Ye

(2007) argues that, generically, indicative schemes will not induce strictly monotonic first-stage bids in equilibrium and so will not tend to guarantee the selection of the bidders that are most likely to have the highest values or the lowest costs in the second stage. In contrast he shows certain types of ERA schemes, like the one we consider, will induce strictly monotonic bidding and so guarantee an efficient selection of entrants. Recently Hendricks and Quint (2013) have shown that, when indicative bids are limited to a discrete set and either entry costs or the number of bidders is large, that the unique indicative bidding equilibrium can involve weakly monotonic bid functions, and that, in computed examples, indicative schemes can outperform standard auctions with free entry. By guaranteeing fully efficient selection of entrants, ERAs should, in theory, be more efficient than indicative schemes when the number of selected entrants is the same.

6 We discuss at the end of the paper several possible reasons why ERAs may be less

5

Some earlier work, such as Krasnokutskaya and Seim (2011), estimates models of endogenous entry into first-price auctions under the informational assumptions of LS. This implies that there is no selection in the entry process, which simplifies estimation because the distribution of values for entrants will be the same as the distribution for potential bidders.

6 Kagel, Pevnitskaya, and Ye (2008) compare indicative schemes and ERAs in the laboratory using student subjects, and find that, because of divergences from equilibrium behavior under both schemes the comparison may be less clear.

Bhattacharya, Roberts, and Sweeting

6 common than indicative schemes or FPAFEs in practice.

The paper is also related to our earlier work on selective entry auction models (Roberts and Sweeting

(2013a), Roberts and Sweeting (2013b)). In those papers we consider selective entry into second-price auctions, where the simpler form of equilibrium bidding strategies simplifies computation and estimation relative to the first-price, low-bid auctions considered in this paper which are the auction format that is most widely used in practice.

7 In Roberts and Sweeting (2013b) we compare the performance of a free entry auction with a sequential bidding process which is more efficient, and can improve the seller’s revenues for a wide range of parameters, even though it allows early bidders to deter entry. Here we consider a different type of design change which retains the simultaneous bidding feature of most standardly used auctions, which is attractive because a sequential process potentially allows the auctioneer to favor certain bidders when determining the order.

Four comments are in order about both our model and the nature of the results. First, when considering the ERA, we model the first-stage auction for entry rights as an all-pay auction . Following Ye (2007), the advantage, from a modeling perspective, of using the all-pay format is that equilibrium first-stage bids are guaranteed to be strictly monotonic in the signal that potential bidders have about their costs, which guarantees the efficient selection of entrants. When the more common discriminatory or uniform price formats are used to sell entry-rights this property may not hold because the expected benefit of being the marginal entrant allowed into the final auction, which determines the bid, may not be monotonic in the signal for all signals, and in fact it is straightforward to find examples where the relationship is not monotonic for the distributions we consider. However, as also shown by Ye, when equilibrium bid functions under discriminatory or uniform schemes are monotonic, the expected costs and efficiency under all three schemes (all pay, discriminatory and uniform) will be identical.

Second, we are comparing our results to a specific form of ERA, where the seller announces an exact value for the number of bidders who will be able to enter the second-stage auction in advance, rather than considering the optimal auction or even the optimal entry rights auction, where the number of allowed entrants into the second-stage might well depend on the bids that are announced in the first stage. We do this partly because it is not straightforward to characterize the optimal form of entry rights auction in our

7

As we assume potential bidders are symmetric it is possible to compute expected efficiency and procurement costs for first-price auctions by exploiting revenue equivalence. However, to use the information contained in all of the submitted bids in estimation it is necessary to be able to solve first-price auctions with selective entry.

Regulating bidder participation in auctions

7 model, but also because even if it was known the optimal mechanism would likely have such a complicated form that it would likely be implausible that any public agency would want to use in practice.

8

In any case, because our bottom-line conclusion is that there may be significant cost and efficiency advantages to using ERAs, this conclusion would only be strengthened if we considered the optimal form of ERA instead.

Third, our results are numerical. When entry is partially selective it is very difficult to solve models analytically outside of some special cases and, as our example shows, efficiency and cost comparisons will often depend on the distributions and parameters that are assumed so it is important to try to be as realistic about these elements as possible.

9 We do investigate how our comparisons change when we vary our parameters around their estimated mean values. By using parameters that are estimated from data, we are also able to provide some guidance as to how large the gains to regulating entry might be.

Finally, while we allow for partially selective entry and a rich model of auction heterogeneity we maintain the independent private values assumption that has been typically made in both empirical studies of public procurement (e.g., Krasnokutskaya and Seim (2011)) and theoretical work on models of two-stage entry rights or indicative bidding procedures (e.g., Hendricks and Quint (2013)). Understanding the effects of trying to regulate entry where costs have both common value and private value components is left to future work.

The paper proceeds as follows. Section 2 presents the model of imperfectly selective entry for FPAFEs and ERAs, and explains how we solve these models. Section 3 describes the data. Section 4 explains our estimation method. Section 5.1 presents and discusses the estimates of the model’s parameters, and the performance of the FPAFE and the ERA are compared in Section 5.2. Section 6 concludes. The

Appendices contains some additional details on our numerical routines and presents some Monte Carlo evidence on the performance of our estimator.

8

Lu and Ye (2013) characterize the optimal entry rights auction format in a setting where there is heterogeneity in potential bidders’ entry costs, but they have no private information on their values prior to entering. Cremer, Spiegel, and Zheng

(2009) characterize the optimal mechanism in this case, which involves a quite complicated sequential search process.

9 For example, Ye (2007) considers a case where bidders are uncertain about a component of costs that can only take on discrete high or low values.

Bhattacharya, Roberts, and Sweeting

2. A MODEL OF ENDOGENOUS BIDDER PARTICIPATION IN AUCTIONS

8

We introduce the general structure of costs and information in our model of procurement and then describe the FPAFE and the ERA. We also present a numerical example illustrating how an efficiency comparison of the two mechanisms will depend on the degree of selection in the entry process.

A procurement agency, which we will call the “procurer” in what follows, wishes to select one of N risk-neutral bidders to complete a project, such as repairing or constructing a bridge. When we go to the data we will allow for N to vary across projects, and for both observed and unobserved heterogeneity in the other parameters of the model. The agency may require that it pay no more than a reserve price r to complete the project, and if no firm submits a bid below this amount, the agency incurs a cost c

0 of procuring the work outside of this specific auction (e.g., this could be the expected cost from re-running the auction at a later date, or negotiating one-on-one with a particular contractor). Potential bidders can complete the project at cost C i distributed F

C

( · ) with compact support [ c, c ] that admits a continuous density f

C

( · ). We assume that bidders are ex ante symmetric and have independent private costs. To participate in the project-allocation auction of any mechanism

10

, a potential bidder must pay an entry cost K . Upon paying K , the bidder learns its private cost for completing the project, and so the entry cost is best interpreted as including some cost of doing the research necessary to learn the cost of completing the bridge, as well as any other participation and bidding costs. We make the assumption that firms cannot participate in the project-allocation auction for the contract without paying K . However, prior to paying the entry cost, each firm observes a signal S i that is correlated with its true cost C i and that is not correlated with any other bidder’s cost. Signals are affiliated with costs in the standard sense: the cost distribution conditional on a signal s first-order stochastically dominates the cost distribution conditional on a signal ˜ .

The two extreme cases of the LS and S models are embedded in this setup. If S i

= C i

, then signals are perfectly informative of costs and the setup reduces to the S model. When S i is independent of C i

, signals contain no information about costs and the setup reduces to the LS model. For many empirical settings, it seems plausible that buyers will have some, but imperfect, information about their costs prior to conducting costly research, consistent with S i being positively, but not perfectly, correlated with C i

.

10 This will be the auction in the FPAFE and the second-stage auction in an ERA.

Regulating bidder participation in auctions

9

2.1. Selective Free Entry into First-Price Auctions

The first model of procurement is the FPAFE which we view as describing the procedure used by the

DoTs in our data and most other government agencies. In the first stage, potential bidders observe their imperfectly informative cost signals. Based on these private signals, they take independent, simultaneous entry decisions and pay K if and only if they enter. In the second stage, the set of entrants compete in a low-bid, first-price auction with reserve price r . We assume that firms that do not pay K cannot participate in the auction and that agents bid without knowing how many of their competitors actually entered the auction, as is done in Li and Zheng (2009). If an entrant learns that his true cost exceeds the reserve price, he will not bid.

We solve for entry decisions and bid functions that define the unique symmetric Bayesian Nash equilibrium with monotone bidding behavior.

11 Potential bidders enter using a cutoff strategy; that is, a bidder enters if and only if he observes a signal s i

< s

0∗ for some critical value of s

0∗

(Gentry and Li

(2012a)). Define H ( c ) to be the probability (not conditional on S i

) that a given bidder either (1) enters the auction and has a cost no smaller than c or (2) does not enter the auction; thus, H ( · ) depends on the value of the signal cutoff used for the entry decision. The equilibrium bid functions β

∗

( · ) are given by the solutions to the optimization problem

β

∗

( c ) ≡ arg max b

( b − c ) H β

∗− 1

( b )

N − 1

.

(1)

The first order condition associated with this optimization problem gives the differential equation

H

0

1 + β

∗− 1

0

( b ) b − β

∗− 1

( b ) ( N − 1)

β

∗− 1

0

( b )

H β ∗− 1

0

( b )

= 0 , (2) with the upper boundary condition

β

∗

( r ) = r.

(3)

The equilibrium critical cutoff values s

0∗ are determined by the indifference condition that any potential bidder who receives a signal of s

0∗ must be indifferent between entering the auction or not paying the

11 Gentry and Li (2012a) establish uniqueness of the symmetric equilibrium in a class of affiliated signal models. As usual in entry games, asymmetric equilibria may exist.

Bhattacharya, Roberts, and Sweeting

10 entry cost at all. This zero-profit condition is thus written

Z r b − β

∗− 1

( b ) f

C b

β

∗− 1

( b ) | s

0∗

H β

∗− 1

( b )

N − 1 db = K, (4) where b ≡ β ( c ), and f

C

( ·| s ) is the conditional density of a bidder’s costs, computed using Bayes’ Rule, given he receives a signal s .

To solve for the bid functions, we use the Mathematical Programming with Equilibrium Constraints approach (Su and Judd (2012)). In a manner similar to that outlined by Hubbard and Paarsch (2009), who use MPEC to solve an FPAFE model where bidders know their values when they decide whether to enter, we express the inverse bid function as a linear combination of the first P Chebyshev polynomials (we use P =25), scaled to the interval [ b, r ]. The choice variables in our programming problem are, therefore,

P Chebyshev coefficients, the signal cutoff, and the value of the low bid ( b ) . We pick a fine grid { x j

} J i =1 with 500 points on the interval [ b, r ]. Then, we solve for the bid functions (more precisely, the Chebyshev coefficients) and the signal cutoff using arg min

{ b,β

∗− 1 ,s

0∗ }

J

X g β

∗− 1

( x j

) j =1

2 s.t. (3) and (4) , (5) where g β

∗− 1

( b ) is defined to be the left-hand side of equation (2). This nonlinear programming problem is solved using the SNOPT solver called from the AMPL programming language. We give more details about the numerical methods in Appendix A.

We note that while we assume ex ante bidder symmetry, this method can also be used to solve for equilibria in auctions with asymmetric potential entrants, for example where there are certain identifiable

‘types’ of bidders (such as small firms and large, well-established firms), although in this setting there could be multiple ‘type-symmetric’ equilibria. Developing methods to solve for all of the equilibria in asymmetric first-price auction models is the topic of ongoing research.

2.2. Entry Rights Auctions

When entry is regulated and potential bidders have some information about how competitive they will be, a straightforward way to choose the firms that will be allowed to bid for the contract is to hold an entry rights auction (Ye (2007)). In the first stage of our ERA, each of the N potential bidders learns

Regulating bidder participation in auctions

11 his signal, and the procurer announces the number of bidders, n , that it will choose to participate in a second auction where the contract will be allocared. Upon receiving this signal s i

, each bidder i submits and pays a non-negative bid given by a function γ ( s i

) for the right to participate in the second-stage auction, with the highest n bidders being selected. The procurer pays each of the selected entrants K to cover their entry costs, which we assume that all of the selected entrants must pay so that they find out their true costs. The procurer also announces the value of the ( n + 1) st highest bid to all of the entering firms.

12

It then holds the second-stage auction using the first-price format where the reserve price r is announced before the first-stage begins. However, as discussed by Ye (2007) any standard auction format, i.e., ones where the firm with the lowest bid wins the contract and a firm with the highest possible cost pays nothing, used in the second-stage will generate the same expected payoffs for all of the parties and the same level of efficiency as long as entrants have symmetric beliefs about their rivals and the first-stage has selected the firms with the lowest costs. We assume a first-price format because it reveals an interesting difference in bidding behavior between the FPAFE and the second-stage of an ERA with at least two entrants, but we exploit payoff equivalence by using a second-price format to compute expected second-stage payoffs more quickly and with a higher degree of numerical accuracy as we do not need to solve for the Chebyshev-approximated bid functions.

The first-stage auction is therefore a binding all-pay auction with an entry subsidy . Some explanation is required as to why we choose to model the ERA in this way. As shown by Ye (2007) in his Proposition

5, in an all-pay auction there will be a unique, symmetric equilibrium where the first-stage bid function

γ ( s ) is a strictly decreasing function of the cost signal (so that a firm with a lower cost signal will bid more) when the entry subsidy is sufficiently large.

13 This monotonicity property is important because it guarantees that the selection of entrants into the second-stage will be efficient. An entry subsidy equal to the due diligence cost K is sufficient for this property to hold, and, by Ye’s Proposition 6, the level of the subsidy does not affect either the procurer’s or the bidders’ total payoffs from the mechanism as long as first-stage bids are strictly decreasing. The intuition is that when entry subsidies are increased,

12 Publicly revealing the ( n + 1) st highest bid in the first-stage, but not the winning bids, ensures that bidders enter the second-stage with symmetric beliefs about the distribution of their rival entrants’ costs, which helps to ensure the efficiency of the second-stage allocation and the revenue equivalence of different second-stage formats. Otherwise beliefs about rivals’ costs will depend on an entrant’s own signal and beliefs will not be symmetric. As pointed out by a referee, it is unclear whether the procurement cost or efficiency performance of the ERA would change significantly if this information was not revealed, but it would certainly complicate our analysis.

13 Fullerton and McAfee (1999) also show how an all-pay auction with an entry subsidy induces efficient entry into the related mechanism of a research tournament.

Bhattacharya, Roberts, and Sweeting

12 firms increase their bids to participate in the second-stage auction in a way that exactly offsets the cost of increasing the subsidy. A corollary of this argument is that none of the our efficiency or procurement cost comparisons would change if we considered slightly lower subsidies that also gave rise to monotonic first-stage bid functions.

In contrast, first-round bid functions might not be strictly decreasing if we considered a first-round format that operated as either a uniform price auction (e.g., the n selected entrants would each pay the

( n + 1) t h highest bid) or a discriminatory auction (e.g., the n selected entrants pay their bid). As shown by Ye (2007) in his Proposition 3, first-round bid functions will be strictly decreasing in these formats only if, in the first-stage, a bidder’s expected gain from participating in the second-stage conditional on being the marginal (i.e., n th

) selected entrant is strictly decreasing in the cost signal. Whether this will be the case will depend on both n and the specifics of the cost and signal distributions considered, and we have found ranges of signals over which this montonicity property does not hold for some of the parameters that we consider in this paper. However, if the property does hold then both efficiency and the expected payoffs of all parties should be the same as under the all-pay format so, once again, it is still insightful to use the all-pay format even if it is used less frequently than other formats in the real-world.

We are interested in a symmetric Bayesian Nash equilibrium where the first-round bidding strategy

γ ( · ) is monotone in the signal and the second-round bid function β ( · ) is monotone in the cost for every possible value of the revealed ( n + 1) st highest first-round bid. Note that in such an equilibrium, revealing the ( n + 1) st highest bid is equivalent to revealing the ( n + 1) st lowest signal and we will denote the random variable corresponding to this signal by S and refer to particular realizations of it by s . Thus, entrants in the second round use Bayes’ Rule to compute the distribution of the costs of any one of their opponents as F

C | S ≤ sa

( · ), with density f

C | S ≤ s

( · ).

When the second-stage auction operates in a first-price format, the second-stage bid function β ( · ; s ) solves the differential equation 14

β

0

( c ; s ) = ( n − 1) ( β ( c ; s ) − c ) f

C | S ≤ s

( c )

1 − F

C | S ≤ s

( c )

, (6) with the boundary condition β ( r ; s ) = r . To solve for the bid function γ ( · ), note that the profit of a

14 If it operates in a second-price format then we assume that bidders will use their dominant strategy of bidding their cost in the second-stage.

Regulating bidder participation in auctions

13 bidder with cost c who is invited to enter the auction, when the ( n + 1) st signal is s , is, in the case of a second-round auction operating in a first-price format,

Π( c ; s ) = ( β ( c ; s ) − c ) 1 − F

C | S ≤ s

( c ) n − 1

, (7) where we are exploiting the fact that the entry cost is being fully subsidized by the procurer.

Assuming that first-round bidding follows a symmetric monotone equilibrium γ

∗

( · ), a given bidder i who receives a signal s in the first round chooses g to solve max g

Z s max g

∗− 1

( s )

Z c c

Π( c ; s ) dF

C | S = s

( c ) dF

S, − i

( s ) − g, where F

C | S = s

( · ) is the conditional distribution of the cost given a signal of s and F

S, − i

( s ) is the distribution of the n th highest signal of the remaining N − 1 bidders. Solving this maximization problem and imposing that in equilibrium g = γ

∗

( s ) gives the differential equation dg

∗

( s ) ds

= −

Z c

Π( c ; s ) f

S, − i

( s ) dF

C | S = s

( c ) , c

(8) with boundary condition g

∗

( s max

) = 0.

As entry is subsidized the integrand on the right-hand size is always positive (a bidder cannot lose money by entering) so the first-stage bid function will indeed be monotonically decreasing in s . The total procurement cost will be equal to the price paid in the second round plus the sum of the entry costs for the n entrants less the sum of the first-stage bids.

The value of n , the number of firms selected for the second-stage, plays an important role in the ERA, and is chosen by the procurer. So that the first-stage is meaningful we assume that 1 ≤ n<N . If n = 1, then the procurer selects one firm in the first-stage, and, effectively, makes that a firm a take-it-or-leave it offer to complete the project at the reserve price r (we assume that this firm would still have to pay the entry cost in order to be able to sign and complete the project). While this case may seem somewhat trivial, this is exactly what the procurer would like to do when bidders are well-informed about their costs because the value of having any firm in addition to the one with the lowest signal pay the entry cost should be small. Therefore for the rest of this section we allow for the possibility that n = 1. However,

Bhattacharya, Roberts, and Sweeting

14 when we turn to our empirical application in Section 5.2, we will restrict our attention to the case where

2 ≤ n<N . As we find that the ERA generally outperforms the FPAFE, changing this constraint would only strengthen our conclusion.

2.3. Efficiency of Bidder Entry

We now perform a comparison of the efficiency of the ERA and the FPAFE using an example to illustrate how the advantage of fixing the number of entrants depends on how well-informed bidders are about their values before they pay the entry cost. A mechanism is more efficient when the cost of completing the project, which is either the cost of the auction winner or c

0 if there is no winner, plus the sum of entry costs is lower than in the other mechanism. Recall from the discussion in the Introduction that while entry strategies in the FPAFE are optimal from an efficiency perspective whether or not signals are affiliated with true costs (Levin and Smith (1994), Gentry and Li (2012a)) if a potential bidder’s entry stratgy is only allowed to depend on that firm’s private information, there may be an efficiency advantage to fixing the number of entrants, for example through an ERA, to overcome the problem that conditioning only a bidder’s own private information will result in uncoordinated entry, and possibly either too many or too few entrants.

In fact it is straightforward to show that an ERA must be more efficient when either signals are completely uninformative, as in the LS model, or completely informative, as in the S model. In the former case, Milgrom (2004), p. 225-227, shows this result, which is due to the fact that, when signals are uninformative, the sum of the expected cost of the lowest cost entrant plus total entry costs will be convex in the number of realized entrants either in an FPAFE or when a fixed number of entrants is chosen randomly by the procurer.

15 In the case of the S model, the efficient entry decision will be achieved when only the firm with the lowest cost signal enters and only the lowest entry cost is paid, which can be achieved through an ERA (assuming the first-stage bid function is strictly decreasing in the cost signal), but not the FPAFE because entry decisions cannot be coordinated.

The situation is more complicated when signals are partially informative. An ERA, where the number

15

This result follows from the convexity of the expectation of the lowest order statistic of a sample with respect to the number of draws taken. In the case of the LS model the expected cost of the lowest cost firm given n entrants will be the same irrespective of whether entry occurred through the equilibrium of an FPAFE, where n firms are chosen randomly by the procurer. Note that when firms are exactly identical, random selection would be just as efficient as any mechanism where firms submit first-stage bids.

Regulating bidder participation in auctions

15 of entrants is fixed in advance, cannot exploit the fact that the bidders have private information that indicates how valuable their entry is likely to be. For example, for a given entry cost, if all but one of the firms have signals that indicate that their costs are likely to be high then it may be efficient for only one firm to enter, whereas if two or three firms have signals that indicate low costs then it may be desirable for more than one firm to enter. Therefore, the FPAFE, where entry decisions and the amount of entry are a function of this private information, potentially has an advantage, even though it retains the disadvantage that entry decisions are uncoordinated.

An example shows that for moderately informative signals it is also possible for the FPA to be preferred to the ERA from an efficiency perspective. Suppose that costs are distributed lognormally with location parameter µ

C

= − 0 .

09 and scale parameter σ

C

= 0 .

2 (implying a mean cost to complete the contract of 0.93 with standard deviation 0.19) and truncated to the interval [0 , 4 .

75]. The cost of entry is 0.02, there are four potential bidders, and S i

= C i

· exp ( i

), where i is distributed normally with mean 0 and standard deviation σ , with i independent across bidders. We assume that c

0 and the reserve in the FPA are equal to 0.85. With this setup, σ 2 controls how much potential buyers know about their costs before deciding whether to enter the mechanism, and in this way it controls the amount of selection in the entry process. Holding σ 2

C fixed, as σ 2 → ∞ , the model will tend towards the informational assumptions of the LS model, and as σ 2 → 0, it tends towards the informational assumptions of the S model. Define

α ≡ σ

2

/ ( σ

2

+ σ

2

C

) as a measure of the informational content of the cost signals, and therefore the degree of selection in the entry process.

16

A value of α near 1 implies that the the signal provides very little information, so that the model tends towards the LS model and the posterior distribution of the cost given the signal will be close to F

C

, whereas if α is close to 0 a signal will predict a firm’s costs very accurately.

[Figure 1 about here.]

Figure 1 plots both the percent difference in efficiency between the ERA and the FPAFE (a positive number means that the ERA is more efficient) as well as the (expected) number of entrants and the expected completion cost in each mechanism as a function of the degree of selection ( α ). Note that the completion cost is the winner’s cost if the object is awarded to a bidder and is c

0 otherwise. Efficiency is the sum of the cost of completion and the entry costs paid, so that lower numbers are more efficient, and

16 s would lognormal with location parameter αµ

C

+ (1 − α ) log( s ) and scale parameter σ

C

·

√

α .

Bhattacharya, Roberts, and Sweeting

16 for any choice of parameters in the ERA, n is set to minimize this quantity. When α is 0 or 1, the ERA is more efficient, as expected based on the arguments above. However, for intermediate levels of selection, the unregulated entry process of the FPAFE yields a more efficient outcome.

For example, when α = 0 .

5, the ERA tends to award the contract to a higher cost firm, so despite the fact that it admits slightly fewer firms to the second stage than tend to enter the FPA (thus saving on entry costs paid), it is the less efficient mechanism. When α = 0 .

6, the reason that the ERA is less efficient than the FPAFE is reversed. Now the ERA awards the contract to a lower cost firm on average, but to do so, the mechanism admits more firms to the second stage than, on average, choose to enter the FPAFE. While these extra entrants do help to lower cost of completion in the ERA, thus improving efficiency, the entry into the ERA can be considered excessive relative to that in the FPA in the following sense: there are more firms that enter the ERA and do not win that would not enter the FPA under free entry (0.492) than vice versa (0.164). We now turn to our empircial application where we estimate how selective the entry process actually is.

3. DATA

We estimate our model using a sample of FPAFE procurement auctions for bridge construction projects conducted by the Oklahoma and Texas Departments of Transportation (DoTs) from March 2000 through

August 2003. This data was originally collected and used by De Silva, Dunne, Kankanamge, and

Kosmopoulou (2008). As they describe, the data is taken from all regions of Oklahoma but only the

North Texas and Panhandle regions of Texas so that to geographic, construction and economic conditions are likely to be similar. Interested bidders are required to purchase plans from the state (the cost of purchasing a plan is on the order of $100), and the list of companies who have purchased a plan is publicly available before bidding takes place so it is reasonable to treat the number of bidders as fixed. These plans contain detailed descriptions of the project, including an engineer’s estimate of the materials required and his estimate of the likely cost of completing the project.

[Table 1 about here.]

Summary statistics for the sample are presented in Table 1.

17 We note that the average number of potential bidders, defined as the set of plan holders (which is reasonable since only plan holders are allowed

17 Appendix B gives details of how we constructed our sample from the original data.

Regulating bidder participation in auctions

17 to participate in the auction), is larger in Texas than Oklahoma, but the entry rates are similar in both states. The seasonally adjusted unemployment rate, which may affect bidders’ expected labor costs, is also systematically higher in Texas than in Oklahoma. The engineer’s estimate for projects is higher in

Texas than in Oklahoma, and the winning bids, normalized by the engineer’s estimates, are also slightly higher in Texas.

Since we will apply the model presented in Section 2 to these data, we briefly discuss how far its major assumptions are appropriate in this setting. The assumption of independent private costs is common in the literature studying highway procurement auctions; examples of papers using the IPV paradigm in highway procurement auctions include Krasnokutskaya and Seim (2011), Li and Zheng (2009) and

Jofre-Benet and Pesendorfer (2003). Because our data do not include exogenous bidder characteristics that allow us to group bidders into different types, we also make the common (see, for example, Li and

Zheng (2009)) assumption that bidders are ex-ante symmetric. That only about 65% of those firms who purchase a plan actually submit a bid provides evidence that, even for planholders, there must be some additional and non-trivial costs associated with submitting bids. Lastly, our model assumes that there is a publicly observable reserve price in order to avoid the problem that with no reserve price, if there is some probability that other bidders do not enter, it will be optimal to submit an infinite bid, and of course, in practice, a DoT would reject bids that were too large even when it does not impose an explicit reserve price in advance of the auction, which is the case in our sample. We therefore impose a reserve of 1.5

times the engineer’s estimate for each auction and drop (see Appendix B) the 5 out of the 471 auctions in which the declared winner bid more than this cutoff.

18

4. ESTIMATION

In this section, we describe how we estimate the model and detail the additional parametric assumptions we make on the specification of the model in order to take it to the data. Appendix C provides a Monte

Carlo study that analyzes the performance of the estimator.

18 An alternate approach (e.g., Li and Zheng (2009)) is to assume that the government acts as an additional bidder if only one firm enters.

Bhattacharya, Roberts, and Sweeting

4.1. Model Setup and Specification

18

We estimate a fully parametric version of the model from Section 2.1. We normalize the cost parameters by the engineer’s estimate of costs, so that if a firm has a cost of 0.9 this means 90% of the engineer’s estimate, and an entry cost of 0.03 means that the entry cost for an auction is 3% of the engineer’s estimate.

19 Allowing for the a subscripts to denote a particular auction, we assume that costs in an auction a are drawn from a truncated lognormal distribution, F

Ca

( · ), with location parameter µ

Ca and scale parameter σ

Ca

. The distribution is upper-truncated to the interval [0 , c ], but we set c = 4 .

75, or almost five times the engineer’s estimate, so that this truncation has very little effect on the distribution for plausible location and scale parameters. Furthermore, we assume S i

= C i

· exp ( i

), where i is distributed normally with mean 0 and standard deviation σ a

, with i independent across bidders. As in the model in the example in Section 2.3, σ 2 a controls the amount of selection in the entry process and a useful measure of the informational content of the signal is α a

. Given σ

Ca

, α a is a strictly monotonic function of σ a

.

Rather than estimating σ a directly, we choose to estimate α a because it has a natural intepretation as a measure of the selectivity of the entry process.

As implicit in the notation, we allow for auction-specific heterogeneity in the parameters ( µ

Ca

, σ

Ca

,

α a

, and K a

). Both observed and unobserved heterogeneity enter parametrically. For each auction a , we observe a set of covariates X a

. The specific parameters for auction a are drawn from truncated normal distributions with a mean parameter that depends on X a

. Let T RN ( µ, σ 2 , c, c ) denote a normal distribution with mean µ and variance σ 2 truncated to the interval [ c, c ]. We then assume that

µ

Ca

∼ T RN X a

β

µ

C

, ω

2

µ

C

, c

µ

C

, c

µ

C

σ

Ca

∼ T RN X a

β

σ

C

, ω

2

σ

C

, c

σ

C

, c

σ

C

α a

∼ T RN X a

β

α

, ω

2

α

, c

α

, c

α

K a

∼ T RN X a

β

K

, ω

2

K

, c

K

, c

K

.

(9)

As researchers we choose the truncation points. For the cost parameters we choose very wide bounds that should have little effect on the parameter distributions for plausible parameters. For example, we

19

It is natural to assume that larger projects will require more due-diligence and may also require the bidders to post larger bonds that guarantee that they will complete the work. Of course, the imposed linearity of the entry cost with the engineer’s estimate is an assumption.

Regulating bidder participation in auctions

19 truncate K a at 0.04% and 16% of the engineer’s estimate, where the industry literature (cited in our discussion of the results below) indicates that values in the range 1–2% are likely.

20

The parameters of interest are Γ ≡ β

µ

C

, β

σ

C

, β

α

, β

K

, ω 2

µ

C

, ω 2

σ

C

, ω 2

α

, ω 2

K

. Previous research (e.g. Krasnokutskaya (2011) and

Bajari, Hong, and Ryan (2010)) has found significant unobserved heterogeneity in the mean of the cost distribution for road construction contracts, and it is plausible that the variance of these distributions and the level of entry costs should also vary across auctions. Note, however, that we assume that within an auction bidders are symmetric in that they draw their costs and signal noise from the same distributions and that they have exactly the same costs of entering an auction.

In our setting, X a is a vector consisting of a constant, a dummy variable that is 1 if the state is Texas and the % unemployment rate in the county where the project is located in the month that the auction is held. We denote the coefficients for these covariates as β

0 , ×

, β

1 , ×

, and β

2 , ×

, respectively, where × can be any of µ

C

, σ

C

, α , or K . We only allow unemployment to affect the location parameter of the cost distribution, setting β

1 ,σ

C

= β

1 ,α

= β

1 ,K

≡ 0. The Texas dummy variable could capture differences in costs or the procurement process across state lines.

4.2. Importance Sampling

We estimate the model using a simulated method-of-moments estimator and we use importance sampling to approximate the moments predicted by a given set of parameters (Ackerberg (2009)). The motivation behind using importance sampling is that it is that it would be costly to repeatedly solve a large number of FPAFE games each time one of the parameters was changed. Instead, with importance sampling, we can solve a very large number of games once, and then weight the outcomes to calculate the moments, and only change the weights, which can be calculated almost instantaneously, as we change the parameters.

To explain the way that we use importance sampling, let y e = f ( X a

, θ a

) denote an expected outcome of an auction (e.g., an indicator for whether the winning bid lies between 0.8 and 0.9 times the engineer’s estimate) with observed characteristics X a where θ a denotes the collection of the structural parameters

θ a

= { µ

Ca

, σ

Ca

, α a

, K a

} . In our setting calculating y e is expensive because we have to solve the FPAFE and then simulate outcomes. The density of θ , given X a and Γ (which we want to estimate) is φ ( θ | X a

, Γ).

20 As noted by Ackerberg (2009), we would not be able to use importance sampling as part of our estimation approach if the truncation points depended on the parameters that we are estimating.

Bhattacharya, Roberts, and Sweeting

Given Γ the expected value of the outcome given X a and Γ, which we can label E ( y e | X a

, Γ), is

E ( y e | X a

, Γ) =

Z f ( X a

, θ a

) φ ( θ a

| X a

, Γ) dθ,

20

(10)

As this integral does not generally have an analytic form, one could approximate it for a given value of

Γ using simulation by drawing S samples of θ from the distribution φ ( θ | X a

, Γ) and computing f ( X a

, θ sa

) for each draw θ sa so that

E ( y e | X a

, Γ) ≈

1

S

S

X f ( X a

, θ sa

) .

s =1

(11)

However, this calculation would require solving a fresh-set of S auctions whenever one of the parameters in Γ was changed.

The importance sampling methodology exploits the fact that

Z f ( X a

, θ a

) φ ( θ a

| X a

, Γ) dθ =

Z f ( X a

, θ a

)

φ ( θ a

| X a

, Γ)

ψ ( θ a

| X a

) dθ,

ψ ( θ a

| X a

)

(12) where ψ ( θ a

| X a

) is an importance density that has the same support as φ ( θ a

| X a

, Γ), but does not depend on unknown parameters. In this case we can calculate f ( X a

, θ sa

) for a large number of draws taken from

ψ ( θ a

| X a

), and then calculate an approximation to E ( y e | X a

, Γ) where we only have to re-compute the weights

φ ( θ a

| X a

, Γ)

ψ ( θ a

| X a

)

, which is computationally cheap, when Γ changes, i.e.,

E ( y e | X a

, Γ) ≈

1

S

S

X f ( X a

, θ a s )

φ ( θ sa

| X a

, Γ)

ψ ( θ sa

| X a

)

.

s =1

(13)

To estimate the parameters, we group the auctions observed in the data into 32 groups, based on the interaction of the number of plan-holders ( N = 4 , .., 11), the state (OK or TX) and whether the unemployment rate in the county is above or below the median level for the state in month that the auction was held. For each of these groups, we create moments that measure the difference between average observed outcomes ( y a

) and expected outcomes ( E ( y e | X a

, Γ)) for auctions in the group. The outcomes consist of indicator variables for the number of firms submitting bids, indicators for whether the winning bid lies in one of 15 discrete bids (we divide the interval [0,1.5] into 15 equally sized bins) and the proportion of all potential bidders’ bids that lie in each of these bins (as some potential bidders

Regulating bidder participation in auctions

21 do not enter, the sum of the proportions across the bins for a given group will be less than 1). We estimate Γ by minimizing the squared sum, across groups and moments, of these differences, which is a consistent, but inefficient, method of moments procedure where the different moments and the different groups receive equal weighting. We use ψ ( θ a

| X a

) ≡ φ ( θ a

| X a

, Γ), where ˜ φ ( · ) denotes the truncated normal model given in equation (9), as our importance sampling density. For each group of auctions, we take N S draws of θ from this distribution and solve the associated FPAFE model, calculating y e

= f ( X a

, θ a

) using simulation, where N S is equal to 100 times the number of auctions in the group.

21

We choose 50 of these simulated auctions for each of our auctions to calculate E ( y e | X a

, Γ), as part of the estimation procedure.

We calculate standard errors using a bootstrap procedure. For a given bootstrap replication, we re-draw auctions from each of our 32 groups with replacement, and, for each of these auctions, we choose a new set of 50 simulated auctions, from our sample of N S for that group, to use in the importance sampling calculation. In this way the reported standard errors should account for the fact that our estimates are dependent on the particular set of importance sampling draws that we use to calculate the moments.

[Table 2 about here.]

Appendix C reports the results of Monte Carlo experiments that indicate that this estimator can accurately recover the parameters even when we use a smaller number of importance sampling draws.

4.3. Identification

Due to data limitations and the need to account for selection, we take a fully parametric estimation approach. Gentry and Li (2012b) study non-parametric identification in a class of selective entry auction models that includes ours, focusing on the case where there is no unobserved auction heterogeneity. The clearest intuition for identification comes when there are exogenous shifters in the threshold signal that firms use to decide whether they will enter or not. For example, when a firm faces few competitors or the reserve price is very high (in a procurement setting) firms may enter with very high probability and the distribution of observed bids will reflect the distribution of observed bids that we would observe

21

This importance sampling density was chosen after a number of initial runs, based on much more diffuse importance densities, indicated that these parameters would lead to mean values for each of the parameters that were quite close to those that we were going to estimate in both states, even though we were not including a Texas specific effect. As is well known, approximations that use importance sampling are more accurate when the importance distribution is more similar to the true distribution.

Bhattacharya, Roberts, and Sweeting

22 in an environment with no selection where we know that the cost distribution is identified from the distribution of bids. With no selection, observing multiple bids from each auction identifies both the cost distribution and the effect of unobserved heterogeneity which will cause bids to be correlated across auctions (Krasnokutskaya and Seim (2011) show non-parametric identification of a first-price auction model with a non-selective entry stage when there is unobserved heterogeneity in costs). In our setting we observe considerable variation in the number of potential bidders, although, as already mentioned, no explicit reserve price is observed. If the cost distribution is identified, from auctions where there is no, or very little, selection then the mean level of entry costs and the degree of selection (or equivalently the informativeness of potential bidders’ signals) will directly affect how much entry occurs and the way in which the observed bid distributions are truncated. For example, when signals are quite uninformative bids will be drawn from a cost distribution that is quite similar to the distribution of costs in the population of potential entrants even when several firms do not enter, whereas when signals are informative, bidders with high costs will not be entering. Of course our parametric assumptions are likely to play a more important role given that our model also allows for unobserved heterogeneity across auctions in both entry costs and the informativeness of signals.

5. RESULTS

In this section we present the estimation results and our counterfactual analysis of the impact of switching from the observed FPAFE format to an ERA format.

5.1. Parameter Estimates

The left-hand columns of Table 3 present the parameter estimates.

[Table 3 about here.]

To interpret the location and scale parameters together, the right-hand columns report the average across auctions in each state of (going down the columns) the mean of the cost distribution, the standard deviation of the cost distribution, the value of α and the entry cost. The mean of the cost distributions are similar in the two states, at around 91.5% of the engineer’s estimate. As the average winning bids are higher than this, it suggests that the mark-ups may be quite substantial. We will return to why

Regulating bidder participation in auctions

23 this happens, and the possible role that an ERA might play in reducing mark-ups, in discussing our counterfactual results.

Mean entry costs are 1.5% of the engineer’s estimate of the cost of the project in Oklahoma and 1.9% in Texas. While in many settings in IO it is hard to know what entry costs are really reasonable, in this setting we are able to compare them with estimates from manuals that are used in the industry.

Halpin (2005) estimates that the cost of researching and preparing bids is around 0.25% to 2% of total project costs. Park and Chapin (1992) estimate that these costs are typically about 1% of the total bid.

Our estimates therefore appear quite reasonable. In contrast, estimates of entry costs based on models with no selection are typically higher. For example, based on highway paving contracts in California,

Krasnokutskaya and Seim (2011) estimate mean entry costs to be approximately 3% of the engineer’s estimate, while Bajari, Hong, and Ryan (2010) estimate them to be 4.5% of the engineer’s estimate. It makes sense that estimated entry costs should be lower once we allow for selection. Taking the probability that other firms enter as given, the attractiveness of entry depends on the firm’s expected profit if it enters, which will reflect the probability that it wins and its mark-up, and its entry cost. A model with selection can explain why some firms do not enter even when their entry costs are low in terms of them having bad expectations about their costs, whereas a model without selection must have a relatively high entry cost to be able to explain why firms do not enter.

The estimates of α suggest that entry is partially selective in this setting, with α around 0.50 in

Oklahoma and 0.61 in Texas; these levels of α correspond to the scale parameter σ of the error distribution being approximately σ

C in Oklahoma and 1.3 times σ

C in Texas. Thus, despite the fact that the amount of entry into these auctions is volatile (for example, in Oklahoma auctions with six plan-holders we see as few as one and as many as 6 firms bidding), the fact that we find that bidders are partially informed of their costs before doing further research suggests that the government may not want to regulate participation in these auctions as it risks unnecessarily including firms that are aware they are likely to have high costs, or exclude an additional firm that it would have been desirable to include in order to reduce the expected cost of completing the contract. We now turn to analyzing this possibility.

Bhattacharya, Roberts, and Sweeting

24

5.2. Regulating Bidder Entry with an Entry Rights Auction

In this subsection, we use our parameter estimates from Section 5.1 to predict how procurement costs and efficiency would change if the DoTs were to switch from procuring contracts using FPAFEs to regulating the number of bidders that can participate using ERAs. Throughout what follows we will use the cost of procurement to refer to what the net amount that the procurer would have to pay to get the project completed, i.e., the winning bid (or c

0

) in the FPAFE and the winning second-round bid (or c

0

) plus all entry costs less first-round bids in the ERA. We will use efficiency to refer to the sum of entry costs and the cost of the firm that completes the project, where a lower number is more desirable. Unless we state otherwise, we assume that, the cost to the DoTs of completing a project outside the auction is 1.5 times the engineer’s estimate and this is used as the reserve price in the FPAFE and the second-stage auction of the ERA.

In presenting the results we assume the DoTs would choose the number of entrants in the all-pay ERA to minimize the cost of procurement, a choice we denote n

∗ cost

. We will comment briefly below about how the results are similar when we use the number, n

∗ eff

, that maximizes efficiency. The choice of n

∗ cost balances several competing effects on the procurement cost. In equilibrium, the procurer pays the full entry cost of an additional entrant, and increasing the number of entrants will reduce the amount that any firm bids in the all-pay auction. On the other hand, when there are more entrants, bidding in the final auction will be more competitive and the additional entrant may be more efficient than the other entrants, which will also tend to reduce the price that the procurer has to pay in the final auction. With the exception of the additional entry cost, the size of these different effects will depend on the degree of selection. When there is more selection an additional entrant will tend to have higher costs than the existing entrants, so that it will be less likely to win the auction, and will also have less effect on the bids and values of entering the second-stage auction of the inframarginal entrants.

Tables 4 and 5 present the results. Table 4 compares the outcomes of interest (procurement costs, efficiencies and bidder profits), while Table 5 looks at the average cost of the winner, the average winning bid and the average amount of entry under each mechanism. For the ERA it also shows the average total expenditure on entry costs (paid for by the procurer) and the average revenue from first-stage bids. In each case, we use a range of different parameter values. The baseline case (top row) corresponds to the mean parameters for auctions in Oklahoma (unemployment rate 4.14%) and approximately the mean

Regulating bidder participation in auctions

25 number of potential bidders in that state (7). In the remaining rows we change the number of potential bidders up and down by one standard deviation and each parameter, in turn, to be equal to either the 10 th or the 90 th percentiles of its distribution. The value that is changed from the baseline is listed in italics .

With the exception of the case when entry costs are low, the ERA gives both a more efficient outcome

(for example, in the base case our efficiency measure is 2.4% lower under the ERA (recall, a lower number is good here)) and lower expected procurement costs (in the base case, procurement costs are also 2.4% lower under the ERA). This result that the ERA does better both in terms of efficiency and procurement costs also holds when we compute the total gain across all of the projects in our sample in both OK and

TX: our model predicts that the sum of expected project completion costs and entry costs would have been $[] million ([]%) lower using an ERA (i.e., a more efficient outcome), and that DoT procurement costs would have been $[] million ([]%) lower. Of course, the DoTs in our sample account for a small proportion of the public construction projects allocated using FPAFEs, so a broader switch could lead to much larger monetary gains.

Table 5 shows that the ERA is more efficient in these cases (i.e., excluding the Low K case) because both the expected number of entrants, and so the total expenditure on entry costs, and the costs of completing the project (i.e. the cost of the winner) are lower under the ERA. The fact that both of these properties hold simultaneously reflects the fact that entry in an FPAFE is uncoordinated, so that while there is higher entry on average, there are also occasions when no firms or only one firm enters and an additional entrant, who will always be in the ERA as n

∗ cost

= 2 for these parameters, would have been valuable.

22 Most of the efficiency advantage comes from lower entry costs in the ERA (for example, 91% of the efficiency gain in the base case). This fact helps to explain why the one case where the FPA is more efficient is when entry costs are very low ( Low K where K equals just 0.2% of the engineer’s estimate) as in this case, the fact that there is more entry into the FPAFE only increases total costs slightly, while tending to slightly reduce the expected cost of the firm that is chosen to complete the project. However in the Low K case, the efficiency advantage of the FPAFE is small (total costs are 0.16% lower under the

FPAFE than under the ERA).

22

For the base parameters, there is no entrant into the FPAFE in 1% of simulated auctions in which case the procurement cost is 1.5. This would only happen if both of the selected firms had costs above 1.5 in the ERA and this happens with almost zero probability, and never in our simulations. If we condition our comparison on cases where at least one firm enters the FPAFE, so that we ignore simulations where the outcome of the FPAFE is particularly inefficient, the expected completion costs under the two mechanisms are very similar, but the differences in overall efficiencies, procurement costs and bidder profits are almost identical to those shown in Table 5.

Bhattacharya, Roberts, and Sweeting

[Table 4 about here.]

26

[Table 5 about here.]

How are the rewards of the ERA’s greater efficiency split between the DoTs and the firms? As can be seen in from Table 4, the gains to using an ERA accrue mainly to the procurer with procurement costs typically falling slightly more than efficiency increases. At the same time bidder profits tend to fall. The exception is when entry costs are high ( High K ), as the bidders also benefit under the ERA, but even in this case the procurer benefits more than the bidders do. It also important to note that these reductions in procurement costs under an ERA are large compared with the reductions that might come from design changes that are often considered, such as setting an optimal reserve price. For example, in the Base case, switching to the ERA reduces procurement costs by 3.14%, whereas using an optimal reserve price in an

FPAFE only lowers procurement costs by 0.2%.

23

To understand why procurement costs tend to be lower under the ERA recall that in an FPAFE the procurer only pays the winning bid whereas in an ERA it pays the winning bid and the entry costs of all of the firms admitted in the first stage, but also receives the first-stage bids of all of the firms. Table 5 shows that whether the sum of first-stage bids exceeds the expenditure on entry costs depends on the parameters, but in general the numbers are close, so that the reduction in procurement costs comes primarily from the fact that the winning bid tends to be significantly lower in the ERA than the FPAFE even though the winner’s costs are similar, and are sometimes higher in the FPAFE than the ERA (e.g., the High σ

C and

High α cases).

Figure 2 plots the distribution of the winner’s costs as well as the procurement costs (including the first-round indicative bid in the ERA) for both mechanisms at the baseline parameters.

24 The distribution of the winner’s costs for the FPAFE and the ERA are similar in shape, with the distribution for the

FPAFE concentrated on slightly lower values than that for the ERA. The distributions of procurement costs, however, show a marked difference in shape: the distribution for the FPAFE has a more pronounced right tail in comparison to that of the ERA. This tail exists because the markups (defined as the ratio of the bid to the cost, less 1) when the winner’s cost is relatively high in the FPAFE are larger than those in

23

We compute the optimal reserve price by searching over an interval of reserve prices with 0.01 spacing. For each reserve price we simulate procurement costs using 500,000 simulations.

24 These parameters correspond to the base case.

Regulating bidder participation in auctions

27 the second-stage ERA, even though, on average, a bidder faces fewer competitors in the second-stage of the ERA.

25

[Figure 2 about here.]

The difference in markups can be seen in Figure 3, which compares the FPAFE bid function and the average bids from an ERA with the same base parameters and n

∗ cost

= 2.

26

The diagrams include the pdfs of the cost distribution of a typical entrant into each mechanism to indicate which parts of the bid curve are the most empirically relevant. The markups in the ERA are slightly larger than in the FPAFE for low-cost bidders, but they are much smaller for higher costs, which the majority of bidders, and even a large share of winning bidders (Figure 2 shows this density), have. This leads to the result that the average mark-up for the winning bidder in the FPAFE (ignoring cases where there is no sale) is 9.4%, whereas the average mark-up in the second-stage of the ERA is only 6.8%.

In the FPAFE bidders with moderately high costs (for example, in the range 0.9 to 1.1, where the average completion cost in the population of potential bidders is 0.91) submit bids with large markups over costs because the number of entrants is uncertain, and, as a bidder’s cost rises, it becomes optimal to focus on winning only when there are no other entrants, in which case the optimal bid should be equal to the reserve price.

27 In contrast, in an ERA with n

∗ cost

= 2 a bidder knows that he can only win if he submits a lower bid than his competitor. Therefore, as a bidder’s cost increases he will tend to bid more aggressively (i.e., with a lower mark-up), as can be seen in Figure 3, where the average mark-up is montonically decreasing in the bidder’s cost in an ERA for all cost realizations below 1.2.

28

[Figure 3 about here.]

25

The left end of the procurement cost distribution in Figure 2(b) is also of note. In an FPAFE there is a single bid function for given parameter values ( θ a

) because entrants do not know how many other planholders are bidding. This bid function maps costs to the interval [ b, r ], and the sharp cutoff in the kernel density for the FPA corresponds to b . On the other hand, bid functions in the ERA depend on the realization of s , and this causes b to vary depending on signals even when the parameters of the auction are fixed. As a result, there is no sharp cutoff in the density for the ERA.

26 Note that the average bid function in the ERA is defined by β ( c ) ≡

( n + 1) st lowest signal.

R β ( c ; s ) f

S

( s ) ds , where f

S

( · ) is the density of the

27 That high-cost bidders bid close to the reserve suggests that the reserve may play an important role in determining the expected markups of the winner and of the representative entrant. Figure 2 shows that the distribution of the winner’s cost is away from the region where bids are close to the reserve. Indeed, the winner’s bid is usually much closer to b , which is still a function of r but only tends to shift slightly as r changes. For the base parameters, changing the reserve price over a wide range—from 1.1 to 4.75 (the maximum of the support of the cost distribution)—only changes the winner’s markups from 8.3% to 10.9%.

28 Above 1.2, the mark-up increases reflecting the shape of the upper tail of the cost distribution. However, it is very rare for the winner in the second-stage of an ERA to have a cost above 1.2, so this has little effect on procurement costs.

Bhattacharya, Roberts, and Sweeting

28

We can also compare the FPAFE and ERA assuming that the the number of selected firms in the ERA is chosen to maximize efficiency ( n

∗ e f f

).

n

∗ eff is set to balance the entry cost associated with an additional entrant against the possible gain that will come from lowering the cost of completing the project.

Given our estimates, it turns out that using n

∗ eff or n

∗ cost does not change the results very much because, given the integer constraint, n

∗ eff and n

∗ cost are usually the same. This is the case for all of the parameters in Tables 4 and 5. As an observation, in all the numerical experiments we have performed, based on a much wider range of parameters, we have never found a case where n

∗ e f f

< n

∗ cost

, whereas there are examples where another entrant is allowed to enter the second-stage of the ERA in order to improve efficiency. The intuition here is that allowing an additional entrant reduces the bids that firms make in the first-stage of the ERA, and a procurer that seeks to minimize procurement costs therefore has an incentive to exert market power by reducing the number of entry slots. This effect outweighs the effect that reducing the number of second-stage entrants will tend to increase the mark-up in the winner’s bid.

Given the sizable gains of using an ERA that we estimate above, a potential puzzle is why they are not used more often for the sort of public procurement we study here. There are several possible explanations.

First, it may be the case that the reduced bidder surplus in the ERA is actually harmful for the seller in the long-run. In most real-world settings, the seller expects to procure many contracts over a number of years, and so will benefit when there are a large number of firms that are active in the industry and are potentially interested in bidding. The need to ensure that bidders get sufficient surplus will be even more important when many potential bidders are small firms who may face liquidity constraints, that may also be tightened by having to make binding bids in ERAs.

29 This would be particularly true if they have to operate in an all-pay format to work efficiently.

30

A second possibility is that bidders may be risk averse. There are competing effects of risk aversion on procurement costs in the standard auction and in the ERA. While entrants will bid more aggressively, which serves to lower procurement costs, potential entrants will also be more reluctant to pay sunk entry costs to participate in an auction with an uncertain outcome, and this will lead to lower participation in both mechanisms, as well as lower indicative bids in the ERA. To evaluate whether risk aversion was a

29

Government agencies typically have targets for the value of contracts that should be allocated to small or minority-owned businesses which often leads them to use set-asides or bid subsidies (e.g., Krasnokutskaya and Seim (2011), Athey, Coey, and Levin (2013)).

30 While we assume an all-pay format to ensure the monotonicity property that makes the ERA model tractable, it is possible that, in practice, an ERA might work well with a more standard first-stage format.

Regulating bidder participation in auctions

29 possible explanation for why ERAs are not used more widely in these auctions we computed procurement costs based on our model’s estimated parameters and a variety of assumptions about the risk preferences of bidders with CARA utility. We found that for plausible levels of bidder risk aversion, the ERA was still more efficient and lowered expected procurement costs.

31 Thus, risk aversion is unlikely to be the primary reason why ERAs are not more widely used.

As a third possibility, Hendricks and Quint (2013) suggest that ERAs may provide sellers with perverse incentives to try to sell off contracts that, once they undertake research, bidders will find to be worthless.

This seems unlikely to be a concern for state agencies that need to procure a large number of contracts and so are likely to have strong incentives to maintain a reputation for probity.

A fourth possibility is suggested by Kagel, Pevnitskaya, and Ye (2008), who find that in a laboratory setting that the strategic complexity of formulating offers in an ERA can lead to overbidding and bankruptcies. Whether this would hold in the field is an open question.

Finally, it may of course be that some of the standard assumptions used to model procurement auctions, which we have maintained here, such as independent private values, simultaneous entry decisions and independent signals (e.g., Athey, Levin, and Seira (2011), Krasnokutskaya and Seim (2011) or Li and

Zheng (2011)) are incorrect and affect the comparison of the different designs. For example, if bidder signals are correlated or entry decisions are made sequentially, entry into the FPA may be less volatile, mitigating some of the gains to using the ERA.

6. CONCLUSION

In procurement settings, where it is expensive for potential bidders to learn their costs of completing a project, unregulated entry can lead to volatile amounts of bidder participation. If bidders have either no information or perfect information about their costs before entering, then fixing the number of entrants, say by auctioning off entry slots as is done in an entry rights auction, will raise efficiency. Motivated by the importance of public procurement we consider whether a simple scheme that regulates entry can raise efficiency and reduce procurement costs.

When bidders’ pre-entry information about their cost of completion is either very precise (so that bidders know their completion costs prior to bidding) or very imprecise, the entry rights auction dominates.

31 Previous versions of this paper contained these simulation results and they are available on request.

Bhattacharya, Roberts, and Sweeting

30

However, if bidders have some pre-entry information about their costs, this need not be the case. The reason is that, unlike a standard auction into which there is free entry, the number of entrants selected by an entry rights auction is not conditioned on bidders’ private pre-entry information.

Therefore, in computing the gains to regulating entry this observation highlights the importance of allowing for bidders to be imperfectly informed of their value of participation prior to entering. We do so when modeling bidder participation into bridge-building contracts in Texas and Oklahoma. We estimate that there is an intermediate amount of selection in the entry process, opening up the possibility that regulating entry may actually hurt efficiency in these auctions despite the fact that bidder participation is variable across different projects in the data. However, when we compute the gains to implementing a standard entry rights auction instead of the current free-entry first-price auction, which to our knowledge is the first quantification of the impact of using an entry rights scheme based on real-world parameters, we predict that efficiency will indeed be improved by regulating entry. We also predict that an entry rights auction will reduce procurement costs, partly by encouraging more aggressive bidding behavior by high cost bidders. Therefore, while in the current procurement mechanism used by the Texas and

Oklahoma DoTs, firms’ independent participation decisions maximize the possible efficiency afforded by such a mechanism, we predict that—despite bidders having imperfect information about their costs prior to entering—efficiency could be even greater were the DoTs to regulate entry by employing an entry rights auction.

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Regulating bidder participation in auctions

A. NUMERICAL METHODS TO SOLVE THE FIRST-PRICE AUCTION

33

In this Appendix we detail the numerical methods used to solve for equilibrium in a first-price auction with partially selective entry. Section A.1 provides parameters, such as the degree of the polynomial expansion and the type of grid, used in the computations. It also enumerates additional constraints used in practice. Section A.2 discusses how we check the accuracy of our solutions.

A.1. Details of the Optimization Problem

The choice of grid { x i

} N i =1 has a minor effect on the solution to the programming problem. Hubbard and

Paarsch (2009) use an N -point Gauss-Lobatto grid on [ b, r ],

32 but we have found in our experiments that an evenly spaced grid is often more efficient. Throughout the paper, we use N = 500 grid points and

P = 25 polynomials in the expansion; these choices solve the optimization problem reliably and efficiently.

Bid functions are monotonic in the cost of the bidder, so we follow Hubbard and Paarsch (2009) and impose that β

∗− 1 ( x i

) ≥ β

∗− 1 ( x i − 1

) for 2 ≤ i ≤ N . Furthermore, we impose the rationality constraint that an agent never bids less than his cost: β

∗− 1 ( x i

) ≤ x i for all i . Finally, we replace the constraint in equation

(4) by the condition that the two sides cannot differ by more than 0.1% of K . The integral in equation (4) is replaced by an approximation using the trapezoidal rule on the N -point grid defined previously.

A.2. Checks on the Solution

We implement a simple consistency check on the solved bid function β

∗ by solving equation (1) by simulation; this check is similar to the one used by Gayle and Richard (2008), who develop a numerical method to solve for bid functions in a first-price auction in the presence of collusion. We choose a fine grid

B of “bids.” For a given cost c , we compute the argument of the right hand side of equation (1) for each b ∈ B by simulation, fixing β

∗− 1 ( · ) and s

0∗ to the values determined by the optimizer. We of course expect the optimum to be β

∗

( c ). This check amounts to computing the best response of a player conditional on all other players using the bid functions computed by the optimizer and an entry rule dictated by the computed value of s

0∗

. Figure 4 plots a sample instance of this check where B has step size equal to 0.003, using the baseline parameters in the Base case in Table 4: the computed functions (solid) and the best

32 A Gauss-Lobatto grid on [ − 1 , 1] is defined to be the points y k grid on other intervals, we simply scale these points linearly.

= cos [ kπ/ ( N − 1)] for k ∈ { 0 , 1 , . . . , N − 1 } . To define a

Bhattacharya, Roberts, and Sweeting

34 response (dashed) match each other quite closely. The major discrepancies between the bid function and the best response occur at low costs and at especially high costs. There are too few draws of bidders with low or high costs for the simulation to return meaningful results.

[Figure 4 about here.]

We can check the computed value of s

0∗ by computing the profits of a marginal entrant. Bayes’ Rule gives the distribution of costs of an entrant who receives a signal s

0∗

. Fixing the bid functions and entry thresholds of all agents to the computed ones and assuming that the marginal entrant will enter, we use simulation to compute the profits of the marginal entrant in the bidding stage. We expect this number to be close to the entry cost. In the baseline case again, using 500,000 simulations shows that the expected profit and the entry cost differ by less than 1%.

Another method to check the value of s

0∗ is to use the result from Gentry and Li (2012b) that the competitive s

0 coincides with the social planner’s choice of s

0

. Given an s

0

, the efficiency of an auction where bidders enter if and only if their signals are below s

0 can be determined by computing an integral.

The value of s

0 that minimizes this integral is the same as the value of s

0∗

. In the baseline case, the computed value of s

0∗ and the value that minimizes the integral agree to four decimal places.

B. DETAILS OF SAMPLE CONSTRUCTION

De Silva, Dunne, Kankanamge, and Kosmopoulou (2008) study auctions from January 1998 through

August 2003. However, we only focus on auctions after March 2000 due to an important change in policy in Oklahoma: prior to March 2000 they did not disclose engineer’s estimates to bidders. Throughout this paper, we restrict our attention to auctions with between 4 and 11 potential bidders since, as discussed in

Section 4, there are very few auctions with other numbers of potential bidders. From the set of bridge construction contracts let during this period, we remove auctions from the dataset in which there was no winner or the winning bid was extremely high or low. In some auctions, companies submitted bids but the DoT did not allocate the contract to any of the bidding companies. There is no indication in the dataset to explain why the DoT did not accept any of the bids and so we drop these auctions. We remove all auctions with a winning bid less than 70% of the engineer’s estimate from the dataset. Only 3 of the 161 auctions in Texas in which a winner was declared were removed due to this criterion, but 47

Regulating bidder participation in auctions

35 of the 310 auctions in Oklahoma were removed. We justify this cutoff by noting that over one-third of the auctions that were dropped in Oklahoma (16 of the 47 auctions) were won by one of two companies, however, these companies only won 4 of the remaining 263 auctions in the sample. Furthermore, there is no noticeable observable difference between these auctions and the rest of the sample in terms of project characteristics. As described in the text, we also drop auctions with winning bids greater than 150% of the engineer’s estimate.

C. MONTE CARLO STUDIES

To test the performance of the estimator, we conduct a set of Monte Carlo experiments using data simulated from the specification in equation (9). We draw the number of potential entrants N uniformly at random from the set { 4 , 6 , 8 } . We set the vector of observed characteristics for auction t to be X a

≡ (1 , x a

) where x a is drawn uniformly at random from the set { 0 , 0 .

2 , 0 .

4 , 0 .

6 , 0 .

8 , 1 } . The remaining parameters in the specification are taken to be β

µ

C

= (0 , 0 .

65)

0

, β

σ

C

= (0 .

05 , 0)

0

, β

α

= (0 .

5 , 0)

0

, β

K

= (0 .

04 , 0 .

05)

0

,

ω

µ

C

= 0 .

05, ω

σ

C

= 0 .

02, ω

α

= 0 .

05, and ω

K

= 0 .

015. We denote this “true” parameter vector, consisting of all β and ω parameters, as Γ

0 throughout this section. The truncation points are taken to be c

µ

C

= − 0 .

4, c

µ

C

= 1, c

σ

C

= 0 .

005, c

σ

C

= 0 .

995, c

α

= 0 .

1, c

α

= 0 .

9, c

K

= 0 .

0001, and c

K

= 0 .

16. These truncation points are chosen to ensure that the solver can determine a correct solution for the auction with any set of parameters drawn from the above distribution. The reserve price is set of 4.75, a value that is beyond the

99% quantile for most ( µ

C

, σ

C

) pairs drawn from the specification. As a result, the reserve is relatively non-binding. For each set of parameters drawn from this distribution, we solve for the equilibrium bid functions and entry decisions and generate a single simulation. We assume that the observable data are the potential number of entrants, x a

, the entry decision of each potential entrant, and the bids of all agents who chose to enter, as is the case in our empirical application.

We separate the data by groups consisting of the same N and x a values. For each such group of auctions, we calculate

1. the number of potential entrants who enter but do not submit a bid (since their valuations are above the reserve),

2. the distribution of the number of entrants,

Bhattacharya, Roberts, and Sweeting

36

3. the distribution of the minimum bid, and

4. the distribution of all bids.

The distribution of the number of entrants is essentially an ( N + 1)-element vector where the k th element gives the proportion of auctions in which k − 1 bidders entered. The distribution of the bids are similarly discretized as well. In particular, for the distribution of the minimum bid, we divide the interval [0 .

4 , 2 .

65] into bins of width 0 .

075. For the distribution of all bids, we subdivide the interval [0 .

4 , 4 .

2] into bins of width 0.1.

We perform two experiments. First, we compute the infeasible estimator by using the true distribution as the importance sampling density. Using notation from Section 4.2, ψ ( θ a

| X a

) ≡ φ ( θ a

| X a

, Γ

0

). We begin by drawing 25,000 values of N and x a

. These variables are drawn uniformly from the discrete distributions specified above. For each of these draws, a single θ a is drawn from the distribution specified by Γ

0

; the draw of θ a of course depends on the draw of x a

. Finally, for each draw of θ a

, we simulate a single auction and record the information detailed above.

The Monte Carlo experiment proceeds by first selecting 500 of these auctions as data. We then choose

2,500 other auctions as importance sampling simulations for importance sampling, which corresponds to using 5 importance sampling simulations per data auction. The simulated moments are calculated for a particular choice of Γ by appropriately weighting the moments from this simulation set. The parameter

Γ is then chosen to minimize the distance between the moments for the data and the moments of the importance sampling simulation, with the identity as the weighting matrix.

33 This procedure is repeated

100 times by reselecting both the data and simulations and subsequently recalculating Γ. Column (A) in

Table 6 shows both the means and the standard deviations over the 100 runs. When sampling from the true distribution, all parameters are recovered accurately.

[Table 6 about here.]

We also run a second experiment where the importance sampling simulations are drawn from a wider distribution Γ

0 where ω

µ

C

= 0 .

1, ω

σ

C

= 0 .

04, ω

α

= 0 .

1, and ω

K

= 0 .

03 and all other parameters are the same as in Γ

0

. The data runs are still drawn from the distribution Γ

0 specified earlier. We generate

33

We have found that the objective function is smooth and we conduct this minimization procedure using Matlab’s built-in optimization routine lsqnonlin . The initial guess is set to the true value, but we have found through experimentation that initial guesses on the same order of magnitude return optimal points that are numerically identical.

Regulating bidder participation in auctions

37

75,000 simulations from this distribution. For each Monte Carlo run, we choose 500 auctions from the true distribution as the data runs and 2,500 auctions from the wider distribution as importance sampling simulations. The results over 100 Monte Carlo runs are given in column (B). We repeat the Monte Carlo runs using 15,000 simulation auctions for each run (corresponding to 30 simulations per auction), and the results over 100 runs are given in column (C). Even with 5 importance sampling simulations per auction, the Monte Carlos effectively recover the true parameters for the constant and the value of the dependence on x a

. Estimates of the standard deviations ω are biased slightly when using 5 simulations per auction, but they become closer to the true values when using 30 simulations per auction.

Bhattacharya, Roberts, and Sweeting

FIGURES

38

Figure 1: Comparing efficiency, amount of entry and the expected completion cost in the FPA and in the

ERA when F

C

( c ) ∼ LN ( − 0 .

09 , 0 .

2), K = 0 .

02, N = 4, r = c

0

= 0 .

85, and S i

= C i

· exp ( i

).

Regulating bidder participation in auctions

39

(a) Winner’s cost (b) Procurement Cost

Figure 2: Kernel densities for (a) the winner’s cost and (b) total procurement costs in both the FPAFE and the ERA, using the base parameters. The costs to the procurer in the ERA includes entry costs paid less the revenue from the first-stage indicative bids. Densities are plotted based on 500,000 simulations, and simulations of the FPAFE in which no bidders entered are ignored.

Figure 3: Comparing the bid function for a FPAFE (solid line) with the average bid function in an ERA with n = 2 (dashed line). The parameters are µ

C

= − 0 .

0963, σ

C

N = 7, and a reserve price r = 1 .

5. The dotted line is the 45

◦

= 0 .

0705, α = 0 .

4979, K = 0 .

0147, line, for comparison. Densities for the cost of a typical entrant into both the FPAFE and the ERA are also plotted.

Bhattacharya, Roberts, and Sweeting

40

Figure 4: Computed bid function (solid) and simulated best response (dashed) for a first-price auction with parameters equal to those in the Base case in Table 4. The dotted line is the 45

◦ line. We use

500,000 simulations to compute the best response.

Regulating bidder participation in auctions

TABLES

Variable

Potential Bidders

Probability of Entry

Unemployment

Engineer Estimate

Win Bid

Relative Win Bid

Mean Std. Dev.

25 th -tile 50 th -tile 75 th -tile

6.882

0.649

4.136

1.997

0.198

0.971

5

0.500

3.2

7

0.667

4.1

8

0.800

4.6

491,120 543,662 226,338 320,755 523,139

451,800 520,945 207,146 288,317 471,151

0.917

0.133

0.826

0.888

0.980

Potential Bidders

Probability of Entry

Unemployment

7.909

0.626

5.641

2.027

0.192

1.214

6

0.500

4.4

8

0.600

5.6

10

0.750

6.5

Engineer Estimate 1,409,158 1,185,331 550,537 967,817 1,977,169

Win Bid 1,366,487 1,177,504 530,084 892,737 1,847,180

Relative Win Bid 0.976

0.134

0.892

0.965

1.038

Potential Bidders

Probability of Entry

Unemployment

Engineer Estimate

Win Bid

Relative Win Bid

7.262

0.640

4.693

2.067

0.196

1.291

6

0.500

3.8

7

0.625

4.4

9

0.778

5.7

830,971 949,131 258,485 446,273 914,745

790,410 936,689 240,553 406,155 887,277

0.939

0.136

0.844

0.914

1.006

41

Table 1: Summary statistics for various quantities, subdivided into state groups. Win Bid and Engineer

Estimate are measured in dollars. Relative Win Bid is the winning bid divided by the engineer’s estimate for the project. See the text for more details of the sample.

µ

C

σ

C

α

K

β

0

β

1

β

2

ω c c

Constant Unemployment Texas Std. Dev.

Lower Trunc.

Upper Trunc.

− 0 .

07

0.05

0.5

0.015

− 0 .

0

0

0

005 0

0

0

0

0.012

0.025

0.25

0.01

−

10

0

0.1

.

4

0.0095

− 4

1.0

0.995

0.9

0.16

by the functional form given in equation (9).

Bhattacharya, Roberts, and Sweeting

42

µ

σ

C

C

β

0

β

1

β

2

ω

Constant Unemployment Texas Std. Dev.

− 0 .

0868

(0 .

0324)

0 .

0687

(0 .

0196)

− 0 .

0023

(0 .

0064)

0 .

0154

(0 .

0132)

− 0 .

0117

(0 .

0213)

0

(0 .

0091)

0 .

.

0142

0304

(0 .

0132)

α 0 .

4979

(0 .

0972)

K − 0 .

0018

(0 .

0406)

0 .

1115

(0 .

0943)

0 .

0105

(0 .

0191)

0 .

1284

(0 .

0764)

0 .

0189

(0 .

0109)

OK Mean TX Mean

91 .

4%

(0 .

84%)

6 .

62%

(0 .

97%)

0 .

4979

(0 .

0770)

1 .

35%

(0 .

26%)

91 .

9%

(1 .

89%)

5 .

33%

(0 .

87%)

0 .

6094

(0 .

0752)

1 .

84%

(0 .

28%)

Table 3: Parameter estimates for Γ. Standard errors on the parameters are computed using the nonparametric bootstrap described in the text. The state means are (going down the columns) the mean of costs, the mean standard deviation of costs, the mean α and the mean entry cost, relative to engineer’s estimate, for auctions in each state. Their standard errors are also computed using the bootstrap.

Regulating bidder participation in auctions

43

Bhattacharya, Roberts, and Sweeting

44

Regulating bidder participation in auctions

45

Parameter

Location Parameter ( µ

C

)

Variable True Value

Constant 0 x a

ω

0.65

0.05

Scale Parameter ( σ

C

Degree of Selection (

Entry Cost ( K )

α

) Constant

) x

ω

Constant x a

ω

Constant x a

ω a

0.05

0

0.02

0.5

0

0.05

0.04

0.05

0.015

0 .

0494

(0 .

0074)

− 0 .

0002

(0 .

0113)

0 .

0192

(0 .

0035)

0 .

4994

(0 .

0205)

0 .

0000

(0 .

0308)

0 .

0479

(0 .

0078)

0 .

0388

(0 .

0047)

0 .

0516

(0 .

0074)

0 .

0143

(0 .

0025)

A

0 .

0000

(0 .

0100)

0 .

6507

(0 .

0174)

0 .

0488

(0 .

0044)

Experiment

B

0 .

0015

(0 .

0186)

0 .

6533

(0 .

0282)

0 .

0575

(0 .

0073)

C

− 0 .

0043

(0 .

0170)

0 .

6602

(0 .

0247)

0 .

0504

(0 .

0059)

0 .

0463

(0 .

0152)

0 .

0085

(0 .

0204)

0 .

0295

(0 .

0094)

0 .

0416

(0 .

0126)

0 .

0195

(0 .

0194)

0 .

0225

(0 .

0077)

0 .

4985

(0 .

0354)

− 0 .

0102

(0 .

0607)

0 .

0803

(0 .

0190)

0 .

0361

(0 .

0118)

0 .

0559

(0 .

0173)

0 .

0209

(0 .

0063)

0 .

4984

(0 .

0231)

− 0 .

0130

(0 .

0571)

0 .

0561

(0 .

0164)

0 .

0387

(0 .

0085)

0 .

0529

(0 .

0121)

0 .

0145

(0 .

0038)

Table 6: Monte Carlo simulations, with standard deviations of the estimates in parentheses. All Monte

Carlo runs use 500 auctions as data. Column A corresponds to the infeasible estimator that uses the true distribution as the importance sampling distribution, with 2,500 simulation auctions per run. Column B corresponds to using a wider distribution as the sampling distribution, with 2,500 simulation auctions per run. Column C corresponds to using the wider distribution, with 15,000 simulation auctions per run. Note that 2,500 simulations per run corresponds to using 5 simulations per auction in the data (and 15,000 simulations per run correspond to using 30 simulations per auction) since simulations are drawn without replacement from the pool of all simulations computed.