Simultaneous First-Price Auctions with Preferencesover Combinations: Identification, Estimation and

Application∗

Matthew Gentry† Tatiana Komarova‡ Pasquale Schiraldi§

October 2014

(Preliminary and incomplete)

AbstractMotivated by the empirical prevalence of simultaneous bidding across a

wide range of auction markets, we develop and estimate a structural modelof strategic interaction in simultaneous first-price auctions when objects areheterogeneous and bidders have preferences over combinations. We begin byproposing a general theoretical model of bidding in simultaneous first priceauctions, exploring properties of best responses and existence of equilibriumwithin this environment. We then specialize this model to an empirical frame-work in which bidders have stochastic private valuations for each object andstable incremental preferences over combinations; this immediately reduces tothe standard separable model when incremental preferences over combinationsare zero. We establish non-parametric identification of the resulting modelunder standard exclusion restrictions, thereby providing a basis for both test-ing on and estimation of preferences over combinations. We then apply ourmodel to data on Michigan Department of Transportation highway procure-ment auctions, we quantify the magnitude of cost synergies and assess possibleefficiency losses arising from simultaneous bidding in this market.

∗We are grateful to Philip Haile, Ken Hendricks, Paul Klemperer, and Balazs Szentes for theircomments and insight. We also thank seminar participants at the University of Wisconsin (Madi-son), the University of Zurich, University of Leuven, Cardiff University, Oxford University, CornellUniversity, the University of East Anglia, and Universitie Paris 1 for helpful discussion.†London School of Economics, m.l.gentry@lse.ac.uk‡London School of Economics, t.komarova@lse.ac.uk§London School of Economics and CEPR, p.schiraldi@lse.ac.uk

1

1 Introduction

Simultaneous bidding in multiple first-price auctions is a commonly occurring but

rarely discussed phenomenon in many real-world auction markets. In environments

where values over combinations are non-additive in the set of objects won, bidders

must account for possible combination wins at the time of bidding. This in turn sub-

stantially alters the strategic bidding problem compared to the standard first price

auction with ambiguous welfare implications depending on the importance of syner-

gies (either positive1 or negative2) among objects. As a first step toward exploring

this issue, we develop a structural model of bidding in simultaneous first-price auc-

tions and study identification and estimation in this framework. We then apply our

results to estimate cost synergies arising in Michigan Department of Transportation

(MDOT) highway procurement auctions, using the resulting estimates to quantify

efficiency losses arising from simultaneous bidding in this application.

To underscore the prevalence of simultaneous bidding in applications, note that

most widely studied first-price marketplaces in fact exhibit simultaneous bids. Con-

crete examples include markets for highway procurement in most US states (Jofret-

Bonet and Pesendorfer 2003, Krasnokutskaya 2009, Krasnokutskaya and Seim 2004,

Somaini 2013, Li and Zheng 2009, Groeger 2014, many others), snow-clearing in

Montreal (Flambard and Perrigne 2006), recycling services in Japan (Kawai 2010),

cleaning services in Sweden (Lunander and Lundberg 2012), oil and drilling rights

in the US Outer Continental Shelf (Hendricks and Porter 1984, Hendricks, Pinkse

and Porter 2003), and to a lesser extent US Forest Service timber harvesting (Lu

and Perrigne 2008, Li and Zheng 2012, Li and Zhang 2010, Athey, Levin and Siera

2011, many others). Furthermore, in many of these applications we expect bidders

to have non-additive preferences over combinations; due, for instance, to capacity ef-

fects in highway procurement (Jofret-Bonet and Pesendorfer 2003), distance between

contracts in snow clearing and waste collection, or information spillovers in Outer

Continental Shelf drilling (Hendricks and Porter 1984). In such cases strategic simul-

1These commonly arise from cost savings in procurement auctions.2These may arise when bidders have resource and capacity constraints or when winning alter-

native items offered can meet the same need.

2

taneity may substantially influence both bidder behavior and market performance.

To illustrate the policy questions arising in simultaneous multi-object auctions,

note that given a set of L heterogeneous objects for sale, bidders could in general as-

sign either positive or negative synergies to winning multiple objects. Consequently,

bidder i’s preference structure could in principle be as complex as a complete 2L-

dimensional set of signals describing the valuations i assigns to each of the 2L possible

subsets of objects. Meanwhile, the simultaneous first-price mechanism allows bidders

to submit (at most) L individual bids on the L objects being sold. Consequently, the

simultaneous first-price auction format is necessarily inefficient – the “message space”

(standalone bids) permitted is insufficiently rich to allow bidders to express their true

preferences (over combinations). On the other hand, while allowing combinatorial

bids would alleviate the “message space” problem, due to strategic considerations

even this is not guaranteed to produce an efficient allocation (see for example the dis-

cussions in Cantillon and Pesendorfer 2006, Crampton at al. 2006). Meanwhile, the

combinatorial first-price mechanism may impose substantial practical costs on both

bidders (the combinatorial bidding problem) and the seller (in allocating objects; the

combinatorial “winner determination problem”). A simultaneous first-price auction

sidesteps both concerns at the cost of potential losses in efficiency or revenue re-

sulting from inability to express combinatorial preferences. Hence in evaluating the

relative merit of the simultaneous first-price format it is first necessary to assess the

magnitude of losses to to simultaneous bidding, a question about which very little is

presently known.3

Despite the prevalence of simultaneous bidding across a variety of first-price mar-

kets, there presently exists very little work assessing the revenue and efficiency im-

plications of such simulteneity in practice. This dearth of research is not surprising

given that there presently exists no known method for estimating preferences over

combinations in simultaneous first-price auctions. In turn, without any means to

assess the magnitude of such preferences in practice, meaningful statements on the

3In environments with substitutes it is known that any Bayes-Nash equilibrium of the simulta-neous first-price auction achieves at least half of optimal social welfare – see Feldman et al. (2012)discussed in more detail below. Apart from this, however, very little is known about efficiency inpractice, and even this (wide) bound becomes meaningless when objects are complements.

3

economic consequences of simultaneous bidding become virtually impossible.

Motivated by this gap in the literature, we develop an empirical model of bidding

in simultaneous first-price auctions when objects are heterogeneous and bidders have

non-additive preferences over combinations, to our knowledge the first such in the

literature. This model turns on a novel decomposition of bidder preferences, which

we outline briefly here. We represent the total value i assigns to a given combina-

tion as the sum of two components: the sum of the standalone valuations i assigns

to winning each object in the combination individually, plus a combination-specific

complementarity (either positive or negative) capturing the change in value i as-

signs to winning objects in combination.4 We interpret standalone valuations as

private information drawn independently across bidders conditional on observables,

but require incremental preferences over combinations to be stable in the sense that

complementarities are unknown functions of observables.5 We find this framework

natural in a variety of procurement contexts – when, for instance, non-additivity in

preferences can be represented as realizations of a “combination shock” realized after

a multiple win. Furthermore – and crucially – our framework collapses immediately

to the standard separable model when complementarities are zero, supporting formal

testing of this hypothesis. We apply this model to data on Michigan Department of

Transportation (MDOT) highway procurement auctions and evaluate the efficiency

losses due to simultaneous bidding in this market. In so doing, we make three main

contributions to the literature on structural analysis of auction markets.

First, we propose a structural model of simultaneous first-price auctions permit-

ting identification of non-additive preferences over combinations. Identification in

this framework rests on two key assumptions. First, as noted above, we assume that

bidders’ incremental preferences over combinations are stable functions of observ-

ables. Second, we assume that the marginal distributions of i’s standalone valua-

tions are invariant either to the characteristics of i’s rivals or to the characteristics

4Note that this decomposition is without loss of generality; the key identifying restriction is thestructure we impose on complementarities.

5Note that this structure does not restrict dependence between i’s standalone valuations fordifferent objects in the market. We view this flexibility as critical, as in practice we expect i’sstandalone valuations to be positively correlated.

4

of other objects on which i is bidding.6 We show that optimal behavior in this en-

vironment yields an inverse bidding system non-parametrically identified up to the

function describing incremental preferences over combinations, with the resulting

system collapsing to the standard inverse bidding function of Guerre, Perrigne and

Vuong (2000) auction by auction when incremental preferences over combinations are

zero. Building on this inverse bidding system, we translate the exclusion restrictions

outlined above into a system of identifying restrictions on model primitives, with

excludable variation in the characteristics of i’s rivals yielding non-parametric iden-

tification and excludable variation in characteristics yielding semiparametric identi-

fication of i’s primitives.7 Taken together, these results provide a formal basis for

structural analysis of simultaneous first-price auctions with non-additive preferences

over combinations, to our knowledge the first such in the literature.

Second, building on our identification argument, we develop a three-step proce-

dure yielding empirical estimates of primitives in our structural model. First, in Step

1, we estimate the multi-variate joint distribution of bids as a function of bidder-

and auction-level characteristics. Due to the high-dimensional nature of this estima-

tion problem, we follow several prior studies (e.g. Cantillon and Pesendorfer 2006

and Athey, Levin and Siera 2011) by employing a parametric approximation to the

observed bid density in implementation of this step. Next, in Step 2, we parametrize

preferences over combinations as a function of bidder- and combination-specific co-

variates8 and estimate parameters in this function by minimization of a simulated

analog to our semiparametric identification criterion. As in practice we parametrize

6The first of these exclusion restrictions is widely invoked in the literature; see, e.g. Guerre, Per-rigne and Vuong (2009) or Somaini (2012). The second is specific to the combinatorial environmentwe consider, although we believe it is natural in a variety of applications.

7As it is difficult to rule out ties a priori in the fully general simultaneous model, we alsoextend our partial identification results (see Appendix C) to accommodate potential mass pointsin equilibrium bids. We show that in this case primitives are partially identified, with MonteCarlo analysis suggesting that identified sets are tight in practice. While we do not believe ties areimportant in the application —they are never observed in the data — we consider, this analysishelps to underscore robustness of our results.

8In our application, combination-specific covariates might include the sum of engineer’s estimatesacross projects in a combination, distance between projects in a combination, and indicators forwhether projects in a combination are of the same type, among others.

5

preferences over combinations as a linear function of observables, this reduces to a

quadratic minimization problem solvable by OLS. Finally, in the third step, we map

estimates derived in Step 2 through the inverse bidding system derived in Step 1

to obtain estimates of the distribution of private costs rationalizing observed bid-

ding behavior. We conclude this part by showing the performance of our estimation

procedure in a Monte Carlo study.

Finally, we apply the framework developed above to analyze simultaneous bidding

in Michigan Department of Transportation (MDOT) highway procurement markets.

We view this market as prototypical of our target application: large numbers of

projects are auctioned simultaneously (an average of 33 per letting round in our

2002-2009 sample period), more than half of bidders bid on at least two projects

simultaneously (with an average of 2.65 bids per round across all bidders in the

sample), and combination and contingent bidding are explicitly forbidden. Within

this marketplace, we show that factors such as size of other projects, number of bid-

ders in other auctions, and the relative distance between projects have substantial

reduced-form impacts on i’s bid in auction l. This finding is expected when bidders

have non-trivial preferences over combinations, but difficult to rationalize within

either the standard separable Independent Private Values model or typical exten-

sions of it (e.g. affiliated values, unobserved heterogeneity, and endogenous entry

among others). We then apply the three-step estimation algorithm described above

to recover structural estimates of primitives for bidders competing for two-auction

combinations, with results suggesting that winning two auctions together leads to

cost savings for moderately sized projects but cost increases for large projects: 4.5

percent savings for a two-auction combination of median size, transitioning to 2.3

percent increase for one at the 90th percentile. Finally, we use our estimation results

to analyze the implications of the auction design chosen by the MDOT. Specifically,

we use our costs estimates to assess the degree of inefficiency in MDOT highway pro-

curement auctions, to estimate the extent of the exposure problem,9 and to determine

what share of the expected total surplus is collected by the auctioneer.

9The exposure problem in auctions of multiple items involves the risk of bidders winning un-wanted items, i.e. winning items at prices above bidders’ values for them.

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Related literature While we are not aware of a detailed structural exploration

of bidding in simultaneous first-price auctions, there is a small but growing empiri-

cal literature on combinatorial first-price auctions : i.e. first-price auctions in which

participants are allowed to submit package bids. The seminal paper in this literature

is Cantillon and Pesendorfer (2006), who analyze simultaneous first-price sealed-bid

auctions in which bidders can submit bids on combinations. The authors address the

problem of non-parametric identification of distributions of bidders valuations when

all the bids and all the identities of the bidders are observed. They also suggest an

estimation procedure which they apply to data from the London bus routes market,

but find little evidence of cost synergies. Allowing bidders to submit combinatorial

bids, substantially alters analysis of the bidding problem; in the language of mech-

anism design, the “message space” (standalone bids) is now sparse relative to the

type space (preferences over combinations), and therefore it leads to a fundamentally

different identification problem. More recently, Kim, Olivers and Weintraub (2014)

extend the methodology of Cantillon and Pesendorfer (2006) to large-scale combina-

torial auctions, using additional structure on combinatorial preferences to circumvent

the substantial curse of dimensionality involved. They then apply methodology to

data on combinatorial auctions used in procurement of Chilean school meals, finding

that the combinatorial first-price auction performs well in terms of both efficiency

and revenue despite the presence of substantial cost synergies in the marketplace.

Paralleling these two structural studies, there is also a small reduced-form liter-

ature seeking to quantify the role of preferences over combinations in multi-object

auctions. For instance, Ausubel, Cramton, McAfee and McMillan (1997) and More-

ton and Spiller (1998) use regression analysis to measure synergy effects in the recent

experience in FCC spectrum auctions. Lunander and Lundberg (2012) use data from

public procurement auctions of internal regular cleaning services in Sweden to show

that firms inflate their standalone bids in combinatorial first-price auctions relative

to their corresponding bids in simultaneous first-price auctions. Despite this, they

do not find significant differences in the procurer’s cost between these two types of

auctions. Lunander and Nilsson (2004) experimentally compare simultaneous, se-

quential, and combinatorial first-price sealed-bid auctions and show that, despite

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potential for synergies, revenue comparison between formats produced statistically

insignificant differences.

From a more theoretical perspective, there exist several studies analyzing strate-

gic interaction in stylized models involving simultaneous first-price auctions. For

instance, Szentes and Rosental (1996) study a first-price sealed-bid acution for iden-

tical objects with two identical players and complete information, while Ghosh (2012)

examines a simultaneous sealed-bid first-price auction for two identical objects with

budget-constrained symmetric bidders. While these models confirm that interaction

between auctions can have substantial effects on behavior and welfare, the stylizy fo-

cus on stylized models which are not well suited to support empirical analysis. There

is also a substantial literature analyzing properties of various combinatorial auction

mechanisms: Ausbel and Milgrom (2002), Ausbel and Cramton (2004), Cramton

(1998, 2002, 2006), Krishna and Rosenthal (1996), Klemperer (2008, 2010), Milgrom

(2000a, 2000b), and Rosenthal and Wang (1996), to mention just a few. Detailed

surveys of this literature are given in de Vreis and Vorha (2003), and Cramton et al.

(2006). While these studies are similar in that they explicitly consider settings where

bidders have preferences over combinations, the theoretical and empirical problems

encountered in the simultaneous first-price format differ substantially from those

encountered in a combinatorial context.10

Finally, approaching the problem from a substantially different angle, there is a

growing literature on simultaneous first-price auctions within computer science. This

literature is focused almost exclusively on theoretical worst-case efficiency bounds,

termed in computer science the “Bayesian price of anarchy.” For instance, working

within essentially the same model we consider here, Feldman et al. (2012) show that

when preferences are subadditive expected social welfare of any BNE is at least 12

of optimal social welfare. Meanwhile, Syrgkanis (2012) considers a more restrictive

class of “fractionally subadditive” valuations, showing that in this case there exists a

10Though only tangentially related to our problem, there is also a growing literature on multi-unitdiscriminatory auctions of homogeneous objects. Reny (1999, 2011), Athey (2001), and McAdams(2006) address issues of existence of equilibria and equilibrium properties in such auctions. Mean-while, Hortacsu and Puller (2008), Hortacsu and McAdams (2010), and Hortacsu (2011) providemore empirical perspectives on multi-unit auctions.

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tighter worst-case efficiency bound. Note that these studies analyze bidder behavior

only tangentially, with existence of BNE never directly addressed. Furthermore, even

these (wide) bounds no longer convey any restrictions when objects are complements;

in this case, virtually nothing is known about efficiency of the simultaneous first-price

mechanism.

2 A general model of simultaneous first-price auc-

tions

Consider a one-shot game in which N risk-neutral bidders compete for L prizes

allocated via separate simultaneous first-price auctions. Apart from the possibility

of simultaneous bidding, auctions are run according to standard first-price rules: the

high bidder in auction l wins object l and pays the amount bid. Bids are binding,

bidders may not submit combination bids, bids in one auction may not be made

contingent on outcomes in any other, and there is no resale. For the moment, assume

ties are broken randomly and independently across auctions.

Allocations and outcomes In analyzing this environment, we adopt the following

notation and definitions. An allocation a is an L× 1 vector whose lth element gives

the identity of the bidder winning object l. An outcome from the perspective of

bidder i is an L×1 indicator vector ω with a 1 in the lth place if object l is allocated

to bidder i (al = i) and a 0 in the lth place otherwise. An outcome matrix Ω is a

2L × L matrix whose rows contain (transposes of) each possible outcome ω: e.g. if

L = 2,

ΩT =

[0 0 1 1

0 1 0 1

]Since in practice we will normalize preferences over the “win nothing” outcome to

zero, we could equivalently omit the row corresponding to ω = 0 from Ω.

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Bidder preferences Bidders have preferences over outcomes but are indifferent to

allocations conditional on outcome: i.e. given any set of objects won, i is indifferent

to how remaining objects are distributed among rivals. Let Y ωi be the combinatorial

valuation bidder i assigns to outcome ω, and Yi be the 2L × 1 vector describing

the combinatorial valuations i assigns to all possible outcomes (with elements of Yi

corresponding to rows in Ω). Yi is known to bidder i at the time of bidding but

unknown to rivals or the auctioneer; Yi thus represents bidder i’s private type in the

bidding game. We maintain the following two assumptions on private types:

Assumption 1 (Independent Private Values). Each bidder i draws private type Yi

from an absolutely continuous c.d.f. FY,i with support on a compact, convex set

Yi ⊂ R2L, with Yi private information, FY,i common knowledge, and types drawn

independent across bidders: Yi ⊥ Yj for all i, j.

Assumption 2 (Values Normalized and Non-decreasing). Y 0i = 0 and Y ω

i is non-

decreasing in the vector of objects won: ω′ ≥ ω implies Y ω′i ≥ Y ω

i .

Actions and strategies Formally, i’s action space is the set of non-negative L×1

vectors of the form bi ≡ (bi1, ..., biL), with bil denoting i’s bid in auction l. Let Bibe the set of feasible (non-negative) bids for bidder i; note that under Assumption

1 we may take Bi to be compact without loss of generality, and under Assumption

2 Bi need not include a null bid (since i always prefers to win any additional object

at bid 0). As usual, a pure strategy for bidder i is a function si : Yi → Bi. Following

Milgrom and Weber (1985), we define a distributional strategy for bidder i as a

probability measure σi on Yi × Bi whose marginal on Yi is Fi. Let s = (s1, ..., sN)

and σ = (σ1, ..., σN) denote pure and distributional strategies respectively.

Standalone valuations and complementarities Let Vil denote the valuation i

assigns to the outcome “i wins object l alone”: Vil ≡ Y eli , where el (the lth unit

vector) is a vector of zeros with a one in the lth place. We call Vil bidder i’s stan-

dalone valuation for object l, and let the L × 1 vector Vi ≡ (Vi1, ..., ViL) describe

i’s standalone valuations over all L objects in the marketplace. We then define the

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complementarity bidder i associates with outcome ω as the difference between i’s

combinatorial valuation for ω and the sum of i’s standalone valuations for objects

won in ω:

Kωi ≡ Y ω

i − ωTVi.

Let Ki be the 2L×1 vector describing i’s complementarities over all possible outcomes

ω:

Ki ≡ Yi − ΩVi.

Note that Ki = 0 if and only if i’s preferences over outcomes are additively separable

in the set of objects won.

Marginal and combination probabilities Given our distinction between stan-

dalone valuations and complementarities, we will frequently wish to distinguish be-

tween the marginal and combination win probabilities generated by bid vector b.

Toward this end, let P (b;σ−i) be the 2L × 1 vector describing the probability distri-

bution over outcomes arising when i submits bid b facing rival strategies σ−i, with

P ω(b;σ−i) the element of P (b;σ−i) describing the probability of outcome ω. Simi-

larly, let Γ(b;σ−i) be the L × 1 vector describing marginal win probabilities arising

when i submits bid vector b facing rival strategies σ−i, with Γl(b;σ−i) the marginal

probability i wins auction l. Note that Γ(b;σ−i) is related to P (b;σ−i) by

Γ(b;σ−i) = ΩTP (b;σ−i).

Furthermore, if ties are broken randomly across auctions then Γl(b;σ−i) depends only

on bid bl, and if ties occur with probability zero then Γl(b;σ−i) is the c.d.f. of the

maximum rival bid in auction l.

2.1 Best-response bidding

As Milgrom (1999) and others have noted in the context of simultaneous ascending

auctions, strategic analysis of bidder behavior in environments with complementar-

ities is complicated enormously by the so-called “exposure problem” – when bids

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are submitted auction by auction but preferences are defined over combinations, a

rational bidder may ultimately regret winning the set of objects allocated to him at

the prices he must pay. Put another way, the “message space” (Bi ⊂ RL) permitted

by the simultaneous first-price auction game is insufficient to allow bidders to fully

communicate their preferences (Yi ⊂ R2L) over outcomes. This turns out to inval-

idate much classical intuition regarding bidding behavior; for instance, as we show

below, a strict increase in type (y′ > y in all elements) can yield a strict decrease in

best-response bid (b′ < b in all elements). In this subsection, we explore what can

(and cannot) be said about behavior in the simultaneous first-price auction model.

Toward this end, consider the problem of bidder i with type realization yi compet-

ing against rivals who bid according to strategy profile σ−i. Conditional on winning

outcome ω at bid vector b, i receives net payoff Y ωi − ωT b, with the 2L × 1 vector

yi − Ωb describing i’s net payoffs over all possible outcomes. Bidder i then chooses

b ∈ Bi to maximize

πi(b;σ−i) ≡ (yi − Ωb)TP (b;σ−i).

Let vi be the standalone valuation vector corresponding to i’s type realization yi,

and ki ≡ yi − Ωvi be i’s corresponding complementarities. Applying the identity

Γ(b;σ−i) = ΩTP (b;σ−i), we can then rewrite πi(b;σ−i) as follows:

πi(b;σ−i) = (yi − Ωb)TP (b;σ−i)

= (Ωvi − Ωb)TP (b;σ−i) + kTi P (b;σ−i)

= (vi − b)TΓ(b;σ−i) + kTi P (b;σ−i). (1)

Note that if i’s preferences over combinations are additive, then ki = 0 and (1)

reduces to the standard separable form

πi(b;σ−i) =L∑l=1

(vil − bl)Γl(bl;σ−i).

So long as ties are broken independently across auctions, bids in one auction will

then have no effect on payoffs in any other, and applying standard auction theory

12

auction-by-auction will characterize equilibrium in the overall bidding game.

Robust predictions of best-response bidding As noted above, the “exposure

problem” arising when preferences over combinations are not additively separable

substantially complicates strategic analysis of the simultaneous first-price auction

game. This problem is, if anything, more acute in simultaneous first price auctions

than in simultaneous ascending auctions, since in the first-price case bidders cannot

update bids as the auction proceeds. This in turn invalidates key predictions of the

classical auction model such as monotonicity of best responses in types. Rather,

applied to our setting, standard best-response arguments yield only the following

(weak) “monotone probability” property:

Lemma 1. Consider bidder i competing against rival strategy profile σ−i. Let y′ and

y′′ be any elements of Yi, and let b′ and b′′ be any corresponding best responses by i

to σ−i. Then

(y′′ − y′) · [P (b′′;σ−i)− P (b′;σ−i)] ≥ 0.

In other words, under best-response bidding, P (·;σ−i) must move in the same dot-

product direction as y.

Proof. By definition of best responses,

(y′ − Ωb′)T · P (b′;σ−i) ≥ (y′ − Ωb′′)T · P (b′′;σ−i)

⇔ y′ · [P (b′;σ−i)− P (b′′;σ−i)] ≥ b′ · ΩTP (b′;σ−i)− b′′ · ΩTP (b′′;σ−i),

and analogously

y′′ · [P (b′′;σ−i)− P (b′;σ−i)] ≥ b′′ · ΩTP (b′′;σ−i)− b′ · ΩTP (b′;σ−i).

Combining these last two inequalities yields the desired result.

Observe that in contrast to the standard single-auction model, increases in a given

element of P (b;σ−i) could correspond to either increases or decreases in the elements

of b. In other words, “monotonicity” of P (b;σ−i) with respect to y (in the sense

13

defined above) says little about monotonicity of b with respect to y. For instance,

suppose that bidder i’s type changes from y′ to y′′ such that y′′ corresponds to a

larger standalone valuation for object l. All else equal, bidder i would then wish to

increase probability weight on the outcome “i wins object l alone.” But (in principle)

bidder i could increase probability weight on this outcome either by increasing bid

in auction l or by reducing bids in other auctions. In extreme cases, this in turn

raises the possibility that best responses could be strictly decreasing in type:

Example 1: Best responses can be strictly decreasing in type Consider the

case of two bidders and two auctions. For bidder 1,

K(1,1) = −1

8v1 −

15

16v2 +

5

4,

and (v1, v2)T ∈ (0, 0)T , (0, 2)T , (2, 2)T. That is, bidder 1 has types y′ = (0, 0, 0, 54)T ,

y′′ = (0, 0, 2, 118

)T , and y′′′ = (0, 2, 2, 258

)T .

Suppose that bidder 2’s fixed strategy is to bid either in auction 1 (with proba-

bility 12) or in auction 2 (with probability 1

2), drawing bids from the uniform U [0; 1]

distribution in either case.

Bidder 1’s types correspond to the following profit functions:

π′ = −b1 ·(

1

2− 1

2·maxmin1, b2, 0

)− b2 ·

(1

2− 1

2·maxmin1, b1, 0

)+ (

5

4− b1 − b2) · maxmin1, b1, 0+ maxmin1, b2, 0

2,

π′′ = −b1 ·(

1

2− 1

2·maxmin1, b2, 0

)+ (2− b2) ·

(1

2− 1

2·maxmin1, b1, 0

)+ (

11

8− b1 − b2) · maxmin1, b1, 0+ maxmin1, b2, 0

2

π′′′ = (2− b1) ·(

1

2− 1

2·maxmin1, b2, 0

)+ (2− b2) ·

(1

2− 1

2·maxmin1, b1, 0

)+ (

25

8− b1 − b2) · maxmin1, b1, 0+ maxmin1, b2, 0

2

14

yielding best response bids b′ =(

18, 1

8

)T, b′′ =

(0, 3

16

)T, and b′′′ =

(116, 1

16

)T, respec-

tively. Ignoring the first component, which corresponds to the case of winning no

auctions, we see that y′′′ is strictly greater than y′. Thus a strict increase in type

(from y′ to y′′′) can generate a strict decrease in i’s best response bid (from b′ to b′′′).

This is what was to be shown.

Monotonicity of bl in vl Obviously, a key feature driving the example above

is that while moving from y′ to y′′′ increases the absolute utility i assigns to all

outcomes, it also decreases the marginal change in payoff (i.e. complementarity) i

assigns to the combination outcome “win 1 and 2 together.” This raises a natural

followup question: holding ki and other elements of vi fixed, must an increase in vil

(weakly) increase i’s best-response bid for object l? Applying Lemma 1, the answer

turns out to be yes:

Lemma 2. Let y′ and y′′ be any elements of Yi such that v′′l > v′l, v′′k = v′k for k 6= l,

and k′′ = k′. For any rival strategy profile σ−i, if b′ is a best response to σ−i at y′

and b′′ is a best response to σ−i at y′′, then b′′l ≥ b′l.

Proof. Setting k = k′ = k′′, we have y′ = Ωv′ + k and y′′ = Ωv′′ + k. Hence applying

Lemma 1,

(y′′ − y′) · [P (b′′;σ−i)− P (b′;σ−i] = (Ωv′′ − Ωv′)T [P (b′′;σ−i)− P (b′;σ−i)]

= (v′′l − v′l) [Γl(b′′l ;σ−i)− Γl(b

′l;σ−i)] ≥ 0.

Thus v′′l > v′l implies Γl(b′′l ;σ−i) ≥ Γl(b

′l;σ−i). If Γl(b

′′l ;σ−i) > Γl(b

′l;σ−i), then

we must have b′′l > b′l since Γl(·;σ−i) is weakly increasing in bl. Alternatively, if

Γl(b′′l ;σ−i) = Γl(b

′l;σ−i), then either b′′l = b′l or b′′l < b′l and Γl(·;σ−i) is flat on

an open interval below b′l. But in the latter case bidder i could slightly reduce b′lwithout changing the probability distribution over objects won, which contradicts

the definition of best response. Hence we must have b′′l ≥ b′l.

Monotone cross-auction effects When does an increase in vl increase best-

response bids in other auctions? The answer turns out to depend on a strong condi-

15

tion which we call supermodular complementarities :

Definition 1 (Supermodular complementarities). For any Yi ∈ Yi and any outcomes

ω1, ω2, Y ω1∧ω2i + Y ω1∨ω2

i ≥ Y ω1i + Y ω2

i , where ω1 ∧ ω2 denotes the meet of ω1, ω2 and

ω1 ∧ ω2 denotes the join of ω1, ω2.

Note that this condition is substantially stronger than either of the following

plausible but insufficient alternatives:

• Positive complementarities: For all Yi ∈ Yi, Yi ≥ ΩVi, where Vi is the stan-

dalone valuation vector corresponding to Yi.

• Superadditive complementarities: For all Yi ∈ Yi and any outcomes ω1, ω2, ω3

such that ω1 = ω2 + ω3, Y ω1i ≥ Y ω2

i + Y ω3i .

While these conditions both reflect cases where bidder i wants to win objects “to-

gether”, they turn out to be insufficient for cross-auction monotonicity because they

fail to provide enough structure on higher-order combinations. Hence it is not hard

to construct examples satisfying both positive and superadditive complementarities

in which an increase in vl decreases some bk.11 Supermodular complementarities

turns out to be exactly the condition required to rule out such problematic cases,

leading to the following result:

Lemma 3. Suppose that Yi satisfies supermodular complementarities, and let y′ and

y′′ be any elements of Yi such that v′′l > v′l, v′′k = v′k for k 6= l, and k′′ = k′. For any

rival strategy profile σ−i, if b′ is a best response to σ−i at y′ and b′′ is a best response

to σ−i at y′′, then b′′ ≥ b′.

Proof. See Appendix.

Two further comments are worth noting here. First, we have also explored con-

ditions under which an increase in vl would necessarily lead to a decrease in bk for

11For instance, a bidder could derive a positive complementarity from winning two objects to-gether but no additional complementarity from winning three. This structure would be consistentwith superadditive complementarities, but an increase in (say) v3 could lead bidder i to shift frombidding aggressively in auctions 1 and 2 to bidding aggressively in auctions 2 and 3.

16

some k 6= l. Not surprisingly, a sufficient condition for this turns out to be submod-

ular complementarities, defined as above but with Y ω1∧ω2i + Y ω1∨ω2

i ≤ Y ω1i + Y ω2

i .

Second, when L = 2, complementarities are supermodular or submodular as ki R 0.

Hence if L = 2 then cross-auction effects have the same sign as ki: best-response b1

is increasing in v2 if ki > 0 and decreasing in v2 if ki < 0.

2.2 Existence of equilibrium

One key message of Lemma 3 is that monotonicity of best responses in the simultane-

ous first-price auction game depends on strong (and in our view typically implausible)

assumptions on the underlying preference space. Insofar as possible, therefore, we

seek to analyze equilibrium existence without invoking monotonicity. Toward this

end, we here consider two alternative approaches to establishing existence. First,

following Milgrom and Weber (1985), we show that an equilibrium in pure strate-

gies exists if the bid space Bi is discrete. Since in practice bids are virtually always

specified up to some minimum currency unit, this in turn is sufficient to guarantee

existence in virtually all applications. Second, building on Jackson, Simon, Swinkels

and Zame (2002), we show that there exists an equilibrium in the “communication

extension” of the simultaneous first-price auction game in which bidders send “cheap

talk” signals which the auctioneer uses to break ties. If ties occur with probability

zero, this in turn corresponds to a full equilibrium in distributional strategies.

Existence with discrete bid space First suppose the bid space is discrete; e.g.

bids can be specified up to a minimum increment of one cent. Then applying the

arguments of Milgrom and Weber (1985) under Assumptions 1 and 2, we establish:

Proposition 1 (Equilibrium with discrete bid space). If the bid space B = ×iBi is

discrete, then there exists a pure strategy Bayesian Nash Equilibrium of the simul-

taneous first-price auction game. If in addition complementarities are stable and

supermodular, then there exists a pure strategy Bayesian Nash Equilibrium for any

compact convex B.

17

Existence with endogenous tiebreaking Alternatively, consider any bid space

B, and let the outcome correspondence P : B → ∆2L describe the set of allocation

rules permissible at bid profile b ∈ B. Following Jackson, Simon, Swinkels and Zame

(2002), we define the communication extension of the simultaneous first-price auction

game as the game that results when the auctioneer allows each bidder i to submit

both a bid b ∈ Bi and a signal s ∈ Si ≡ Yi indicating his private type, where signals

s1, ..., sN may be used to break ties but are otherwise “cheap talk” in that conditional

on allocation they have no effect on payoffs. Then applying Theorem 1 in Jackson,

Simon, Swinkels and Zame (2002), we obtain the following result:

Proposition 2 (Equilibrium with endogenous tiebreaking). There exists an equilib-

rium with endogenous tiebreaking in the communication extension of the simultaneous

first-price auction game: that is, a profile of distributional strategies σ∗ = (σ∗1, ..., σ∗N)

and a tiebreaking rule p∗ : S ×B → ∆2L selected from P such that bidders truthfully

communicate types and σ∗ represents an equilibrium given tiebreaking rule p∗. If in

at least one such equilibrium ties occur with probability zero, then there exists an

equilibrium in distributional strategies in the original auction game.

3 An empirical framework for simultaneous first-

price auctions

Having outlined the key building blocks of our structural model, we turn to the

main objective of this paper – development of an empirical framework supporting

structural analysis of simultaneous first-price auction markets. Toward this end, we

assume the econometrician has access to a “typical” simultaneous first-price auction

sample, interpreted as a sample of T auction rounds drawn from some stable un-

derlying data generating process. In each round, the auctioneer offers Lt objects for

auction to Nt bidders active in the marketplace (though in general not all bidders

need be active in all auctions). Bidders then simultaneously submit sealed bids on

the set of auctions in which they are active, with the set of bidders active in each

18

auction common knowledge to all participants.12 For each round t, we assume the

econometrician observes data on all bidders i and auctions l present in the market.

Asymptotic statements should be interpreted as applying when T →∞.

For each bidder i present at time t, let Sit be the set of auctions in which i

bids at time t, with bit the corresponding vector of i’s bids. For each round t, the

econometrician observes Sit, bit, and a vector of bidder-specific characteristics Zit for

all bidders i active in the round; to simplify notation, we will adopt the convention

that Zit includes Sit. For future reference, let Lit denote the cardinality of Sit,

and define Zt ≡ (Z1t, ..., ZNt,t). Extending our theoretical model, we assume bidders’

private information is drawn independently conditional on Zt and the auction-related

observables described below.

Meanwhile, on the auction side, we partition the econometrician’s information

into two sets of covariates. First, for each object l auctioned at time t, the econo-

metrician observes a vector of covariates Xlt; for future reference, we define Xt ≡(X1t, ..., XLt,t). The econometrician may also observe a vector of market-level char-

acteristics Wt taken to affect combinatorial valuations but not standalone valuations;

we formalize this restriction in Assumption 4 below. In a highway procurement con-

text, Xt would include factors like project size, project location, and type of work

in each project, whereas Wt could include distance between projects, interaction be-

tween project sizes, and other factors assumed irrelevant for Vil after conditioning

on Zit and Xlt. Note that our identification analysis permits Wt to be null.

3.1 Identifying assumptions

Even cursory analysis of the simultaneous first-price problem suggests a major em-

pirical obstacle: whereas in general the model could involve up to 2Lit − 1 unknown

combinatorial valuations for a given bidder, the data generating process yields only

Lit observed bids corresponding to these unobservables. To obtain a viable empirical

model for simultaneous first-price auctions, it is therefore imperative to specialize

12While we do not model entry formally here, our analysis can be readily extended to incorporateendogenous participation along the lines of Levin and Smith (1994) and Athey, Levin and Siera(2011)). We outline this extension in more detail below.

19

the model before taking it to data. Obviously, whether a given specialization is

plausible will depend crucially on the problem at hand, and no one assumption is

likely to be suitable for all applications. Since our main interest here is procurement,

however, we here propose two restrictions on primitives which we find natural in

many procurement contexts. As we go on to show, these turn out to be sufficient for

non-parametric identification of primitives given data of the form above. We thereby

provide a formal basis for empirical analysis of simultaneous bidding in a wide range

of applications of practical and policy interest. To our knowledge this is the first

such framework proposed in the literature, opening the door for empirical analysis

of factors never previously explored.

Assumption 3 (Stochastic Vi, stable Ki). For all i and t, Kit = κi(Zt,Wt, Xt), with

Vit is distributed according to joint c.d.f. Fi(·|Zt,Wt, Xt).

Assumption 4 (Exclusion). Fi(·|Zt,Wt, Xt) = Fi(·|Zit, Xt) and κi(Zt,Wt, Xt) =

κi(Zit,Wt, Xt).

Assumption 3 says that complementarities are a stable function of bidder-, auction-

, and combination-specific observables. This assumption is motivated by our inter-

pretation of Ki as a pure combination effect; i.e. an incremental cost or benefit

derived from winning two objects together. We find this structure reasonable for ap-

plications such as procurement contracting, where bidders are obligated to perform

all projects won. It would be less plausible in a setting like unit demand with resale,

where the object ultimately resold would depend on i’s idiosyncratic preferences.

Note that κi(·) can also be interpreted as an expectation over a combination-specific

utility shock realized after a multiple win. Also, and importantly, Assumption 3

formally nests the hypothesis of additively separable preferences: κi(Zt,Wt, Xt) = 0.

Assumption 4 imposes two exclusion restrictions: own primitives Fi, κi are in-

variant to rival characteristics Z−it, and standalone valuations Vi are invariant to

combination characteristics Wt given Zi, Xt. The former is widely invoked in the

non-parametric auction literature (e.g. Haile, Hong and Shum (2003), Guerre, Per-

rigne and Vuong (2009), Somaini (2014)), while the latter formalizes the exclusion

restriction underlying the definition of Wt. Note that the first part of Assumption

20

4 can be justified within a market-wide entry and bidding model along the lines

Levin and Smith (1994); where, for instance, auction-level entry decisions depend on

auction-specific entry cost draws, with realizations of Vi discovered after entry.13

In addition to the fundamental restrictions on primitives captured in Assumptions

3 and 4, we maintain two regularlity conditions on the underlying auction process:

Assumption 5. In each letting t, observed bids are generated by play of an equilib-

rium with endogenous tiebreaking in the JSSZ communication extension to the simul-

taneous first-price auction game. Furthermore, for any t, t′ such that (Zt,Wt, Xt) =

(Zt′ ,Wt′ , Xt′), the equilibrium played at t is the same as the equilibrium played at t′.

Assumption 6. For each (Zt,Wt, Xt) ∈ Z × W × X and each bidder i active in

letting t, the joint distribution of bids submitted by i is absolutely continuous.

The assumption that a single equilibrium is played is widely invoked in the litera-

ture; see, e.g., Somaini (2014) and references therein. In light of our existence results,

Assumption 5 formally interprets this equilibrium to be of the JSSZ type. Assump-

tion 6 is a weak regularity condition on the observed distribution of bids which we

expect to hold in any equilibrium such that bidders do not bid atoms. Note that

when Ki = 0 any combination of strategies which would represent an equilibrium

auction-by-auction will also be an equilibrium in the bidding game, and under stan-

dard regularity conditions (e.g. Assumption 1) strategies in any such equilibrium

will satisfy the smoothness requirements in Assumption 1. Hence Assumption 6 is

without loss of generality when Ki = 0, implying that all results below formally nest

this hypothesis as a special case.

3.2 Non-parametric Identification

Under Assumptions 3 and 4, model primitives are the distribution of standalone val-

uations Fi(·|Zit, Xt) and the complementarity function κi(Zit,Wt, Xt) for each bidder

13In practice, we expect more flexible models of auction entry (e.g. Roberts and Sweeting (2013),Gentry and Li (2014)) to better reflect true participation decisions. Under the null hypothesisof zero complementarities, however, these will not help to rationalize the cross-auction biddingpatterns noted below. We therefore focus on exogenous participation as a baseline, leaving detailedanalysis of cross-auction entry for future research.

21

i. Taking i as given, the identification problem is thus to recover Fi(·|Zit, Xt) and

κi(Zit, Xt) from observed bidding behavior. Let Gi(·|Zt,Wt, Xt) be the c.d.f. of the

joint distribution of the Lit × 1 bid vector bi submitted by bidder i at observables

(Zt,Wt, Xt); note that under Assumption 5 Gi(·|Zt,Wt, Xt) is identified directly from

observables for all i and t. Consistent with Assumption 1, we permit arbitrary cor-

relation between elements of Vi, but assume vectors Vi, Vj are drawn independently

across bidders; we relax this structure to accommodate auction-level unobserved het-

erogeneity below. Independence of Vi, Vj implies independence of bi, bj, so knowledge

of G1, ..., GNt is sufficient to characterize the joint distribution of all bids submitted

by all bidders at time t.

Inverse Bid Function Let P−i(·|Zt,Wt, Xt) : Bi → ∆2Lit be the probability distri-

bution over outcomes facing bidder i at observables (Zt,Wt, Xt) taking rival strate-

gies as given, and Γ−i(·|Zt,Wt, Xt) ≡ ΩTP−i(·|Zt,Wt, Xt) be the Lit × 1 vector of

marginal win probabilities corresponding to P−i(·|Zt,Wt, Xt). Note that identifica-

tion of G1, ..., GNt implies identification of P−i,Γ−i for all i and (Zt,Wt, Xt). Given

any realization vi of Vi and any vector of complementarities Ki, we can therefore

write the problem facing bidder i at observables (Zt,Wt, Xt) in terms of directly

identified objects as follows:

maxb∈Bi(vi − b) · Γ−i(b|Zt,Wt, Xt) + P−i(b|Wt, Zt, Xt)

TKi.

Temporarily suppose that i’s objective is differentiable at b∗ ∈ int(Bi); we show below

that under Assumption 6 this holds almost surely with respect to the measure on

Bi induced by Gi. Then by hypothesis of equilibrium play, b∗ must satisfy necessary

first-order conditions for an interior optimum:

∇bΓ−i(b∗|Zt,Wt, Xt)(vi − b∗) = Γ−i(b

∗|Zt,Wt, Xt)−∇bP−i(b∗|Wt, Zt, Xt)

TKi. (2)

Note that these first order conditions do not require an equilibrium in pure strategies;

they depend only on continuity of G1, ..., GNt . Assumption 6 formally guarantees this

22

condition holds, but recall this is without loss of generality when Ki = 0.

Let Kit denote the following (2Lit − Lit − 1)-dimensional subspace of R2Lit :

Kit = k ∈ R2Lit : k1 = k2 = . . . = kLit+1 = 0.

That is, Kit contains 2Lit-dimensional vectors whose first Lit + 1 components are

equal to zero. These zero components correspond to the cases of bidder i winning at

most one object (ω = (0, . . . , 0) or ω′ω = 1).

Taking Ki ∈ Kit as given, the first order conditions (2) generate for each b∗ ∈int(Bi) an Lit × 1 system of equations in the Lit × 1 vector of unknown standalone

valuations vi. We now establish that this system may be inverted for vi at almost

every bi submitted by i. Recall that Γ−i(b|Zt,Wt, Xt) is an Lit × 1 vector whose

lth element describes the probability that bid vector b wins auction l, and under

Assumption 6 this is simply the probability that the maximum rival bid in auction l

is below bl. Hence ∇bΓ−i(b|Zt,Wt, Xt) will be a diagonal matrix with (l, l)th element

given by the p.d.f. of the maximum rival bid in auction l. In equilibrium this p.d.f.

must be positive at (almost) every b∗ ∈ int(Bi), and again invoking Assumption 6

this will be (almost) every bid submitted. We therefore conclude:

Proposition 3 (Inverse Bidding Function). Let K be any vector in Kit, (Z,W,X)

be any realization in Z ×W ×X , and maintain Assumptions 1-6. Then for almost

every bi drawn from Gi(·|Z,W,X), there exists a unique vector v ∈ RLit satisfying

the first-order system (2) at bi given (K;Z,W,X). This v can be expressed in terms

of bi via the inverse bidding function

v = ξi(bi|K;Z,W,X),

where ξi(·|·;Z,W,X) : Bit ×Kit → RLit is defined by

ξi(b|K;Z,W,X) ≡ b+ [∇bΓ−i(b|Z,W,X)]−1

×[Γ−i(b|Z,W,X)−∇bP−i(b|Z,W,X)TK

], (3)

23

and the right-hand expression is identified up to K.

Interpreted as a function of bi, ξi(bi|K;Z,W,B) describes the unique vector of

candidate standalone valuations at which bi could be a best response under the

hypothesis K = κi(Z,W,X). If in fact K = κi(Z,W,X), then under Assumptions 5

and 6 the first-order system (2) describes the true equilibrium bidding relationship

and hence we must have vi = ξi(bi|K;Z,W,B) almost surely. Otherwise, there may or

may not exist v rationalizing bi at K, but regardless ξi(bi|K;Zt,Wt, Bt) will represent

the unique candidate for vi at which bi satisfies first order necessary conditions for a

best response.14 Note that at K = 0 the lth element of ξi(·) reduces to

ξil(b|0;Zt,Wt, Xt) = bil +Γ−i,l(bil|Zt,Wt, Xt)

dΓ−i,l(bil|Zt,Wt, Xt)/dbl;

i.e. the standard inverse bidding function of Guerre, Perrigne and Vuong (2000)

defined auction by auction.

Identifying restrictions We next apply Assumptions 3 and 4 to translate the

inverse bidding relationship (3) into a system of identifying restrictions on model

primitives κ and F . For ease of exposition, we suppress the subscript t in notation.

First enforce Assumption 3: Ki = κi(Z,W,X) for all i. By Proposition 3, we

then must have for almost every bi drawn from Gi(·|Z,W,X),

vi = ξi(bi|κi(Z,W,X);Z,W,X).

In practice, of course, κi(Z,W,X) is unknown. But for any vectorK ∈ Kit, ξi(·|K;Z,W,X)

becomes an identified map from bi to the unique vi consistent with bi under the hy-

pothesis κi(Z,W,X) = K:

vi = ξi(bi|K;Z,W,X) if κi(Z,W,X) = K. (4)

14Obviously, imposing sufficient conditions for bi to be a best response – by, for instance, requiringsecond-order conditions to hold at ξ(bi|K;Zt,Wt, Bt) – can only improve identification.

24

But recall that this relationship must hold for almost every bi in the support of

Gi(·|Z,W,X). Under the hypothesis κi(Z,W,X) = K, Equation 4 thus implies a

unique identified candidate Fi(·|K;Z,W,X) for the unknown c.d.f. Fi(·|Z,W,X):

Fi(v|K;Z,W,X) =

∫Bi

1[ξi(Bi|K;Z,W,X) ≤ v]Gi(dBi|Z,W,X). (5)

Now enforce Assumption 4: κi(Z,W,X) = κi(Zi,W,X), Fi(·|Z,W,X) = Fi(·|Zi, X).

Letting κ0 ≡ κi(Zi,W,X), we then must have for any (Z−i,W ):

Fi(·|κ0;Zi, Z−i,W,X) = Fi(·|Z,W,X) = Fi(·|Zi, X). (6)

The right-hand side of this final expression is invariant to (Z−i,W ), with κ0 also

invariant to Z−i. Holding (Zi, X) fixed, we thereby obtain two classes of identifying

restrictions on the unknown function κi(·): the first induced by variation in Z−i and

leading to the possibility of non-parametric identification, the second induced by W

and leading to the possibility of semiparametric identification. We develop each of

these in turn.

Non-parametric identification of κi based on variation in Z−i To understand

how variation in Z−i identifies κi(·), consider a simple two-auction example. Holding

(Zi,W,X) fixed, define κ0 ≡ κi(Zi,W,X) as above. Starting from some initial

competition structure Z−i, let Z ′−i be the competition structure derived from Zi

by adding one additional bidder to Auction 2. Then the marginal probability that i

wins Auction 1 will be similar at Z−i and Z ′−i, but the probability of the combination

outcome “i wins both 1 and 2” will differ. Furthermore, under Assumption 4, this

change in combination win probabilities is the only way changing Z−i matters for

i’s strategy in Auction 1. Therefore to the extent that moving from competition

structure Z−i to competition structure Z ′−i matters for i’s behavior in Auction 1,

it can be only through κ0; if moving from Z−i to Z ′−i has no effect, then we must

have κ0 = 0. The number of feasible “experiments” is limited only by the support

of Z−i, with each experiment inducing a continuum of non-linear equations in the

25

finite vector κ0. Under weak regularity conditions this system will have the unique

(overdetermined) solution κ0 = κi(Zi,W,X). Noting that (Zi,W,X) is arbitrary,

iteration of the argument then yields identification of κi(·) for any (Zi,W,X).

We now formalize this intuition. By linearity of ξi(Bi|K;Z,W,X) in K, note

that for any K ∈ Kit and any (Z,W,X) we can write

EBi [ξi(Bi|K;Z,W,X)|Z,W,X] = Υi(Z,W,X)−Ψi(Z,W,X) ·K, (7)

where Υi(Z,W,X) is an identified Lit × 1 vector defined by

Υi(Z,W,X) =

∫Bi

(Bi +∇bΓ−i(Bi|Z,W,X)−1Γ−i(Bi|Z,W,X)

)Gi(dBi|Z,W,X)

and Ψi(Z,W,X) is an identified Lit × 2Lit matrix defined by

Ψi(Z,W,X) =

∫Bi∇bΓ−i(Bi|Z,W,X)−1∇bP−i(Bi|Z,W,X)T Gi(dBi|Z,W,X).

Furthermore, by Equation (6) and invariance of Fi(·|Zi, X) in Z−i we must have for

any Z−i, Z′−i:

EBi [ξ(Bi|κ0;Zi, Z−i,W,X)|Z,W,X] = EBi [ξ(Bi|κ0;Zi, Z′−i,W,X)|Z ′,W,X]. (8)

Substituting (7) into (8), we thereby obtain an Li × 1 system of linear restrictions

in the 2Li × 1 vector κ0 = κ(Zi,W,X):

(Υi(Z,W,X)−Υi(Z′,W,X))− (Ψi(Z,W,X)−Ψi(Z

′,W,X)) · κ0 = 0. (9)

For a single Z−i, Z′−i pair, this system will typically be rank-deficient and thus will

not uniquely determine κ0. But the underlying equality restriction must hold for

every Z−i, Z′−i ∈ Z−i. Pooling these restrictions, we therefore conclude:

Proposition 4. For any (Zi,W,X) ∈ Zi × W × X , suppose there exist vectors

26

Z−i,0, Z−i,1, ..., Z−i,J in the support of Z−i|Zi,W,X such that the JLi × 2Li matrix

MΨ ≡

Ψi(Zi, Z−i,1,W,X)−Ψi(Zi, Z−i,0,W,X)

...

Ψi(Zi, Z−i,J ,W,X)−Ψi(Zi, Z−i,0,W,X)

has full column rank when projected onto Kit. Then κi(Zi,W,X) is identified.

Recall that the expectations criterion (8) exploits only equality of first moments

of Fi(·|Zi, X) across Z−i, whereas the underlying invariance restriction (6) requires

equality (and, under Assumption 1, finiteness) of all moments. The system of equa-

tions in Proposition 4 merely provides a simple and directly verifiable sufficient con-

dition guaranteeing that the underlying system of functional identities has a unique

solution. Note further that variation in, e.g., number of rivals in other auctions will

produce exactly the kind of variation in Ψi needed for full column rank of MΨ: in-

tuitively, changes in combination win probabilities relevant for cross-auction bidding

only through κ0. We thus view full column rank of MΨ as a weak regularity condition

guaranteeing identification of model primitives.

Parametric identification of κi based on variation in (Z−i,W ) While non-

parametric identification is useful as an ideal, in applications we will typically wish

to impose some parametric structure on κi(·). Toward this end, suppose that to the

assumptions maintained above we add the hypothesis that κi(·) is of the form

κi(Zi,W,X) = Ci(Zi,W,X, θ0i), (10)

where Ci(Zi,W,X, θ0i) is a known transformation of (Zi,W,X, θ0i), and θ0i ∈ Θi ⊂Rpi . For simplicity, we consider the case when κi(Zi,W,X) is linear in parameters

– that is, κi(Zi,W,X) = Ci(Zi,W,X)θ0i. Then Assumption 4 gives us that for any

(Z,W,X), (Z ′,W ′, X) with Zi = Z ′i, Equation (8) reduces to the linear-in-parameters

form

(Υi −Υ′i)− (ΨiCi −Ψ′iC′i) · θ0i = 0,

27

where Υi,Ψi,Ci are identified functions of (Z,W,X) and Υ′i,Ψ′i,C

′i are identified

functions of (Z ′,W ′, X). Given an appropriate collection of

(Zi,j, Z−i,j,Wj, Xj), (Zi,j, Z′−i,j,W

′j , Xj)Jj=1,

we can then express θ0i as the solution to the following L2-minimization problem

minθ∈Θi

J∑j=1

(Υij −Υ′ij − (ΨijCij −Ψ′ijC

′ij) · θ

)T (Υij −Υ′ij − (ΨijCij −Ψ′ijC

′ij) · θ

),

(11)

with identification of θ0i implied by a standard rank condition on the difference

(ΨijCij − Ψ′ijC′ij) across j. Note that given Υij,Υ

′ij,Ψij,Ψ

′ijJj=1, the problem of

finding θ0i reduces to intercept-free least squares of differences (Υij −Υ′ij) on differ-

ences (ΨijCij −Ψ′ijC′ij) across j.

The L2-minimization problem described above can be written in an analogousintegral form if we consider all possible

((Zi,j, Z−i,j,Wj, Xj), (Zi,j, Z

′−i,j,W

′j , Xj)

).

Namely, the L2 criterion would have the form∫(Zi,X)

∫(Z−i,W )

∫(Z′−i,W

′)

(Υi −Υ′i − (ΨiCi −Ψ′iC′i) · θ)

T(Υi −Υ′i − (ΨiCi −Ψ′iC

′i) · θ)

dFZ−iW (Z ′−i,W′|Zi, X) dFZ−iW (Z−i,W |Zi, X)dFZiX(Zi, X)

This latter criterion will typically induce a richer set of restrictions on θ0i than will

the discrete version above.

Identification of Fi To complete the argument, it only remains to note that un-

der Assumptions 3-6, identification of κi(Zi,W,X) for any (Zi,W,X) implies non-

parametric identification of Fi(·|Zi, X) at (Zi, X):

Fi(v|Zi, X) ≡ Fi(v|κi(Zi,W,X);Z,W,X) for all (Zi,W,X),

with Fi(·|·;Z,W,X) a directly identified function of observables. The conditions

above yield identification of κi(Zi,W,X) for arbitrary (Zi,W,X), from which we

28

conclude that Fi(·|Zi, X) is identified.

4 Application: Simultaneous Bidding in Michigan

Highway Procurement

As noted in the Introduction, simultaneous bidding arises in a wide variety of fre-

quently studied procurement markets: for highway construction, snow clearing, recy-

cling, cleaning, and offshore drilling among others. Our goal in the last section was

to develop an empirical framework suitable for structural analysis across many such

markets. As one illustration of the novel empirical insights made possible by this

framework, we now turn to consider a particular application: the marketplace for

Michigan Department of Transportation (MDOT) highway construction and main-

tenance contracts. As common in similar procurement contexts, MDOT allocates

contracts for a wide range of highway construction and maintenance services via low-

price sealed-bid auctions. Contracts are auctioned across 15 to 20 “letting dates”

per year, with multiple contracts auctioned on each letting date: an average of 33

per letting date across our sample period (2002-2009), with a maximum of 83 on a

single date. More than half (56 percent) of bidders submit bids on multiple contracts

within any given letting date, with a median of 3 and a mean of 3.96 contracts bid

among bidders in this subset. Bids are submitted to MDOT auction by auction,

with combination and contingent bidding explicitly forbidden by MDOT auction

rules. Bidders may amend bids up to the letting date, but once announced letting

results are legally binding, with winning bidders held liable for failure to complete

contracts won (though they may subcontract up to 60 percent of contract work).

The instutional framework of the MDOT auction marketplace thus closely parallels

our simultaneous first-price structure, with factors like capacity constraints, subcon-

tracting costs, project location, and project type inducing potential non-additivity

in project payoffs.

29

4.1 Data and descriptive statistics

MDOT provides detailed records on contracts auctioned, bids received, and letting

outcomes on its letting website (http://www.michigan.gov/mdot). Building on these

records, we observe data on (almost) all contracts auctioned by MDOT over the

sample period October 2002 to March 2009.15 Our sample includes a total of 5532

auctions over X letting dates, where for each auction the following information is

observed: project description, project location, prequalification requirements, the

internal MDOT engineer’s estimate of the total cost of the project, and the list of

participating firms and their bids. Based on this information, we classify projects

into several project types, leading to a final distribution of projects across types

summarized in Table 1. As evident from Table 1, roughly 60 percent of contracts

are for road and bridge construction and maintenance broadly defined, with the

remainder mainly for safety and other miscellaneous construction.

The data contains information on a total of 629 unique bidders active in the

MDOT marketplace over our sample period, which we subclassify by size and scope

of activity as follows. We define “regular” bidders to be those who have submitted

more than 100 bids in the sample period. This yields a total of 35 regular bidders

in the sample, with all remaining bidders classified as “fringe”. For the subsample

of regular bidders, we also collect data on number and location of plants by firm.

This data is derived from a variety of sources: OneSource North America Business

Browser, Dun and Bradstreet, Hoover’s, Yellowpages.com and firms’ websites. Based

on this information, we further subclassify regular bidders as “large” or “small” by

number of plants in Michigan, with “large” regular bidders defined as those with at

least 6 plants. We thus obtain a final classification of 9 large regular bidders, 26

small regular bidders, and 594 fringe bidders in the MDOT marketplace.

4.1.1 Summary statistics

Tables 2 and 3 summarize several key measures of market structure and bidder be-

havior. Table 2 surveys the auction side of the marketplace. The first key feature

15For a small number of contracts MDOT records are incomplete.

30

Table 1: Summary of Projects by Type

Contract Type Frequency

New Construction 0.46Preventive Maintenance 12.00Resurfacing 14.99Road Reconstruction 9.07Road Rehabilitation 7.70Roadside Facilities 6.42Safety 11.28Traffic Operations 6.12Bridge Construction 0.36Bridge Reconstruction 11.37Bridge Rehabilitation 3.58Miscellaneous 16.65

emerging from this table is, not surprisingly, the large number of contracts auctioned

simultaneously in the market: a mean of 33 per letting date across the whole letting

date, with a maximum of 83 on a single letting date (note that smaller “supple-

ments” lettings are occasionally held two or three weeks after the main letting in

a given month). The number of bidders submitting bids on any given contract is

small relative to the total number of bidders active in the marketplace, with about

five bids per contract received on average across the sample (approximately 2.5 of

these on average by regular bidders). For each contract, MDOT prepares an inter-

nal “Engineer’s Estimate” of expected procurement cost released to bidders before

bidding; the log of this estimate is summarized in Table 2, with the dispersion in

this measure indicating the substantial variation in size and complexity of projects

in the marketplace (from tens of thousands to hundreds of millions of dollars if mea-

sured in levels rather than logs). The statistic “Money Left on the Table” measures

the percent difference between lowest and second-lowest bids; on average this is 7.4

percent or roughly $112,000 per contract, suggesting the presence of substantial un-

certainty in the marketplace. Finally, to provide a more complete picture of bidder

activity, we also report data on project subcontracting. As evident from the last

two rows of Table 2, this represents an important component of the MDOT auction

31

Table 2: Auction Level Summary Statistics

Mean St. Dev. Min Max

Auctions per Round 33.279 21.776 1.000 83.000Bidders per Auction 4.893 3.109 1.000 19.000Large Bidders per Auction 0.900 0.961 0.000 5.000Medium Bidders per Auction 1.563 1.709 0.000 8.000Fringe Bidders per Auction 2.520 2.476 0.000 16.000Log Engineer’s Estimate 14.296 1.543 9.482 18.923Money Left on the Table 0.074 0.084 0.000 0.931Project Duration (in days) 119.808 147.801 6.000 910.000Fraction Sub-Contracted 0.153 0.198 0.000 0.917Sub-Contractors per Project 2.985 4.079 0.000 49.000

Table 3: Bidder Level Summary Statistics

Mean St. Dev. Min MaxBids by Round 2.65 2.45 1.000 26.000Bids by Round if Large 4.873 1.283 1.000 26.000Bids by Round if Small 3.161 0.583 1.000 17.000Participation Rate 0.010 0.023 0.000 0.242Participation Rate if Large 0.060 0.068 0.024 0.242Participation Rate if Small 0.060 0.034 0.006 0.127

marketplace, with approximately three subcontractors and 15 percent of contract

value subcontracted on average.

Table 3 reframes the auction-level participation variables in Table 2 to provide a

clearer picture of bidder behavior in the MDOT auction marketplace. Again, the key

pattern emerging from Table 3 is the prevalence of simultaneous bidding in MDOT

procurement auctions, with the average bidder competing in roughly 2.5 auctions per

round and regular bidders competing in substantially more (3.16 for small regular

and 4.87 for large regular bidders respectively). Again note that number of bids

submitted in any given auction is small relative to the number of bidders in the

marketplace, with regular bidders competing in about six percent of total auctions

on average.

32

Figure 1: Distribution of Simultaneous Bids Submitted

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Number of Simultaneous Bids

Fra

ctio

n of

Bid

ders

in S

ampl

e

As a graphical perspective on the scope of simultaneous bidding in the MDOT

marketplace, Figure 1 plots the distribution of the number of bids by round submitted

by all bidders in the sample. As evident from Figure 1, more than 55 percent of

bidders in our sample submit multiple bids in the same round, with the median

bidder in this subsample bidding on 3 contracts simultaneously. Despite this, it is

uncommon for a representative bidder to compete in a large number of auctions;

almost 90 percent of bidders in our sample bid in 5 or fewer auctions and only 2

percent bid in more than 10. Not surprisingly, the outliers in this respect are almost

exclusively large regular bidders, although even these bid in fewer than 5 auctions

on average.

4.1.2 Reduced-form regressions

To assess the potential implications of simultaneous bidding on bidder behavior and

auction outcomes, we first explore several simple reduced-form regressions. The

33

unit of analysis in these regressions is a bidder-auction-round combination, with the

dependent variable log of bid submitted by bidder i in auction l in letting t. We

regress log bids on a vector of regressors indended to capture effects of own-auction

and cross-auction characteristics on i’s bid in auction l at time t. We describe

these regressors in detail below. Note that for bidders competing in many auctions

simultaneously there are a very large number of ways to measure potential cross-

auction effects, with effects potentially non-linear across combinations. To preserve

transparency of the empirical specification, therefore, we focus on bidders competing

in two or three auctions.

Regression specification Based on prior work on highway contracting markets,

we expect three auction-level characteristics to be of primary importance in deter-

mining i’s bid in auction l: the size of auction l, as proxied by the MDOT engineer’s

estimate of expected project cost, the level of competition i faces in auction l, and

the distance between project l and i’s base of operations. To control for the first two

effects, we include log of engineer’s estimate and number of rivals in all regression

specifications. Meanwhile, as a proxy for the third, we construct for each bidder-

project pair the minimum straight-line distance (in miles) between any of i’s plants

and the centroid of the county in which project l is located. We then control for the

log of this distance in our baseline specifications. As elsewhere in the literature, we

expect project costs to be increasing almost one-to-one in project size, aggressiveness

to be increasing in competition, and project costs (to i) to be increasing in distance.

We therefore expect a positive cost on log of distance to project, a negative coefficient

on number of rivals, and a coefficient on log engineer’s estimate of close to one.

To explore potential cross-auction interaction in the MDOT marketplace, we seek

a set of covariates relevant for bidding in auction l only through κ: i.e. factors shifting

combination payoffs but irrelevant for standalone valuations after conditioning on

characteristics of auction l. We expect (at least) the following factors to be relevant in

this respect: combination size (due to potential capacity effects), number of rivals in

other auctions (shifts combination win probabilities), distance to alternative projects

(as a proxy for substitution between auctions), and whether projects are of the same

34

type. We construct measures of each of these as follows.

As one measure of bid effects due to cross-auction competition, we consider the

average number of rivals across all auctions except l played by bidder i. The effects

of cross-auction competition on strategies in auction l are theoretically ambiguous,

depending both on the sign of κ and on strategic responses by bidders in both

auctions. A priori, however, if κ is negative, we expect greater competition in auction

j to increase marginal returns to winning auction l.

To proxy for effects of combination size, we consider the log of the average engi-

neer’s estimate across all auctions but l in which i is competing. Insofar as marginal

costs to be increasing in capacity utilization (e.g. Jofret-Bonet and Pesendorfer

(2003)), we expect the coefficients on this variable to be positive.

In principle, complementarities arising between similar projects may differ from

those arising between different projects. To allow for this possibility, we also con-

sider log of average engineer’s estimates in other probjects of the same type, defined

similarly to ln aeng other but only averaging across other projects of the same type

as l. In principle, the sign of this effect could be either positive or negative, with a

negative sign interpreted as a relative complementarity between projects of the same

type.

Finally, as an additional proxy for substitutability between projects, we define

log of distance to the neareast other projects, measured as the log of the minimum

distance (in miles) between bidder i and the nearest project other than l. Insofar as

two projects close to i are more substitutable than two projects further away from

i, we expect this variable to have a negative sign.

Regression results Table 4 reports OLS estimates for our baseline regression

specifications: logs bids by bidder, round, and auction on the own- and cross-auction

characteristics defined above. All regression specifications include a full set of bidder

type, project type, and letting date indicators, with standard errors clustered at the

bidder-round level to allow for correlation within bidder i’s bids. Columns in Table 4

report results of this procedure for bidders competing in two auctions, three auctions,

and either two or three auctions respectively.

35

Estimated effects of own-auction characteristics correspond closely both to our

priors and to findings elsewhere in the literature. As expected, bids are increasing

almost one for one in project size, with the coefficient on log engineer’s estimate

exceeding 0.98 in all specifications. Similarly, the negative coefficient on number of

rivals suggests that competition increases bidder aggressiveness, with one additional

competitor associated with a 0.5 percent decrease in average bids. Finally, the coef-

ficient on log distance to project suggests that a one percent increase in i’s distance

from the project leads to about a 0.8 percent increase in i’s bid on average.

More importantly, estimated cross-auction effects are also highly significant, with

magnitudes stable across specifications and signs broadly consistent with our prior

expectations. In particular, the positive coefficient on log of average engineer’s es-

timates in other auctions suggests that competing in a larger auction k leads to a

substantial decrease in aggressiveness by bidder i in auction l, with the negative

(though absolutely small) coefficient on log of average engineer’s estimates in other

same-type projects suggesting that this effect is ameliorated slightly when the two

projects are of the same type. Similarly, the coefficient on average number of rivals

in other auctions suggests that facing more competition in auction k leads bidder

i to bid less aggressively in auction l; while slightly surprising in light of our pri-

ors, this effect runs strongly counter to the standard separable model. Finally, the

negative sign on log minimum distance to other projects indicates that increasing

distance to project k makes project k less substitutable with auction l. While the

latter effect is only significant at the 10 percent level, it corroborates the hypothesis

that simultaneous bidding induces strategic spillovers.

4.2 Structural estimation

Building on the identification results in Section 3.2, we now turn to consider struc-

tural estimation of the complementarity vector κ(·). In principle, the results in

Section 3.2 support fully non-parametric estimation. In practice, of course, the di-

mensionality of the problem renders this infeasible. We therefore implement estima-

tion of κ(·) in two steps. First, following Athey, Levin and Siera (2011) and Cantillon

36

Table 4: OLS Estimates of Cross-Auction Effects

y = ln(bid) L = 2 L = 2 L = 2, 3 L ≥ 2

log of engineer’s estimate 0.983*** 0.983*** 0.986*** 0.980***(0.00364) (0.00363) (0.00253) (0.00156)

number of rivals in the auction -0.00577*** -0.00574*** -0.00582*** -0.00591***(0.00113) (0.00113) (0.000777) (0.000488)

log of minimum distance tothe county centroid 0.00959** 0.00960** 0.0127*** 0.0167***

(0.00373) (0.00373) (0.00254) (0.00168)average number of rivals acrossall other auctions played 0.00229** 0.00224** 0.00294*** 0.00236***

(0.00109) (0.00109) (0.000823) (0.000621)log of average engineer’s estimatein all other auctions played 0.0156*** 0.0192* 0.0152*** 0.0162***

(0.00267) (0.0104) (0.00234) (0.00173)log of average engineer’s estimatein all other same type auctionsplayed -0.00373*** -0.00373*** -0.00372*** -0.00312***

(0.00109) (0.00109) (0.000844) (0.000427)(log of average engineer’s estimatein all other auctions played)2 -0.000310

(0.000804)log of minimum distance tothe nearest other project -0.00894** -0.00906** -0.0106*** -0.00702***

(0.00367) (0.00369) (0.00301) (0.00164)Constant 0.0407 0.0302 0.0226 0.0511**

(0.0398) (0.0501) (0.0294) (0.0242)

Observations 4,193 4,193 7,946 22,359R-squared 0.974 0.974 0.972 0.973

Unit of analysis is bidder-auction-round, with standard errors clustered by bidder within each round.Variables log of engineer’s estimate, number of rivals in the auction and log of minimum distance tothe county centroid measure size, strength of competition, and distance to project l respectively. Re-maining variables proxy for cross-auction characteristics: average number of rivals, average engineer’sestimate, and distance to auctions other than l in which i is competing.

37

and Pesendorfer (2006) among others, we estimate a parametric approximation to the

equilibrium distribution Gi of bids submitted by each bidder i appearing in bidder i’s

problem. Second, we translate these estimates through the first-order condition (2)

to obtain a minimum-distance criterion paralleling Equation (8). Under the auxiliary

assumption that κ(·) is linear in parameters, the solution to this minimum-distance

problem has a simple OLS closed form, with the resulting parameters yielding a

minimum-distance estimator of the unknown function κ.

As a first take on the problem of estimating κ, we focus on bidders competing

in two simultaneous auctions. Our motivation for this is twofold. First, for the

two-auction problem, the list of primitive-relevant characteristics is relatively clear,

and the dimensionality of these is such that most can be explicitly conditioned on.

Second, larger auction sets will require approximating a higher-dimensional joint

distribution of bids, with cross-auction effects potentially non-monotone. This will

require substantially more challenging tradeoffs between parsimony and flexibility,

with (at present) little guidance on an appropriate technical balance. The two-

auction case allows us to abstract somewhat from these challenging technical issues,

thereby substantially increasing both transparency and credibility of the resulting

estimates.

Specification for Gi Building on our reduced-form analysis, we model i’s bid in

auction l as depending on the following observables: i’s type, characteristics Xilt

influencing i’s standalone valuation for contract l, characteristics Wilt relevant for

i’s preferences over combinations involving auction l, competition in auction l, and

competition in other auctions for which i is competing. As above, we use (log)

number of rivals as a proxy for strength of competition in auction l, and (log) average

number of rivals in other auctions where i bids as a proxy for strength of cross-auction

competition. We describe construction of Xilt and Wilt in detail below.

We follow Cantillon and Pesendorfer (2006) in specifying a joint log-normal ap-

proximation to the bid vector bi submitted by each bidder i. In particular, for bidder

38

i facing market structure (Zt,Wt, Xt), we estimate the following first-step model:

ln(bit) ∼ MVN(·|µ(Zt,Wt, Xt),Σ(Zt,Wt, Xt)).

As typical in applications, we take µ(·) to be a linear function of observables:

µilt = βDµilt,

where Dµilt is a subset of (Zt,Wt, Xt) which includes the following elements: log engi-

neer’s estimate for project l, log number of rivals in auction l, log sum of engineer’s

estimates across other projects in which i bids, log number of rivals in i’s other

auctions, an indicator for submitting multiple bids, and a constant term.

We break our specification for Σ(Zt,Wt, Xt) into two parts: one for the marginal

variance σ2l (Zt,Wt, Xt) of bilt, the other for correlations ρkl(Zt,Wt, Xt) between bikt

and bilt. In particular, for variance terms σ2l (Zt,Wt, Xt) we specify

σ2l (Zt,Wt, Xt) = exp(αDσ

ilt),

and for correlation terms ρkl(Zt,Wt, Xt) we specify

ρkl(Zt,Wt, Xt) =exp(γDρkl

it − 1)

exp(γDρklit + 1)

where Dσilt and Dρkl

it are known transformations of (Zt,Wt, Xt). In our baseline spec-

ification, Dσilt includes log engineer’s estimate in auction l, log number of rivals in

auction l, log sum of engineer’s estimate across other auctions, and log number of

rivals across other auctions. Meanwhile, Dρklit includes the product of log engineer’s

estimates for pair kl, the product of number of rivals in pair kl, and indicators for

projects of the same type and projects in the same county.

39

Specification for κ For the moment, we adopt the following simple linear speci-

fication for κ(·): for each outcome ω involving at least two projects,

κω(Z,W,X) = θ10 + θ2

0 · ln sum engω, (12)

where ln sum engω denotes the log sum of engineer’s estimates across projects won

at ω. Note that in the L = 2 case (our main focus here) this reduces to a log sum of

engineer’s estimates in auctions played by bidder i.

Estimation algorithm Having specified κ to be linear in parameters θ0, we im-

plement estimation of θ0 based on L2-type criterion (11) derived in Section 3.2. If

all Γj, Γ′j, Ψj, Ψ′j, Cj, C′j in the criterion (11) were known or their consistent esti-

mators were available, we could immediately obtain a θ for θ0 by conducting least

squares based on (11). To construct a criterion of the form (11), however, we must

first resolve several practical issues: choosing a set of counterfactual quadruples

(Zi,j, Z−i,j,Wj, Xj), (Zi,j, Z′−i,j,W

′j , Xj)Jj=1 at which to evaluate (11), accounting

for the deterministic link between X1,j, X2,j and ln sum eng(1,1)j in choice of these

counterfactual quadruples, and simulation of the functions Υj,Υ′j,Ψj,Ψ

′j at the coun-

terfactual quadruples chosen. We address these implementation issues as follows.

First consider choice of the set of counterfactuals (Zi,j, Z−i,j,Wj, Xj), (Zi,j, Z′−i,j,W

′j , Xj)Jj=1

at which to evaluate the criterion (11). In principle, any choice of such that “regres-

sors” ΨjCj − Ψ′jC′j in (11) satisfy a standard rank condition will be sufficient to

construct an estimator for θ0. In practice, however, we expect selections approxi-

mating the empirical distribution of (Z,W,X) to improve estimation performance.

For the set of baseline points (Zj,Wj, Xj)Jj=1 in the criterion (11), we thus em-

ploy all realizations of (Z,W,X) in the subset of bidders with L = 2. For each

(Zj,Wj, Xj) in this collection, we then construct a counterfactual pair (Z ′j,W′j , X

′j)

as follows. First, we randomly draw one auction within the pair; label this Auction 2

WLOG. For this auction, we then draw counterfactual realizations of the number of

rivals and log engineer’s estimate from their empirical distributions among projects

of the same type as Auction 2, holding all other characteristics fixed. We thereby

40

obtain a counterfactual point (Z ′j,W′j , X

′j) for comparison with (Zj,Wj, Xj) in the

criterion (11).

Second, observe that in this construction drawing a new realization of Wj (size of

combination) necessarily involves drawing a new realization of X2,j (size of project

2). We therefore consider estimation under the following slight strengthening of

Assumption 4:

Assumption 7. For all bidders i, auctions l, and rounds t, the marginal distribution

of Vilt depends on Xt only through Xlt:

Fl(·|Zi, X) = Fl(·|Zi, Xl).

In other words, standalone valuations in auction l depend only on characteristics

in auction l. Hence shifting from (Zj,Wj, Xj) to (Z ′j,W′j , X

′j), where X1,j = X ′1,j but

X2,j and X ′2,j can be different, will give us the following equation:

(Υ1,j −Υ′1,j)− (Ψ1,jCj −Ψ′1,jC′j) · θ0 = 0,

where Υ1,j denotes the first element of the L×1 vector Υj, and Ψ1,j denotes the first

row of the L × 2L matrix Ψj, and as above C(Z,W,X) denotes the transformation

of observables entering linearly in κ.16

Applying the logic behind our system criterion (11), we thus obtain a final esti-

mation criterion based on invariance of standalone valuations in Auction 1 only:

θ0 = arg minθ

J∑j=1

Mj∑mj=1

(Υ1,j −Υ

(mj)1,j −

(Ψ1,jCj −Ψ

(mj)1,j C

(mj)j

)· θ)2

. (13)

16In general, when L is greater than 2, shifting from (Zj ,Wj , Xj) to (Z ′j ,W′j , X

′j), where for some

l0 we have Xl,j = X ′l,j for l 6= l0 but Xl0,j and X ′l0,j can be different, gives us the following systemof (L− 1) equations:

(Υ−l0,j −Υ′−l0,j)− (Ψ−l0,jCj −Ψ′−l0,jC′j) · θ0 = 0,

where Υ−l0,j denotes the (L− 1)× 1 vector obtained from the L× 1 vector Υj by eliminating thel0-th component Υl0,j , and Ψ−l0,j denotes the (L−1)×2L matrix obtained from the L×2L matrixΨj by eliminating the l0-th row.

41

Here, for each (Zj,Wj, Xj) we considerMj counterfactual points(Z

(mj)j ,W

(mj)j , X

(mj)j

),

mj = 1, . . . ,Mj, that have the properties described above for (Z ′j,W′j , X

′j). The sim-

plest case is when Mj = 1, j = 1, . . . , J .

Finally, given the L2-type criterion of the form (13), we need to simulate (Υj,Ψj)

and (Υ(mj)j ,Ψ

(mj)j ) for each (Zj,Wj, Xj) and

(Z

(mj)j ,W

(mj)j , X

(mj)j

), mj = 1, . . . ,Mj,

compared. For a given realization (Z,W,X), we accomplish this in three steps. First,

we draw a size-R random sample of bid vectors briRr=1 from the joint distribution

Gi(·|Z,W,X) implied by our first-step estimates for Gi. Next, for each realization

bri of Bi, we compute corresponding realizations for Γi(bri |Z,W,X), ∇Γi(b

ri |Z,W,X),

and∇Pi(bri |Z,W,X) based on our first-step estimates for the bid distributions G1(·|Z,W,X),

. . . , GNt(·|Z,W,X) played by i’s rivals (with∇Pi(bri |Z,W,X) approximated via finite

differences). Finally, we approximate Ψ and Υ by averaging appropriate products of

these functions across draws briRr=1:

Υ =1

R

R∑r=1

bri +∇Γi(bri |Z,W,X)−1 Γi(b

ri |Z,W,X);

Ψ =1

R

R∑r=1

∇Γi(bri |Z,W,X)−1∇Pi(bri |Z,W,X)T .

Repeating this procedure for each set of pairs (Zj,Wj, Xj) and(Z

(mj)j ,W

(mj)j , X

(mj)j

),

mj = 1, . . . ,Mj, appearing in (13), we ultimately obtain the simulated L2-type

criterionJ∑j=1

Mj∑mj=1

(Υ1,j − Υ

(mj)1,j −

(Ψ1,jCj − Ψ

(mj)1,j C

(mj)j

)· θ)2

.

The minimization of this criterion with respect to θ will yield a closed-form estimate

θ for θ0 expressed by the formula for the OLS estimator. Practically, however, first-

step approximation error in Gi typically produces a small number of substantial

outliers in estimated pseudo-values (typically two or three draws at each estimation

iteration).17 To avoid undue influence of these obvious outliers on estimates of θ, we

17More precisely, first-step approximation error in Gi leads to a small number of bid draws at

42

therefore implement estimation using Stata robust regression in place of simple OLS.

4.3 Estimation results

We now report results from applying this structural estimation procedure in Section

4.2 to the sample of bidders competing in two simultaneous MDOT auctions. We first

report results from our first-step estimation of bid distributions Gi for all bidders,

then discuss estimates of κ(·) for the two-bidder sample derived from these through

the algorithm outlined above.

Estimates of Gi Table 5 reports results from first-step maximum likelihood esti-

mation of i’s bid distribution Gi based on the log-normal approximation described

in Section 4.2. As above, we subclassify these into effects on mean parameters

µ(Z,W,X) and effects on variance parameters Σ(Z,W,X), with point estimates in-

terpreted as follows.

The first panel of Table 5 reports estimates β for parameters β affecting mean

parameters µ(·). Not surprisingly, these are qualitatively similar to those in our

reduced-form specifications 4, with number of rivals increasing aggressiveness, size of

other contracts decreasing aggressiveness, and a coefficient on log engineer’s estimate

of approximately 0.98. The most notable differences is that number of rivals in other

auctions now increases aggressiveness in auction l; while the model is theoretically

ambiguous, this was our prior when auctions are substitutes.

In the next two panels of Table 5, we present estimates for parameters in the

variance-covariance matrix Σ(Z,W,X). Variance parameters (Panel 2) suggest that

bidders facing more competition and competing in larger auctions submit less dis-

persed bids; while we have no strong priors on these effects, the direction seems

natural. More interestingly, covariance parameters suggest several broad patterns in

bidding across auctions. First, not surprisingly, bidder i bids relatively more similarly

in similar auctions: i.e. in the same county and of the same type. Second, competing

very large quantiles of the maximum rival bid. Since the equilibrium inverse bid function involvesdivision by g−i(·), these in turn produce artificially large pseudo-values for virtually all choices ofθ.

43

Table 5: First-Step MLE Estimates of Gi

Mean µl β MLE SEs 95% CI

Constant 0.20278 0.00894 0.18527 0.22030Log engineer’s estimate 0.98079 0.00101 0.97881 0.98277

Log rivals in auction -0.02816 0.00231 -0.03268 -0.02364Log sum engineer’s (across l) 0.01142 0.00161 0.00826 0.01458

Log sum rivals (across l) -0.00506 0.00220 -0.00938 -0.00075Multiple auction -0.14973 0.02155 -0.19198 -0.10749

Variance σl α MLE SEs 95% CI

Constant -1.03484 0.05439 -1.14144 -0.92823Log engineer’s estimate -0.19618 0.00619 -0.20831 -0.18404

Log rivals in auction -0.26642 0.01848 -0.30265 -0.23020Log sum engineer’s (across l) -0.02824 0.00305 -0.03422 -0.02225

Log sum rivals (across l) -0.10595 0.01309 -0.13160 -0.08029

Covariance ρkl γ MLE SEs 95% CI

Constant 0.68384 0.08874 0.50991 0.85777Same county 0.30562 0.03110 0.24466 0.36658

Same type 0.19214 0.02260 0.14784 0.23643Product of log engineer’s -1.38328 0.53391 -2.42973 -0.33682

Product of log rivals -0.06264 0.01542 -0.09286 -0.03241

in larger projects tends to decrease correlation in i’s bids. In other words, bidders

competing in two large projects tend to compete in one relatively more aggressively

than the other. We interpret this as consistent with the presence of increasing costs

to multiple wins. Finally, stronger competition within the pair tends to decrease

correlation in bids. Since more competition obliges bidder i to compete more aggres-

sively, and thereby decrease markups conditional on winning, we again interpret this

as consistent with substitution between projects.

Estimates of κ Building on the first-step estimates in Table 5, we now apply the

two-step algorithm outlined in Section 4.2 to obtain estimates of the structural pa-

44

rameters θ0 appearing in κ(·). As above, we focus on the sample of bidders competing

in two auctions, with results interpreted as applying to this subsample. Expectations

in the criterion (13) are simulated based on 30 draws from Gi(·|Z,W,X), where to

reduce simulation noise we use the same latent random draws for both (Zj,Wj, Xj)

and (Z ′j,W′j , X

′j).

Table 6 reports estimates θ derived from this procedure. Results suggest that

κ(W ) is characterized by a positive intercept (point estimate 110.29), with projects

of the same type and projects with sum engineer’s estimates below 1000 having

larger intercepts on average. As sum of engineer’s estimates increases, however,

projects become more and more substitutable (point estimate -0.0503), with the small

estimated quadratic effect failing to reverse this sign. The mean of sum engineer’s

estimates in our two-auction sample is 2300, with a median of 1200 and a 90-10 range

of [364, 5000] (units in all cases in thousands of dollars). Evaluating the coefficients

in 6 at these values yields point estimates of κ(W ) = 0.42 at the mean and κ(W ) =

51.51 at the median of product sizes, with a range [108.66,−113.71] corresponding

to the 90-10 range of sum engineer’s estimates.

To interpret these numbers, first recall that in the MDOT marketplace bidder i

is solving a low-price sealed bidding problem:

maxb

(b− vi)Γi(b|Z,W,X) + Pi(b|Z,W,X)Tκ(Z,W,X),

The key source of private information in this problem is cost of contract completion,

so a positive point estimate for κ(·) here implies lower cost, i.e. complementarity

in the language of our model. The estimates in Table 6 thus suggest that bidders

view small projects as complements but large projects as substitutes. While com-

bination effects are approximately zero on average, they are substantially positive

(approximately $51,000) at the median and substantially negative (approximately

$-113,000) at the 90th percentile of project sizes. Our estimates thus suggest that

a joint win leads to cost savings are roughly 4.25 percent of combination size at the

median ($51,000 / $1,200,000), transitioning to cost increases of approximately 2.27

percent of combination size (-$113,000 / $5,000,000) at the 90th percentile.

45

Table 6: Estimates of θ0, Two-AuctionSubsample

θ SE

Constant 110.29 90.14Sum of EE -0.0503 0.0040

(Sum of EE)2 / 1000 0.0011 0.0003(Sum of EE) ≤ 1000 16.34 6.03

Projects of same type 215.16 91.73

Units are in thousands of dollars, positiveκ means higher cost. “Sum of EE” is sumof engineer’s estimates. Mean sum of EEis 2300, mean bid is 1180. Standard errorsare not yet corrected for first-step and sim-ulation error, hence are underestimated.

Taken together, these numbers highlight both the potential importance of com-

binatorial preferences and the fact that these may differ qualitatively across both

auctions and bidders. While declining complementarities between larger projects is

natural and consistent with prior findings in the literature (e.g. Jofre-Bonet and

Pesendorder (2003)), the changing sign of κ(·) is both novel and of considerable eco-

nomic interest. We view this pattern as consistent with an underlying U-shaped cost

curve, with average completion costs falling until firm resources are fully employed

and rising substantially thereafter.

5 Counterfactuals [Preliminary and Incomplete]

To illustrate how non-additive preferences enrich our understanding of auction design

in multi-object markets, we turn now to consider a policy simulation: how would total

procurement costs and social efficiency change if, in place of the current simulatenous

first-price auction format, MDOT allocated contracts via a combinatorial Vickery-

Clarke-Groves (VCG) mechanism?

As our preliminary structural estimates focused exclusively on bidders compet-

46

ing in at most two auctions, we cannot (yet) answer this question fully; the results

reported in Section 4 are silent about complementarities for bidders competing in

more than two auctions. As a first pass illustrating the types of counterfactuals we

ultimately hope to explore, however, we here quantify the effects of switching to a

VCG mechanism for the subsample of auctions selected such that each participant

in every auction participates in no more than two auctions. This yields a counter-

factual sample of roughly 500 auctions over the entire estimation period, averaging

one to two auctions per letting round. Although this sample is not representative

– in particular, it tends to select smaller bidders and (therefore) smaller auctions

– it provides an internally consistent thought experiment within which to explore

the potential welfare costs of the simultaneous first-price formation. This in turn

motivates our extension of the structural analysis to bidders competing in more than

two auctions, currently in progress.

Combinatorial VCG mechanism The combinatorial Vickery-Clarke-Groves (VCG)

mechanism operates as follows. First, bidders report cost vectors yi ≡ ΩT ci + κi de-

scribing their private costs over all combinations of projects for which they submit

bids. Second, given the cost reports submitted by all bidders, the auctioneer chooses

a final allocation a∗ to minimize total social costs of completing all projects; i.e.

chooses a∗ to solve

mina∈A

N∑j=1

yj(a),

where yj(a) denotes bidder i’s total cost at allocation a ∈ A. Finally, the payments

from MDOT to each winning bidder are determined by the usual VCG payment rule:

pi(yi, y−i) = mina∈A\i

∑j 6=i

yj(a)−∑j 6=i

yj(a∗).

In other words, MDOT pays each bidder the positive externality (reduction in total

completion costs) generated by their presence in the auction. As usual, given this

payment rule it is a weakly dominant strategy for bidders to report their values

truthfully. For purposes of the counterfactual, we restrict attention to the equilibrium

47

involving truthful bids.

Counterfactual implementation As a first take on the counterfactual we have

in mind, we compare performance of the simultaneous first-price auction to that of

the combinatorial VCG mechanism in the counterfactual subsample described above:

the set of all auctions such that all players in each auction participate in no more

than two auctions. This leaves us with a sample of approximately 500 auctions on

which to base the counterfactual analysis.

Taking this sample as given, we implement our counterfactual comparison as

follows. First, for each player in the counterfactual sample, we draw a sample of

bids briRr=1 from the bid distribution Gi(·) estimated for that player in Step 1 of

our structural analysis. Second, for each bid vector bri drawn for each player i, we

recover the corresponding standalone valuation vector vri implied by the inverse bid

function (2), taking as given the point estimates κi(·) for κi(·) obtained in Step 2 of

our structural analysis. For each replication r ∈ 1, ..., R, we proceed as follows:

• Simulate the allocation arFPA and procurement cost CrFPA arising under the

simulatenous first-price auction format given bid realizations briNi=1 for each

bidder in the sample; i.e. awarding each auction to the bidder submitting the

lowest standalone bid. Then given estimated complementarities κi(·)Ni=1 and

valuation vectors virNi=1 corresponding to bid draws birNi=1, simulate total

social costs of project complection SrFPA to winning bidders.

• Simulate the allocation arV CG and social cost SrV CG induced by the (weakly

dominant) truthful strategy equilibrium of the VCG mechanism given valuation

draws vri Ni=1 and estimated complementarities κi(·)Ni=1. Then compute the

corresponding total procurement cost CrV CG by applying the VCG payment

rule for each bidder.

Averaging CrFPA and SrFPA across replications r = 1, ..., R should then approximate

expected procurement and social costs arising under the simultaneous first-price auc-

tion, while averaging CrV CG and SrV CG across r approximates expected procurement

48

and social costs arising under the counterfactual VCG mechanism. Comparing these

objects forms the basis for the policy counterfactual of interest.

Counterfactual results The results of our first-take counterfactual analysis are

very favorable for the combinatorial VCG mechanism: relative to the simultaneous

first-price auction, the combinatorial VCG mechanism reduces MDOT procurement

by approximately 7.1 percent and total project completion costs by approximately

14.7 percent. This suggests that the choice of multi-object mechanism can have

potentially large welfare implications, underscoring the importance of the empirical

analysis conducted above.

The estimated magnitude of total social cost savings (14.7 percent) generated by

the VCG auction, while substantial, is likely driven at least in part by our counter-

factual sample definition: requiring that all bidders in every auction participate in

no more than two auctions tends to select auctions in which all participants were

relatively small bidders. These in turn tend to be relatively small projects, for

which our structural procedure estimates relatively large positive complementarities.

These estimated complementarities are likely to be significantly larger than average

complementarities across all auctions, leading to a relatively larger welfare gain from

switching from the inefficient FPA to the efficient VCG mechanism. We expect coun-

terfactual analysis on the overall sample [in progress] to yield smaller total welfare

gains.

The estimated magnitude of MDOT procurement cost reduction (7.1 percent)

is driven by the interaction between two competing effects. First, switching from a

simultaneous FPA to a combinatorial VCG mechanism substantially reduces total

social costs, which (all else equal) tends to reduce MDOT procurement costs. Second,

when projects exhibit positive complementarities, the “exposure problem” inherent

in the simultaneous FPA format creates a strategic incentive for bidders to bid more

aggressively (shade bids downward) to increase their chances of a multiple win. This

incentive disappears in the combinatorial VCG mechanism, leading (all else equal)

to an increase in total procurement costs. In the preliminary counterfactual sample

we consider, the former (cost-savings) effect dominates the latter (strategic shading)

49

effect, with MDOT capturing about half of total social savings generated by the

VCG mechanism. This result underscores the potential efficiency costs of simulta-

neous bidding in procurement markets, further underscoring the contribution of the

structural investigation pursued here. As noted above, however, the counterfactual

sample we consider is also quite restrictive. We thus interpret these preliminary

results as no more than suggestive; a more inclusive counterfactual is currently in

progress.

6 Conclusion

Motivated by an institutional framework common in procurement applications, we

develop and estimate a structural model of bidding in simultaneous first-price auc-

tions, to our knowledge the first such in the literature. We first explore a general

theoretical model of the simultaneous first-price problem, showing that sparsity of

the “message space” relative to the preference space can substantially alter behavior

in the bidding problem. We then specialize this theoretical model to an empirical

framework in which standalone valuations are stochastic but incremental preferences

over combinations are stable functions of observables, establishing non-parametric

identification of the resulting model under standard exclusion restrictions. Finally,

we apply this framework to data on Michigan Department of Transportation highway

construction and maintenance auctions, with preliminary results suggesting substan-

tial complementarities between small projects and substantial substitution between

large projects in this market.

As our primary interest in this paper is simultaneity per se, we do not explic-

itly address dynamics in our structural bidding model. Under some simplifying

assumptions, however, our results have a formal dynamic interpretation. In particu-

lar, suppose that dynamic aspects of bidder behavior are well-approximated by the

Oblivious Equilibrium (OE) solution concept of Weintraub, Benkard, and Van Roy

(2008): bidders forecast evolution of their own states in detail but form expectations

over rival behavior based only on a long-run market average. Then estimates of

κ(·) as we derive them above will formally capture both within- and across-period

50

complementarities associated with winning each combination of auctions. In princi-

ple, we could then further decompose κ(·) into dynamic and static elements, but for

purposes of our analysis here this is inessential; we primarily seek to assess impli-

cations of preferences over combinations on bidding behavior and auction outcomes,

and at least within round t these are likely to be similar regardless of how such

preferences arise.18 Note that the large number of projects auctioned in the MDOT

market, the large number of bidders competing for these auctions, and the relatively

low participation rates by even large bidders together suggest that market evolution

will in fact be well-approximated by a long-run average. Thus while OE may not

perfectly capture all aspects of strategic interaction over time, we find it a plausible

representation in the application considered here.

Appendix A: Complementarities depending on V

In this appendix, we explore prospects for generalizing the conclusions in Sections 2 and 3above to the case where complementarities are additively separable and/or affine functionsof standalone valuations. Such a case could arise if, for instance, winning two auctionstogether increases i’s valuation for one or both objects by a fixed percentage.

Notation and definitions Let el denote the L-dimensional lth unit vector. We saycomplementarities are additively separable in v if for each ω that contains at least twonon-zero components (that is, ωTω ≥ 2), the complementarity function is a function of thevector of standalone valuations v = (v1, v2, . . . , vL)T such that

Kω(v) =∑

l:ωT el=1

φl(vl) + Kω (14)

for some functions φl, l = 1, . . . , L. If each function φl(·) is linear in its argument vl, thenwe obtain the special case of complementarities affine in v:

Kω(v) =∑

l:ωT el=1

δlvl + Kω, if ωTω ≥ 2. (15)

18Obviously, for predicting evolution of behavior across rounds a precise decomposition of κ(·)would be of first importance.

51

As usual, if ω contains at most one component equal to one (that is, ωTω ≤ 1), then weset Kω(v) ≡ 0.

An important special case of (15) is when all δl are identical and Kω = 0 for any ω.This case describes the situation of a constant relative complementarity – that is, whenKω(v) is a constant ratio of the additive valuation.

Now assume that complementarities are affine in v, and define an L × 1 vector δ andan L× L matrix D(δ) as follows:

δ ≡ (δ1, δ2, . . . , δL)T

D(δ) ≡ diag(δ1, δ2, . . . , δL).

Let A denote the 2L × 2L matrix such that its (2L − L − 1) × (2L − L − 1) submatrix(aij)i,j=L+2,...,2L coincides with the identity matrix of size 2L − L − 1, with all the otherelements of A are 0. We then have

K(v) = AΩD(δ)v + K,

where K denotes the 2L× 1 vector of constant components Kω in the functions of comple-mentarities. Clearly, the first L + 1 elements in K are zero since they correspond to thecases of ω such that ωTω ≤ 1.

Monotonicity of best responses Analysis of own- and cross-auction monotonicitybecomes much more complex when we try to extend the results of Lemma 2 to more generalforms of complementarities.

Consider the case of affine complementarities. Suppose that v′′ and v′ differ only inthe l’s component, and that the collection Kω is fixed. Then, using techniques similarto the ones in the proof of Lemma 2 we can show that for any rival strategy profile σ−i, ifb′ is a best response to σ−i at v′ and b′′ is a best response to σ−i at v′′, then

(1 + δl)(v′′l − v′l)[Γl(b

′′l ;σ−i)− Γl(b

′l;σ−i)

]≥ δl(v′′l − v′l)

(P el(b′′;σ−i)− P el(b′;σ−i)

).

The right-hand side of this inequality contains the probability of winning auction l onlyand, thus, depends on the whole vectors of best responses b′ and b′′ rather than just theirl’s components. Not much can be inferred from this inequality, even though some resultscan still be obtained. Suppose that v′′l > v′l. If −1 < δl < 0 and if the best response inauction l decreases – that is, b′′l < b′l, – then at least for one other component m 6= l thebest response decreases too – that is, b′′m < b′m. The inequality can be rewritten as

Γl(b′′l ;σ−i)− Γl(b

′l;σ−i) ≥ −δl

∑ω:ωT el=1, ωTω≥2

(Pω(b′′;σ−i)− Pω(b′;σ−i)

).

Alternatively, suppose that δl > 0. Then b′′l < b′l implies that at least for one other

52

component m 6= l it holds that b′′m > b′m.Example 2 below illustrates that with affine complementarities it is possible to have

situations when a strict increase in vl with other standalone valuations unchanged canproduce a strict decrease in bl.

Example 2. Consider the case of two bidders and two auctions. For bidder 1,

K(1,1) = −1

8v1 −

7

6v2 +

11

4,

and (v1, v2)T ∈ (0, 0)T , (0, 2)T . That is, bidder 1 has types y′ = (0, 0, 0, 114 )T and y′′ =

(0, 0, 2, 2912)T .

Suppose that bidder 2’s fixed strategy is to bid either in auction 1 (with probability 12)

or in auction 2 (with probability 12), drawing bids from the uniform U [0; 1] distribution in

either case.Bidder 1’s types correspond to the following profit functions:

π′ = −b1 ·(

1

2− 1

2·maxmin1, b2, 0

)− b2 ·

(1

2− 1

2·maxmin1, b1, 0

)+

(11

4− b1 − b2

)· maxmin1, b1, 0+ maxmin1, b2, 0

2,

π′′ = −b1 ·(

1

2− 1

2·maxmin1, b2, 0

)+ (2− b2) ·

(1

2− 1

2·maxmin1, b1, 0

)+

(29

12− b1 − b2

)· maxmin1, b1, 0+ maxmin1, b2, 0

2

yielding best response bids b′ =(

78 ,

78

)Tand b′′ =

(0, 17

24

)T. Thus, even though v2 increases

and v1 remains fixed, the best response in auction 2 decreases. Notice that y′′ is not greaterthan y′ (in the component-wise sense) as the last component in y′′ is strictly smaller thanthat in y′. Such a situation was impossible in the case of constant complementaritiesbecause there an increase in one standalone valuation vl weakly increased the 2L-vector ofvaluations y and, moreover, for the components of y that increased strictly the changeswere identical and equal to (v′′l − v′l). Now the change is equal to (v′′l − v′l) for ω = el andis equal to (1 + δl)(v′′l − v′l) for any ω such that ωT el = 1 and ωTω ≥ 2. These changes willeven be of different signs if δl < −1, as was illustrated in this example.

The lemma below considers additively separable complementarities in the form of (14)and shows that the supermodularity condition for these complementarities is satisfied ifand only if it is satisfied for their constant components Kω.

Lemma 4. Let complementarities be defined as in (14). Then for any outcomes ω1, ω2,

Y ω1∧ω2i + Y ω1∨ω2

i − Y ω1i − Y

ω2i = Kω1∧ω2 + Kω1∨ω2 − Kω1 − Kω2 . (16)

53

Hence, complementarities are supermodular if and only if

Kω1∧ω2 + Kω1∨ω2 ≥ Kω1 + Kω2 .

Proof. See Appendix B.

Equation (16) and the fact that its the right-hand side does not depend on standalonevaluations allows us to generalize the result of Lemma 3 to the case of additively separablecomplementarity functions.

Lemma 5. Suppose complementarities are defined as in (14) and are supermodular. Sup-pose that the vectors v′′ and v′ of standalone valuations are such that v′′l > v′l, v

′′k = v′k for

k 6= l. For any rival strategy profile σ−i, if b′ is a best response to σ−i at the correspondingy′ and b′′ is a best response to σ−i at the corresponding y′′, then b′′ ≥ b′.

Non-parametric identification Our next step is to generalize the non-parametricidentification result above to the case when complementarities are affine functions of stan-dalone valuations. Namely, for a given subset ω containing at least two elements,

Kω = Kω(vi, Zi,W,X) =∑

l:ωT el=1

δl(Zi,W,X)vi,l + Kω(Zi,W,X).

When ωTω ≤ 1, the complementarity is set to 0. As can be seen, the functional formof complementarities does not depend on Z−i. As we show below, under weak conditionsthere is enough variation in Z−i |Zi,W,X to determine the linear (in vi,l) part of comple-mentarities as well as the constant part.

Define the Li × 1 vector δ(Zi,W,X) as

δ(Zi,W,X) = (δ1(Zi,W,X), δ2(Zi,W,X), . . . , δLi(Zi,W,X))T ,

and the D(δ(Zi,W,X)) as the Li × Li matrix

D(δ(Zi,W,X)) = diag(δ1(Zi,W,X), δ2(Zi,W,X), . . . , δLi(Zi,W,X)).

ThenK(vi, Zi, X) = AiΩiD(δ(Zi,W,X))vi + K(Zi,W,X),

where K(Zi,W,X) denotes the 2Li × 1 vector of constant components in the complemen-tarities (obviously, K(Zi,W,X) ∈ Ki). Matrix Ai denotes the 2Li × 2Li matrix such thatits (2Li − Li − 1)× (2Li − Li − 1) submatrix (aij)i,j=Li+2,...,2Li coincides with the identity

matrix of size 2Li − Li − 1, and all the other elements of Ai are 0. Clearly, the rank ofmatrix AiΩi is equal to Li.

54

Using the first-order condition and taking into account the form of K(vi, Zi,W,X),obtain

vi = bi + [∇bΓ−i(bi|Z,W,X)]−1 Γ−i(bi|Z,W,X)

− [∇bΓ−i(bi|Z,W,X)]−1∇bP−i(bi|Z,W,X)T[AiΩiD(δ(Zi,W,X))vi + K(Zi,W,X)

],

where, as before, Z = (Zi, Z−i). Denote

Π(bi, δ, Z,W,X) = ILi + [∇bΓ−i(bi|Z,W,X)]−1∇bP−i(bi|Z,W,X)TAiΩiD(δ).

For given Zi, Z−i, W and X, define ∆(Zi, Z−i,W,X) as the set of δ ∈ <Li such that

Π(bi, δ, Z,W,X) is non-singular for almost all bi.

E.g., ∆(Zi, Z−i,W,X) 3 0. If δ = δ(Zi,W,X) ∈ ∆(Zi, Z−i,W,X), then

vi = Π(bi, δ(Zi,W,X), Z,W,X)−1bi

+ Π(bi, δ(Zi,W,X), Z,W,X)−1 [∇bΓ−i(bi|Z,W,X)]−1 Γ−i(bi|Z,W,X)

−Π(bi, δ(Zi,W,X), Z,W,X)−1 [∇bΓ−i(bi|Z,W,X)]−1∇bP−i(bi|Z,W,X)T K(Zi,W,X).

Assuming that δ =∈ ∆(Zi, Z−i,W,X), let us denote

D1(δ, Zi, Z−i,W,X) = EBi[Π(Bi, δ, Z,W,X)−1Bi

∣∣Z,W,X]+ EBi

[Π(Bi, δ, Z,W,X)−1 [∇bΓ−i(Bi|Z,W,X)]−1 Γ−i(Bi|Z,W,X)

∣∣Z,W,X] ,D2(δ, Zi, Z−i,W,X) = EBi

[Π(Bi, δ, Z,W,X)−1 [∇bΓ−i(Bi|Z,W,X)]−1∇bP−i(Bi|Z,W,X)T

∣∣Z,W,X] ,Keeping Zi,W,X fixed, let us draw another value Z ′−i from the support of Z−i|Zi,W,X,

and denote Z ′ = (Zi, Z′−i). Due to the assumptions made on the distribution of the

standalone valuations, E[Vi|Z,W,X] = E[Vi|Z ′,W,X]. Therefore, for δ = δ(Zi,W,X) ∈∆(Zi, Z−i,W,X) ∩∆(Zi, Z

′−i,W,X),

D1(δ(Zi,W,X), Zi, Z′−i,W,X)−D1(δ(Zi,W,X), Zi, Z−i,W,X) =(

D2(δ(Zi,W,X), Zi, Z′−i,W,X)−D2(δ(Zi,W,X), Zi, Z−i,W,X)

)K(Zi,W,X).

For fixed Zi,W,X, this system has 2Li − 1 unknowns (Li in δ(Zi,W,X) and 2Li − Li − 1in K(Zi,W,X)) and Li equations. This gives us the following result.

Proposition 5. Suppose that for (Zi,W,X) ∈ Zi × W × X , there exist J + 1 ≥ (2Li −1)/Li + 1 vectors Z−i,0, Z−i,1, ..., Z−i,J in the support of Z−i|Zi,W,X such that there is a

unique δ ∈⋂Jj=0 ∆(Zi, Z−i,j ,W,X) and a unique κ ∈ Ki that solve the system of J · Li

55

equations

D1(δ, Zi, Z−i,j ,W,X)−D1(δ, Zi, Z−i,0,W,X) =

(D2(δ, Zi, Z−i,j ,W,X)−D2(δ, Zi, Z−i,0,W,X))κ, j = 1, . . . , J.(17)

Then the values of δ(Zi,W,X) and K(Zi,W,X) are identified, and thus, the complemen-tarity function is identified for these values of Zi, W , X.

System (17) is non-linear in δ. However, for each fixed δ ∈⋂Jj=0 ∆(Zi, Z−i,j ,W,X), this

system is linear in κ. Proposition 5 implies that in the case of identification it is not possibleto have a situation when for different δ1 and δ2, where δ1, δ2 ∈

⋂Jj=0 ∆(Zi, Z−i,j ,W,X),

system (17) has solutions κ1 ∈ Ki and κ2 ∈ Ki, respectively. Thus, in this sense thequestion of identification of δ(Zi,W,X) and K(Zi,W,X) comes down to the question ofexistence of solutions to systems of linear equations: (17) can have a solution κ for oneδ only, and for that δ it has to be unique. Using the Kronecker-Capelli theorem, whichgives the necessary and sufficient conditions for the existence of a solution to a system oflinear equations, and also the necessary and sufficient conditions for the uniqueness of sucha solution, we formulate the identification result in the Proposition 6 below.

Before we proceed to Proposition 6, let Ei denote the 2Li × (2Li −Li − 1) matrix suchthat its (2Li − Li − 1)× (2Li − Li − 1) submatrix (eij)i=Li+2,...,2Li , j=1,...,2Li−li−1 coincides

with the identity matrix of size 2Li − Li − 1, and all its other elements (that is, all theelements in the first Li + 1 rows) are equal to zero. For every κ ∈ Ki there is a unique

κ ∈ R2Li−Li−1 such thatκ = Eiκ.

Obviously, this κ is formed by the last 2Li − Li − 1 values in κ. System (17) can beequivalently written as

D1(δ, Zi, Z−i,j ,W,X)−D1(δ, Zi, Z−i,0,W,X) =

((D2(δ, Zi, Z−i,j ,W,X)−D2(δ, Zi, Z−i,0,W,X))Ei) κ, j = 1, . . . , J,(18)

with κ ∈ R2Li−Li−1. For a fixed δ, system (17) is linear in κ, has the J · Li × 2Li matrixof coefficients, and imposes restrictions on the solution κ by requiring that κ ∈ Ki. Itsequivalent representation (18) is linear in κ for a fixed δ, has the J · Li × (2Li − Li − 1)

matrix of coefficients, and does not impose any restrictions on the solution κ ∈ R2Li−Li−1.This allows us to apply the Kronecker-Capelli theorem to system (18) in a straightforwardway.

Proposition 6. Suppose that for (Zi,W,X) ∈ Zi × W × X , there exist J + 1 ≥ (2Li −1)/Li + 1 vectors Z−i,0, Z−i,1, ..., Z−i,J in the support of Z−i|Zi,W,X such that there is a

unique δ ∈⋂Jj=0 ∆(Zi, Z−i,j ,W,X) that satisfies the following two conditions:

56

1. First,

rank ([M1(δ, Zi,W,X) |M2(δ, Zi,W,X)]) = rank (M2(δ, Zi,W,X)) , (19)

where M2(δ, Zi,W,X) denotes the J · Li × (2Li − Li − 1) matrix

M2(δ, Zi,W,X) ≡

(D2(δ, Zi, Z−i,1,W,X)−D2(δ, Zi, Z−i,0,W,X))Ei...

(D2(δ, Zi, Z−i,J ,W,X)−D2(δ, Zi, Z−i,0,W,X))Ei

,and M1(δ, Zi,W,X) denotes the J · Li × 1 vector

M1(δ, Zi,W,X) ≡

D1(δ, Zi, Z−i,1,W,X)−D1(δ, Zi, Z−i,0,W,X)...

D1(δ, Zi, Z−i,J ,W,X)−D1(δ, Zi, Z−i,0,W,X)

.2. Moreover, this δ is such that M2(δ, Zi,W,X) has full column rank:

rank (M2(δ, Zi,W,X)) = 2Li − Li − 1. (20)

Then the values of δ(Zi,W,X) and K(Zi,W,X) are identified, and thus, the complemen-tarity function is identified for these values of Zi, W , X.

Condition (19) requires that in system (18), the rank of the matrix of coefficientsM2(δ, Zi,W,X) is equal to the rank of the augmented matrix [M1(δ, Zi,W,X) |M2(δ, Zi,W,X)]for one δ only. The Kronecker-Capelli theorem guarantees then that (18) has a solution κfor that δ only. Condition (20) then guarantees this κ is determined uniquely, and, thus,κ = Eiκ is determined uniquely.

Note that all the identification conditions in Proposition 6 are formulated in terms of δ.The closed form for δ(Zi,W,X) cannot be found but in practice one can find δ(Zi,W,X)and K(Zi,W,X) by solving, e.g., the following optimization problem:

minδ∈

⋂Jj=0 ∆(Zi,Z−i,j ,W,X), κ∈R2Li−Li−1

Q(δ, κ, Zi,W,X),

where

Q(δ, κ, Zi,W,X) ≡ (M1(δ, Zi,W,X)−M2(δ, Zi,W,X)κ)T (M1(δ, Zi,W,X)−M2(δ, Zi,W,X)κ) .

57

Appendix B: Proofs

Proof of Lemma 3. Suppose that K satisfies supermodular complementarities, and let b′,b′′

be any two feasible bids. Let Bl−i be the (Nl − 1) × 1 vector of rival bids in auction l,

B−i = (B1′−i, ..., B

L′−i)′ be the (

∑Nl − l)× 1 vector of all rival bids across all auctions, and

G−i(B−i; s−i) be the joint distribution of B−i given strategies σ−i. Let k(b;B−i) be playeri’s expected complementarity given own bid vector b and the complete rival bid vector B−i:

k(b;B−i) = Eω[Kω|b, B−i].

The expectation here is over ties; otherwise, bidder i will win each auction in which his bidis the highest, ω = ω(b, B−i) will be deterministic, and k(b;B−i) = Kω(b,B−i). Note thatKTP (b) is the expectation of k(b;B−i) with respect to G−i(·):

KTP (b) = Eω[Kω|b]= EB−i [Eω[Kω|b, B−i]]

=

∫k(b;B−i) dG−i(B−i).

We next show that KTP (b) is supermodular in b. By definition, KTP (b) is supermodularin b if for all b′, b′′ we have

KTP (b′′ ∨ b′) +KT p(b′′ ∧ b′) ≥ KT p(b′′) +KT p(b′),

where b′′ ∨ b′ and b′′ ∧ b′ denote the meet and join of (b′, b′′) respectively.Obviously, if b′′ ≥ b′, then b′′ ∨ b′ = b′′ and b′′ ∧ b′ = b′, so the inequality holds trivially

(and likewise if b′ ≥ b′′). Thus suppose that the vectors are not ordered: i.e. that b′′ b′

and b′ b′′. Now rewrite the supermodularity condition in terms of the integrals above:∫k(b′′ ∨ b′;B−i) dG−i(B−i) +

∫k(b′′ ∧ b′;B−i) dG−i(B−i)

≥∫k(b′′;B−i) dG−i(B−i) +

∫k(b′′;B−i) dG−i(B−i).

Rearranging terms yields the equivalent integral expression∫ [k(b′′ ∨ b′;B−i) + k(b′′ ∧ b′;B−i)− k(b′′;B−i)− k(b′;B−i)

]dG−i(B−i) ≥ 0. (21)

58

In the case of no ties for (b′′ ∨ b′;B−i), (b′′ ∧ b′;B−i), (b′′;B−i) and (b′;B−i), we have

ω(b′′ ∨ b′;B−i) = ω(b′;B−i) ∨ ω(b′′;B−i),

ω(b′′ ∧ b′;B−i) = ω(b′;B−i) ∧ ω(b′′;B−i),

and thus, by the supermodularity of K,

k(b′′ ∨ b′;B−i)− k(b′′;B−i) −[k(b′;B−i)− k(b′′ ∧ b′;B−i)

]≥ 0.

If for b′′ ∨ b′, b′′ ∧ b′, b′′ and b′, ties occur with probability zero (with respect to thedistribution G−i(·) of B−i), then the last inequality implies (21). Thus, supermodular Kimplies supermodularity of KTP (b) in b.

To complete the argument, note that supermodularity of KTP (b) implies that an in-crease in any element of b will weakly increase marginal returns to other elements of b. Weknow from Lemma 2 that an increase in vl will weakly increase bl, and by supermodularityof KT p(b) this will also increase bids in other auctions. This establishes the claim.

Proof of Lemma 4. Note that

Y ω1∧ω2i + Y ω1∨ω2

i =∑

l: (ω1∧ω2)T el=1

(vl + φl(vl)) + Kω1∧ω2 +∑

l: (ω1∨ω2)T el=1

(vl + φl(vl)) + Kω1∨ω2

=∑

l:ωT1 el=1, ωT2 el=0

(vl + φl(vl)) +∑

l:ωT2 el=1, ωT2 el=0

(vl + φl(vl)) + 2∑

l: (ω1∧ω2)T el=1

(vl + φl(vl))

+ Kω1∧ω2 + Kω1∨ω2 .

Also,

Y ω1i + Y ω2

i =∑

l:ωT1 el=1

(vl + φl(vl)) + Kω1 +∑

l:ωT2 el=1

(vl + φl(vl)) + Kω2 =∑

l:ωT1 el=1, ωT2 el=0

(vl + φl(vl))+

+∑

l:ωT2 el=1, ωT1 el=0

(vl + φl(vl)) + 2∑

l: (ω1∧ω2)T el=1

(vl + φl(vl)) + Kω1 + Kω2 .

The result follows.

Proof of Lemma 5. Analogously to the proof of Lemma 3, for player i’s given vector ofstandalone valuations v, own bid vector b and the complete rival bid vector B−i, considerthe expected complementarity

k(v, b;B−i) = Eω[Kω(v)|v, b, B−i].

59

As in the proof of Lemma 3, the expectation here is over ties. Note that K(v)TP (b) is theexpectation of k(v, b;B−i) with respect to G−i(·):

K(v)TP (b) = Eω[Kω(v)|v, b] = EB−i [Eω[Kω(v)|v, b, B−i]] =

∫k(v, b;B−i) dG−i(B−i).

Let b′ and b′′ be any two feasible bid vectors. Let us establish that K(v)TP (b) is super-modular in b, that is, for a given v,

K(v)TP (b′′ ∨ b′) +K(v)T p(b′′ ∧ b′) ≥ K(v)T p(b′′) +K(v)T p(b′).

This supermodularity condition van be rewritten as∫k(v, b′′ ∨ b′;B−i) dG−i(B−i) +

∫k(v, b′′ ∧ b′;B−i) dG−i(B−i)

≥∫k(v, b′′;B−i) dG−i(B−i) +

∫k(v, b′′;B−i) dG−i(B−i),

or equivalently, as∫ [k(v, b′′ ∨ b′;B−i) + k(v, b′′ ∧ b′;B−i)− k(v, b′′;B−i)− k(v, b′;B−i)

]dG−i(B−i) ≥ 0.

(22)In the case of no ties for (b′′∨b′;B−i), (b′′∧b′;B−i), (b′′;B−i) and (b′;B−i), from Lemma

4 we have that

k(v, b′′ ∨ b′;B−i) + k(v, b′′ ∧ b′;B−i)− k(v, b′′;B−i)− k(v, b′;B−i)

= Kω(b′′∨b′;B−i) + Kω(b′′∧b′;B−i) − Kω(b′;B−i) − Kω(b′′;B−i)

= Kω(b′;B−i)∨ω(b′′;B−i) + Kω(b′;B−i)∧ω(b′′;B−i) − Kω(b′;B−i) − Kω(b′′;B−i) ≥ 0.

If for b′′ ∨ b′, b′′ ∧ b′, b′′ and b′, ties occur with probability zero (with respect to thedistribution G−i(·) of B−i, then the last inequality implies (22).

Proof of Proposition 3. The proof of Proposition 3 rests on two key claims. First, thefirst-order system (2) must be well-defined for almost every bi submitted by i, i.e. almosteverywhere with respect to the measure induced by Gi(·|Z,W,X). Second, at almost everybi at which first order conditions hold, the first-order system (2) must be invertible. Weestablish each claim in turn.

First show that the first order system (2) is well-defined for almost every bi submittedby i. Recall that we can write bidder i’s objective as

π(vi, b|K;Z,W,X) = (Ωvi +K − Ωb)TP−i(b|Z,W,X).

60

where vi and K are given at the time of maximization. Note that the system (2) necessarilyholds at any best respose where π(vi, ·|K;Z,W,X) is differentiable and that Assumption5 implies that each observed bi is a best response. Hence the system (2) will be welldefined for almost every bi submitted by i if and only if π(vi, ·|K;Z,W,X) is differentiablealmost everywhere with respect to the measure on Bi induced by Gi(·|Z,W,X). But underAssumption 6 Gi(·|Z,W,X) is absolutely continuous. To establish the claim, it is thussufficient to prove differentiability of π(vi, ·|K;Z,W,X) a.e. with respect to the Lebesguemeasure on Bi.

Clearly (Ωvi +K −Ωb) is differentiable in b for any vi,K ∈ RLit ×K. Thus differentia-bility of π(vi, ·|K;Z,W,X) at b is equivalent to differentiability of P−i(·|Z,W,X) at b. LetB−i be the Li × 1 random vector describing maximum rival bids in the set of auctions inwhich i participates. Again applying Assumption 6 to rule out ties, the probability i winscombination ω at bid b is

Pω(b|Z,W,X) = Pr(∩l:ωl=10 ≤ B−i,l ≤ bi,l ∩ ∩l:ωl=0bi,l ≤ B−i,l <∞).

For each ω ∈ Ωi, let bω be the (∑ω) × 1 sub-vector of b describing i’s bids for objects in

ω, Bω−i be the (

∑ω) × 1 sub-vector of B−i describing maximum rival bids for objects in

ω, and Gω−i(bω|Z,W,X) be the equilibrium joint c.d.f. of Bω

−i at (Z,W,X). Applying theformula for a rectangular probability and simplifying, we can then represent P−i(·|Z,W,X)in the form

Pω−i(b|Z,W,X) =∑ω′∈Ω

aωω′Gω′−i(b

ω′ |Z,W,X),

where each aωω′ is a known scalar (determined by ω, ω′) taking values in −1, 0, 1. Butby absolute continuity each c.d.f. Gω−i(·|Z,W,X) is differentiable a.e. (Lebesgue) in its

support, and interpreted as a function from Bi to RLi , each bω′is continuously differentiable

in b. Thus interpreted as a function from Bi to R, each Gω′−i(b

ω′ |Z,W,X) is differentiable on

a set of full Lebesgue measure in B−i. The set of points in Bi at which all Gω′−i(b

ω′ |Z,W,X)

are differentiable is the intersection of points in Bi at which each Gω′−i(b

ω′ |Z,W,X) isdifferentiable, i.e. the intersection of a finite collection of sets of full Lebesgue measure inBi. But from above differentiability of Gω

′−i(b|Z,W,X) for all ω′ implies differentiability of

Pω−i(b|Z,W,X). Hence Pω−i(·|Z,W,X) is differentiable on a set of full Lebesgue measurein Bi. This in turn implies differentiability of π(vi, ·|K;Z,W,X) a.e. with respect to themeasure on Bi induced by Gi(·|Z,W,X), as was to be shown.

We next establish that the first-order system (2) must yield a unique solution v foralmost every bi submitted by i. Let Bi be the set of points in Bi at which π(·, ·|K;W,Z,X)is differentiable in b; from above, Bi is a subset of full Lebesgue measure in Bi. Choosingany b ∈ Bi and rearranging (2) yields

∇bΓ−i(b|Z,W,X)v = ∇bΓ−i(b|Z,W,X)b+ Γ−i(b|Z,W,X)−∇bP−i(b|W,Z,X)TKi.

61

Hence uniqueness of v is equivalent to invertibility of the Li×Li matrix∇bΓ−i(b|Z,W,X).Recall that Γ−i(b|Z,W,X) is an Li × 1 vector whose lth element describes the probabilitythat bid vector b wins auction l. Note that b ∈ Bi rules out ties at b. Thus for b ∈ Bi thelth element of Γ−i(b|Z,W,X) is the marginal c.d.f. Gl−i(b|Z,W,X) of B−i,l, from whichit follows that ∇bΓ−i(b|Z,W,X) is a diagonal matrix whose l, lth element is the marginalp.d.f. gl−i(b|Z,W,X) of B−i,l. Hence ∇bΓ−i(b|Z,W,X) will be invertible at b if and only ifgl−i(b|Z,W,X) > 0 for all l.

But by hypothesis each submitted bid bi is a best response to rival play at (Z,W,X)for some (v,K). Suppose that there exists an ε > 0 such that gl−i(·|Z,W,X) = 0 on(bil − ε, bi]. Then player i could infintesimally reduce bil without affecting either Γ−ior P−i, a profitable deviation for any (v,K). Hence we must have gl−i(·|Z,W,X) > 0almost everywhere (Lebesgue) in the support of Bi. By Assumption 6, this in turn impliesgl−i(·|Z,W,X) > 0 for almost every bi submitted by i. Since l was arbitrary, we must have∇bΓ−i(bi|Z,W,X) invertible for almost every bid bi submitted by i. Hence for almost everybi submitted by i there will exist a unique v satisfying (2) at bi, given by

v = bi +∇bΓ−i(bi|Z,W,X)−1Γ−i(bi|Z,W,X)

+∇bΓ−i(bi|Z,W,X)−1∇bP−i(bi|W,Z,X)TK.

The RHS of this expression is identified up to K, establishing the claim.

Appendix C: Partial identification with general Gi

The point identification result for the complementarity function κi(Zi,W,X) and the con-ditional distribution of Vi|Zi,W,X relied on the equations in the first order conditionsobtained from bidder’s optimization of the payoff function. To derive those equations weemployed the absolute continuity of bid distribution functions Gi. That, in particular,eliminated the possibility of bidders playing atoms in the equilibrium. In this appendix,we want to illustrate an approach to the identification question when no continuity re-strictions are imposed on Gi. Our identification method is based on using inequalities forbidder’s best responses and employing the exclusion restrictions in Assumption 4 to obtainbounds on the complementarity function and the distributions of standalone valuations.Throughout our analysis here we continue to impose Assumptions 1-5.

Let us fix (Zi,W,X) ∈ Zi ×W × X . As before, Fi(·|Zi,W,X) denotes the L-variatedistribution of the vector of standalone valuations Vi of the bidder conditional on Zi,W,X.As Fac(Rp) we denote the set of all absolutely continuous cumulative distribution functionson Rp.

Observable are the distribution functions Gi(b|Z,W,X), where Z = (Zi, Z−i), of thebids in the equilibrium played by the bidders for any Z−i ∈ Z−i|Z,W,X, i = 1, . . . , N .

62

The identification set is defined as the set

Hi(Zi,W,X) = (Fi(·|Zi,W,X), κi(Zi,W,X)) : Conditions (C1), (C2), (C3) satisfied .

When writing the identification set in this form, we already make us of the exclusionrestrictions in Assumption 4. The conditions in the definition of Hi(Zi,W,X) are thefollowing:

(C1) Fi(·|Zi,W,X) ∈ Fac(RLi), and the support of Fi(·|Zi,W,X) is a compact, convexset in RLi (consistent with Assumption 1).

(C2) κi(Zi,W,X) ∈ Ki.

(C3) For each Z−i ∈ Z−i|Zi,W,X, bidder’s behavior is consistent with the maximizationof the payoff function

π(vi, b;Z,W,X) = vTi Γ−i(b|Z,W,X)−bTΓ−i(b|Z,W,X)+P−i(b|Z,W,X)Tκi(Zi,W,X)

with respect to b ∈ Bi. That is, for each Z−i ∈ Z−i|Zi, X,W , every bidder i’s bidvector bi observed in the equilibrium satisfies the inequality

vTi Γ−i(bi|Z−i)− bTi Γ−i(bi|Z−i) + P−i(bi|Z−i)Tκ(Zi,W,X) ≥vTi Γ−i(b|Z−i)− bTΓ−i(b|Z−i) + P−i(b|Z−i)Tκ(Zi,W,X)

∀ b ∈ Bi, (23)

where for notational simplicity we wrote Γ−i(·|Z−i) and P−i(bi|Z−i) instead of Γ−i(bi|Z,W,X)and P−i(bi|Z,W,X) respectively, thus omitting fixed (Zi,W,X) from the notation.

Let Hi,κ(Zi,W,X) stand for the projection of the identification set onto the secondcomponent – on the set of κi(Zi,W,X). Let Hi,F (Zi,W,X) stand for the projectionof the identification set onto the first component – on the set of Fi(·|Zi,W,X); and letHi,Fl(Zi,W,X) stand for the projection of Hi,F (Zi,W,X) onto the marginal distributionof Vil conditional on Zi,W,X – that is, on the set of univariate distribution functionsFil(·|Zi,W,X). Even though it is very difficult to obtain a closed form description of thesets Hi,F (Zi,W,X) and Hi,κ(Zi,W,X), it is possible to give closed form characterizationsof their supersets. These supersets are given in Proposition 7 below.

First, we introduce some notations. Let ∆+ε,l[f(u)] and ∆−ε,l[f(u)] denote differences in

the values of f(·) associated with adding ε and −ε to the lthe component of u respectively:

∆+ε,l[f(u)] = f(u+ εel)− f(u),

∆−ε,l[f(u)] = f(u− εel)− f(u),

where el denotes the Li-dimensional lth unit vector.

63

Suppose there exist known scalars v ≥ 0 and v < ∞ such that v ≤ vil ≤ v for anyl; note that these could be strictly outside the support of Vil. For each bi ∈ Bi, defineI−ε,l(bi|Z−i), I

+ε,l(bi|Z−i) as follows:

I−ε,l(bi|Z−i) =

v if ∆−ε,l[Γ−i(bi|Z−i)] = 0,

∆−ε,l[bTi Γ−i(bi|Z−i)]

∆−ε,l[Γ−i,l(bi|Z−i)]else

;

I+ε,l(bi|Z−i) =

v if ∆+

ε,l[Γ−i(bi|Z−i)] = 0,∆+ε,l[b

Ti Γ−i(bi|Z−i)]

∆+ε,l[Γ−i,l(bi|Z−i)]

else

.

Also, for each bi ∈ Bi, define the following S−ε,l(bi|Z−i) and S+ε,l(bi|Z−i):

S−ε,l(bi|Z−i) =

0 if ∆−ε,l[Γ−i(bi|Z−i)] = 0,

∆−ε,l[P−i(bi|Z−i)]∆−ε,l[Γ−i,l(bi|Z−i)]

else

;

S+ε,l(bi|Z−i) =

0 if ∆+

ε,l[Γ−i(bi|Z−i)] = 0,∆+ε,l[P−i(bi|Z−i)]

∆+ε,l[Γ−i,l(bi|Z−i)]

else

.

For any K ∈ Ki, let F−il (·|K;Z−i) denote the c.d.f. of

supε>0

(I−ε,l(bi|Z−i)− S

−ε,l(bi|Z−i)

TK),

and let F+il (·|K;Z−i) denote the c.d.f. of

infε>0

(I+ε,l(bi|Z−i)− S

+ε,l(bi|Z−i)

TK)

.Hereafter we assume that ties are broken independently across auctions.

Proposition 7. a) A superset of Hi,κ(Zi,W,X) can be found in the following way:

H(1)i,κ (Zi,W,X) =

L⋂l=1

Ki,l,

where Ki,l is defined as

Ki,l = K ∈ Ki∣∣ F+

il (·|K;Z−i) ≤ F−il (·|K;Z ′−i) ∀Z−i, Z ′−i ∈ Z−i|Zi,W,X.

b) A superset of Hi,Fl(Zi,W,X) can be found as the set of univariate functions Fil(·) ∈

64

Fac(R) such that for any η ∈ R,

Fil(η) ∈⋂

κ0∈H(1)i,κ(Zi,W,X)

⋂Z−i,Z′−i∈Z−i|Zi,W,X

[F+il (η|κ0;Z−i), F

−il (η|κ0;Z ′−i)].

Let us denote this superset as H(1)i,Fl

(Zi,W,X).c) A superset of Hi,F (Zi,W,X) can be found as the set of Li-variate functions Fi(·) ∈

Fac(RLi) such that each lth marginal distribution function generated by Fi(·) belongs to

H(1)i,Fl

(Zi,W,X), l = 1, . . . , Li.Moreover, for any η = (η1, . . . , ηLi),

Fi(η) ≤ minl=1,...,Li

infκ0∈H(1)

i,κ(Zi,W,X)

infZ−i∈Z−i|Zi,W,X

F+il (ηl|κ0;Z−i),

Fi(η) ≥ max

Li∑l=1

supκ0∈H(1)

i,κ(Zi,W,X)

supZ−i∈Z−i|Zi,W,X

F−il (ηl|κ0;Z−i)− Li + 1, 0

.

Proof. Let ∆δ[f(u)] = (f(u + δ) − f(u)) denote changes in f(u) induced by adding thevector δ to the vector u. For notational simplicity, let κi(Zi,W,X) = κ0. Then we canequivalently restate (23) as follows:

vTi ∆δ[Γ−i(bi|Z−i)] − ∆δ[bTi Γ−i(bi|Z−i)] + ∆δ[P−i(bi|Z−i)T ]κ0 ≤ 0 ∀ δ ∈ Bi − bi, (24)

where the difference Bi − bi is understood as the Minkowski difference.System (24) is linear in vi and κ0. The set of solutions of (vi, κ0) to this system can

be shown to be convex. Its intersection with [v, v]L × Ki is convex as well. In a specialcase when Bi consists of a finite number of points, the set of solutions is a convex (closed)

polyhedron in <2L−1.Any subsystem of a finite number of inequalities from system (24) can easily be resolved

to give an upper bound (a lower bound) on the lth component vil in terms of the minimum(maximum) of some linear functions with of κ0 with known coefficients. That could beused to get bounds on the distribution functions of standalone valuations in terms of κ0.However, the formulas for the bounds on each component obtained from a finite system oflinear inequalities are quite complicated. An easier way to obtain bounds on each vil is toconsider only those alternative bids b that differ from bi in the lth coordinate.

Since the ties are broken independently across auctions at bi, then a change in the lthcomponent of bi affects only the lth component of Γ−i. Noting that Γ−i,l is monotone in

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bil and rearranging, we thus must have for any ε such that bi − εel ∈ Bi and bi + εel ∈ Bi

∆−ε,l[bTi Γ−i(bi|Z−i)]

∆−ε,l[Γ−i,l(bi|Z−i)]−

∆−ε,l[P−i(bi|Z−i)T ]

∆−ε,l[Γ−i,l(bi|Z−i)]κ0 ≤ vil

≤∆+ε,l[b

Ti Γ−i(bi|Z−i)]

∆+ε,l[Γ−i,l(bi|Z−i)]

−∆+ε,l[P−i(bi|Z−i)

T ]

∆+ε,l[Γ−i,l(bi|Z−i)]

κ0. (25)

If for a given ε, we have that bi − εel /∈ Bi or bi + εel /∈ Bi, then at least one of∆−ε,l[Γ−i,l(bi|Z−i)] and ∆+

ε,l[Γ−i,l(bi|Z−i)] is equal to 0, and then in order to bound vil wecan use our prior knowledge that vil ∈ [v, v].

Using our notations above, we can say that for any bi in the support of Gi:

I−ε,l(bi|Z−i)− S−ε,l(bi|Z−i)

Tκ0 ≤ vil ≤ I+ε,l(bi|Z−i)− S

+ε,l(bi|Z−i)

Tκ0,

and hence,

supε>0

(I−ε,l(bi|Z−i)− S

−ε,l(bi|Z−i)

Tκ0

)≤ vil ≤ inf

ε>0

(I+ε,l(bi|Z−i)− S

+ε,l(bi|Z−i)

Tκ0

). (26)

Then by the inequalities in (26), we must have for any Z−i ∈ Z−i|Zi,W,X:

F+il (·|κ0;Z−i) ≤ Fil(·) ≤ F−il (·|κ0;Z−i). (27)

Pooling information across Z−i, it follows that we can have K = κ0 only if

F+il (·|K;Z−i) ≤ F−il (·|K;Z ′−i) ∀Z−i, Z ′−i ∈ Z−i|Zi,W,X. (28)

This gives part a) of this theorem. Part b) immediately follows from (27). Part c) fol-lows from (27) and the well known result on sharp Frechet-Hoeffding bounds for jointdistributions.

The next proposition provides an expectations version of the partial identification ar-gument. Even though the supersets it gives are larger than those in Proposition 7, com-putationally they are easier to obtain. Before formulating this proposition, let us defineLi × 1 vectors Ψ−ε (Z−i), Ψ+

ε (Z−i) and Li × 2Li matrices χ−ε (Z−i), χ+ε (Z−i) as follows:

Ψ−ε (Z−i) ≡[E[I−ε,l(Bi|Z−i)|Z−i]

]Lil=1

Ψ+ε (Z−i) ≡

[E[I+

ε,l(Bi|Z−i)|Z−i]]Lil=1

χ−ε (Z−i) ≡[E[S−ε,l(Bi|Z−i)|Z−i]

T]Lil=1

χ+ε (Z−i) ≡

[E[S+

ε,l(Bi|Z−i)|Z−i]T]Lil=1.

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Proposition 8. A superset of Hi,κ(Zi,W,X) can be found in the following way:

H(2)i,κ (Zi,W,X) =

⋂ε>0

Kεi ,

where Kεi is defined as

Kεi ≡K ∈ Ki

∣∣∣ (Ψ−ε (Z−i)−Ψ+ε (Z ′−i)

)−(χ−ε (Z−i)− χ+

ε (Z ′−i))K ≤ 0 for all Z−i, Z

′−i ∈ Z−i|Zi,W,X

.

Results analogous to parts b) and c) in Proposition 8 hold as well with H(2)i,κ (Zi,W,X)

replacing H(1)i,κ (Zi,W,X).

Proof. Taking expectations of (26) across bil, we obtain

E[I−ε,l(Bi|Z−i)|Z−i]− E[S−ε,l(Bi|Z−i)|Z−i]κ0 ≤ E[Vil|Z−i]≡ E[Vil] ≤ E[I+

ε,l(Bi|Z−i)|Z−i]− E[S+ε,l(Bi|Z−i)|Z−i]κ0. (29)

Then pooling restrictions of the form (29) across Z−i, Z′−i and l, we obtain

Ψ−ε (Z−i)− χ−ε (Z−i)κ0 ≤ Ψ+ε (Z ′−i)− χ+

ε (Z ′−i)κ0 ∀ Z−i, Z ′−i ∈ Z−i|Zi,W,X. (30)

Set H(2)i,κ (Zi,W,X) is larger than H(1)

i,κ (Zi,W,X) because first-order stochastic domi-nance implies inequalities for expectations.

Note two features of H(2)i,κ (Zi,W,X). First, it can be represented as the intersection of

a set of half-spaces in Ki, where half-spaces are bounded by hyperplanes involving slopevectors (χ−ε,l(Z−i)− χ

+ε,l(Z

′−i)) and intercepts (Ψ−ε,l(Z−i)−Ψ+

ε,l(Z′−i)), and the intersection

is taken over collections of (Z−i, Z′−i, ε, l).

Second, if Gi is absolutely continuous, then H(2)i,κ (Zi,W,X) is a singleton, and as we

show below, the analysis of H(2)i,κ (Zi,W,X) essentially becomes our identification strategy

in the case of point identification. Indeed, bidder i’s objective function is differentiable atalmost every observed bi. Hence as ε→ 0 we will have for all l

limε→0

∆−ε,lbiΓ−i(bi|Z−i)∆−ε,lΓ−i,l(bi|Z−i)

= limε→0

∆−ε,lbiΓ−i(bi|Z−i)/ε∆−ε,lΓ−i,l(bi|Z−i)/ε

=∂(biΓ−i(bi|Z−i))/∂bildΓ−i,l(bi|Z−i))/dbil

,

and therefore Ψ−ε (·)→ Ψ(·). Analogously, it is straightforward to show that Ψ+ε (·)→ Ψ(·),

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χ−ε → χ(·), and χ+ε → χ(·). Hence the restriction (30) implies

Ψ(Z−i)− χ(Z−i)κ0 ≤ Ψ(Z ′−i)− χ(Z ′−i)κ0 ∀ Z−i, Z ′−i ∈ Z−i|Zi,W,X.

Noting that Z−i, Z′−i are interchangeable, we thus have for any Z−i, Z

′−i ∈ Z−i|Zi,W,X:

Ψ(Z−i)− χ(Z−i)κ0 ≤ Ψ(Z ′−i)− χ(Z ′−i)κ0

Ψ(Z ′−i)− χ(Z ′−i)κ0 ≤ Ψ(Z−i)− χ(Z−i)κ0,

or equivalently

Ψ(Z−i)− χ(Z−i)κ0 = Ψ(Z ′−i)− χ(Z ′−i)κ0 ∀ Z−i, Z ′−i ∈ Z−i|Zi,W,X.

But this is exactly the identification restriction invoked in Proposition 3 in the currentpaper. Thus we can strictly generalize our existing identification results (which depend ondifferentiability a.e.) to partial identification for arbitrary Gi.

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