Simultaneous First-Price Auctions with Preferences
over Combinations: Identification, Estimation and
Application∗
Matthew Gentry†
Tatiana Komarova‡
Pasquale Schiraldi§
October 2014
(Preliminary and incomplete)
Abstract
Motivated by the empirical prevalence of simultaneous bidding across a
wide range of auction markets, we develop and estimate a structural model
of strategic interaction in simultaneous first-price auctions when objects are
heterogeneous and bidders have preferences over combinations. We begin by
proposing a general theoretical model of bidding in simultaneous first price
auctions, exploring properties of best responses and existence of equilibrium
within this environment. We then specialize this model to an empirical framework in which bidders have stochastic private valuations for each object and
stable incremental preferences over combinations; this immediately reduces to
the standard separable model when incremental preferences over combinations
are zero. We establish non-parametric identification of the resulting model
under standard exclusion restrictions, thereby providing a basis for both testing on and estimation of preferences over combinations. We then apply our
model to data on Michigan Department of Transportation highway procurement auctions, we quantify the magnitude of cost synergies and assess possible
efficiency losses arising from simultaneous bidding in this market.
∗
We are grateful to Philip Haile, Ken Hendricks, Paul Klemperer, and Balazs Szentes for their
comments and insight. We also thank seminar participants at the University of Wisconsin (Madison), the University of Zurich, University of Leuven, Cardiff University, Oxford University, Cornell
University, 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 design of 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 substantially alters the strategic bidding problem compared to the standard first price
auction with ambiguous welfare implications depending on the importance of synergies (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 auctions 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. Concrete examples include markets for highway procurement in most US states (JofretBonet 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 effects 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 simul1
These commonly arise from cost savings in procurement auctions.
These may arise when bidders have resource and capacity constraints or when winning alternative items offered can meet the same need.
2
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 assign 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 combinatoric 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 discussions in Cantillon and Pesendorfer 2006, Crampton at al. 2006). Meanwhile, the
combinatorial first-price mechanism may impose substantial practical costs on both
bidders (the combinatoric 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 resulting from inability to express combinatoric 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 markets, there presently exists very little work assessing the revenue and efficiency implications 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
3
In environments with substitutes it is known that any Bayes-Nash equilibrium of the simultaneous 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 in
practice, 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 combination 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 assigns 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 permitting 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 observables. Second, we assume that the marginal distributions of i’s standalone valuations are invariant either to the characteristics of i’s rivals or to the characteristics
4
Note that this decomposition is without loss of generality; the key identifying restriction is the
structure we impose on complementarities.
5
Note that this structure does not restrict dependence between i’s standalone valuations for
different objects in the market. We view this flexibility as critical, as in practice we expect i’s
standalone valuations to be positively correlated.
4
of other objects on which i is bidding.6 We show that optimal behavior in this environment 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 identification and excludable variation in characteristics yielding semiparametric identification 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 procedure 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 bidderand auction-level characteristics. Due to the high-dimensional nature of this estimation 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 covariates8 and estimate parameters in this function by minimization of a simulated
analog to our semiparametric identification criterion. As in practice we parametrize
6
The first of these exclusion restrictions is widely invoked in the literature; see, e.g. Guerre, Perrigne and Vuong (2009) or Somaini (2012). The second is specific to the combinatoric environment
we consider, although we believe it is natural in a variety of applications.
7
As it is difficult to rule out ties a priori in the fully general simultaneous model, we also
extend our partial identification results (see Appendix C) to accommodate potential mass points
in equilibrium bids. We show that in this case primitives are partially identified, with Monte
Carlo analysis suggesting that identified sets are tight in practice. While we do not believe ties are
important in the application —they are never observed in the data — we consider, this analysis
helps to underscore robustness of our results.
8
In our application, combination-specific covariates might include the sum of engineer’s estimates
across projects in a combination, distance between projects in a combination, and indicators for
whether 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 bidding 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 bidders 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 extensions 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 procurement 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.
9
The exposure problem in auctions of multiple items involves the risk of bidders winning unwanted items, i.e. winning items at prices above bidders’ values for them.
6
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 empirical 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 mechanism 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 combinatoric auctions, using additional structure on combinatoric preferences to circumvent
the substantial curse of dimensionality involved. They then apply methodology to
data on combinatoric auctions used in procurement of Chilean school meals, finding
that the combinatoric 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 literature seeking to quantify the role of preferences over combinations in multi-object
auctions. For instance, Ausubel, Cramton, McAfee and McMillan (1997) and Moreton 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, sequential, and combinatorial first-price sealed-bid auctions and show that, despite
7
potential for synergies, revenue comparison between formats produced statistically
insignificant differences.
From a more theoretical perspective, there exist several studies analyzing strategic interaction in stylized models involving simultaneous first-price auctions. For
instance, Szentes and Rosental (1996) study a first-price sealed-bid acution for identical 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 focus 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
10
Though only tangentially related to our problem, there is also a growing literature on multi-unit
discriminatory auctions of homogeneous objects. Reny (1999, 2011), Athey (2001), and McAdams
(2006) address issues of existence of equilibria and equilibrium properties in such auctions. Meanwhile, Hortacsu and Puller (2008), Hortacsu and McAdams (2010), and Hortacsu (2011) provide
more empirical perspectives on multi-unit auctions.
8
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 auctions
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 × 1 matrix whose rows contain (transposes of) each possible outcome ω: e.g. if
L = 2,
"
#
0
0
1
1
ΩT =
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.
9
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 Yiω be the combinatoric
valuation bidder i assigns to outcome ω, and Yi be the 2L × 1 vector describing
the combinatoric 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
L
Yi ⊂ R2 , 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). Yi0 = 0 and Yiω is non0
decreasing in the vector of objects won: ω 0 ≥ ω implies Yiω ≥ Yiω .
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 Bi
be 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 ≡ Yiel , where el (the lth unit
vector) is a vector of zeros with a one in the lth place. We call Vil bidder i’s standalone 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
10
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:
Kiω ≡ Yiω − ω T Vi .
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 standalone valuations and complementarities, we will frequently wish to distinguish between 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 distribution 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. Similarly, 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 ) = ΩT P (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 complementarities is complicated enormously by the so-called “exposure problem” – when bids
11
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
L
communicate their preferences (Yi ⊂ R2 ) over outcomes. This turns out to invalidate much classical intuition regarding bidding behavior; for instance, as we show
below, a strict increase in type (y 0 > y in all elements) can yield a strict decrease in
best-response bid (b0 < 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 competing against rivals who bid according to strategy profile σ−i . Conditional on winning
outcome ω at bid vector b, i receives net payoff Yiω − ω T 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)T P (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 ) = ΩT P (b; σ−i ), we can then rewrite πi (b; σ−i ) as follows:
πi (b; σ−i ) = (yi − Ωb)T P (b; σ−i )
= (Ωvi − Ωb)T P (b; σ−i ) + kiT P (b; σ−i )
= (vi − b)T Γ(b; σ−i ) + kiT 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
X
(vil − bl )Γl (bl ; σ−i ).
l=1
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 0 and
y 00 be any elements of Yi , and let b0 and b00 be any corresponding best responses by i
to σ−i . Then
(y 00 − y 0 ) · [P (b00 ; σ−i ) − P (b0 ; σ−i )] ≥ 0.
In other words, under best-response bidding, P (·; σ−i ) must move in the same dotproduct direction as y.
Proof. By definition of best responses,
(y 0 − Ωb0 )T · P (b0 ; σ−i ) ≥ (y 0 − Ωb00 )T · P (b00 ; σ−i )
⇔ y 0 · [P (b0 ; σ−i ) − P (b00 ; σ−i )] ≥ b0 · ΩT P (b0 ; σ−i ) − b00 · ΩT P (b00 ; σ−i ),
and analogously
y 00 · [P (b00 ; σ−i ) − P (b0 ; σ−i )] ≥ b00 · ΩT P (b00 ; σ−i ) − b0 · ΩT P (b0 ; 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 0 to y 00 such that y 00 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,
15
5
1
K (1,1) = − v1 − v2 + ,
8
16
4
and (v1 , v2 )T ∈ {(0, 0)T , (0, 2)T , (2, 2)T }. That is, bidder 1 has types y 0 = (0, 0, 0, 54 )T ,
y 00 = (0, 0, 2, 11
)T , and y 000 = (0, 2, 2, 25
)T .
8
8
Suppose that bidder 2’s fixed strategy is to bid either in auction 1 (with probability 12 ) or in auction 2 (with probability 21 ), drawing bids from the uniform U [0; 1]
distribution in either case.
Bidder 1’s types correspond to the following profit functions:
1 1
1 1
π = −b1 ·
− · max{min{1, b2 }, 0} − b2 ·
− · max{min{1, b1 }, 0}
2 2
2 2
max{min{1, b1 }, 0} + max{min{1, b2 }, 0}
5
+ ( − b1 − b2 ) ·
,
4 2
1 1
1 1
00
π = −b1 ·
− · max{min{1, b2 }, 0} + (2 − b2 ) ·
− · max{min{1, b1 }, 0}
2 2
2 2
11
max{min{1, b1 }, 0} + max{min{1, b2 }, 0}
+ ( − b1 − b 2 ) ·
8
2 1
1
1 1
000
π = (2 − b1 ) ·
− · max{min{1, b2 }, 0} + (2 − b2 ) ·
− · max{min{1, b1 }, 0}
2 2
2 2
25
max{min{1, b1 }, 0} + max{min{1, b2 }, 0}
+ ( − b1 − b 2 ) ·
8
2
0
14
T
3 T
1 1 T
, and b000 = 16
, 16 , respecyielding best response bids b0 = 18 , 18 , b00 = 0, 16
tively. Ignoring the first component, which corresponds to the case of winning no
auctions, we see that y 000 is strictly greater than y 0 . Thus a strict increase in type
(from y 0 to y 000 ) can generate a strict decrease in i’s best response bid (from b0 to b000 ).
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 0 to y 000 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 0 and y 00 be any elements of Yi such that vl00 > vl0 , vk00 = vk0 for k 6= l,
and k 00 = k 0 . For any rival strategy profile σ−i , if b0 is a best response to σ−i at y 0
and b00 is a best response to σ−i at y 00 , then b00l ≥ b0l .
Proof. Setting k = k 0 = k 00 , we have y 0 = Ωv 0 + k and y 00 = Ωv 00 + k. Hence applying
Lemma 1,
(y 00 − y 0 ) · [P (b00 ; σ−i ) − P (b0 ; σ−i ] = (Ωv 00 − Ωv 0 )T [P (b00 ; σ−i ) − P (b0 ; σ−i )]
= (vl00 − vl0 ) [Γl (b00l ; σ−i ) − Γl (b0l ; σ−i )] ≥ 0.
Thus vl00 > vl0 implies Γl (b00l ; σ−i ) ≥ Γl (b0l ; σ−i ). If Γl (b00l ; σ−i ) > Γl (b0l ; σ−i ), then
we must have b00l > b0l since Γl (·; σ−i ) is weakly increasing in bl . Alternatively, if
Γl (b00l ; σ−i ) = Γl (b0l ; σ−i ), then either b00l = b0l or b00l < b0l and Γl (·; σ−i ) is flat on
an open interval below b0l . But in the latter case bidder i could slightly reduce b0l
without changing the probability distribution over objects won, which contradicts
the definition of best response. Hence we must have b00l ≥ b0l .
Monotone cross-auction effects When does an increase in vl increase bestresponse bids in other auctions? The answer turns out to depend on a strong condi15
tion which we call supermodular complementarities:
Definition 1 (Supermodular complementarities). For any Yi ∈ Yi and any outcomes
ω1 , ω2 , Yiω1 ∧ω2 + Yiω1 ∨ω2 ≥ Yiω1 + Yiω2 , 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 standalone valuation vector corresponding to Yi .
• Superadditive complementarities: For all Yi ∈ Yi and any outcomes ω1 , ω2 , ω3
such that ω1 = ω2 + ω3 , Yiω1 ≥ Yiω2 + Yiω3 .
While these conditions both reflect cases where bidder i wants to win objects “together”, 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 0 and
y 00 be any elements of Yi such that vl00 > vl0 , vk00 = vk0 for k 6= l, and k 00 = k 0 . For any
rival strategy profile σ−i , if b0 is a best response to σ−i at y 0 and b00 is a best response
to σ−i at y 00 , then b00 ≥ b0 .
Proof. See Appendix.
Two further comments are worth noting here. First, we have also explored conditions under which an increase in vl would necessarily lead to a decrease in bk for
11
For instance, a bidder could derive a positive complementarity from winning two objects together but no additional complementarity from winning three. This structure would be consistent
with superadditive complementarities, but an increase in (say) v3 could lead bidder i to shift from
bidding 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 submodular complementarities, defined as above but with Yiω1 ∧ω2 + Yiω1 ∨ω2 ≤ Yiω1 + Yiω2 .
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 simultaneous 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 strategies 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 = ×i Bi is
discrete, then there exists a pure strategy Bayesian Nash Equilibrium of the simultaneous 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
L
B, and let the outcome correspondence P : B → ∆2 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 equilibrium with endogenous tiebreaking in the communication extension of the simultaneous
∗
)
first-price auction game: that is, a profile of distributional strategies σ ∗ = (σ1∗ , ..., σN
∗
2L
and a tiebreaking rule p : S × B → ∆ 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 firstprice 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 underlying 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 econometrician 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 characteristics Wt taken to affect combinatoric valuations but not standalone valuations;
we formalize this restriction in Assumption 4 below. In a highway procurement context, Xt would include factors like project size, project location, and type of work
in each project, whereas Wt could include distance between projects, interaction between 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 empirical obstacle: whereas in general the model could involve up to 2Lit − 1 unknown
combinatoric 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
12
While we do not model entry formally here, our analysis can be readily extended to incorporate
endogenous 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 interpretation 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 applications 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 invariant 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, Perrigne 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 equilibrium with endogenous tiebreaking in the JSSZ communication extension to the simultaneous first-price auction game. Furthermore, for any t, t0 such that (Zt , Wt , Xt ) =
(Zt0 , Wt0 , Xt0 ), the equilibrium played at t is the same as the equilibrium played at t0 .
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 literature; 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. Assumption 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 standard 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 valuations Fi (·|Zit , Xt ) and the complementarity function κi (Zit , Wt , Xt ) for each bidder
13
In 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 hypothesis
of zero complementarities, however, these will not help to rationalize the cross-auction bidding
patterns noted below. We therefore focus on exogenous participation as a baseline, leaving detailed
analysis 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 correlation between elements of Vi , but assume vectors Vi , Vj are drawn independently
across bidders; we relax this structure to accommodate auction-level unobserved heterogeneity 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.
L
Inverse Bid Function Let P−i (·|Zt , Wt , Xt ) : Bi → ∆2it be the probability distribution over outcomes facing bidder i at observables (Zt , Wt , Xt ) taking rival strategies as given, and Γ−i (·|Zt , Wt , Xt ) ≡ ΩT P−i (·|Zt , Wt , Xt ) be the Lit × 1 vector of
marginal win probabilities corresponding to P−i (·|Zt , Wt , Xt ). Note that identification 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:
max{(vi − b) · Γ−i (b|Zt , Wt , Xt ) + P−i (b|Wt , Zt , Xt )T Ki }.
b∈Bi
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:
∇b Γ−i (b∗ |Zt , Wt , Xt )(vi − b∗ ) = Γ−i (b∗ |Zt , Wt , Xt ) − ∇b P−i (b∗ |Wt , Zt , Xt )T Ki . (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.
L
Let Kit denote the following (2Lit − Lit − 1)-dimensional subspace of R2 it :
Lit
Kit = {k ∈ R2
: 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 0 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 ∇b Γ−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 ṽ ∈ RLit satisfying
the first-order system (2) at bi given (K; Z, W, X). This ṽ can be expressed in terms
of bi via the inverse bidding function
ṽ = ξi (bi |K; Z, W, X),
where ξi (·|·; Z, W, X) : Bit × Kit → RLit is defined by
ξi (b|K; Z, W, X) ≡ b + [∇b Γ−i (b|Z, W, X)]−1
× Γ−i (b|Z, W, X) − ∇b P−i (b|Z, W, X)T K ,
23
(3)
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 vector K ∈ Kit , ξi (·|K; Z, W, X)
becomes an identified map from bi to the unique vi consistent with bi under the hypothesis κi (Z, W, X) = K:
vi = ξi (bi |K; Z, W, X) if κi (Z, W, X) = K.
14
(4)
Obviously, imposing sufficient conditions for bi to be a best response – by, for instance, requiring
second-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 F̃i (·|K; Z, W, X) for the unknown c.d.f. Fi (·|Z, W, X):
Z
1[ξi (Bi |K; Z, W, X) ≤ v] Gi (dBi |Z, W, X).
F̃i (v|K; Z, W, X) =
(5)
Bi
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 ):
F̃i (·|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
0
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
0
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
0
structure Z−i to competition structure Z−i
matters for i’s behavior in Auction 1,
0
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
Z
Bi + ∇b Γ−i (Bi |Z, W, X)−1 Γ−i (Bi |Z, W, X) Gi (dBi |Z, W, X)
Υi (Z, W, X) =
Bi
and Ψi (Z, W, X) is an identified Lit × 2Lit matrix defined by
Z
Ψi (Z, W, X) =
∇b Γ−i (Bi |Z, W, X)−1 ∇b P−i (Bi |Z, W, X)T Gi (dBi |Z, W, X).
Bi
Furthermore, by Equation (6) and invariance of Fi (·|Zi , X) in Z−i we must have for
0
:
any Z−i , Z−i
0
EBi [ξ(Bi |κ0 ; Zi , Z−i , W, X)|Z, W, X] = EBi [ξ(Bi |κ0 ; Zi , Z−i
, W, X)|Z 0 , 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 0 , W, X)) − (Ψi (Z, W, X) − Ψi (Z 0 , W, X)) · κ0 = 0.
(9)
0
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
0
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


Ψi (Zi , Z−i,1 , W, X) − Ψi (Zi , Z−i,0 , W, X)


..

MΨ ≡ 
.


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 equations in Proposition 4 merely provides a simple and directly verifiable sufficient condition 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 : intuitively, 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 nonparametric 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) = C̃i (Zi , W, X, θ0i ),
(10)
where C̃i (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 0 , W 0 , X) with Zi = Zi0 , Equation (8) reduces to the linear-in-parameters
form
(Υi − Υ0i ) − (Ψi Ci − Ψ0i C0i ) · θ0i = 0,
27
where Υi , Ψi , Ci are identified functions of (Z, W, X) and Υ0i , Ψ0i , C0i are identified
functions of (Z 0 , W 0 , X). Given an appropriate collection of
0
, Wj0 , Xj )}Jj=1 ,
{(Zi,j , Z−i,j , Wj , Xj ), (Zi,j , Z−i,j
we can then express θ0i as the solution to the following L2 -minimization problem
min
θ∈Θi
J
X
Υij − Υ0ij − (Ψij Cij − Ψ0ij C0ij ) · θ
T
Υij − Υ0ij − (Ψij Cij − Ψ0ij C0ij ) · θ ,
j=1
(11)
with identification of θ0i implied by a standard rank condition on the difference
(Ψij Cij − Ψ0ij C0ij ) across j. Note that given {Υij , Υ0ij , Ψij , Ψ0ij }Jj=1 , the problem of
finding θ0i reduces to intercept-free least squares of differences (Υij − Υ0ij ) on differences (Ψij Cij − Ψ0ij C0ij ) across j.
The L2 -minimization problem described above can be written in an analogous
0
integral form if we consider all possible (Zi,j , Z−i,j , Wj , Xj ), (Zi,j , Z−i,j
, Wj0 , Xj ) .
Namely, the L2 criterion would have the form
Z
Z
Z
T
(Υi − Υ0i − (Ψi Ci − Ψ0i C0i ) · θ) (Υi − Υ0i − (Ψi Ci − Ψ0i C0i ) · θ)
0 ,W 0 )
(Zi ,X) (Z−i ,W ) (Z−i
0
dFZ−i W (Z−i
, W 0 |Zi , X) dFZ−i W (Z−i , W |Zi , X)dFZi X (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 under Assumptions 3-6, identification of κi (Zi , W, X) for any (Zi , W, X) implies nonparametric identification of Fi (·|Zi , X) at (Zi , X):
Fi (v|Zi , X) ≡ F̃i (v|κi (Zi , W, X); Z, W, X) for all (Zi , W, X),
with F̃i (·|·; 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 frequently studied procurement markets: for highway construction, snow clearing, recycling, 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 maintenance contracts. As common in similar procurement contexts, MDOT allocates
contracts for a wide range of highway construction and maintenance services via lowprice 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, subcontracting 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 behavior. Table 2 surveys the auction side of the marketplace. The first key feature
15
For a small number of contracts MDOT records are incomplete.
30
Table 1: Summary of Projects by Type
Contract Type
New Construction
Preventive Maintenance
Resurfacing
Road Reconstruction
Road Rehabilitation
Roadside Facilities
Safety
Traffic Operations
Bridge Construction
Bridge Reconstruction
Bridge Rehabilitation
Miscellaneous
Frequency
0.46
12.00
14.99
9.07
7.70
6.42
11.28
6.12
0.36
11.37
3.58
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 “supplements” 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 internal “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 measured 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 uncertainty 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
Auctions per Round
Bidders per Auction
Large Bidders per Auction
Medium Bidders per Auction
Fringe Bidders per Auction
Log Engineer’s Estimate
Money Left on the Table
Project Duration (in days)
Fraction Sub-Contracted
Sub-Contractors per Project
Mean
33.279
4.893
0.900
1.563
2.520
14.296
0.074
119.808
0.153
2.985
St. Dev.
21.776
3.109
0.961
1.709
2.476
1.543
0.084
147.801
0.198
4.079
Min
1.000
1.000
0.000
0.000
0.000
9.482
0.000
6.000
0.000
0.000
Max
83.000
19.000
5.000
8.000
16.000
18.923
0.931
910.000
0.917
49.000
Table 3: Bidder Level Summary Statistics
Bids by Round
Bids by Round if Large
Bids by Round if Small
Participation Rate
Participation Rate if Large
Participation Rate if Small
Mean
2.65
4.873
3.161
0.010
0.060
0.060
St. Dev.
2.45
1.283
0.583
0.023
0.068
0.034
Min
1.000
1.000
1.000
0.000
0.024
0.006
Max
26.000
26.000
17.000
0.242
0.242
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
0.45
0.4
Fraction of Bidders in Sample
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
Number of Simultaneous Bids
20
25
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 crossauction 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 determining 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 bidderproject 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 engineer’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 consider 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 coefficient 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 estimates 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 priors, 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 structural estimation of the complementarity vector κ(·). In principle, the results in
Section 3.2 support fully non-parametric estimation. In practice, of course, the dimensionality of the problem renders this infeasible. We therefore implement estimation 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)
log of engineer’s estimate
number of rivals in the auction
log of minimum distance to
the county centroid
average number of rivals across
all other auctions played
log of average engineer’s estimate
in all other auctions played
log of average engineer’s estimate
in all other same type auctions
played
L=2
L=2
L = {2, 3}
L≥2
0.983***
(0.00364)
-0.00577***
(0.00113)
0.983***
(0.00363)
-0.00574***
(0.00113)
0.986***
(0.00253)
-0.00582***
(0.000777)
0.980***
(0.00156)
-0.00591***
(0.000488)
0.00959**
(0.00373)
0.00960**
(0.00373)
0.0127***
(0.00254)
0.0167***
(0.00168)
0.00229**
(0.00109)
0.00224**
(0.00109)
0.00294***
(0.000823)
0.00236***
(0.000621)
0.0156***
(0.00267)
0.0192*
(0.0104)
0.0152***
(0.00234)
0.0162***
(0.00173)
-0.00373***
(0.00109)
-0.00373***
(0.00109)
-0.00372***
(0.000844)
-0.00312***
(0.000427)
(log of average engineer’s estimate
in all other auctions played)2
log of minimum distance to
the nearest other project
Constant
Observations
R-squared
-0.000310
(0.000804)
-0.00894**
(0.00367)
0.0407
(0.0398)
-0.00906**
(0.00369)
0.0302
(0.0501)
-0.0106***
(0.00301)
0.0226
(0.0294)
-0.00702***
(0.00164)
0.0511**
(0.0242)
4,193
0.974
4,193
0.974
7,946
0.972
22,359
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 to
the county centroid measure size, strength of competition, and distance to project l respectively. Remaining variables proxy for cross-auction characteristics: average number of rivals, average engineer’s
estimate, 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 twoauction 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 approximation 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:
µ
µilt = βDilt
,
µ
is a subset of (Zt , Wt , Xt ) which includes the following elements: log engiwhere Dilt
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 σl2 (Zt , Wt , Xt ) of bilt , the other for correlations ρkl (Zt , Wt , Xt ) between bikt
and bilt . In particular, for variance terms σl2 (Zt , Wt , Xt ) we specify
σ
σl2 (Zt , Wt , Xt ) = exp(αDilt
),
and for correlation terms ρkl (Zt , Wt , Xt ) we specify
ρkl (Zt , Wt , Xt ) =
exp(γDitρkl − 1)
exp(γDitρkl + 1)
σ
where Dilt
and Ditρkl are known transformations of (Zt , Wt , Xt ). In our baseline specσ
ification, Dilt
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, Ditρkl 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 specification for κ(·): for each outcome ω involving at least two projects,
κω (Z, W, X) = θ01 + θ02 · ln sum eng ω ,
(12)
where ln sum eng ω 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 implement estimation of θ0 based on L2 -type criterion (11) derived in Section 3.2. If
all Γj , Γ0j , Ψj , Ψ0j , Cj , C0j in the criterion (11) were known or their consistent estimators were available, we could immediately obtain a θ̂ 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
0
{(Zi,j , Z−i,j , Wj , Xj ), (Zi,j , Z−i,j
, Wj0 , Xj )}Jj=1 at which to evaluate (11), accounting
(1,1)
in choice of these
for the deterministic link between X1,j , X2,j and ln sum engj
counterfactual quadruples, and simulation of the functions Υj , Υ0j , Ψj , Ψ0j at the counterfactual quadruples chosen. We address these implementation issues as follows.
0
, Wj0 , Xj )}Jj=1
First consider choice of the set of counterfactuals {(Zi,j , Z−i,j , Wj , Xj ), (Zi,j , Z−i,j
at which to evaluate the criterion (11). In principle, any choice of such that “regressors” Ψj Cj − Ψ0j C0j in (11) satisfy a standard rank condition will be sufficient to
construct an estimator for θ0 . In practice, however, we expect selections approximating 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 employ 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 (Zj0 , Wj0 , Xj0 )
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 (Zj0 , Wj0 , Xj0 ) 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
0
in auction l. Hence shifting from (Zj , Wj , Xj ) to (Zj0 , Wj0 , Xj0 ), where X1,j = X1,j
but
0
X2,j and X2,j can be different, will give us the following equation:
(Υ1,j − Υ01,j ) − (Ψ1,j Cj − Ψ01,j C0j ) · 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 estimation criterion based on invariance of standalone valuations in Auction 1 only:
θ0 = arg min
θ
Mj J X
X
2
(m )
(m ) (m )
Υ1,j − Υ1,j j − Ψ1,j Cj − Ψ1,j j Cj j · θ .
(13)
j=1 mj =1
In general, when L is greater than 2, shifting from (Zj , Wj , Xj ) to (Zj0 , Wj0 , Xj0 ), where for some
0
l0 we have Xl,j = Xl,j
for l 6= l0 but Xl0 ,j and Xl00 ,j can be different, gives us the following system
of (L − 1) equations:
16
(Υ−l0 ,j − Υ0−l0 ,j ) − (Ψ−l0 ,j Cj − Ψ0−l0 ,j C0j ) · θ0 = 0,
where Υ−l0 ,j denotes the (L − 1) × 1 vector obtained from the L × 1 vector Υj by eliminating the
l0 -th component Υl0 ,j , and Ψ−l0 ,j denotes the (L − 1) × 2L matrix obtained from the L × 2L matrix
Ψj by eliminating the l0 -th row.
41
(m )
(m )
(m )
we consider Mj counterfactual points Zj j , Wj j , Xj j
the properties described above for (Zj0 , Wj0 , Xj0 ). The sim-
Here, for each (Zj , Wj , Xj )
,
mj = 1, . . . , Mj , that have
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 )
(m )
(m )
(m )
(m )
(m )
and (Υj j , Ψj j ) for each (Zj , Wj , Xj ) and Zj j , Wj j , Xj 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 {bri }R
r=1 from the joint distribution
Ĝi (·|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 (bri |Z, W, X),
and ∇Pi (bri |Z, W, X) based on our first-step estimates for the bid distributions Ĝ1 (·|Z, W, X),
. . . , ĜNt (·|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 {bri }R
r=1 :
R
1X r
Υ̂ =
bi + ∇Γi (bri |Z, W, X)−1 Γi (bri |Z, W, X);
R r=1
R
1X
Ψ̂ =
∇Γi (bri |Z, W, X)−1 ∇Pi (bri |Z, W, X)T .
R r=1
(m )
(m )
(m )
Repeating this procedure for each set of pairs (Zj , Wj , Xj ) and Zj j , Wj j , Xj j ,
mj = 1, . . . , Mj , appearing in (13), we ultimately obtain the simulated L2 -type
criterion
Mj J X
2
X
(mj )
(mj ) (mj )
Υ̂1,j − Υ̂1,j − Ψ̂1,j Cj − Ψ̂1,j Cj
·θ .
j=1 mj =1
The minimization of this criterion with respect to θ will yield a closed-form estimate
θ̂ for θ0 expressed by the formula for the OLS estimator. Practically, however, firststep 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
17
More 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 estimation 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 interpreted as follows.
The first panel of Table 5 reports estimates β̂ 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 dispersed 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 involves
division by g−i (·), these in turn produce artificially large pseudo-values for virtually all choices of
θ.
43
Table 5: First-Step MLE Estimates of Gi
Mean µl
β̂
MLE SEs
95% CI
0.00894
0.00101
0.00231
0.00161
0.00220
0.02155
0.18527 0.22030
0.97881 0.98277
-0.03268 -0.02364
0.00826 0.01458
-0.00938 -0.00075
-0.19198 -0.10749
α̂
MLE SEs
95% CI
-1.03484
-0.19618
-0.26642
-0.02824
-0.10595
0.05439
0.00619
0.01848
0.00305
0.01309
γ̂
MLE SEs
95% CI
0.08874
0.03110
0.02260
0.53391
0.01542
0.50991 0.85777
0.24466 0.36658
0.14784 0.23643
-2.42973 -0.33682
-0.09286 -0.03241
Constant 0.20278
Log engineer’s estimate 0.98079
Log rivals in auction -0.02816
Log sum engineer’s (across l) 0.01142
Log sum rivals (across l) -0.00506
Multiple auction -0.14973
Variance σl
Constant
Log engineer’s estimate
Log rivals in auction
Log sum engineer’s (across l)
Log sum rivals (across l)
Covariance ρkl
Constant 0.68384
Same county 0.30562
Same type 0.19214
Product of log engineer’s -1.38328
Product of log rivals -0.06264
-1.14144
-0.20831
-0.30265
-0.03422
-0.13160
-0.92823
-0.18404
-0.23020
-0.02225
-0.08029
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 aggressively, 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 pa44
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 (Zj0 , Wj0 , Xj0 ).
Table 6 reports estimates θ̂ 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:
max(b − vi )Γi (b|Z, W, X) + Pi (b|Z, W, X)T κ(Z, W, X),
b
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 combination 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-Auction
Subsample
θ̂
SE
Constant 110.29 90.14
Sum 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, positive
κ means higher cost. “Sum of EE” is sum
of engineer’s estimates. Mean sum of EE
is 2300, mean bid is 1180. Standard errors
are not yet corrected for first-step and simulation error, hence are underestimated.
Taken together, these numbers highlight both the potential importance of combinatorial 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 economic 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
Conclusion
Motivated by an institutional framework common in procurement applications, we
develop and estimate a structural model of bidding in simultaneous first-price auctions, 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
46
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 substantial 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 explicitly address dynamics in our structural bidding model. Under some simplifying
assumptions, however, our results have a formal dynamic interpretation. In particular, 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
complementarities associated with winning each combination of auctions. In principle, 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 implications 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.
18
Obviously, for predicting evolution of behavior across rounds a precise decomposition of κ(·)
would be of first importance.
47
Appendix A: Complementarities depending on V
In this appendix, we explore prospects for generalizing the conclusions in Sections 2 and 3
above to the case where complementarities are additively separable and/or affine functions
of standalone valuations. Such a case could arise if, for instance, winning two auctions
together 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 say
complementarities are additively separable in v if for each ω that contains at least two
non-zero components (that is, ω T ω ≥ 2), the complementarity function is a function of the
vector of standalone valuations v = (v1 , v2 , . . . , vL )T such that
X
K ω (v) =
φl (vl ) + K̄ ω
(14)
l: ω T el =1
for some functions φl , l = 1, . . . , L. If each function φl (·) is linear in its argument vl , then
we obtain the special case of complementarities affine in v:
X
K ω (v) =
δ l vl + K̄ ω , if ω T ω ≥ 2.
(15)
l: ω T el =1
As usual, if ω contains at most one component equal to one (that is, ω T ω ≤ 1), then we
set K ω (v) ≡ 0.
An important special case of (15) is when all δ l are identical and K̄ ω = 0 for any ω.
This case describes the situation of a constant relative complementarity – that is, when
K ω (v) is a constant ratio of the additive valuation.
Now assume that complementarities are affine in v, and define an L × 1 vector δ and
an L × L matrix D(δ) as follows:
δ ≡ (δ 1 , δ 2 , . . . , δ L )T
D(δ) ≡ diag(δ 1 , δ 2 , . . . , δ L ).
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 other
elements of A are 0. We then have
K(v) = AΩD(δ)v + K̄,
where K̄ denotes the 2L × 1 vector of constant components K̄ ω in the functions of complementarities. Clearly, the first L + 1 elements in K̄ are zero since they correspond to the
cases of ω such that ω T ω ≤ 1.
48
Monotonicity of best responses Analysis of own- and cross-auction monotonicity
becomes much more complex when we try to extend the results of Lemma 2 to more general
forms of complementarities.
Consider the case of affine complementarities. Suppose that v 00 and v 0 differ only in
the l’s component, and that the collection {K̄ ω } is fixed. Then, using techniques similar
to the ones in the proof of Lemma 2 we can show that for any rival strategy profile σ−i , if
b0 is a best response to σ−i at v 0 and b00 is a best response to σ−i at v 00 , then
(1 + δ l )(vl00 − vl0 ) Γl (b00l ; σ−i ) − Γl (b0l ; σ−i ) ≥ δ l (vl00 − vl0 ) P el (b00 ; σ−i ) − P el (b0 ; σ−i ) .
The right-hand side of this inequality contains the probability of winning auction l only
and, thus, depends on the whole vectors of best responses b0 and b00 rather than just their
l’s components. Not much can be inferred from this inequality, even though some results
can still be obtained. Suppose that vl00 > vl0 . If −1 < δ l < 0 and if the best response in
auction l decreases – that is, b00l < b0l , – then at least for one other component m 6= l the
best response decreases too – that is, b00m < b0m . The inequality can be rewritten as
X
P ω (b00 ; σ−i ) − P ω (b0 ; σ−i ) .
Γl (b00l ; σ−i ) − Γl (b0l ; σ−i ) ≥ −δ l
ω: ω T el =1, ω T ω≥2
Alternatively, suppose that δ l > 0. Then b00l < b0l implies that at least for one other
component m 6= l it holds that b00m > b0m .
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 can
produce a strict decrease in bl .
Example 2. Consider the case of two bidders and two auctions. For bidder 1,
1
7
11
K (1,1) = − v1 − v2 + ,
8
6
4
T
00
and (v1 , v2 )T ∈ {(0, 0)T , (0, 2)T }. That is, bidder 1 has types y 0 = (0, 0, 0, 11
4 ) and y =
29 T
(0, 0, 2, 12 ) .
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.
49
Bidder 1’s types correspond to the following profit functions:
1 1
1 1
0
π = −b1 ·
− · max{min{1, b2 }, 0} − b2 ·
− · max{min{1, b1 }, 0}
2 2
2 2
max{min{1, b1 }, 0} + max{min{1, b2 }, 0}
11
+
− b1 − b2 ·
,
4
2
1 1
1 1
00
− · max{min{1, b2 }, 0} + (2 − b2 ) ·
− · max{min{1, b1 }, 0}
π = −b1 ·
2 2
2 2
max{min{1, b1 }, 0} + max{min{1, b2 }, 0}
29
+
− b1 − b2 ·
12
2
T
17 T
yielding best response bids b0 = 78 , 78 and b00 = 0, 24
. Thus, even though v2 increases
and v1 remains fixed, the best response in auction 2 decreases. Notice that y 00 is not greater
than y 0 (in the component-wise sense) as the last component in y 00 is strictly smaller than
that in y 0 . Such a situation was impossible in the case of constant complementarities
because there an increase in one standalone valuation vl weakly increased the 2L -vector of
valuations y and, moreover, for the components of y that increased strictly the changes
were identical and equal to (vl00 − vl0 ). Now the change is equal to (vl00 − vl0 ) for ω = el and
is equal to (1 + δ l )(vl00 − vl0 ) for any ω such that ω T el = 1 and ω T ω ≥ 2. These changes will
even be of different signs if δ l < −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 if
and only if it is satisfied for their constant components K̄ ω .
Lemma 4. Let complementarities be defined as in (14). Then for any outcomes ω1 , ω2 ,
Yiω1 ∧ω2 + Yiω1 ∨ω2 − Yiω1 − Yiω2 = K̄ ω1 ∧ω2 + K̄ ω1 ∨ω2 − K̄ ω1 − K̄ ω2 .
(16)
Hence, complementarities are supermodular if and only if
K̄ ω1 ∧ω2 + K̄ ω1 ∨ω2 ≥ K̄ ω1 + K̄ ω2 .
Proof. See Appendix B.
Equation (16) and the fact that its the right-hand side does not depend on standalone
valuations allows us to generalize the result of Lemma 3 to the case of additively separable
complementarity functions.
Lemma 5. Suppose complementarities are defined as in (14) and are supermodular. Suppose that the vectors v 00 and v 0 of standalone valuations are such that vl00 > vl0 , vk00 = vk0 for
k 6= l. For any rival strategy profile σ−i , if b0 is a best response to σ−i at the corresponding
y 0 and b00 is a best response to σ−i at the corresponding y 00 , then b00 ≥ b0 .
50
Non-parametric identification Our next step is to generalize the non-parametric
identification result above to the case when complementarities are affine functions of standalone valuations. Namely, for a given subset ω containing at least two elements,
X
δ l (Zi , W, X)vi,l + K̄ ω (Zi , W, X).
K ω = K ω (vi , Zi , W, X) =
l: ω T el =1
When ω T ω ≤ 1, the complementarity is set to 0. As can be seen, the functional form
of complementarities does not depend on Z−i . As we show below, under weak conditions
there is enough variation in Z−i | Zi , W, X to determine the linear (in vi,l ) part of complementarities as well as the constant part.
Define the Li × 1 vector δ(Zi , W, X) as
δ(Zi , W, X) = (δ 1 (Zi , W, X), δ 2 (Zi , W, X), . . . , δ Li (Zi , W, X))T ,
and the D(δ(Zi , W, X)) as the Li × Li matrix
D(δ(Zi , W, X)) = diag(δ 1 (Zi , W, X), δ 2 (Zi , W, X), . . . , δ Li (Zi , W, X)).
Then
K(vi , Zi , X) = Ai Ωi D(δ(Zi , W, X))vi + K̄(Zi , W, X),
where K̄(Zi , W, X) denotes the 2Li × 1 vector of constant components in the complementarities (obviously, K̄(Zi , W, X) ∈ Ki ). Matrix Ai denotes the 2Li × 2Li matrix such that
its (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 of
matrix Ai Ωi is equal to Li .
Using the first-order condition and taking into account the form of K(vi , Zi , W, X),
obtain
vi = bi + [∇b Γ−i (bi |Z, W, X)]−1 Γ−i (bi |Z, W, X)
− [∇b Γ−i (bi |Z, W, X)]−1 ∇b P−i (bi |Z, W, X)T Ai Ωi D(δ(Zi , W, X))vi + K̄(Zi , W, X) ,
where, as before, Z = (Zi , Z−i ). Denote
Π(bi , δ, Z, W, X) = ILi + [∇b Γ−i (bi |Z, W, X)]−1 ∇b P−i (bi |Z, W, X)T Ai Ωi D(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 .
51
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)−1 bi
+ Π(bi , δ(Zi , W, X), Z, W, X)−1 [∇b Γ−i (bi |Z, W, X)]−1 Γ−i (bi |Z, W, X)
− Π(bi , δ(Zi , W, X), Z, W, X)−1 [∇b Γ−i (bi |Z, W, X)]−1 ∇b P−i (bi |Z, W, X)T K̄(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)−1 Bi Z, W, X
h
i
+ EBi Π(Bi , δ, Z, W, X)−1 [∇b Γ−i (Bi |Z, W, X)]−1 Γ−i (Bi |Z, W, X)Z, W, X ,
h
i
D2 (δ, Zi , Z−i , W, X) = EBi Π(Bi , δ, Z, W, X)−1 [∇b Γ−i (Bi |Z, W, X)]−1 ∇b P−i (Bi |Z, W, X)T Z, W, X ,
0 from the support of Z |Z , W, X,
Keeping Zi , W, X fixed, let us draw another value Z−i
−i i
0 ). Due to the assumptions made on the distribution of the
and denote Z 0 = (Zi , Z−i
standalone valuations, E[Vi |Z, W, X] = E[Vi |Z 0 , W, X]. Therefore, for δ = δ(Zi , W, X) ∈
0 , W, X),
∆(Zi , Z−i , W, X) ∩ ∆(Zi , Z−i
0
D1 (δ(Zi , W, X), Zi , Z−i
, W, X) − D1 (δ(Zi , W, X), Zi , Z−i , W, X) =
0
D2 (δ(Zi , W, X), Zi , Z−i , W, X) − D2 (δ(Zi , W, X), Zi , Z−i , W, X) K̄(Zi , W, X).
For fixed Zi , W, X, this system has 2Li − 1 unknowns (Li in δ(Zi , W, X) and 2Li − Li − 1
in K̄(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
T
unique δ ∈ Jj=0 ∆(Zi , Z−i,j , W, X) and a unique κ ∈ Ki that solve the system of J · Li
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 K̄(Zi , W, X) are identified, and thus, the complementarity function is identified for these values of Zi , W , X.
T
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 possible
TJ
to have a situation when for different δ1 and δ2 , where δ1 , δ2 ∈ j=0 ∆(Zi , Z−i,j , W, X),
system (17) has solutions κ1 ∈ Ki and κ2 ∈ Ki , respectively. Thus, in this sense the
question of identification of δ(Zi , W, X) and K̄(Zi , W, X) comes down to the question of
existence of solutions to systems of linear equations: (17) can have a solution κ for one
52
δ only, and for that δ it has to be unique. Using the Kronecker-Capelli theorem, which
gives the necessary and sufficient conditions for the existence of a solution to a system of
linear equations, and also the necessary and sufficient conditions for the uniqueness of such
a 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 such
that its (2Li − Li − 1) × (2Li − Li − 1) submatrix (ẽij )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 the
elements in the first Li + 1 rows) are equal to zero. For every κ ∈ Ki there is a unique
L
κ̌ ∈ R2 i −Li −1 such that
κ = Ei κ̌.
Obviously, this κ̌ is formed by the last 2Li − Li − 1 values in κ. System (17) can be
equivalently 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 ) κ̌,
j = 1, . . . , J,
(18)
L
with κ̌ ∈ R2 i −Li −1 . For a fixed δ, system (17) is linear in κ, has the J · Li × 2Li matrix
of coefficients, and imposes restrictions on the solution κ by requiring that κ ∈ Ki . Its
equivalent representation (18) is linear in κ̌ for a fixed δ, has the J · Li × (2Li − Li − 1)
L
matrix of coefficients, and does not impose any restrictions on the solution κ̌ ∈ R2 i −Li −1 .
This allows us to apply the Kronecker-Capelli theorem to system (18) in a straightforward
way.
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
T
unique δ ∈ Jj=0 ∆(Zi , Z−i,j , W, X) that satisfies the following two conditions:
1. First,
rank ([M1 (δ, Zi , W, X) | M2 (δ, Zi , W, X)]) = rank (M2 (δ, Zi , W, X)) ,
where M2 (δ, Zi , W, X) denotes the J · Li × (2Li − Li − 1) matrix


(D2 (δ, Zi , Z−i,1 , W, X) − D2 (δ, Zi , Z−i,0 , W, X)) Ei


..
M2 (δ, Zi , W, X) ≡ 
,
.
(D2 (δ, Zi , Z−i,J , W, X) − D2 (δ, Zi , Z−i,0 , W, X)) Ei
53
(19)
and M1 (δ, Zi , W, X) denotes the J · Li × 1 vector


D1 (δ, Zi , Z−i,1 , W, X) − D1 (δ, Zi , Z−i,0 , W, X)


..
M1 (δ, Zi , 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 K̄(Zi , W, X) are identified, and thus, the complementarity function is identified for these values of Zi , W , X.
Condition (19) requires that in system (18), the rank of the matrix of coefficients
M2 (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 κ̌
for that δ only. Condition (20) then guarantees this κ̌ is determined uniquely, and, thus,
κ = Ei κ̌ 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 K̄(Zi , W, X) by solving, e.g., the following optimization problem:
min
T
2Li −Li −1
δ∈ J
j=0 ∆(Zi ,Z−i,j ,W,X), κ̌∈R
Q(δ, κ̌, Zi , W, X),
where
Q(δ, κ̌, Zi , W, X) ≡ (M1 (δ, Zi , W, X) − M2 (δ, Zi , W, X)κ̌)T (M1 (δ, Zi , W, X) − M2 (δ, Zi , W, X)κ̌) .
Appendix B: Proofs
Proof of Lemma 3. Suppose that K satisfies supermodular complementarities, and let b0 ,b00
l be the (N − 1) × 1 vector of rival bids in auction l,
be any two feasible bids. LetPB−i
l
10
L0
0
B−i = (B−i , ..., B−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 player
i’s expected complementarity given own bid vector b and the complete rival bid vector B−i :
k(b; B−i ) = Eω [K ω |b, B−i ].
The expectation here is over ties; otherwise, bidder i will win each auction in which his bid
is the highest, ω = ω(b, B−i ) will be deterministic, and k(b; B−i ) = K ω(b,B−i ) . Note that
54
K T P (b) is the expectation of k(b; B−i ) with respect to G−i (·):
K T P (b) = Eω [K ω |b]
= EB−i [Eω [K ω |b, B−i ]]
Z
= k(b; B−i ) dG−i (B−i ).
We next show that K T P (b) is supermodular in b. By definition, K T P (b) is supermodular
in b if for all b0 , b00 we have
K T P (b00 ∨ b0 ) + K T p(b00 ∧ b0 ) ≥ K T p(b00 ) + K T p(b0 ),
where b00 ∨ b0 and b00 ∧ b0 denote the meet and join of (b0 , b00 ) respectively.
Obviously, if b00 ≥ b0 , then b00 ∨ b0 = b00 and b00 ∧ b0 = b0 , so the inequality holds trivially
(and likewise if b0 ≥ b00 ). Thus suppose that the vectors are not ordered: i.e. that b00 b0
and b0 b00 . Now rewrite the supermodularity condition in terms of the integrals above:
Z
00
0
Z
k(b ∨ b ; B−i ) dG−i (B−i ) +
k(b00 ∧ b0 ; B−i ) dG−i (B−i )
Z
Z
00
≥ k(b ; B−i ) dG−i (B−i ) + k(b00 ; B−i ) dG−i (B−i ).
Rearranging terms yields the equivalent integral expression
Z
00
k(b ∨ b0 ; B−i ) + k(b00 ∧ b0 ; B−i ) − k(b00 ; B−i ) − k(b0 ; B−i ) dG−i (B−i ) ≥ 0.
(21)
In the case of no ties for (b00 ∨ b0 ; B−i ), (b00 ∧ b0 ; B−i ), (b00 ; B−i ) and (b0 ; B−i ), we have
ω(b00 ∨ b0 ; B−i ) = ω(b0 ; B−i ) ∨ ω(b00 ; B−i ),
ω(b00 ∧ b0 ; B−i ) = ω(b0 ; B−i ) ∧ ω(b00 ; B−i ),
and thus, by the supermodularity of K,
k(b00 ∨ b0 ; B−i ) − k(b00 ; B−i ) − k(b0 ; B−i ) − k(b00 ∧ b0 ; B−i ) ≥ 0.
If for b00 ∨ b0 , b00 ∧ b0 , b00 and b0 , ties occur with probability zero (with respect to the
distribution G−i (·) of B−i ), then the last inequality implies (21). Thus, supermodular K
implies supermodularity of K T P (b) in b.
To complete the argument, note that supermodularity of K T P (b) implies that an increase in any element of b will weakly increase marginal returns to other elements of b. We
know from Lemma 2 that an increase in vl will weakly increase bl , and by supermodularity
55
of K T p(b) this will also increase bids in other auctions. This establishes the claim.
Proof of Lemma 4. Note that
X
Yiω1 ∧ω2 + Yiω1 ∨ω2 =
X
(vl + φl (vl )) + K̄ ω1 ∧ω2 +
l: (ω1 ∨ω2 )T el =1
l: (ω1 ∧ω2 )T el =1
X
=
l: ω1T el =1,
(vl + φl (vl )) + K̄ ω1 ∨ω2
X
(vl + φl (vl )) +
ω2T el =0
l: ω2T el =1,
X
(vl + φl (vl )) + 2
(vl + φl (vl ))
l: (ω1 ∧ω2 )T el =1
ω2T el =0
+ K̄ ω1 ∧ω2 + K̄ ω1 ∨ω2 .
Also,
Yiω1 + Yiω2 =
X
l: ω1T el =1
X
+
X
(vl + φl (vl )) + K̄ ω1 +
l: ω2T el =1
X
(vl + φl (vl )) + 2
l: ω2T el =1, ω1T el =0
X
(vl + φl (vl )) + K̄ ω2 =
(vl + φl (vl ))+
l: ω1T el =1, ω2T el =0
(vl + φl (vl )) + K̄ ω1 + K̄ ω2 .
l: (ω1 ∧ω2 )T el =1
The result follows.
Proof of Lemma 5. Analogously to the proof of Lemma 3, for player i’s given vector of
standalone valuations v, own bid vector b and the complete rival bid vector B−i , consider
the expected complementarity
k(v, b; B−i ) = Eω [K ω (v)|v, b, B−i ].
As in the proof of Lemma 3, the expectation here is over ties. Note that K(v)T P (b) is the
expectation of k(v, b; B−i ) with respect to G−i (·):
Z
K(v)T P (b) = Eω [K ω (v)|v, b] = EB−i [Eω [K ω (v)|v, b, B−i ]] = k(v, b; B−i ) dG−i (B−i ).
Let b0 and b00 be any two feasible bid vectors. Let us establish that K(v)T P (b) is supermodular in b, that is, for a given v,
K(v)T P (b00 ∨ b0 ) + K(v)T p(b00 ∧ b0 ) ≥ K(v)T p(b00 ) + K(v)T p(b0 ).
56
This supermodularity condition van be rewritten as
Z
Z
k(v, b00 ∨ b0 ; B−i ) dG−i (B−i ) + k(v, b00 ∧ b0 ; B−i ) dG−i (B−i )
Z
Z
≥ k(v, b00 ; B−i ) dG−i (B−i ) + k(v, b00 ; B−i ) dG−i (B−i ),
or equivalently, as
Z
k(v, b00 ∨ b0 ; B−i ) + k(v, b00 ∧ b0 ; B−i ) − k(v, b00 ; B−i ) − k(v, b0 ; B−i ) dG−i (B−i ) ≥ 0.
In the case of no ties for
4 we have that
(b00 ∨b0 ; B−i ),
(b00 ∧b0 ; B
−i ),
(b00 ; B
−i )
and
(b0 ; B
(22)
−i ), from Lemma
k(v, b00 ∨ b0 ; B−i ) + k(v, b00 ∧ b0 ; B−i ) − k(v, b00 ; B−i ) − k(v, b0 ; B−i )
= K̄ ω(b
0
00 ∨b0 ;B )
−i
= K̄ ω(b ;B−i )∨ω(b
+ K̄ ω(b
00 ;B )
−i
00 ∧b0 ;B
−i )
0
0
00 ;B )
−i
− K̄ ω(b ;B−i ) − K̄ ω(b
+ K̄ ω(b ;B−i )∧ω(b
00 ;B )
−i
0
− K̄ ω(b ;B−i ) − K̄ ω(b
00 ;B )
−i
≥ 0.
If for b00 ∨ b0 , b00 ∧ b0 , b00 and b0 , ties occur with probability zero (with respect to the
distribution 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, the
first-order system (2) must be well-defined for almost every bi submitted by i, i.e. almost
everywhere with respect to the measure induced by Gi (·|Z, W, X). Second, at almost every
bi at which first order conditions hold, the first-order system (2) must be invertible. We
establish each claim in turn.
First show that the first order system (2) is well-defined for almost every bi submitted
by i. Recall that we can write bidder i’s objective as
π(vi , b|K; Z, W, X) = (Ωvi + K − Ωb)T P−i (b|Z, W, X).
where vi and K are given at the time of maximization. Note that the system (2) necessarily
holds at any best respose where π(vi , ·|K; Z, W, X) is differentiable and that Assumption
5 implies that each observed bi is a best response. Hence the system (2) will be well
defined for almost every bi submitted by i if and only if π(vi , ·|K; Z, W, X) is differentiable
almost everywhere with respect to the measure on Bi induced by Gi (·|Z, W, X). But under
Assumption 6 Gi (·|Z, W, X) is absolutely continuous. To establish the claim, it is thus
sufficient to prove differentiability of π(vi , ·|K; Z, W, X) a.e. with respect to the Lebesgue
measure on Bi .
Clearly (Ωvi + K − Ωb) is differentiable in b for any vi , K ∈ RLit × K. Thus differentiability of π(vi , ·|K; Z, W, X) at b is equivalent to differentiability of P−i (·|Z, W, X) at b. Let
57
B−i be the Li × 1 random vector describing maximum rival bids in the set of auctions in
which i participates. Again applying Assumption 6 to rule out ties, the probability i wins
combination ω at bid b is
P ω (b|Z, W, X) = Pr({∩{l:ωl =1} 0 ≤ B−i,l ≤ bi,l } ∩ {∩{l:ωl =0} bi,l ≤ B−i,l < ∞}).
P
For each ω ∈ ΩiP
, let bω be the ( ω) × 1 sub-vector of b describing i’s bids for objects in
ω be the (
ω, B−i
ω) × 1 sub-vector of B−i describing maximum rival bids for objects in
ω at (Z, W, X). Applying the
ω, and Gω−i (bω |Z, W, X) be the equilibrium joint c.d.f. of B−i
formula for a rectangular probability and simplifying, we can then represent P−i (·|Z, W, X)
in the form
X
0
0
ω
P−i
(b|Z, W, X) =
aωω0 Gω−i (bω |Z, W, X),
ω 0 ∈Ω
where each aωω0 is a known scalar (determined by ω, ω 0 ) taking values in {−1, 0, 1}. But
by absolute continuity each c.d.f. Gω−i (·|Z, W, X) is differentiable a.e. (Lebesgue) in its
0
support, and interpreted as a function from Bi to RLi , each bω is continuously differentiable
0
0
in b. Thus interpreted as a function from Bi to R, each Gω−i (bω |Z, W, X) is differentiable on
0
0
a set of full Lebesgue measure in B−i . The set of points in Bi at which all Gω−i (bω |Z, W, X)
0
0
are differentiable is the intersection of points in Bi at which each Gω−i (bω |Z, W, X) is
differentiable, i.e. the intersection of a finite collection of sets of full Lebesgue measure in
0
Bi . But from above differentiability of Gω−i (b|Z, W, X) for all ω 0 implies differentiability of
ω (b|Z, W, X). Hence P ω (·|Z, W, X) is differentiable on a set of full Lebesgue measure
P−i
−i
in Bi . This in turn implies differentiability of π(vi , ·|K; Z, W, X) a.e. with respect to the
measure 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 ṽ for
almost every bi submitted by i. Let B̃i be the set of points in Bi at which π(·, ·|K; W, Z, X)
is differentiable in b; from above, B̃i is a subset of full Lebesgue measure in Bi . Choosing
any b ∈ B̃i and rearranging (2) yields
∇b Γ−i (b|Z, W, X)ṽ = ∇b Γ−i (b|Z, W, X)b + Γ−i (b|Z, W, X) − ∇b P−i (b|W, Z, X)T Ki .
Hence uniqueness of ṽ is equivalent to invertibility of the Li ×Li matrix ∇b Γ−i (b|Z, W, X).
Recall that Γ−i (b|Z, W, X) is an Li × 1 vector whose lth element describes the probability
that bid vector b wins auction l. Note that b ∈ B̃i rules out ties at b. Thus for b ∈ B̃i the
lth 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 which
it follows that ∇b Γ−i (b|Z, W, X) is a diagonal matrix whose l, lth element is the marginal
l (b|Z, W, X) of B
p.d.f. g−i
−i,l . Hence ∇b Γ−i (b|Z, W, X) will be invertible at b if and only if
l
g−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)
l (·|Z, W, X) = 0 on
for some (v, K). Suppose that there exists an > 0 such that g−i
58
(bil − , bi ]. Then player i could infintesimally reduce bil without affecting either Γ−i
l (·|Z, W, X) > 0
or P−i , a profitable deviation for any (v, K). Hence we must have g−i
almost everywhere (Lebesgue) in the support of Bi . By Assumption 6, this in turn implies
l (·|Z, W, X) > 0 for almost every b submitted by i. Since l was arbitrary, we must have
g−i
i
∇b Γ−i (bi |Z, W, X) invertible for almost every bid bi submitted by i. Hence for almost every
bi submitted by i there will exist a unique ṽ satisfying (2) at bi , given by
ṽ = bi + ∇b Γ−i (bi |Z, W, X)−1 Γ−i (bi |Z, W, X)
+ ∇b Γ−i (bi |Z, W, X)−1 ∇b P−i (bi |W, Z, X)T K.
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 conditional distribution of Vi |Zi , W, X relied on the equations in the first order conditions
obtained from bidder’s optimization of the payoff function. To derive those equations we
employed 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 restrictions are imposed on Gi . Our identification method is based on using inequalities for
bidder’s best responses and employing the exclusion restrictions in Assumption 4 to obtain
bounds 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-variate
distribution 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 functions
on Rp .
Observable are the distribution functions Gi (b|Z, W, X), where Z = (Zi , Z−i ), of the
bids in the equilibrium played by the bidders for any Z−i ∈ Z−i |Z, W, X, i = 1, . . . , N .
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 exclusion
restrictions in Assumption 4. The conditions in the definition of Hi (Zi , W, X) are the
following:
(C1) Fi (·|Zi , W, X) ∈ Fac (RLi ), and the support of Fi (·|Zi , W, X) is a compact, convex
set in RLi (consistent with Assumption 1).
59
(C2) κi (Zi , W, X) ∈ Ki .
(C3) For each Z−i ∈ Z−i |Zi , W, X, bidder’s behavior is consistent with the maximization
of the payoff function
π(vi , b; Z, W, X) = viT Γ−i (b|Z, W, X)−bT Γ−i (b|Z, W, X)+P−i (b|Z, W, X)T κi (Zi , W, X)
with respect to b ∈ Bi . That is, for each Z−i ∈ Z−i |Zi , X, W , every bidder i’s bid
vector bi observed in the equilibrium satisfies the inequality
viT Γ−i (bi |Z−i ) − bTi Γ−i (bi |Z−i ) + P−i (bi |Z−i )T κ(Zi , W, X) ≥
viT Γ−i (b|Z−i ) − bT Γ−i (b|Z−i ) + P−i (b|Z−i )T κ(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 second
component – on the set of κi (Zi , W, X). Let Hi,F (Zi , W, X) stand for the projection
of the identification set onto the first component – on the set of Fi (·|Zi , W, X); and let
Hi,Fl (Zi , W, X) stand for the projection of Hi,F (Zi , W, X) onto the marginal distribution
of Vil conditional on Zi , W, X – that is, on the set of univariate distribution functions
Fil (·|Zi , W, X). Even though it is very difficult to obtain a closed form description of the
sets Hi,F (Zi , W, X) and Hi,κ (Zi , W, X), it is possible to give closed form characterizations
of their supersets. These supersets are given in Proposition 7 below.
−
First, we introduce some notations. Let ∆+
,l [f (u)] and ∆,l [f (u)] denote differences in
the values of f (·) associated with adding and − to the lthe component of u respectively:
∆+
,l [f (u)] = f (u + el ) − f (u),
∆−
,l [f (u)] = f (u − el ) − f (u),
where el denotes the Li -dimensional lth unit vector.
Suppose there exist known scalars v ≥ 0 and v̄ < ∞ such that v ≤ vil ≤ v̄ for any
l; note that these could be strictly outside the support of Vil . For each bi ∈ Bi , define
−
+
I,l
(bi |Z−i ), I,l
(bi |Z−i ) as follows:
(
−
I,l
(bi |Z−i ) =
v if ∆−
,l [Γ−i (bi |Z−i )] = 0,
(
+
I,l
(bi |Z−i )
=
v̄ if
∆+
,l [Γ−i (bi |Z−i )]
60
= 0,
T
∆−
,l [bi Γ−i (bi |Z−i )]
∆−
,l [Γ−i,l (bi |Z−i )]
T
∆+
,l [bi Γ−i (bi |Z−i )]
∆+
,l [Γ−i,l (bi |Z−i )]
)
else ;
)
else .
−
+
Also, for each bi ∈ Bi , define the following S,l
(bi |Z−i ) and S,l
(bi |Z−i ):
(
−
S,l
(bi |Z−i )
=
0 if
∆−
,l [Γ−i (bi |Z−i )]
= 0,
(
+
S,l
(bi |Z−i )
=
0 if
∆+
,l [Γ−i (bi |Z−i )]
= 0,
∆−
,l [P−i (bi |Z−i )]
∆−
,l [Γ−i,l (bi |Z−i )]
∆+
,l [P−i (bi |Z−i )]
∆+
,l [Γ−i,l (bi |Z−i )]
)
else ;
)
else .
For any K ∈ Ki , let F̃il− (·|K; Z−i ) denote the c.d.f. of
−
−
sup I,l
(bi |Z−i ) − S,l
(bi |Z−i )T K ,
>0
and let F̃il+ (·|K; Z−i ) denote the c.d.f. of
+
+
(bi |Z−i )T K
(bi |Z−i ) − S,l
inf I,l
>0
.
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:
(1)
Hi,κ (Zi , W, X)
=
L
\
K̃i,l ,
l=1
where K̃i,l is defined as
0
K̃i,l = {K ∈ Ki F̃il+ (·|K; Z−i ) ≤ F̃il− (·|K; Z−i
)
0
∀ 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 (·) ∈
Fac (R) such that for any η ∈ R,
\
\
0
)]}.
Fil (η) ∈
[F̃il+ (η|κ0 ; Z−i ), F̃il− (η|κ0 ; Z−i
0
κ0 ∈Hi,κ (Zi ,W,X)Z−i ,Z−i ∈Z−i |Zi ,W,X
(1)
(1)
Let us denote this superset as Hi,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
(1)
Hi,Fl (Zi , W, X), l = 1, . . . , Li .
61
Moreover, for any η = (η1 , . . . , ηLi ),
Fi (η) ≤
min
inf
inf
l=1,...,Li κ ∈H(1) (Z ,W,X) Z−i ∈Z−i |Zi ,W,X
0
i
i,κ
Fi (η) ≥ max

Li
X

l=1
F̃il+ (ηl |κ0 ; Z−i ),
sup
sup
(1)
κ0 ∈Hi,κ (Zi ,W,X)
Z−i ∈Z−i |Zi ,W,X
F̃il− (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 the
vector δ to the vector u. For notational simplicity, let κi (Zi , W, X) = κ0 . Then we can
equivalently restate (23) as follows:
viT ∆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 special
case when Bi consists of a finite number of points, the set of solutions is a convex (closed)
L
polyhedron in <2 −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 be
used 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 of
linear inequalities are quite complicated. An easier way to obtain bounds on each vil is to
consider 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 lth
component of bi affects only the lth component of Γ−i . Noting that Γ−i,l is monotone in
bil and rearranging, we thus must have for any such that bi − el ∈ Bi and bi + el ∈ Bi
T
∆−
,l [bi Γ−i (bi |Z−i )]
∆−
,l [Γ−i,l (bi |Z−i )]
−
T
∆−
,l [P−i (bi |Z−i ) ]
∆−
,l [Γ−i,l (bi |Z−i )]
≤
κ0 ≤ vil
T
∆+
,l [bi Γ−i (bi |Z−i )]
∆+
,l [Γ−i,l (bi |Z−i )]
−
T
∆+
,l [P−i (bi |Z−i ) ]
∆+
,l [Γ−i,l (bi |Z−i )]
κ0 . (25)
If for a given , we have that bi − el ∈
/ Bi or bi + el ∈
/ Bi , then at least one of
+
∆−
[Γ
(b
|Z
)]
and
∆
[Γ
(b
|Z
)]
is
equal
to
0,
and
then
in order to bound vil we
,l −i,l i −i
,l −i,l i −i
can use our prior knowledge that vil ∈ [v, v].
62
Using our notations above, we can say that for any bi in the support of Gi :
−
−
+
+
I,l
(bi |Z−i ) − S,l
(bi |Z−i )T κ0 ≤ vil ≤ I,l
(bi |Z−i ) − S,l
(bi |Z−i )T κ0 ,
and hence,
−
−
+
+
sup I,l
(bi |Z−i ) − S,l
(bi |Z−i )T κ0 ≤ vil ≤ inf I,l
(bi |Z−i ) − S,l
(bi |Z−i )T κ0 .
>0
>0
(26)
Then by the inequalities in (26), we must have for any Z−i ∈ Z−i |Zi , W, X:
F̃il+ (·|κ0 ; Z−i ) ≤ Fil (·) ≤ F̃il− (·|κ0 ; Z−i ).
(27)
Pooling information across Z−i , it follows that we can have K = κ0 only if
0
0
) ∀ Z−i , Z−i
∈ Z−i |Zi , W, X.
F̃il+ (·|K; Z−i ) ≤ F̃il− (·|K; Z−i
(28)
This gives part a) of this theorem. Part b) immediately follows from (27). Part c) follows from (27) and the well known result on sharp Frechet-Hoeffding bounds for joint
distributions.
The next proposition provides an expectations version of the partial identification argument. Even though the supersets it gives are larger than those in Proposition 7, computationally they are easier to obtain. Before formulating this proposition, let us define
+
Li matrices χ− (Z ), χ+ (Z ) as follows:
Li × 1 vectors Ψ−
−i
−i
(Z−i ), Ψ (Z−i ) and Li × 2
Li
−
Ψ−
(Z−i ) ≡ E[I,l (Bi |Z−i )|Z−i ] l=1
Li
+
Ψ+
(Z−i ) ≡ E[I,l (Bi |Z−i )|Z−i ] l=1
−
T Li
χ−
(Z−i ) ≡ E[S,l (Bi |Z−i )|Z−i ] l=1
+
T Li
χ+
(Z−i ) ≡ E[S,l (Bi |Z−i )|Z−i ] l=1 .
Proposition 8. A superset of Hi,κ (Zi , W, X) can be found in the following way:
(2)
Hi,κ (Zi , W, X) =
\
K̂i ,
>0
where K̂i is defined as
n
+
0
K̂i ≡ K ∈ Ki Ψ−
(Z−i ) − Ψ (Z−i )
o
+
0
0
− χ−
(Z−i ) − χ (Z−i ) K ≤ 0 for all Z−i , Z−i ∈ Z−i |Zi , W, X .
63
(2)
Results analogous to parts b) and c) in Proposition 8 hold as well with Hi,κ (Zi , W, X)
(1)
replacing Hi,κ (Zi , W, X).
Proof. Taking expectations of (26) across bil , we obtain
−
−
E[I,l
(Bi |Z−i )|Z−i ] − E[S,l
(Bi |Z−i )|Z−i ]κ0 ≤ E[Vil |Z−i ]
+
+
≡ E[Vil ] ≤ E[I,l
(Bi |Z−i )|Z−i ] − E[S,l
(Bi |Z−i )|Z−i ]κ0 . (29)
0 and l, we obtain
Then pooling restrictions of the form (29) across Z−i , Z−i
−
+
0
+
0
Ψ−
(Z−i ) − χ (Z−i )κ0 ≤ Ψ (Z−i ) − χ (Z−i )κ0
(2)
0
∀ Z−i , Z−i
∈ Z−i |Zi , W, X.
(30)
(1)
Set Hi,κ (Zi , W, X) is larger than Hi,κ (Zi , W, X) because first-order stochastic dominance implies inequalities for expectations.
(2)
Note two features of Hi,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 slope
+
−
+
0
0
vectors (χ−
,l (Z−i ) − χ,l (Z−i )) and intercepts (Ψ,l (Z−i ) − Ψ,l (Z−i )), and the intersection
0
is taken over collections of (Z−i , Z−i , , l).
(2)
Second, if Gi is absolutely continuous, then Hi,κ (Zi , W, X) is a singleton, and as we
(2)
show below, the analysis of Hi,κ (Zi , W, X) essentially becomes our identification strategy
in the case of point identification. Indeed, bidder i’s objective function is differentiable at
almost every observed bi . Hence as → 0 we will have for all l
lim
∆−
,l bi Γ−i (bi |Z−i )
→0 ∆− Γ−i,l (bi |Z−i )
,l
= lim
∆−
,l bi Γ−i (bi |Z−i )/
→0 ∆− Γ−i,l (bi |Z−i )/
,l
=
∂(bi Γ−i (bi |Z−i ))/∂bil
,
dΓ−i,l (bi |Z−i ))/dbil
+
and therefore Ψ−
(·) → Ψ(·). Analogously, it is straightforward to show that Ψ (·) → Ψ(·),
+
χ−
→ χ(·), and χ → χ(·). Hence the restriction (30) implies
0
0
Ψ(Z−i ) − χ(Z−i )κ0 ≤ Ψ(Z−i
) − χ(Z−i
)κ0
0
∀ Z−i , Z−i
∈ Z−i |Zi , W, X.
0 are interchangeable, we thus have for any Z , Z 0 ∈ Z |Z , W, X:
Noting that Z−i , Z−i
−i i
−i
−i
0
0
Ψ(Z−i ) − χ(Z−i )κ0 ≤ Ψ(Z−i
) − χ(Z−i
)κ0
0
0
Ψ(Z−i
) − χ(Z−i
)κ0 ≤ Ψ(Z−i ) − χ(Z−i )κ0 ,
or equivalently
0
0
Ψ(Z−i ) − χ(Z−i )κ0 = Ψ(Z−i
) − χ(Z−i
)κ0
64
0
∀ Z−i , Z−i
∈ Z−i |Zi , W, X.
But this is exactly the identification restriction invoked in Proposition 3 in the current
paper. Thus we can strictly generalize our existing identification results (which depend on
differentiability a.e.) to partial identification for arbitrary Gi .
65
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