Are Structural Estimates of Asymmetric First-Price Auction
Models Credible?
Semi & Nonparametric Scrutinizations
Kirill Chernomaz∗
and
Hisayuki Yoshimoto†
August 28, 2014
Abstract
Structural auction estimation methods, which rigorously build on the assumption of Bayesian
Nash equilibrium, have enabled researches to recover underlying latent valuations and have provided prolific empirical findings. However, due to the latent nature of underlying valuations in
first-price auctions, the assumption of Bayesian Nash equilibrium (BNE) is not feasibly testable
with field data, a fact that evokes harsh criticism and skepticism about the findings in empirical
auction literature. To respond to such criticism regarding credibility, we provide a focused answer
by scrutinizing estimates derived from experimental asymmetric first-price auction data in which
researchers observe valuations. By using the semi and nonparametric structural estimations, we
test the statistical equivalence between the estimated and true value distributions. We then jointly
assess (1) Bayesian Nash equilibrium assumption and (2) other testable hypotheses, including riskaverse bidders and affiliated valuations. We find that although bidders do not strictly adhere to
BNE, the Modified Kolmogorov-Smirnov test fails to reject the distributional equivalence between
the estimated and observed valuations, strongly supporting the accuracy and potential of structural
asymmetric auction estimates.
Keywords and Phrases:
Empirical Auction
Asymmetric Auction
Structural Estimation
Risk Aversion
Semiparametric Estimation
Nonparametric Estimation
Nonparametric Identification
JLE Classifications:
C13 - Estimation: General
D44 - Auctions
∗
†
San Francisco State University, Department of Economics, [email protected]
University of Glasgow, Adam Smith Business School, [email protected]
1
1
Introduction
By scrutinizing the estimates derived from the experimental auction data in which researchers
observe laboratory-assigned valuations, and by statistically testing the equivalence between estimated and observed valuations, this research establishes the credibility of broadly used asymmetric
first-price auction estimates, both structural and nonparametric, which has not been reported in
the literature.1
In empirical first-price auction literature, researchers are interested in describing strategic interactions among bidders for understanding and designing auction markets based on underlying economic incentives. Traditionally, reduced-from regressions had been used, despite the restriction on
linearity had been the hindrance to describe strategic behaviors and to obtain profound insights.
In order to overcome the linearity restriction, structural2 and nonparametric estimation methods
arose and have been widely used over the last twenty years for investigating testable implications
and drawing policy recommendations supported by counterfactual experiments.3 In addition, due
to the ubiquity of asymmetry among bidders, the estimation methods are extended to asymmetric auctions. In such empirical auction research, estimated valuations are particularly important
to both researchers and industry practitioners as market designs, such as setting reserve prices
or detecting collusions, crucially depend on empirical estimates. As a result, asymmetric auction
estimates now come to serve as the vital foundations of many auction market research.
However, while more and more asymmetric first-price auction estimates are reported in the literature, there is a fundamental difficulty in evaluating the performance of these estimates. Structural
methods, in which researchers strictly assume that observed bids are derived from the Bayesian
Nash equilibrium, estimate the bidders’ valuations. However, despite the fact that the comparison between estimated and true valuations is essential in accurately measuring the performance of
estimates, the truth is bidders’ valuations are latent in empirical first-price auctions. This latent
nature of bidders’ valuations in empirics lead to an infeasible comparison between estimated and
true valuations.4 Such an infeasible comparison, along with the rigid Bayesian Nash equilibrium
assumptions for describing bidders’ behavior, then becomes the target of harsh criticism and skepticism on empirical auction estimates.5
1
In this paper, we repeatedly use the terminology of “performance” and “accuracy” to mean the results of statistical
tests that compare the distributions of estimated and true valuations. Specifically, we use the two-sample Modified
Kolmogorov-Smirnov test.
2
Given the assumption of Bayesian Nash Equilibrium (BNE), the structural elements are potentially heterogeneous
von-Neumann-Morgenstern (vNM) payoff functions and potentially affiliated bidders’ value distributions.
3
The cornerstone work in the dawn of empirical and structural first-auction literature should be credited. To the best
of our knowledge, the literature was initiated by the contribution made by the Ph.D. thesis of Paarsch (1992) [55] with
parametric models. Donald and Paarsch (1993, 1996) [20] [21], Elyakime, Laffont, Loisel, and Vuong (1994) [22], Laffont,
Ossard, and Vuong (1995) [41] established statistically rigorous yet flexible parametric estimation methods. The survey
paper of Hendricks and Paarsch (1995) [27] and Perrigne and Vuong (1999) [57] concisely illustrate the early contributions
in the literature.
4
This difficulty has been widely recognized form the beginning of the literature. The survey of Hendricks and Paarsch
(1995) [27] concisely summarizes the difficulty as “The difficulty with field data.....is that neither the valuations of potential
buyers nor the probability law determining these valuations is observed by the researcher.” In addition, McAfee and Vincent
(1992) [48] commented the difficulty of unobserved valuation (signal) as “The most obvious roadblock to test auction theory
is the heavy use made of unobservables in the theory. Bidders choose optimal bids based on signals that are not observed
by econometrician studying auction behavior.”
5
There are at least two sorts of reported criticism. The first is made by the group of mechanism design researchers
who have the skeptical view on the bidders’ abilities to find Beyesian Nash Equilibrium, especially in asymmetric auctions,
that is the solutions of intricate best-response functions. These theoretical researchers claim bidders are not able to find
BNE and suggest to use weaker equilibrium concepts, such as the (prior distribution-free) iterations of dominant-strategy
2
Figure 1: Research Outline
Observed Bids
&
(Experimentally Assigned) True Valuations
Bid (Only) Data
Semi & Nonparametrically
Estimating Valuations
(See Figure 3 for Details)
Estimated Valuations
True Valuations
Testing Distributional Equivalence
between Estimated and True Valuations
Against such criticism and skepticism, Bajari and Hortaçsu (2005) [10] provide a concrete and
focused6 response by using symmetric first-price auction data from laboratory study in which researchers observe experimentally-assigned true valuations, as depicted in Figure 1.7 By using the
valuations observed in the experiment as a benchmark, they compare the estimates generated by
various structural models with nonparametric estimation methods. Their analyses shows that estimates based on the Constant Relative Risk Averse (CRRA) Bayes-Nash model can recover the
distributions of latent valuations in symmetric auctions with a statistically acceptable degree of
accuracy,8 at least under but absolutely not limited to a laboratory environment, and demonstrates
the great potential of structural and nonparametric auction estimation methods.
This research contributes to the empirical auction literature by extending the seminal symmetric
first-price auction study of Bajari and Hortaçsu (2005) [10] to an asymmetric auction framework.
Given the ubiquity of asymmetry among bidders in real-world auctions, prevalence of structural
asymmetric first-price auction studies over the last 20 years,9 and difficulties reported in theoretical
eliminations in second price auctions, to design auction markets. See Wilson (1987) [63] for the critique on BNE. The
second criticism comes from the group of labor economics researchers who claim assumptions made for structural analyses
are implausibly strong. For example, Angrist and Pischke (2010) [2] describe structural elements as “superstructure of
assumptions” and “industrial disorganization” and suggest to use reduced-from analyses of field-experiment data to obtain
policy implications. As Bajari and Hortaçsu (2005) [10] mention, these skepticism are not without merit.
6
Here, we use the word “focused” as Bajari and Hortaçsu (2005) [10] use laboratory experimental data, which has a
more simplified framework compared to field auctions.
7
In a typical auction laboratory experiment, valuations of an object is exogenously and randomly assigned to bidders,
often by using computer-generated random numbers, and researchers are able to observe valuations.
8
In Bajari and Hortaçsu (2005) [10], the accuracy of estimates is statistically supported by the two-sample Modified
Kolmogorov-Smirnov test.
9
The nonparametric auction estimation methods are extended to an asymmetric auction framework by Campo, Perrigne, and Vuong (2003) [15]. Numerous empirical asymmetric auction studies followed. The incomplete list of such
empirical asymmetric auction studies includes Flambard and Perrigne (2006) [23] for Canadian snow removal contracts,
Marion (2007) [49] for bid preference in Californian highway procurement, Lu and Perrigne (2008) [47] for US Forest
Service timbers, Campo (2009, Working Paper) [?] for Californian construction procurement, Krasnokutskaya (2011) [38]
for Californian highway procurement, Krasnokutskaya and Seim (2011) [39] for bid preference in Californian highway
procurement, Nakabahashi (2013) [54] for small businesses set-aside in Japanese construction procurement, and Balat
(2013, Working Paper) [12] for dynamic analysis of Californian construction procurement. All of the empirical asymmetric auction studies use estimated latent valuations to investigate the policy implications in which performances (accuracy)
of estimated valuables are crucial for answering important policy questions. By establishing the credibility of asymmetric
3
asymmetric auction literature,10 it is our belief that testing the performance and pointing out the
potential improvements are invaluable, as they crucially matter to auction research and market
design policy recommendations. Following the precedent set by Bajari and Hortaçsu (2005) [10],
we likewise use laboratory data as it is able to capture insightful views on the performance of
auction estimates. Both strengths and shortcomings of estimates are explicitly detected with laboratory data, and such findings are essential for improving the performance of asymmetric first-price
auction estimates. Also, despite the fact that asymmetries among bidders are ubiquitous in real
world, to the best of our knowledge, direct evaluations of asymmetric auction estimates have not
previously been investigated in the literature.
Specifically, we investigate the performance of de-facto-standardly used asymmetric first-price auction estimations with semi and nonparametric methods introduced by Isabelle Perrigne, Quang
Vuong, and their coauthors. Following the cornerstone work of Guerre, Perrigne, and Vuong (2000,
henceforth GPV2000) [24] that proposes a method to be used for symmetric first-price auctions,
Campo, Perrigne, and Vuong (2003, henceafter CPV2003) [15] extend the nonparametric estimation
method to asymmetric auctions in which bidders draw their valuations from asymmetric distributions. In addition, Guerre, Perrigne, and Vuong (2009, henceforth GPV2009) [25] and Campo,
Guerre, Perrigne, and Vuong (2011, henceafter CGPV2011) [16] broaden the nonparametric estimation method to allow risk-averse preferences among bidders. Due to the versatility in allowing
heterogeneous bidder types and computational tractablity, these asymmetric auction estimation
methods are now the standard used by numerous empirical works for investigating the auction
markets.
Particularly, we use the unique data from the asymmetric private value first-price auctions collected
in the laboratory experiment conducted by Chernomaz (2012) [18]. The data contains submitted
bids and experimentally-assigned valuations for each bidder. Additionally, Chernomaz (2012) [18]
investigates the effects caused by asymmetry among bidders under exogenously changing auction
environments,11 while the mojority of bidder valuations remain fixed before and after such changes.
Given such exogeneity, the bottom line idea in this research is exploiting such exogenous change in
auction environment to identify both bidders payoff functions and underlying values distributions,
as bid distributions varies before and after the change while value distributions remain unchanged.
Specifically, these particular data characteristics enable us not only to estimate risk-averse parameters but also to compare estimated and true value distributions.
The primary analytic methodology employed in our research straightforwardly follows those implemented in Bajari and Hortaçsu (2005) [10] , as depicted in Figure 1, yet we newly extend their
analyses to several empirically important dimensions. The first extension is that we investigate
asymmetry among bidders and associated market efficiencies. Second, we allow then test potential
affiliations among underlying value distributions as empirical researchers seldom have priori knowledge in independence of underlying valuations. Third, in addition to the semi-parametric models,
we investigate the nonparametric model with concave von-Neumann-Morgenstern functions that
allow the most flexibility to bidder’s risk preferences. The last but not least extension is an error
decomposition analyses to verify the importance of modeling bidder heterogeneity.
auction estimates, this study indirectly supports these empirical studies and findings.
10
Krishna (2009) [40], one of the most popular auction-theory textbooks, describes the theoretical difficulties in extending symmetric auction results to asymmetric environments as “Much of theory developed in the symmetric case is
fragile and does not extend to situations in which bidders are asymmetric.” In this research, we demonstrate this negative
statement is not the case of empirical and structural asymmetric auction research.
11
Figure 2 visually illustrates the details of such exogenous changes.
4
Based on the comparisons between estimated variations and true valuations, we report these main
conclusions: (1) the risk-neutral model assumption, which is often assumed for simplicity and
tractability in the literature, tends to inflate estimated valuations; (2) the assumption of riskaverse bidders is indispensable as it drastically improves the performance of estimates; (3) with
the risk-averse model, the two-sample Modified Kolmogoro-Smirnov test fails to reject the statistical equivalence between the nonparametrically estimated and experimentally-assigned value
distributions, positively supporting the many empirical findings reported in the empirical asymmetric auction literature; and (4) estimated value distributions of stochastically-dominated bidders
are relatively more accurate compared to the those of stochastically-dominating bidders.12
The paper is organized as follows: Section 2 illustrates the experimental data that contains both
valuation and bid information; Section 3 explains the theoretical auction models that are the basis
for structural estimations; Section 4 describes the semi and nonparametric asymmetric auction
estimation methods that are used to generate estimates; Section 5 visually reports the estimation
results; Section 6 statistically tests and assesses the performance of asymmetric auction estimates;
and lastly, Section 7 provides the external validity and conclusions.
12
As Bajari and Hortaçsu (2005) [10] mention, and as auctions in the laboratory may differ from those in the real-world
due to the heterogeneity among bidders and other unobserved factors, the strong caveat to the external validity (validity
outside the reach of the experimental laboratory) should be explicitly noted. We will discuss such caveats and external
validity in the conclusion section. In short, the positive findings in this research indicate the great potential of proposed
structural auction estimations and strongly encourage the current and future work on structural auction estimations.
Furthermore, if experimental environments do not differ from real-world auction environments, deduced insights could be
easily used for improving estimates of latent signals in empirical auctions. Such insights could provide an auctioneer with
a great possibility to achieve higher efficiency and revenues by adjusting entry subsidies and bid preference for asymmetric
bidders.
5
2
Data Descriptions
In this section, we illustrate the laboratory auction data used to obtain results described in the empirical section. Key to this section is that bidders in experiments participated in two exogenously
varying formats of auctions while the majority (two out of three, as illustrated in Figure 2) of
their valuations remained unchanged. Such exogeneity provides us with the exclusion restrictions
and enables us to identify both risk-averse vNM payoff functions and value distributions in the
empirical section. With the emphasis on such exogenous change, we first describe the laboratory
auction procedures, then explicate the summary statistics for illustrating the differences in bidding
behavior before and after the change.
The data is from Chernomaz (2012) [18], which investigates the results of joint bids in independent
private value first-price auctions.13 The participants were recruited from undergraduate students
at Ohio State University and paid a $6 show-up fee. There were three experiment runs (denoted
as Experiment Run I, II, and III), conducted at different times, and participants were not allowed
to join more than one experiment run.14 Table 1 summarizes the number of participating bidders
and observed bids in each experiment run. At the beginning of each experiment run, bidder types
(joint and solo) were randomly assigned, and every participant remained the same type throughout
the experiment run. Thus, a participating bidder kept playing the same type throughout an entire
experiment run. In each experiment run, bidders initially experienced two practice rounds, then
they participated in twenty-four rounds involving monetary incentives. Figure 2 depicts the stages
within each round.15 At the beginning of each round, participating bidders were randomly matched
to form a three-bidder group. Out of three within a matched group, two were from the pool of
joint-type bidders, and the remaining one was from the pool of solo-type bidders. Then, valuations
were drawn from i.i.d. uniform distribution U r$0, $18.75s, denoted by v1 to a joint-type bidder, v2
to another joint-type bidder, and v3 to a solo-type bidder, as depicted in Figure 2. Within each
round, there were symmetric- and asymmetric-auction stages. In a symmetric-auction stage, three
bidders submitted one bid each (denoted as b1 , b2 , and b3 ), yet the outcome of a symmetric-stage
auction was not announced until the result-announcement stage. Next, at the beginning of an
asymmetric-auction stage, the two joint-type bidders aggregated their valuations as max tv1 , v2 u.16
In an asymmetric-auction stage, a solo-type bidders submitted a bid bSolo , based on her valuation of
v3 . On the other hand, each joint-type bidder submitted a respective bid, based on the aggregated
valuation of max tv1 , v2 u.17 At an asymmetric-auction stage, these two joint-type bids (denoted
13
A joint bid (also known as a consortium bid) is defined as two or more bidders who form a group and submit one
joint (consortium) bid in an auction. Joint bids were allowed in Mexico and Louisiana Gulf Outer Continental Shelf
(OSC) wildcat auctions, and as a result the implications of joint bids are now intensively scrutinized in empirical auction
literature. Hendricks and Porter (1998) [31], Campo, Perrigne, and Vuong (2003) [15], and Hendricks, Pinkse, and Porter
(2003) [33] investigate joint bids in wildcat auctions and the associated asymmetry in available economic resources. Note
that our laboratory procedures in Figure 2 can be viewed as a miniature of hypothetical wildcat auctions in which both
(non-collusively) individual and joint bids are allowed to submit to an auction where an auctioneer randomly determines
whether or not to allow a joint bid.
14
A computer based laboratory was used for this experiment, and participants interacted through computer screens.
15
In summary, there were three experiment runs (I, II, and III). In each run, there were twenty-four rounds, excluding
the two practice rounds. In each round, there were several stages, including symmetric- and asymmetric-auction stages,
as depicted in Figure 2.
16
This way of value aggregation, adopting a maximum valuation among joint-type bidders, corresponds to the empirical observations that joint (consortium) bidders share their economic resources, such as the most available cost-saving
technology, the closest geographical locations, and the information of best-available resale opportunities. In our experiment, this means that one of the joint-type bidders kept having the same valuation before and after the exogenous value
aggregation, and this invariant nature is exploited in the estimation section.
17
In an asymmetric-auction stage, the two within-a-matched-group joint-type bidders are informed of their aggregated
valuation (i.e. max tv1 , v2 u) through their respective computer screens. However, verbal or textual communication between
6
as b1,Joint and b2,Joint ) were separately submitted by each joint-type bidder; then the experiment
organizer (i.e. the auctioneer) randomly chose one of them with equal probability (described as
50% and 50% in Figure 2) to be the chosen joint-type bid.18 At the result-announcement stage,
within-a-matched-group results, including assigned valuations (v1 , v2 , and v3 ), aggregated valuation (max tv1 , v2 u), bids in each stage (b1 , b2 , b3 , chosen bJoint , and bSolo ), and winning/losing
statuses in each stage were announced to the matched group members, yet the identities of bidders
were kept hidden. In addition, monetary payoffs were calculated and added to each participating
bidder’s account.19,20 Lastly, at the end of each round, a matched group was dissolved, and participants returned to the pool of bidders.
Results were announced only to matched-group members at the end of each round, and the results
of a specific matched group were NOT available to members of any other groups. As a natural
consequence, we observe sizable learning and adjusting behavior in the first half of the rounds in
each experiment run. For investigating the strategic interactions and estimates of valuations without concern for the learning effect, this research excludes the data from the first half of the rounds,
and only the data from the second half of the rounds is used in the empirical investigations. Table
1 summarizes the sample size in each auction stages.
Table 2 lists the summary statistics of observed bids in both symmetric- and asymmetric-auction
stages. In theory, bidders are predicted to bid less in asymmetric-auction stages, as symmetricauction stages consist of three bidders while asymmetric-auction stages consist of only two bidders
(i.e. a chosen joint-type bidder and a solo-type bidder). In most of the experiment runs, both
joint- and solo-type bidders decreased their bids in asymmetric-auction stages, yet we observe a
small increase in bids in Experiment Run II that could negatively affect the performance of structural estimations. Figure 3 depicts the pairs of symmetric-auction stage bids and chosen (and also
announced announced) asymmetric-auction stage bids in each experiment run for which we later
apply apply kernel density estimations. Although we observe the slight rounding-up/down effect
(eg. bidders tend to bid in increments of five -$0, $5, $10, and $15), this does not seem to severely
affect the estimates of distributional functions as, overall, bids appear widely spread out.21
joint-type bidders was strictly forbidden. Therefore, a bid made by a joint-type bidder in an asymmetric stage (i.e. b1,Joint
or b2,Joint in Figure 2) was derived from a single-agent payoff-maximization problem. This single-agent decision nature is
advantageous for estimating preferences among bidders in the estimation section.
18
This randomized choice among joint-type bids (i.e. among b1,Joint and b2,Joint ) with equal probability was designed
for investigating behavioral difference in bidding behavior between a consortium-leader-firm-like joint-type bidder whose
valuation draw was originally max tv1 , v2 u and a consortium-follower-firm-like joint-type bidder whose valuation draw
was originally min tv1 , v2 u. We encourage interested researchers to see Chernomaz (2012) [18] for further details on the
experiment design and behavioral differences.
19
Monetary payoffs were calculated as follows. After a result-announcement stage, the experiment organizer (i.e. the
auctioneer) randomly selected with equal probability an auction stage in which an outcome was actually paid. Note that
as far as bidders’ vNM functions are additively separable, which is usually assumed and accepted in auction literature,
this random selection does not affect bidders’ payoff-maximization problems in each auction stage. This randomized
selection process was empirically motivated by the factual observation of timber auctions in which the U.S. Forest Service
randomized different auction rules. Lu and Perrigne (2008) [47] and Athey, Levin, and Seira (2011) [8] exploit such
randomization of timber auctions for detailed investigations of identifications and bidding behavior.
20
In addition, if an asymmetric-stage auction was selected (by the experimental organizer) and if a chosen (by the
experimental organizer) joint-type bid exceeded a solo-type bid, the payoff of the joint-type bidders was equally split. This
means that, conditional on being selected, chosen, and winning, each joint-type bidder’s payoff was 12 pmax tv1 , v2 u bchosen
Joint q
as illustrated in Figure 2, where bchosen
is
a
chosen
joint-type
bid.
Although
this
splitting
rule
does
not
affect
the
risk
Joint
neutral and CRRA models, it affects the estimation methods of joint-type bidders’ risk preferences for CARA and
nonparametric models. See Appendix for details.
21
As this research emphasizes the empirical perspective of asymmetric auction data, we refrain from examining the
data of laboratory-assigned valuations until Estimation Result section. Nonetheless, the plots of laboratory-assigned
7
Figure 2: Stages within a Round
Valuation-Draw
Stage
Symmetric-Auction
Stage
v2
v1
v1
i.i.d.
Uniform[$0, $18.75 ]
max{v1,v2}
max{v1,v2}
50% 50%
b2
b1
v2
Asymmetric-Auction
Stage
b1,Joint
b2,Joint
Auction
Auction
bSolo
b3
v3
(v1, b1),
(v2, b2),
(v3, b3),
(max{v1,v2 }, chosen {bJoint}),
(v3, bSolo),
and
winning/losing statuses
were announced
v3
v3
Start of a Round:
Random
Group-Member
Matching
Result-Announcement
Stage
Monetary-Payoff
Calculation
Pool of Bidders
End of a Round:
Dissolving
a Matched Group
Type (Joint or Solo) Assignment
: Joint-Type Bidder
Start of an Experiment Run
: Solo-Type Bidder
Table 1: Sample Size
Experiment Run I
Experiment Run II
Experiment Run III
Joint Type
Solo Type
Joint Type
Solo Type
Joint Type
Solo Type
Number of
Participants
14 Bidders
7 Bidders
16 Bidders
8 Bidders
12 Bidders
6 Bidders
Symmetric-Auction
Stage Bids Observed
168 Bids (84 Chosen Bids)
84 Bids
192 Bids (96 Chosen Bids)
96 Bids
142 Bids (72 Chosen Bids)
72 Bids
Asymmetric-Auction
Stage Bids Observed
84 Bids
84 Bids
96 Bids
96 Bids
72 Bids
72 Bids
Table 2: Summary Statistics of Bid Data
Joint Type
Experiment Run I
Solo Type
Joint Type
Experiment Run II
Solo Type
Joint Type
Experiment Run III
Solo Type
Symmetric
Asymmetric
Difference
Symmetric
Asymmetric
Difference
Symmetric
Asymmetric
Difference
Symmetric
Asymmetric
Difference
Symmetric
Asymmetric
Difference
Symmetric
Asymmetric
Difference
Mean
Standard
Deviation
10th
Quantile
25th
Quantile
50th
Quantile
75th
Quantile
90th
Quantile
$08.77
$08.17
$00.60
$07.09
$07.01
$00.08
$09.54
$09.58
-$0.04
$06.79
$06.92
-$0.13
$10.60
$10.22
$00.38
$08.52
$08.43
$00.09
3.13
3.22
05.00
04.22
00.78
01.57
01.50
00.08
05.00
04.75
00.25
01.02
00.75
00.27
03.16
03.16
00.00
02.03
02.03
00.00
06.38
05.53
00.85
03.44
03.49
-0.05
06.28
06.76
-0.49
02.58
02.56
00.01
07.55
07.38
00.17
04.53
04.53
00.00
08.88
07.84
01.04
06.94
06.50
00.44
09.99
09.28
00.71
06.69
07.22
-0.54
11.49
11.25
00.24
08.61
08.31
00.30
10.50
10.38
00.13
10.69
10.28
00.41
12.40
12.34
00.06
11.06
10.93
00.14
14.18
13.64
00.54
12.78
12.75
00.03
12.62
12.55
00.07
12.63
13.15
-0.52
14.98
15.00
-0.02
12.96
13.75
-0.79
15.79
15.63
00.17
15.27
15.08
00.19
4.38
4.47
3.75
3.95
4.60
4.64
4.30
4.39
4.88
4.85
valuations and bids are reported in Appendix, and we observe large overbidding (bidding more than risk neutral BNE)
in our laboratory auction data.
8
Figure 3: Observed Bids: LEFT - Symmetric Auction Stages, RIGHT - Asymmetric-Auction Stages
Observed Pairs Bids in Asymmetric−Auction Stages
18
15
15
Joint-Type Bids (in U.S. Dollars): bAsym
Joint
Own Bids (in U.S. Dollars): bSym
i
Observed Pairs of Bids in Symmetric−Auction Stages
18
12
9
6
3
0
12
9
6
3
0
3
6
9
12
15
Highest Opponent Bids (in U.S. Dollars): bSym
−i
0
18
0
Sym
(a) Experiment Run I: bSym
i and bi
15
15
12
9
6
3
12
9
6
3
0
3
6
9
12
15
Highest Opponent Bids (in U.S. Dollars): bSym
−i
0
18
0
3
6
9
12
15
Solo-Type Bids (in U.S. Dollars): bAsym
Solo
18
Asym
(d) Experiment Run II: bAsym
Solo and bJoint
Sym
(c) Experiment Run II: bSym
i and bi
Observed Pairs of Bids in Symmetric−Auction Stages
Observed Pairs Bids in Asymmetric−Auction Stages
18
18
15
15
Joint-Type Bids (in U.S. Dollars): bAsym
Joint
Own Bids (in U.S. Dollars): bSym
i
18
Observed Pairs Bids in Asymmetric−Auction Stages
18
Joint-Type Bids (in U.S. Dollars): bAsym
Joint
Own Bids (in U.S. Dollars): bSym
i
Observed Pairs of Bids in Symmetric−Auction Stages
12
9
6
3
0
6
9
12
15
Solo-Type Bids (in U.S. Dollars): bAsym
Solo
Asym
(b) Experiment Run I: bAsym
Solo and bJoint
18
0
3
12
9
6
3
0
3
6
9
12
15
Highest Opponent Bids (in U.S. Dollars): bSym
−i
0
18
Sym
(e) Experiment Run III: bSym
i and bi
9
0
3
6
9
12
15
Solo-Type Bids (in U.S. Dollars): bAsym
Solo
18
Asym
(f) Experiment Run III: bAsym
Solo and bJoint
3
Auction Models
This section describes the theoretical models of an affiliated private value (APV) auction that
include an independent private value (IPV) auction as a special case. Although the bid data
used in this research is generated from the experiments of IPV auctions, APV models are initially
employed for the following empirically pragmatic considerations.22 In many empirical investigations,
researchers seldom have enough prior evidence to determine the independence of underlying value
distributions. Given the pervasiveness of insufficient initial information, therefore, it is empirically
prudent for researchers to first estimate latent valuations with a model that can allow potential
affiliations, then test independence. Formal and statistical tests for distributional independence are
demonstrated in the Estimation Result section.23 These cautious model-specification procedures
are also based on the robustness requisition proposed by the well-known and influential critique by
Leamer (1983) [42], which is largely concerned with the credibility of estimates in general empirical
research due to the lack of robustness in changes to model assumptions.24 With the emphasis on
generality in modeling assumptions, we first explain the symmetric auction models, then illustrate
the asymmetric auction models.
3.1
Symmetric Auction Models
A single and indivisible object is sold in an auction to N bidders who have the von-NeumannMorgenstern (vNM) function U pq that is twice differentiable with U 1 pq ¡ 0 and U 2 pq ¤ 0 to allow
potential risk aversion. As a vNM function is unique up to the positive-affine transformation, the
normalization of U p0q 0 and U p1q 1 is imposed without loss of generality. For the same of
clear notations, we use a capital letter for describing a random variable and a lower-case letter for
describing the realization of a random variable. Assuming that N 3 bidders in an auction with
index i P t1, 2, 3u, bidders draw private valuations tv1 , v2 , v3 u from the potentially affiliated joint
distribution FV1 ,V2 ,V3 pv1 , v2 , v3 q. The arguments of the joint distribution are exchangeable in its
N 3 elements, meaning that the model is distributionally symmetric.25 Given that other bidders
employ a symmetric equilibrium strategy, bidder i’s expected payoff maximization problem is
max U pvi bi q FYi |Vi pφpbi q|vi q,
bi
where Yi is a random variable of the highest valuations among opponent bidders with its realization yi maxj i vj , and φpq is the inverse of a symmetric equilibrium bidding function. In
addition, FYi |Vi pyi |vi q denotes the conditional distribution function of Yi given vi . The firstorder necessary condition26 of the above maximization problem, where a bidder equates marginal
22
Surprisingly, APV models provide slightly more precise estimates compared to those derived from IPV models, and
we will investigate the reasons behind this in the Estimation Result section.
23
Given the existence of behavioral bidders who may not strictly play BNE, the independence of observed bids does
not imply the independence of valuations (vice versa).
24
See the recent development on the Leamer Critique argued in Angrist and Pischke (2010) [2] and Sims (2010) [61]
for a detailed description of model robustness.
25
This setting is called the symmetric affiliated private value (APV) model, and its econometric identification and
rationalizability are intensively investigated in Li, Perrigne, and Vuong (2002) [46]. By the Monte Carlo simulations, they
investigate the problem of misspecification, estimating distributions of valuations under the IPV assumption when the true
data-generating process is APV. They report that such misspecification tends to result in overestimation of private values,
specifically in the upper domain of private values. This overestimation is caused by the neglect of modeling a strategic
behavior in an APV environment; if a bidder has a high valuation, other bidders are also likely to have high valuations,
and she needs to bid aggressively. Conversely, we in this research investigate the empirically prudent misspecification with
the laboratory data, estimating value distributions under the APV assumption when the true data-generating process
is IPV. We are happy to report that empirical asymmetric auction estimates in our research are robust (and even more
accurate) against such empirically prudent misspecification, as we explain in the Estimation Result section.
26
Throughout this research, we assume that second-order conditions are satisfied.
10
cost and benefit of changing her bid, can be written in the form of
vi
FYi |Vi pφpbi q|vi q
,
λ 1 F
Yi |Vi pφpbi q|vi q
1
φ pbiq
dyi
loooooooooooooooooomoooooooooooooooooon
bi
Shading Function
where, according to the tradition of empirical auction literature, we define λpq U pq{U 1 pq and
λ1 pq as a corresponding inverse function. We refer to a function λ1 pq as a shading function after
its role of describing the difference between a valuation and a bid. We also refer to the argument
of the shading function as a reciprocal factor.27 Since in field-auction data we cannot empirically
observe latent valuations that appear in the argument of the shading function, we now need to
replace unobserved valuations with observed bids. We denote that Bi is a random variable of a
highest bid among opponent bidders with its realization bi maxj i bj . In addition, we denote
the conditional distribution of Bi given bi as GBi |Bi pbi |bi q and its derivative as gBi |Bi pbi |bi q.
By assuming bidder i also employs a symmetric equilibrium bidding strategy, the probabilistic
relation between observable bids and latent valuations is GBi |Bi px|bi q FYi |Vi pφpxq|vi q.28 Moreover, by the fundamental theorem
of calculus, the conditional density is obtained as gBi |Bi px|bi q dFYi |Vi pφpxq|φpbi qq{dyi φ1 pxq. Then, by using these probabilistic relations that bridge the
unobservable to the observable, the first-order necessary condition can be equivalently written as
vi
bi
GBi |Bi pbi |bi q
λ1
,
gBi |Bi pbi |bi q
(1)
where components of the right-hand side of the equation are observable or estimatable. Furthermore, as the distributional functions in the right hand side of equation (1) share the same
conditional variable, we exploit the definitions of conditional density and distribution functions,
as suggested by Li, Perrigne, and Vuong (2002) [46], gBi |Bi pz |bi q gBi ,Bi pz, bi q{gBi pbi q and
³
x
GBi |Bi px|bi q b gBi ,Bi pz, bi qdz {gBi pbi q where b is the lower bound of bid distribution. Thus,
the equation (1) can be re-written in an unconditional fashion as
vi
³
λ 1 bi
bi
b
gBi ,Bi pz, bi qdz
gBi ,Bi pbi , bi q
.
(2)
For the sake of organized
notations for later empirical use, we introduce the convenient terms of
³x
Γpx, bi |gBi ,Bi q b gBi ,Bi pz, bi qdz and of reciprocal factor as Rrx, y |ΓBi ,Bi , gBi ,Bi s
= Γpx, y |gBi ,Bi q{gBi ,Bi px, y q. Given these simplified notations, we can denote the first-order
necessary condition of symmetric auction as
viSym
bSym
i
,
loooooooooooooooooooooooooomoooooooooooooooooooooooooon
Sym,Affi Sym Sym Sym,Affi Sym,Affi
λ1 bi , bi ΓBi ,Bi , gBi ,Bi
R
(3)
(i)
27
This slightly awkward naming comes after the fact that Guerre, Perrigne, and Vuong (2009) [25] use the notation R
to express this argument (see p.1202 of their paper for details), yet they do not provide a memorable name for this object.
“R”eciprocal factor represents a quotient of “probability of winning” divided by “marginal probability of winning.” It is
actually the reciprocal of semi-elasticity rpdhpxq{dxq{hpxqs where hpxq is a winning probability and dx is a change in
bid.
28
The derivation is as follows: GBi |Bi px|bi q PrpBi ¤ x|bi φ1 pvi qq PrpφpBi q ¤ φpxq|φpbi q vi q PrpYi ¤
φpxq|vi q FYi |Vi pφpxq|vi q.
11
where the upper indices of “Sym” and “Sym,Affi” emphasize that the auction model is symmetric
and of affiliated value. In addition, if a researcher further assumes the independence
of private val³x
uations, the bi-variate functions are simplified as gBi ,Bi pz, bi q gBi pz q and b gBi ,Bi pz, bi qdz GBi pxq. Accordingly, first-order necessary condition equation (3) becomes the well-known equation
in empirical auction literature,
vi
bi
λ1
GBi pbi q
gBi pbi q
Lastly, by denoting a reciprocal factor as Rrx|GBi , gbi s
first-order necessary condition as
.
(4)
GB pxq{gB pxq, we can write the
i
i
viSym
bSym
i
,
looooooooooooooooooooooomooooooooooooooooooooooon
Sym,Inde Sym Sym,Inde Sym,Inde
, gBi
bi GBi
λ1 R
(5)
(ii)
where the upper indices of “Sym” and “Sym,Inde” emphasize that the auction model is symmetric
and of independent value.
3.2
Asymmetric Auction Models
We next introduce asymmetric auction models. In order to be notationally minimalistic, we henceforward focus on the simplest environment, a two-type and two-bidder asymmetric auction on
which our experimental asymmetric auction data is based. We define the index of the bidder type
as t P tJoint, Solou. By exploiting the two-type-two-bidder nature, we use a convenient notation of
t for representing an opponent bidder’s type. We denote Utpq as a vNM function of type t bidder, as we allow for the possibility of joint- and solo-type bidders having different payoff functions.
Bidders draw their private valuations from joint distribution FVt ,Vt pvt , vt q in which arguments
are not exchangeable and valuations are potentially affiliated.29,30 Assuming a type t bidder draws
her valuation of vt and assuming an opponent bidder employs an equilibrium strategy φt pq, her
expected payoff maximization problem is31
max Ut pvt bt q FVt |Vt pφt pbt q|vt q,
(6)
bt
where FVt |Vt pVt |vt q denotes the conditional distribution function of Vt given vt . By differentiating the above expected payoff function with respect to bt , a type t bidder equates the marginal
cost and benefit of changing her bid, and we have the first-order necessary condition, written as
vt
bt
1
λ
t
|
p p q| q
p q| q 1
p q
FVt Vt φ t bt vt
,
dFVt |Vt φt bt vt
φ t bt
dvt
loooooooooooooooooooooooooomoooooooooooooooooooooooooon
p
Shading Function
29
The model explored in this subsection is the simplest version of an asymmetric and affiliated private value auction
model, and its econometric identifications and rationalizability are established by Campo, Perrigne, Vuong (2003) [15].
30
According to theoretical asymmetric auction literature, for example, Maskin and Riley (2000a, 200b) [51] [52], we
assume all types share the common support of private values. We follow Milgrom and Weber (1982) [53] for the definition
of affiliation among private values.
31
Technically speaking, as joint-type bidders equally split a monetary payment, the maximization problem for a jointtype bidder is maxbJoint tUJoint ppvJoint bJoint q{2qFVSolo |VJoint pφSolo pbJoint q|vJoint qu. Here, for simplicity’s sake, we abbreviate explanation as seen in (6) and continue using it for maximization problems of both joint- and solo-type bidders. The detailed
explanations are found in Appendix. Note that this equal-split-payment rule does not affect the discussion of theoretical
rJoint pxq UJoint px{2q), yet it slightly affects the
asymmetric auction models (as we can re-define a vNM function as U
structural estimations.
12
1
where λt pq Ut pq{Ut1 pq for each type of t P tJoint, Solou and λ
t pq is its inverse function (called
as a shading function in this research). Next, similar to the symmetric case, we derive the relations
between empirically unobservable valuations and observable bids. We denote the conditional distribution of an opponent type’s bid Bt given bt as GBt |Bt pbt |bt q and its derivative as gBt |Bt pbt |bt q.
By further assuming a type t bidder employs an equilibrium strategy φt pq, the probabilistic relation
between latent valuations and observable bids is GBt |Bt px|bt q FVt |Vt pφt pxq|vt q.32 In addition,
by the fundamental theorem
of calculus, the conditional density is obtained as gBt |Bt px|bt q 1
dFVt |Vt pφt pxq|vt q{dvt φt pxq. Therefore, using these probabilistic relations, the first-order
necessary condition can be equivalently written as the function of observable bids,
vt
1 GBt |Bt pbt |bt q
λ
.
t
gBt |Bt pbt |bt q
bt
(7)
Furthermore, similar to the symmetric case, by exploiting the definitions of conditional density and
distribution functions, gBt |Bt pz |bt q gBt ,Bt pz, bt q{gBt pbt q and GBt |Bt px|bt q
³
x
b gBt ,Bt
pz, btqdz {gB pbtq
where b is the lower bound of bid distributions, the first order
necessary condition equation (7) can be written in an unconditional fashion as
t
vt
³
1
λ
t
bt
bt
b
gBt ,Bt pz, bt qdz
gBt ,Bt pbt , bt q
.
(8)
³x
To simplify notations, we again introduce the convenient terms of Γpx, bt |gBt ,Bt q b gBt ,Bt pz, bt qdz
and of reciprocal factor as Rrx, y |ΓBt ,Bt , gBt ,Bt s Γpx, y |gBt ,Bt q{gBt ,Bt px, y q. Given these simplified notations, we can denote the first-order necessary condition as
vtAsym
bAsym
t
Asym,Affi Asym,Affi 1 Asym,Affi
λ
bAsym
, bAsym
ΓBt ,Bt , gBt ,Bt
t R
t
t
,
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
(9)
(iii)
where upper indices of “Asym” and “Asym,Affi”emphasize that the auction model is asymmetric
and of affiliated value. In addition, if a researcher further assumes the independence
of private val³x
uations, the bi-variate functions are simplified as gBt ,Bt pz, bt q gBt pz q and b gBt ,Bt pz, bt qdz GBt pxq. Accordingly, equation (8) becomes
vt
bt
1 GBt pbt q
λ
.
t
gBt pbt q
Lastly, by denoting a reciprocal factor as Rt rx|GBt , gBt s
first-order necessary condition as
(10)
GB pxq{gB pxq, we can write the
t
t
vtAsym
bAsym
t
.
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
Asym,Inde Asym,Inde
1 Asym,Inde
bAsym
, gBt
λ
GBt
t R
t
(11)
(iv)
where the upper indices “Asym” and “Asym,Inde” emphasize that the auction model is asymmetric
and of independent value.
The derivation is as follows: GBt |Bt px|bt q PrpBt
PrpVt ¤ φt pxq|vt φt pbt qq FVt |Vt pφt pxq|vt q.
32
¤ x|bt φt 1 pvt qq Prpφt pBt q ¤ φt pxq|φt pbt q vt q
13
Figure 4: Estimation Steps and Methods
Step
Object
Method
Step 1
p s, gps
Γ
p
Gs
Nonparametric Kernel Density
Empirical CDF
Step 1: Estimating Distributional Functions
𝛤s, 𝐺 s, 𝑔s
Step 2: Estimating Shading Functions
(CRRA, CARA, and Nonparametric
vNM Function Models)
𝛤s, 𝐺 s, 𝑔s
Step 2
pq
pq
pqs
OLS
NLLS
Sieve
𝜆−1
𝑡 (⋅)s
Step 3
Step 3: Estimating Valuations
4
p 1 s
CRRA λ
p 1 s
CARA λ
p 1
Nonparametric λ
vps
Substituting the objects
estimated in Step 1
and Step 2 in f.o.c.s
Structural Estimation Methods
In this section, we illustrate the estimation methods for recovering valuations. Estimation procedures are summarized into the three steps as depicted in Figure 4: Step 1 – nonparametrically
estimating distributional functions; Step 2 – by applying semi or nonparametric methods, esti1
mating shading functions (i.e. λ
t pqs); Step 3 – estimating valuations based on estimated distributional functions and shading functions. As the main purpose of this research is to investigate
the accuracy of asymmetric auctions estimates, we primarily recover valuations from bids observed
in asymmetric auctions, and bid data from symmetric auctions is subsidiarily used solely for the
purpose of estimating the shading functions. For the sake of clear notations, we introduce the
following indices: r P t1, . . . , Ru as an auction round index; m P t1, . . . , M u as a (within-a-round)
matched group index; and i P t1, . . . , N u where N 3 as a bidder index in a symmetric-auction
stage. In addition, as estimates are separately calculated for each experiment run, we omit an
index for experiment runs.33
Given the goal of obtaining the estimates of valuations, this section is organized as follows. The
first subsection illustrates Step 1 with descriptions of nonparametric estimation method for distributional functions. The following subsections illustrate Step 2 and Step 3 in the order of risk
neutral, general estimation framework for risk-averse models, constant relative risk averse (CRRA),
constant absolute risk averse (CARA), nonparametric vNM function, and heterogeneous risk averse
attitude models, as the latter models require incrementally advanced estimation methods.
4.1
Nonparametric Estimations of Distributional Functions
Here, as Step 1, we nonparametrically estimate the distributional functions that are the basis for
further estimations of shading functions and valuations. Nonparametric kernel estimation methods
are employed throughout this research, and we use the Gaussian kernel with Silverman’s rule of
thumb bandwidths. For bi-variate distributional function estimations, we use a product kernel.
4.1.1
Affiliated Value Assumption
The distributional functions under affiliated private value assumption are estimated as follows. By
using symmetric-auction stage bid data, we estimate (remind that bi is the highest of opponents’
33
Any estimation results reported in the rest of this research are separately calculated for each experiment run.
14
bids)34,35
p Sym,Affi px, y q Γ
Bi ,Bi
»x
b
{
Sym,Affi
gB
i ,Bi pz, y qdz
1
Sym,Affi RM N
h
Ŗ
1
Γ
M̧
Ņ
1p
bSym
r,m, i
r 1m 1i 1
¤ xq K
bSym
r,m,i y
(12)
,
hSym,Affi
Γ
and by using asymmetric-auction stage bid data, for each type of t P tJoint, Solou, we estimate36
p q
p Asym,Affi x, y
Γ
Bt ,Bt
»x
b
{
Asym,Affi
gB
t ,Bt pz, y qdz
Ŗ
1
1
Asym,Affi RM
h
Γ
M̧
1p
¤ xq K
bAsym
r,m, t
r 1m 1
bAsym
r,m,t y
hAsym,Affi
Γ
(13)
,
where hSym,Affi
cΓ pRM N q1{5 and hAsym,Affi
cΓ pRM q1{5 with cΓ 1.06 σ̂b, and σ̂b is the
Γ
Γ
empirical standard deviation of corresponding observed bids.37 For the bi-variate density functions,
we estimate
Sym,Affi
gpB
i ,Bi
px, yq 1
Sym,Affi
Sym,Affi RM N
hgx
hgy
1
and for each type of t P tJoint, Solou,
Asym,Affi
gpB
t ,Bt
px, yq Ŗ
Ņ
K
r 1m 1i 1
1
Asym,Affi RM
Asym,Affi
hgy
hgx
1
M̧
Ŗ
M̧
r 1m 1
K
bSym
r,m,i x
hSym,Affi
gx
bAsym
r,m,t x
hAsym,Affi
gx
K
K
bSym
r,m,i y
,
hSym,Affi
gy
bAsym
r,m,t y
hAsym,Affi
gy
(14)
,
(15)
where hSym,Affi
cg pRM N q1{6, and hAsym,Affi
cg pRM q1{6 with cg 1.06 σ̂b and σ̂b is the
gx
gx
and hAsym,Affi
empirical standard deviation of corresponding observed bids. In addition, hSym,Affi
gy
gy
are determined in the same manner. Given these estimated distributional functions with affiliated
value assumption, we can calculate reciprocal factors as
R
Sym,Affi
Sym p Sym,Affi Sym,Affi
pBi ,Bi
bSym
r,m,i , br,m,i ΓBi ,Bi , g
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
pxi)
pSym,Affi bSym , bSym
R
r,m,i r,m,i
Bi ,Bi
looooooooooooomooooooooooooon
Shorthand Notation
34
p Sym,Affi pbSym , bSym q
Γ
r,m,i r,m,i
Bi ,Bi
Sym
Sym,Affi Sym
gpB
i ,Bi pbr,m,i , br,m,i q
, (16)
In this subsection (and only in this subsection), we generically use x for a first functional argument, y for a second
functional argument, and z for an integrating variable. Note that a first function argument is a realized random variable of
an opponent who has the highest valuation in a symmetric-auction stage or of an opponent-type bidder in an asymmetricauction stage. A second functional argument is a realized random variable of a bidder i in a symmetric-auction stage or
of a type t bidder in an asymmetric-auction stage.
35
Note that, in round r P t1, . . . , Ru within matched group m P t1, . . . , M u , we use tbr,m1 , br,m,2 , br,m,3 u in a symmetric
auction stage and tbr,m,Joint , bm,r,Solo u in an asymmetric-auction stage for estimating distributional functions. Although
there are two observed joint-type bids in an asymmetric-auction stage (as depicted in Figure 2), only a chosen (and
announced) joint-type bid in an asymmetric-auction stage is used for estimations of distributional functions. In other
words, a non-chosen (and unannounced) joint-type bid in an asymmetric-auction stage is not used for estimations of
distributional functions.
36
As there are only two bids in an asymmetric-auction stage used for estimations, (one is submitted by a solo-type bidder, and the other is submitted by a chosen joint-type bidder) and as we estimate asymmetric-auction stage distributional
functions for each type of bidder, we drop summations over bidder types in the following equations.
37
Following Bajari and Hortaçsu (2005) [10], in this research we use the unbounded-support Gaussian kernel with
rule of thumb, c 2.978 1.06 σ̂b (see Li and Racine (2007) [44] page 26, for example). Note that, some of preceding
and influential works (eg. Li, Perrigne, and Vuong (2002) [46] and Campo, Perrigne, and Vuong (2002) [15]) use the
bounded-support triweight kernel with the rule of thumb, c 2.978 1.06 σ̂b . See Härdle (1990, 1991) [28] [29] for the
detailed description of bandwidth choices.
15
and for each type of t P tJoint, Solou,
R
Asym,Affi
Asym p Asym,Affi Asym,Affi
bAsym
pBt ,Bt
r,m,t , br,m,t ΓBt ,Bt , g
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
py
iii)
pAsym,Affi bAsym , bAsym
R
r,m,t
r,m,t
Bt ,Bt
looooooooooooooomooooooooooooooon
Shorthand Notation
p Asym,Affi pbAsym , bAsym q
Γ
r,m,t
r,m,t
Bt ,Bt
Asym,Affi Asym Asym
gpB
t ,Bt pbr,m,t , br,m,t q
The plots of these affiliated value distributional function estimates based on observed bids are
found in Online Appendix.
4.1.2
Independent Value Assumption
Next, the distributional functions under independent private value assumption are estimated as
follows. By using symmetric-auction stage bid data, we derive the empirical CDF as
p Sym,Inde pxq
G
Bi
1
RM N
Ŗ
M̧
Ņ
r 1m 1i 1
1pbSym
r,m,i ¤ xq,
(18)
and by using symmetric-auction stage bid data, for each type of t P tJoint, Solou, we derive
p Asym,Inde pxq
G
Bt
1
RM
Ŗ
M̧
1pbAsym
r,m,t ¤ xq.
r 1m 1
(19)
For uni-variate density functions, we estimate
Sym,Inde
gpB
i
pxq Ŗ
1
1
Sym,Inde RM N
hg x
M̧
Ņ
K
bSym
r,m,i x
,
hSym,Inde
gx
r 1m 1i 1
(20)
and, for each type of t P tJoint, Solou,
Asym,Inde
gpB
t
px q 1
1
Asym,Inde RM
hg x
Ŗ
M̧
K
bAsym
r,m,t x
r 1m 1
hAsym,Inde
gx
,
(21)
where hSym,Inde
cg pRM N q1{5, and hAsym,Inde
cg pRM q1{5 with cg 1.06 σ̂b and σ̂b is
gx
gx
the empirical standard deviation of corresponding observed bids. Accordingly, we can calculate
reciprocal factors as
R
Sym,Inde
p Sym,Inde Sym,Inde
bSym
, gpBi
r,m,i GBi
loooooooooooooooooooooooomoooooooooooooooooooooooon
pxii)
pSym,Inde bSym
R
r,m,i
Bi
looooooooomooooooooon
Shorthand Notation
p Sym,Inde pbSym q
G
r,m,i
Bi
Sym,Inde Sym
gpB
pbr,m,iq
i
,
(22)
and for each type of t P tJoint, Solou,
R
Asym,Inde
p Asym,Inde Asym,Inde
bAsym
, gpBt
r,m,t GBt
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
py
iv)
pAsym,Inde bAsym
R
r,m,t
Bt
loooooooooomoooooooooon
Shorthand Notation
p Asym,Inde pbAsym q
G
r,m,t
Bt
Asym,Inde Asym
gpB
pbr,m,t q
t
.
(23)
The plots of these independent value distributional function estimates based on observed bids are
found in Online Appendix.
16
.(17)
4.2
Estimation Method for Risk Neutral Model
We begin by assuming bidders behave according to the simplest model, risk neutral (RN) vNM
1
function with Ut pxq x, λt pxq Ut pxq{Ut1 pxq x, and the shading function λ
t py q y. Under
the risk neutral model, by substituting estimated distributional functions, equilibrium first order
condition equations (9) under the APV assumption and (11) under the IPV assumption become
Asym,Affi
vpRN,r,m,t
bAsym
r,m,t
Asym,Inde
vpRN,r,m,t
bAsym
r,m,t
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
py
iii)
Asym,Inde
p Asym,Inde , g
p
RAsym,Inde bAsym
G
r,m,t
Bt
Bt
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
py
iv)
Asym p Asym,Affi Asym,Affi
pBt ,Bt
RAsym,Affi bAsym
r,m,t , br,m,t ΓBt ,Bt , g
(24a)
(24b)
for each type of t P tJoint, Solou. Accordingly, by substituting observed bids with estimated
distributional !functions
we
) obtain the estimates
!
) in the right hand side of the above equations,
Asym,Inde m1, ,M
Asym,Affi m1, ,M
under the IPV
under the APV assumption and vpRN,r,m,t
of valuations vpRN,r,m,t
r 1, ,R
r 1, ,R
assumption for each type of t P tJoint, Solou.
4.3 Semi & Nonparametric Estimations of Type-Homogeneous
Risk Preference Models with Exogenous Variation
In empirical auctions, the assumption of risk neutrality is justified when a bidder can be seen as a
large firm whose wealth is relatively large compared to the to the value of an object under auction.
However, in reality, a bidder is likely to be a representative of a firm whose incentives (eg. individual
bonus, promotion, or opportunity costs) depend on the result of an auction. Under such situation,
bidders are seen to have risk averse preferences, and derived model implications could be largely
different from those derived from the risk neutral model.38 Accordingly, for investigating bidders’
risk averse preferences, we now assume that bidders have preferences with type-homogeneous (i.e.
bidders share the same risk attitude within a same type) risk averse vNM functions,39 Ut pq. In Step
2, we use quantile restrictions to derive compatibility conditions, which in turn are used for semi
1
40
and nonparametrically estimating the shapes of type-homogeneous shading functions, λ
t pqs. We
Sym
introduce the notation for bi,α to denote the αth quantile for distribution of observed symmetricfor the αth
auction stage bids submitted by all types of bidders. Similarly, we denote bAsym
t,α
quantile for distribution of observed asymmetric-auction stage bids submitted by type t bidders.
In addition, we denote vi,α as αth quantile of value distribution among all types of bidders and
vt,α as αth quantile of value distribution among type t bidders. Then, the quantile notations of
equilibrium first order conditions (3) and (5) for the symmetric-auction models are
bSym
i,α
Sym,Inde
vi,α
bSym
i,α
Sym,Affi
vi,α
Sym
Sym,Affi Sym,Affi
λ1 RSym,Affi bSym
i,α , bi,α ΓBi ,Bi , gBi ,Bi
λ1 RSym,Inde
38
Sym Sym,Inde Sym,Inde
bi,α GBi
, gBi
,
(25a)
(25b)
The empirical evidences and estimates of risk averse preferences among bidders are reported by Athey and Levin
(2001) [7], Lu and Perrigne (2008) [47], and Campo, Guerre, Perrigne, and Vuong (2011) [16] in their investigations of
U.S. Forest Service timber auctions and Campo (2012) [14] in her scrutiny of Los Angeles construction contract auctions.
39
To avoid the verbal confusions, in this research, the terminologies of symmetric and asymmetric are used to express
the diversity in value distributions, while the terminologies of homogeneous (type-homogeneous) and heterogeneous are
used to express the diversity in bidders’ risk preferences.
40
Quantile restrictions with the resulting compatibility conditions are proposed by Guerre, Perrigne, and Vuong (2009)
[25], Campo, Guerre, Perrigne, and Vuong (2011) [16], Campo (2012) [14] and also used by Bajari and Hortaçsu (2005)
[10].
17
and of equilibrium first order conditions (9) and (11) for the asymmetric-auction models are
bAsym
t,α
Asym,Inde
vt,α
bAsym
t,α
Asym,Affi
vt,α
Asym
Asym,Affi Asym,Affi
1
λ
RAsym,Affi bAsym
t,α , bt,α ΓBt ,Bt , gBt ,Bt
t
1
λ
RAsym,Inde
t
Asym,Inde Asym,Inde
bAsym
, gBt
.
t,α GBt
(26a)
(26b)
Next, by exploiting the fact that the majority of bidders in experiments did not change their
valuations within a round while they submitted distinct bids in symmetric- and asymmetric-auction
stages, we take advantage of the observed differences of bids between auction stages. Note that,
as the valuations of a solo-type bidder and of one of the joint-type bidders were unchanged in
Sym
both symmetric- and asymmetric-auction stages, we have the equivalence of valuations, vr,m,i
t Asym
vr,m,t
with the slight abuse of notation i t for each type of t P tJoint, Solou, meaning that
a type t bidder did not change her valuation across auction stages. Accordingly, by using this
unchanged nature of valuations, we can apply the quantile restrictions on bidders’ private value
Asym
Sym
Asym
distribution viSym
t,α vt,α , where vit,α and vt,α denote the αth quantiles of type t bidders’
value distribution.41 Thus, for each type t P tJoint, Solou, we can equate the equilibrium first order
condition equations (??) and (26a) under the APV assumption and (??) and (26b) under the IPV
1
assumption. Then, by assuming that a type t bidder reveals the same preference Ut pq and λ
t pq
in different and exogenously changing auction stages, we have the following compatibility condition
equations for each type of t P tJoint, Solou:
Asym
bSym
it,α bt,α
Asym
Asym,Affi Asym,Affi
1
λ
RAsym,Affi bAsym
t
t,α , bt,α ΓBt ,Bt , gBt ,Bt
λt 1
Sym
Sym,Affi Sym,Affi
RSym,Affi bSym
it,α , bit,α ΓBi ,Bi , gBi ,Bi
(27a)
Asym,Inde , gAsym,Inde λ1 RSym,Inde bSym GSym,Inde , gSym,Inde ,
1
Asym,Inde
λt
R
bAsym
t,α GBt
t
it,α Bi
B t
Bi
bSym
i t,α
bAsym
t,α
(27b)
where we use the notation of bSym
it,α for the αth quantile of observed bids made by type t bidders
in symmetric-auction stages.42 In the subsequent semi and nonparametric estimations, we use a
large number of point αs ranging from 25th to 75th quantiles. Quantile points less than the
25th and larger than the 75th quantile are not used in order to avoid the well-known boundary
problems in nonparmetric kernel density estimations. In practice, we choose the sequence of quantile
points tαq uq0, ,Q! where
) αq P r0.25,
! 0.75s), and calculate the quantiles of bid distributions at these
quantile points, bSym
and
bAsym
.43 (see Appendix for a detailed description on the
t,αq
t,αq
q 0, ,Q
q 0, ,Q
constructions of quantile points.) Given the quantiles of bid distributions, in Step 2, the equation
(27a) and (27b) can be estimated by the semi and nonparametric methods explained later in this
p1,Affi pq from the equation (27a)
subsection. Once we obtain the estimates of shading functions, λ
t
p1,Inde pq from the equation (27b), in Step 3, we can obtain the estimates of valuations by
and λ
t
41
Note that, as we exploiting the invariance of laboratory assigned valuations, our quantile restrictions are unconditional. Guerre, Perrigne, and Vuong (2009) [25] Campo, Guerre, Perrigne, and Vuong (2011) [16] and Campo (2012) [14]
establish identifications based on conditional quantile restrictions (that allow, observed characteristics of auction objects,
endogenous participation, and unobserved heterogeneity) that accommodate a broad class of auction models and data
generating processes. Our unconditional restrictions in this subsection are the special case of their conditional restrictions.
42
Precisely describing, within a matched group (of three bidders), a solo-type bidder and one of the joint-type bidders
did not change valuations as depicted in Figure 2 . For the semi and nonparametric estimations of risk-averse vNM
functions, we abandon the bid data of joint-type bidders whose valuations were exogenously changed within a round.
43
To the best of our knowledge, the literature has not settled a method to choose quantile points αs (i.e. how and
how many αs a researcher should use). The recent work of Zincenko (2014) [64] establishes the identification and uniform
consistency of nonparametric λ1 pq function by using the minmax absolute distance estimator in which maximum distance
is chosen over quantile points and minimum distance is chosen over a sieve space.
18
substituting observed the bids and estimated objects in Step 1 and Step 2 in first order necessary
conditions as
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
py
iii)
Asym,Affi
vpr,m,t
bAsym
r,m,t
Asym,Affi
p1,Affi RAsym,Affi bAsym , bAsym Γ
p Asym,Affi , g
pBt ,Bt
λ
t
r,m,t
r,m,t
Bt ,Bt
Asym,Inde
vpr,m,t
bAsym
r,m,t
Asym,Inde
p1,Inde RAsym,Inde bAsym G
p Asym,Inde , g
pBt
λ
.
t
r,m,t
Bt
(28a)
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
py
iv)
(28b)
We now introduce the semi and nonparametric specifications of shading functions.
4.3.1
Semiparametric Estimation for Type-Homogeneous CRRA Model
We assume that bidders have the preference of constant relative risk averse (CRRA) vNM functions
Ut pxq xθt where 0 θt ¤ 1 for t P tJoint, Solou. The CARA model has the advantage, as it
nests the risk neutral model as the the special case of θt 1 that is empirically testable. Under
1
the CARA model, we have λt pxq Ut pxq{Ut1 pxq x{θt and the shading function, λ
t py q θt y.
Then, in Step 2, the compatibility condition equations (27a) and (27b) with estimated distributional
p Gs,
p and g
ps) for each type of t P tJoint, Solou become
functions (Γs,
!
pSym
p
pAsym R
pSym,Affi p
bSym
bAsym
pbAsym
θt RpBAsym,Affi
t,α , bt,α
t,α
it,α , bit,α
B ,B
,B
!
)
Sym
p
pAsym θt R
p
pAsym,Inde p
bSym
bAsym
RpBSym,Inde
b
εt,α ,
t,α
it,α bt,α
i
t,α
B
p
bSym
i t,α
t
t
i
t
)
i
εt,α
!
bAsym
t,αq
)
,Q
q 0,
(29b)
i
where we use the shorthand notations for simplicity. With the bid quantile data of
and
(29a)
!
bSym
t,αq
)
q 0, ,Q
, we can apply the OLS estimation to the equations (29a) and (29b) to obtain
Pt
u
θptAffi and θptInde
type of t
Joint, Solo . Consequently, in!Step 3, we) obtain the estimates of
! for each)m
1, ,M
m1, ,M
Asym,Affi
Asym,Inde
under the APV assumption and vpCRRA,r,m,t
under the IPV
valuations vpCRRA,r,m,t
r 1, ,R
assumption by applying
,R
r 1,
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
py
iii)
Asym p Asym,Inde Asym,Inde
Asym,Inde
R
br,m,t GBt
, gpBt
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
py
iv)
Asym,Affi
vpCRRA,r,m,t
bAsym
r,m,t
Asym p Asym,Affi Asym,Affi
pBt ,Bt
θptAffi RAsym,Affi bAsym
r,m,t , br,m,t ΓBt ,Bt , g
Asym,Inde
vpCRRA,r,m,t
bAsym
r,m,t
θptInde (30a)
(30b)
for each type of t P tJoint, Solou.
4.3.2
Semiparametric Estimation for Type-Homogeneous CARA Model
Next, we assume that bidders have the preference of constant relative risk averse (CARA) vNM
pζt xq . As decisions made by CARAfunctions by modeling bidders payoff functions as Ut pxq 11exp
exppζt q
preference bidders are not affected by the level of their wealth (i.e. no wealth/income effect), the
CARA model has the advantage to control bidders’ heterogeneity in their wealth levels that are
rarely observed in empirical auction research. Given the CARA model, we have λt pxq ζ1t rexppζt xq1s,
1
1
and the shading function has the form of λ
t py q ζt lnp1 ζt y q. Then, in Step 2, the compatibility
19
p Gs,
p and g
ps) for
condition equations (27a) and (27b) with estimated distributional functions (Γs,
each type of t P tJoint, Solou become
pbAsym
ζ1
t,α
p
bSym
i t,α
!
pAsym
pAsym,Affi p
bAsym
ζt R
t,α , bt,α
Bt ,Bt
ln 1
t
pbAsym
t,α
p
bSym
i t,α
1 ! ln 1
ζt
ζt pAsym,Inde p
R
bAsym
Bt
t,α
ln
ln
1
1
ζt pSym
pSym,Affi p
ζt R
bSym
it,α , bit,α
Bi ,Bi
εt,α
(31a)
)
pSym,Inde p
R
bSym
Bi
)
εt,α ,
i t,α
(31b)
where we use the shorthand notations for simplicity. With the bid quantile data of
and
!
bAsym
t,αq
)
!
bSym
t,αq
)
q 0, ,Q
, we can apply the non-linear least square (NLLS) estimation to the equations
,Q
(31a) and (31b) to obtain ζptAffi and ζptInde for! each type) of t P tJoint, Solou. Consequently, in
m1, ,M
Asym,Affi
Step 3, we obtain the estimates of valuations vpCARA,r,m,t
under the APV assumption and
r 1, ,R
!
)m1, ,M
Asym,Inde
vpCARA,r,m,t
under the IPV assumption by applying
r 1, ,R
q 0,
Asym,Affi
vpCARA,r,m,t
bAsym
r,m,t
ln 1
pAffi
1
ζt
Asym,Inde
vpCARA,r,m,t
bAsym
r,m,t
ln 1
pInde
1
ζt
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
py
iii)
Asym p Asym,Affi Asym,Affi
pBt ,Bt
ζptAffi RAsym,Affi bAsym
(32a)
r,m,t , br,m,t ΓBt ,Bt , g
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
py
iv)
Asym,Inde
p Asym,Inde , g
pBt
ζptInde RAsym,Inde bAsym
r,m,t GBt
(32b)
for each type of t P tJoint, Solou.44
4.3.3
Nonparametric Estimation for Type-Homogeneous vNM Function Model
Finally, we estimate the nonparametric (NP) vNM function model that is proposed by Guerre,
Perrigne, and Vuong (2009) [25] as it allows the most flexibility to the shapes of shading functions,
1
λ
t pq. To the best of our knowledge, this is the first applied auction work of their nonparametric
method for identifying and estimating risk-averse preferences. Based on the method indicated
1
in their seminal research, we use the sieve method to estimate the shading functions λ
t pq P
1
1
45,46,47
Λ , where Λ is a set of differentiable and (strict) monotonically increasing
functions.
In
°n
1
k
practice, we choose Λ as the set of polynomial functions Polpy; ηt,n q k1 ηt,k y without
intercept terms, where ηt,n stands for the coefficient vector of nth order polynomial. Then, in Step
2, as polynomials are linear in their coefficients, the compatibility condition equations (27a) and
p Gs,
p and g
(27a) with estimated distributional functions (Γs,
ps) for each type of t P tJoint, Solou
44
As joint-type bidders’ monetary payments are equally split, we need slight modifications to equations (31a), (31b),
(28a), and (28b) for joint-type bidders. See Appendix for details of such slight modifications.
45
Chen (2007) [17] extensively surveys the recent developments of sieve estimations.
1
46
Specifically, as we normalize a vNM function, we impose the theoretical restrictions of Restriction1: λ
t p0q 0 and
d 1
Restriction2: 0 dy λt py q ¤ 1. In programming, for Restriction1, we remove the intercept terms in polynomials. For
d 1
Restriction2, we impose the restrictions ε ¤ dy
λt py q ¤ 1 where ε 108 on the MATLAB fmincon subroutine.
47
This research takes a view that sieve is a nonparametric estimation method. Some researchers use the terminology
“semi-nonparametric” to describe sieve.
20
become
pAsym it,α bt,α
p
bSym
pAsym it,α bt,α
p
bSym
ņ
Affi
ηt,k
"
k 1
ņ
pAsym
pAsym,Affi p
bAsym
R
t,α , bt,α
Bt ,Bt
Inde
ηt,k
"
k 1
pAsym,Inde p
R
bAsym
t,α
Bt
k
k
pSym
pSym,Affi p
R
bSym
it,α , bit,α
Bi ,Bi
pSym,Inde p
R
bSym
it,α
Bi
k *
εt,α
(33a)
k *
εt,α ,
(33b)
where we use the shorthand notations for simplicity. Under this polynomial specification, Λ1 becomes the linear space, and
problems by the least square method with
! we)can solve the!minimization
)
Asym
the bid quantile data of bSym
and
b
.
In practice, we have to set a criterion to
t,αq
t,αq
q 0, ,Q
q 0, ,Q
(
Affi , η
Affi , η
Affi , , η
Affi , pt,2
pt,3
pt,n
choose the number of polynomial terms.48 For selecting ηptAffi among ηpt,1
(
Inde , η
Inde , η
Inde , , η
Inde , , we adopt a cross-validation criterion. Consept,2
pt,3
pt,n
and ηptInde among ηpt,1
!
quently, in Step 3, we obtain the estimates of valuations
!
Asym,Inde
assumption and vpCARA,r,m,t
)m1, ,M
,R
r 1,
Asym,Affi
vpCARA,r,m,t
)m1, ,M
,R
r 1,
under the APV
under the IPV assumption by applying
Asym,Affi
vpNP,r,m,t
Asym,Inde
vpNP,r,m,t
bAsym
r,m,t
bAsym
r,m,t
loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon
py
iii)
Asym p Asym,Affi Asym,Affi
pBt ,Bt
Pol RAsym,Affi bAsym
; ηptAffi r,m,t , br,m,t ΓBt ,Bt , g
looooooooooooooooooooooooooomooooooooooooooooooooooooooon
py
iv)
Asym,Inde
p Asym,Inde , g
pBt
Pol RAsym,Inde bAsym
; ηptInde r,m,t GBt
(34a)
(34b)
for each type of t P tJoint, Solou.49
48
In programming, we choose the maximum number of polynomial terms as 15.
Similar to the CARA model case, as joint-type bidders’ monetary payments are equally split, we need slight modification to equations (33a), (33b), (34a), and (34b) for joint-type bidders. See Appendix for details of such slight
modifications.
49
21
4.4 Semiparametric Estimations for Heterogeneous Preference Models with Exogenous Variation
Until this point, we have assumed bidders are type homogeneous in their risk preference (that is,
bidders have the same preference within their same type). Thus, the models so far do not allow
for heterogeneous risk attitudes among bidders; however, in reality, bidders are likely to be heterogeneous in their risk preferences,50 and the models with heterogeneous risk preferences could
potentially improve the accuracy of estimated valuations. Accordingly, to investigate the heterogeneity in bidders’ risk attitudes and resulting estimates of valuations, this section explains the
model of heterogeneous risk preferences, revealed by each bidder in the different auction stages.
For consistency of notations, we use the index tID P tt01, t02, , tIt u to denote a type t bidder
with a bidder-specific identity number (ID) where It is the total number of type t bidders in an
experiment run.51
In Step 2, heterogeneous risk preferences are easily introduced by modifying the compatibility
condition equations into the forms of bidder-specific IDs,
Asym
bSym
itID,α btID,α
Asym
Asym,Affi Asym,Affi
1
Asym,Affi
λ
bAsym
tID R
tID,α , btID,α ΓBt ,Bt , gBt ,Bt
λtID1
Sym
Sym,Affi Sym,Affi
RSym,Affi bSym
itID,α , bitID,α ΓBi ,Bi , gBi ,Bi
(35a)
Asym Asym,Inde Asym,Inde
1
1 RSym,Inde bSym GSym,Inde , gSym,Inde ,
Asym,Inde
λ
R
b
G
,
g
λ
tID
tID,α
Bt
B t
tID
itID,α
B i
B i
bSym
i tID,α
bAsym
tID,α
(35b)
Asym
where bSym
itID,α is an αth quantle of symmetric-auction stage bids submitted by bidder tID and btID,α
is an αth quantle of asymmetric-auction stage bids submitted by bidder tID that can be estimated
based on the observed bids. Then, after calculating the left hand side and right hand side variables
at each quantile point α P tα0 , α1 , , αQ u, and by modeling the heterogeneous risk preferences
1
with the CRRA payoff specification, UtID pxq xθtID and λ
tID py q θtID y, we can estimate CRRA
risk averse parameters for each bidder by applying the OLS estimation. Similarly, by modeling
pζtID xq
1
1
heterogeneous CARA payoff function, UtID pxq 11exp
exppζtID q and λtID py q ζtID lnp1 ζtID y q, we can
estimate CARA risk averse parameters for each bidder by non-linear least square (NLLS).52 Once
we obtain the heterogeneous risk averse parameters, we can estimate valuations by using equations
loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon
py
iii)
Asym,Affi
vpr,m,tID
bAsym
r,m,tID
Asym,Affi
p1,Affi RAsym,Affi bAsym , bAsym Γ
p Asym,Affi , g
pBt ,Bt
λ
Bt ,Bt
tID
r,m,tID r,m,tID
Asym,Inde
vpr,m,tID
bAsym
r,m,tID
Asym,Inde
p1,Inde RAsym,Inde bAsym G
p Asym,Inde , g
pBt
λ
.
r,m,tID
Bt
tID
loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon
py
iv)
50
(36a)
(36b)
For an example of such heterogeneity with empirical evidence, Cohen and Einav (2007) [19], by using the automobile
insurance deductible choice data in Israel, estimate the CARA risk averse parameters among auto insurance buyers and
report substantial heterogeneity in their risk preferences.
51
The complete list of bidder IDs is found in Appendix.
52
Here and in the next subsection, although it is feasible to do it for all bidders (i.e. 63 distinct bidders in our
dataset), we opt out of applying nonparmetric sieve estimations for heterogeneous risk preferences, as the execution time
of MATLAB fmincon subroutine is too long to search over the functional space of high order polynomials.
22
4.5 Semiparametric Estimations for Heterogeneous Preference Models without Exogenous Variation
This research so far has exploited the exogenous variation of auction environments. If available,
observable exogenous variations, such as exogenous changes in auction rules (as in this research and
Lu and Perrigne (2008) [47]) or exogenous changes in number of bidders (as in Bajari and Hortaçsu
(2005) [10]), are powerful tools for separately identifying risk preferences and value distributions.
However, in reality, exogenous variations are not always available in empirical auction research and
an alternative estimation method is demanded.
In order to estimate heterogeneous and individual bidder-specific risk averse preferences and also
asymmetric value distributions without exogenous variations,53 this subsection illustrates an estimation method that uses only the asymmetric-auction stage bid data.54 Specifically, we exploit the
institutional structure of the auction market. That is, the same type of bidders draw valuations
from the same value distribution, which in turn allows us to construct compatibility conditions
across the same type of bidders. The estimation method illustrated in this subsection is first proposed by Campo, Guerre, Perrigne, and Vuong (2011) [16] and Campo (2012) [14] to overcome the
empirical difficulty in observing exogenous variations, and we straightforwardly apply their method
to our laboratory auction data.55 Henceafter, we use the notation It to denote the number of type
t P tJoint, Solou bidders within each experiment run.56 We also denote Ft as a value distribution
function of type t bidders.
For a type t bidder with an identity tID P tt01, t02, , tIt u who has a bidder-specific payoff
1
function UtID pq and associated shading function λ
tID pq, when she draws the αth quantile of valuation vt,α from the distribution Ft , her equilibrium first order necessary conditions under affiliated
and independent value assumptions are written as
bAsym
tID,α
vt,α bAsym
tID,α
vt α
Asym,Affi Asym,Affi
Asym
1
Asym,Affi
λ
bAsym
tID R
tID,α , btID,α ΓBt ,Bt , gBt ,Bt
1
Asym,Inde
λ
tID R
(37a)
Asym,Inde Asym,Inde
.
,
g
bAsym
G
Bt
Bt
tID,α
(37b)
Next, for another type t bidder with an identity tID1 p tIDq who has a bidder-specific payoff func1
pq, when he draws the αth quantile of valuation
tion UtID1 pq and associated shading function λ
tID1
vt,α from the distribution Ft , his equilibrium first order necessary conditions are written as
bAsym
tID1 ,α
vt,α bAsym
tID1 ,α
vt α
Asym,Affi
1 1 RAsym,Affi bAsym
λtID
, bAsym ΓAsym,Affi , gB
tID1 ,α tID1 ,α Bt ,Bt
t ,Bt
1
RAsym,Inde
λ
tID1
Asym,Inde Asym,Inde
bAsym
G
,
g
.
Bt
Bt
tID1 ,α
(38a)
(38b)
As tID and tID1 are both type t bidders and draw their valuations from the same distribution Ft ,
this distributional structure allows us to take the advantage to equate their valuation quantiles at
a quantile point α. Accordingly, by equating (37a) and (38a) under the affiliated value assumption and (37b) and (38b) under the affiliated value assumption, we obtain intra-type compatibility
53
In order to avoid terminological confusion, this research uses type-homogeneous (homogeneous within a same type) vs
heterogeneous to describe bidders risk preferences, while symmetric vs asymmetric is used to describe value distributions.
54
Stating in a different way, in this subsection we purposefully abandon symmetric-auction stage bid data from our
dataset and derive compatibility conditions used for estimations by using only asymmetric-auction stage bid data.
55
Specifically, Campo (2012) [14] shows the substantial heterogeneity in risk preferences among construction firms who
participated procurement auctions in Los Angles.
56
For example, IJoint 14 and ISolo 7 in Experiment Run I, as listed in Table 1.
23
conditions between a pair of type t bidders as
Asym
bAsym
tID,α btID1 ,α
Asym,Affi Asym,Affi
Asym
1
Asym,Affi
λ
bAsym
tID1 ,α , btID1 ,α ΓBt ,Bt , gBt ,Bt
tID1 R
λtID1
Asym,Affi Asym,Affi
Asym
RAsym,Affi bAsym
tID,α , btID,α ΓBt ,Bt , gBt ,Bt
(39a)
bAsym
tID,α
bAsym
tID1 ,α
Asym,Inde Asym,Inde
1
Asym,Inde
, gBt
λ
bAsym
tID1 R
tID1 ,α ΓBt
λtID1
Asym,Inde Asym,Inde
, gBt
RAsym,Inde bAsym
tID,α ΓBt
.
(39b)
1
1
In the Equation (39a), there are two objects to estimate, [λ
tID pq and λtID1 pq], within one equation. Thus, we have the under-identified condition. However, as the equation (40a) can be derived
for any pair of bidders within the same type, we have It C2 intra-type compatibility conditions for
type t bidders at a specific quantile point α.57 In addition, for a given pair of same type bidders,
Equation (39a) can be calculated at a quantile point α P tα0 , α1 , , αQ u, and we can have pQ 1q
intra-type compatibility conditions. Therefore, we totally have It C2 pQ 1q intra-type compatibility
conditions that are
1
used as over-identify conditions for estimating It shading functions,
1
1
λt01 pq, λt02 pq, , λtIt pq . The identical discussion of over-identifying conditions applies for the
equation (40b).
Here, the semi-parametric CRRA model provides a particularly simple estimation method. As1
sume that a bidder tID has the payoff function of UtID pxq xθtID and associated λ
tID py q θtID y,
1
1
while a bidder tID has the payoff function of UtID1 pxq xθtID1 and associated λtID1 py q θtID1 y.
With the heterogeneous CRRA specification specification and with estimated R functions, the
compatibility conditions (39a) and (39b) becomes
pAsym
pAsym,Affi
tID,α btID1 ,α θtID1 RB ,B
p
bAsym
p
bAsym
tID,α
t
pbAsym
tID1 ,α
t
Asym
p
p
,
b
bAsym
1
1
tID ,α
Asym
pAsym,Inde p
θtID1 R
b
1
Bt
tID ,α
tID ,α
θtID RpBAsym,Affi
,B
θtID t
t
p
bAsym , pbAsym
pAsym,Inde p
R
bAsym
Bt
tID,α
tID,α
tID,α
εtID,tID1 ,α
(40a)
εtID,tID1 ,α ,
(40b)
where we use the shorthand notations and error terms are added. Intuitively speaking, since bidder
tID and tID1 both face the opponent type bidders (i.e type t bidders), and since their responses
p (which primarily consists of the distribution of
to winning probability are already captured by Rs,
opposing type t bidders’ bids), the observed difference in their bids at quantile point α in the left
hand side of the above equations is explained by the difference in their risk preferences.
Once over-identifying compatibility conditions are derived, the equations (40a) and (40b) among It
bidders can be estimated by the It -variate OLS58 for obtaining pθpt01 , θpt02 , , θptIt q, and estimated
valuations for a bidder tID are derived as
Asym,Affi,w.o.Exo
vpCRRA,r,m,tID
Asym,Inde,w.o.Exo
vpCRRA,r,m,tID
bAsym
r,m,tID
bAsym
r,m,tID
Affi,w.o.Exo
Asym
Asym p Asym,Affi Asym,Affi
Asym,Affi
p
θtID
R
br,m,tID , br,m,tID ΓBt ,Bt , gpBt ,Bt
(41a)
loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon
py
iii)
Inde,w.o.Exo
Asym
Asym,Inde
Asym,Inde
p
θptID
RAsym,Inde br,m,tID G
,
g
p
.
(41b)
Bt
Bt
loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon
py
iv)
where the index of “w.o.Exo” emphasizes the estimates are derived without exogenous changes.
For instance as IJoint 14 in Experiment Run I, we have 14 C2 91 equations of (40a) for joint type bidders at a
given quantile point α.
58
The construction of matrices used for estimation are illustrated in Appendix.
57
24
5
Estimation Results
This section reports the results of structural and nonparametric estimations under various modeling
assumptions. Here, we analyze estimates of valuations derived from asymmetric stage bid data for
comparing experimentally-assigned true and estimated valuations, as symmetric stage data are
used solely for the purpose of recovering bidders’ risk-averse preference. Regarding the model
restrictions on vNM payoff functions, we start with the Risk-Neutral model. Then, we discuss
risk-averse models in the order of CRRA, CARA, and Nonparametric models, since latter models
enjoy better fits.
Table 3: OLS Estimates of Type-Homogeneous CRRA Risk-Averse Parameters: Ut pxq xθt
Sample Used
CRRA: θ̂t
under Affiliated PV
Assumption
Bidder Type
Experiment Run I
Experiment Run II
Experiment Run III
Joint Type
Solo Type
Joint Type
Solo Type
Joint Type
Solo Type
0.1954
0.0960
0.0535
0.2271
0.0986
0.1126
(
(
(
(
(
(
0.1843,
0.0649,
0.0414,
0.1637,
0.0924,
0.1008,
0.2064)
0.1272)
0.0656)
0.2905)
0.1049)
0.1244)
CRRA: θ̂t
under Independent PV
Assumption
0.1017
0.0664
0.0330
0.0658
0.0431
0.0646
(
(
(
(
(
(
0.0931,
0.0448,
0.0262,
0.0350,
0.0373,
0.0532,
0.1102)
0.0880)
0.0397)
0.0966)
0.0489)
0.0759)
pζt xq
Table 4: NLLS Estimates of Type-Homogeneous CARA Risk-Averse Parameters: Ut pxq 11exp
exppζt q
Sample Used
Experiment Run I
Experiment Run II
Experiment Run III
Bidder Type
CARA: ζ̂t
under Affiliated PV
Assumption
Joint Type
Solo Type
Joint Type
Solo Type
Joint Type
Solo Type
0.8405
1.0257
4.3771
4.6990
1.5142
2.3819
25
CARA: ζ̂t
under Independent PV
Assumption
0.6811
4.4741
8.5899
7.0760
1.6892
1.7932
Figure 5: Comparison between CRRA and CARA Functions
5
5
Risk Neutral
Risk Neutral
4.5
4.5
4
4
ζ = 0.1250
θ = 0.9000
3.5
C ARA Payoff Function
C RRA Payoff Function
3.5
θ = 0.7500
3
2.5
θ = 0.5000
2
θ = 0.4000
2.5
ζ = 0.5000
2
ζ = 1.0000
1.5
θ = 0.2000
θ = 0.1000
1.5
1
θ = 0.0500
1
ζ = 2.0000
ζ = 4.0000
ζ = 8.0000
0.5
0.5
0
ζ = 0.2500
3
0
0
0.5
1
1.5
2
2.5
x
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
x
3
3.5
4.5
(b) CARA Payoffs: U pxq (a) CRRA Payoffs: U pxq xθ
5
p q
p q
1 exp ζ x
1 exp ζ
5
5
Risk Neutral
Risk Neutral
4.5
4.5
4
4
θ = 0.9000
C ARA λ−1 (Sha ding) Function
C RRA λ−1 (Shading) Function
4
3.5
θ = 0.7500
3
2.5
θ = 0.5000
2
θ = 0.4000
1.5
ζ = 0.1250
3.5
3
ζ = 0.2500
2.5
ζ = 0.5000
2
ζ = 1.0000
1.5
ζ = 2.0000
1
1
θ = 0.2000
0.5
0
ζ = 4.0000
0.5
θ = 0.1000
ζ = 8.0000
θ = 0.0500
0
0.5
1
1.5
2
2.5
y
3
3.5
4
4.5
0
5
0
0.5
1
1.5
2
2.5
y
3
3.5
4
4.5
5
(c) CRRA Shading Functions: λ1 py q θ y (d) CARA Shading Functions: λ1 py q 26
p
q
ln 1 ζ y
ζ
Figure 6: Risk-Neutral Model, Ut pxq x: True vs Estimated Valuations in Asymmetric Auctions
45 Degree Line
A Pair of vJoint and v̂Joint
45 Degree Line
A Pair of vJoint and v̂Joint
33
45 Degree Line
A Pair of vSolo and v̂Solo
33
30
27
27
21
18
15
12
9
24
21
18
15
12
9
Estimated Valuations (in U.S. Dollars): v̂Solo
30
27
Estimated Valuations (in U.S. Dollars): v̂Solo
30
27
24
24
21
18
15
12
9
24
21
18
15
12
9
6
6
6
6
3
3
3
3
0
0
0
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vJoint
(a) Joint Type:
Risk-Neutral
Affiliated PV
Assumption
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vJoint
(b) Joint Type:
Risk-Neutral
Independent PV
Assumption
0
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vSolo
(c) Solo Type:
Risk-Neutral
Affiliated PV
Assumption
45 Degree Line
A Pair of vSolo and v̂Solo
33
30
Estimated Valuations (in U.S. Dollars): v̂Joint
Estimated Valuations (in U.S. Dollars): v̂Joint
33
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vSolo
(d) Solo Type:
Risk-Neutral
Independent PV
Assumption
Figure 7: Type Homogeneous CRRA Model, Ut pxq xθt : True vs Estimated Valuations in Asymmetric
Auctions
39
39
Joint Type Valuation
45 Degree Line
36
39
Solo Type Valuation
45 Degree Line
36
33
30
30
24
21
18
15
12
9
27
24
21
18
15
12
9
Estimated Valuations (in U.S. Dollars): v̂i
33
30
Estimated Valuations (in U.S. Dollars): v̂i
33
30
27
27
24
21
18
15
12
9
27
24
21
18
15
12
9
6
6
6
6
3
3
3
3
0
(a)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Joint Type:
CRRA
Affiliated PV
Assumption
0
(b)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
0
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Joint Type:
(c)
CRRA
Independent PV
Assumption
Solo Type:
CRRA
Affiliated PV
Assumption
27
Solo Type Valuation
45 Degree Line
36
33
Estimated Valuations (in U.S. Dollars): v̂i
Estimated Valuations (in U.S. Dollars): v̂i
36
39
Joint Type Valuation
45 Degree Line
0
(d)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Solo Type:
CRRA
Independent PV
Assumption
Figure 8: Type Homogeneous CARA Model, Ut pxq
Asymmetric Auctions
39
39
Joint Type Valuation
45 Degree Line
p q : True vs Estimated Valuations in
p q
1 exp ζt x
1 exp ζt
39
Joint Type Valuation
45 Degree Line
36
39
Solo Type Valuation
45 Degree Line
36
33
30
30
24
21
18
15
12
9
27
24
21
18
15
12
9
Estimated Valuations (in U.S. Dollars): v̂i
33
30
Estimated Valuations (in U.S. Dollars): v̂i
33
30
27
27
24
21
18
15
12
9
27
24
21
18
15
12
9
6
6
6
6
3
3
3
3
0
0
(a)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Joint Type:
CARA
Affiliated PV
Assumption
0
(b)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
0
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Joint Type:
(c)
CARA
Independent PV
Assumption
Solo Type:
CARA
Affiliated PV
Assumption
Solo Type Valuation
45 Degree Line
36
33
Estimated Valuations (in U.S. Dollars): v̂i
Estimated Valuations (in U.S. Dollars): v̂i
36
0
(d)
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Solo Type:
CARA
Independent PV
Assumption
Figure 9: Type Homogeneous Nonparametric Payoff Function Model: True vs Estimated Valuations in
Asymmetric Auctions
39
36
Joint Type Valuation
45 Degree Line
36
39
Solo Type Valuation
45 Degree Line
36
33
30
30
24
21
18
15
12
9
27
24
21
18
15
12
9
Estimated Valuations (in U.S. Dollars): v̂i
33
30
Estimated Valuations (in U.S. Dollars): v̂i
33
30
27
27
24
21
18
15
12
9
27
24
21
18
15
12
9
6
6
6
6
3
3
3
3
0
(a)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Joint Type:
Nonparametric
Affiliated PV
Assumption
0
(b)
0
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
(c)
Joint Type:
Nonparametric
Independent PV
Assumption
Solo Type:
Nonparametric
Affiliated PV
Assumption
28
Solo Type Valuation
45 Degree Line
36
33
Estimated Valuations (in U.S. Dollars): v̂i
Estimated Valuations (in U.S. Dollars): v̂i
39
39
Joint Type Valuation
45 Degree Line
0
(d)
0
3
6
9
12
15
18
True Valuations (in U.S. Dollars): vi
Solo Type:
Nonparametric
Independent PV
Assumption
6 Tests of Independence, Distributional Equivalence,
and Error Decomposition
In this section, we examine the testable hypotheses that measures the performance of asymmetric auction estimates. We first provide the answer to the nitty-gritty of the dispute on empirical
auctions, the accuracy of asymmetric auction estimates. We then address the results from independency and asymmetry tests.
6.1 Test of Equivalence between True and Estimated Value Distributions
The core question in the empirical auction literature is the accuracy of estimated valuations. If
estimated valuations are statistically different from true ones, we should not expect any policy
insights, such as optimal reserve prices, derived are not credible as such insights are based on in
accurate estimates. For examining the equivalence between true and estimated value distributions,
we here investigate the two-sample Modified KolmogorovSmirnov statistics that is the difference
between two empirical distribution functions.
29
Table 5: :Testing Bivariate Independence: Nonparametric Blum, Kiefer, and Rosenblatt Statistics (nBn )
BKR Statistic
Experiment Run I: RM
84
Experiment Run II: RM
96
Experiment Run III: RM
72
0.0236
(0.5056)
Not Rejected
0.0350
(0.2281)
Not Rejected
0.0254
(0.4496)
Not Rejected
Note: We test the independence between estimated valuations of joint-type bidders (v̂r,m,Joint s) and solo-type bidders (v̂r,mSolo s) based on
the asymmetric stage bid data under the affiliated value assumption. The test statistics are calculated for each experiment run separately.
The Rejected/Not Rejected statuses inside of parentheses indicate the rejection of independence at 5% significance level. As a shading unction
λ 1
pq is monotonic, all risk-neutral, CRRA, CARA, and nonparametric models provide the same BKR statistics for each Experiment Run.
6.2
Test of Independence
In many empirical auction research investigations, the independency of valuations are assumed
beforehand, yet it is empirically diligent to test the independency to avoid potential modelmisspecification problem. In this subsection, we test the null hypothesis of independent valuations
against the alternative of non-independent valuations. Formally, we employ the nonparametric test
of independence proposed by Blum, Kiefer, and°Rosenblatt (1961, henceforth BKR) [13] with the
2
test statistic of 21 π 4 N BN where BN N 4 N
l1 tN1 plq N4 plq N2 plq N3 plqu with the index
l P t1, , RM u where R is the number of rounds in each experiment run and M is the number of (within-a-round) matched groups.59 Here, N1 plq, N2 plq, N3 plq, and N4 plq are defined as the
number of data points that fall in the region of tpx, y q : x ¤ Xl , y ¤ Yl u, tpx, y q : x ¥ Xl , y ¤ Yl u,
tpx, yq : x ¤ Xl , y ¥ Yl u, and tpx, yq : x ¥ Xl , y ¥ Yl u respectively.60 Table 5 reports the results of
the BKR tests.61 Independencies are not rejected in all Experiment Runs.
For instance, RM is equal to 84 (= 12 rounds 7 matched groups) in Experiment Run I.
Campo, Perrigne, and Vuong (2003) [15] use the same test statistics to investigate the affiliation of estimated
valuations Outer Continental Shelf (OCS) wildcat lease auction data and reject the null of independent valuations.
61
The p-value of the BKR statistic with 5% significance level( 2.844) is listed in Table II on p.497 of Blum, Kiefer,
and Rosenblatt (1961) [13].
59
60
30
Figure 10: Comparison between Experimental and Field Auctions
(1) Auctions
with High
Strategic Intricacies
(eg. an auction with strategic entries)
(1) Auctions with Low
Strategic Intricacies
(similar to this experimental auction research)
(3) High
(3) High
Bidder Seriousness / Monetary Stake
Bidder Seriousness / Monetary Stake
Field
Auctions
Field
Auctions
(2) Low
(2) High
(2) High (2) Low
Bidder Sophistication
Bidder Sophistication
This
Experimental
Auction
Research
(3) Low
7
(3) Low
Conclusion
To provide an answer to the criticism and skepticism on empirical asymmetric auction studies, this
research provides laboratory evidence to support the credibility of asymmetric first-price auction
estimates. We have investigated the blind spot of empirical asymmetric auction research, that
being latent valuations, and have provided new statistical and visual evidence of estimate accuracy
by manifesting how the estimates of valuations fit the true valuations. We have also shown that
incorporating the risk-aversion has nonnegligible improvements in estimates. Although the degree
of such improvements will differ by applications, the fact of improvement is translatable to other
empirical research.
7.1
External Validity
Finally, the external validity (i.e. generalizability and translatablity) of our results to other auctions, especially empirical auction research with field auction data, must be addressed. We recognize
that getting one good estimate result in one specific auction environment does not guarantee that
researchers will get a similar result in other situations. However, we can deductively bring conservative yet practical insights by contrasting our experimental auctions to field auctions in the
following ways. The experimental auctions differ from field ones in, at least, three dimensions:
(1) strategic intricacies of auctions; (2) bidder sophistication; and (3) bidder seriousness caused
by associated monetary stakes. The discussion below, as depicted in Figure 10, breaks down (1)
into two parts (i.e. high/advanced and low/similar strategic intricacies compared to this research),
then examines the generalizability of our results regarding (2) and (3).
Asymmetric auctions with high/advanced strategic intricacies, as compared to this research (left
hand side of Figure 10): If an environment of undertaking empirical asymmetric auction research
is more intricate than the one we have discussed in this research, such as endogenous and strategic
participation in auctions or binding reserve prices, our research has the limited external validity
on the accuracy of estimates. Bidders who face such a high degree of strategic intricacies may
behave differently than what we observe in this research. Further investigation on the accuracy
of estimates derived from experimental data, or any field data that directly or indirectly contains
the information of underlying valuations, will extend the results of our research for such intricate
31
auctions.
Asymmetric auctions with low/similar strategic intricacies, as compared to this research (right
hand side of Figure 10): In our experimental asymmetric auctions, the participants were recruited
undergraduate students. Thus, given the lack of their real-world business experience, their degree
of strategic sophistication is reasonably expected to be lower than the ones observed in real-world
competitive business industry (i.e. low degree of (2)). In addition, as the monetary stake in our
experiment is relatively low compared to the stakes observed in real-world auctions, the associated
seriousness among bidders in the experiment laboratory is also expected to be low (i.e. low degree
of (3)).
However, the positive finding of this research is that structural estimates derived from bids submitted by such strategically unsophisticated and less seriousness bidders are statistically shown to be
accurate. Therefore, we can deductively translate the positive finding in the accuracy of estimates
reported in this research into the estimates generated from bids submitted by professional industry
bidders in field auctions by the following reasons: first, professional industry bidders must have
a high degree of strategic sophistication in order to survive harsh industry competition (i.e. high
degree of (2)), and secondly, as the monetary stakes in real-world auctions are high, the associated
seriousness among bidder is also high (i.e. high degree of (3)). It stands to reason then, compared
to our experiment participants, such professional industry bidders are more likely to profoundly
recognize underlying strategic interactions in auctions as prescribed by BNE and less likely to
make optimization errors. Accordingly, because structural estimations are rigidly based on BNE,
the estimates derived from such sophisticated and seriously considered bids are likely to be more
accurate than those reported in this research. Thus, as far as the strategic intricacy of underlying
asymmetric auction market is not vastly different from the one discussed in this research and as far
as industry bidders are maximizing expected payoffs, we can deductively reason that what holds
accurate in our laboratory auctions also holds accurate in a real-world industry setting.
For these reasons, this research not only contributes to providing support for estimates previously reported in the literature but also pushes the credibility of present and future empirical
asymmetric auction research further.
32
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36
8
8.1
Appendix
Usage of Joint-Type Bid Data
Table 6: Used Data for Estimations of Distributional and Shading Functions
Estimating Object
Distributional Functions:
Γs, Gs, and gs
Shading Functions:
1
λ
t pqs
(with compatibility conditions)
Joint-Type Bid Data Used
Note
Chosen and announced (by the
Non-chosen and unannounced
experimental organizer) bids
bids are NOT used
Joint-Type bids made by bidders
Bids made by bidders whose
who had unchanged valuations
valuations were changed
Sym
Asym
Sym
Asym
(i.e. vr,m,it vr,m,t )
(i.e. vr,m,i
t vr,m,t ) are NOT used
8.2 A Joint-Type Bidder’s Asymmetric-Stage Payoff: An Equal
Split
The purpose of this appendix subsection is providing the detailed explanation of joint-type payoffs
in asymmetric-stage auctions on estimation methods. In our experiment, joint-type bidders equally
split their payoffs in asymmetric-auction stages. Thus, a joint-type bidder’s payoff maximization
problem is written as
"
max UJoint
bJoint
vJoint bJoint
2
FV
Solo
*
|VJoint pφSolo pbJoint q|vJoint q .
The first order necessary condition of this maximization problem is written as
vJoint
1 1
2 λ
Joint
2
bJoint
FVSolo |VJoint pφSolo pbJoint q|vJoint q
,
dFVSolo |VJoint pφSolo pbJoint q|vJoint q
1 pbJoint q
φ
Solo
dvSolo
1 pq and λ1 pq is the corresponding inverse function. By applying
where λJoint pq UJoint pq{UJoint
Joint
the same manipulations and notations as in Auction Model section, we have the simplified notation
for first order necessary condition under the APV assumption as
Asym
vJoint
bAsym
Joint
1
1 2 λ
Joint Asym Asym,Affi
Asym,Affi
RAsym,Affi bAsym
, gB
Joint , bJoint ΓB
,B
,B
2 loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon
Solo
Joint
Solo
Joint
(42)
(iii:Joint)
and under the IPV assumption as
Asym
vJoint
bAsym
Joint
1
1 2 λ
Joint Asym,Inde Asym,Inde RAsym,Inde bAsym
, gB
Joint GB
.
2 looooooooooooooooooooooooooomooooooooooooooooooooooooooon
Solo
Solo
(43)
(iv:Joint)
8.2.1
Risk Neutral Model
1
1
With the risk neutral vNM functionon, we have λ
Joint py q y. As 2 and 2 in equations (42) and (43)
are canceled out, estimated valuations are obtained by equations (24a) under the APV assumption
and (24b) under the IPV assumption.
37
8.2.2
CRRA Model
1
With the constant relative risk averse (CRRA) vNM functionon, we have λ
Joint py q θJoint y.
Affi
Similar to the risk neutral case, 2 and 21 in equations (42) and (43) are canceled out. Thus, θpJoint
Inde are obtained by the ordinary least square (OLS) with equations (29a) under the APV
and θpJoint
assumption and (29b) under the IPV assumption. In addition, estimated valuations are obtained
by equations (30a) under the APV assumption and (30b) under the IPV assumption.
8.2.3
CARA Model
1
1
With the constant relative risk averse (CRRA) vNM functionon, we have λ
Joint py q ζJoint lnp1
ζJoint y q. Unlike risk neutral and CRRA cases, 2 and 12 in equations (42) and (43) are NOT canceled
Affi and ζ
pinde are obtained by the non-linear least square (NLLS) with a
out. Thus, in Step 2, ζpJoint
Joint
slightly modified version of compatibility condition equations (31a) under the APV assumption and
(31b) under the IPV assumption as
Asym
bSym
iJoint,α bJoint,α
1
"
ζJoint
2 ln 1
ζJoint pAsym,Affi Asym Asym RBSolo ,BJoint bJoint,α , bJoint,α
2
ln
1
Sym
Sym
pBSym,Affi
ζJoint R
i ,Bi biJoint,α , biJoint,α
*
(44a)
2 ln 1
bSym
i Joint,α
1
"
ζJoint
bAsym
Joint,α
ζJoint pAsym,Inde Asym bJoint,α
RBSolo
2
ln
1
*
Sym
Sym,Inde
p
,
biJoint,α
ζJoint RBi
(44b)
where we use the shorthand notations for simplicity. Accordingly, as Step 3, estimated valuations
are obtained by the slightly modified version of equations (30a) under the APV assumption and
(30b) under the IPV assumption as
Asym,Affi
vpCARA,r,m,Joint
Asym,Inde
vpCARA,r,m,Joint
8.2.4
bAsym
r,m,Joint
bAsym
r,m,Joint
2
Affi
ζpJoint
2
Inde
ζpJoint
ln 1
ln 1
Affi
ζpJoint
2
pAsym,Affi pAsym,Affi Asym
loooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooon
RAsym,Affi bAsym
r,m,Joint , br,m,Joint ΓBSolo ,BJoint , g
BSolo ,BJoint piii:Joint)
Inde
ζpJoint
2
GpAsym,Inde , gpAsym,Inde . (45b)
looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon
RAsym,Inde bAsym
r,m,Joint
BSolo
BSolo
piv:Joint)
{
(45a)
{
Nonparametric vNM Function Model
°
n
1
k
With the nonparametric vNM functionon, we have λ
k1 ηt,k y . Thus,
Joint py q Polpy; ηt,n q Affi
Inde
in Step 2, ηpJoint and ηpJoint are obtained by the seive method with a slightly modified version
of compatibility condition equation (30a) under the APV assumption and (30b) under the IPV
assumption as
bSym
i Joint,α
bAsym
Joint,α
Asym
bSym
iJoint,α bJoint,α
n
¸
#
Affi
ηJoint,k
2
k 1
n
¸
k 1
#
Inde
ηJoint,k
2
1 pAsym,Affi Asym Asym RBSolo ,BJoint bJoint,α , bJoint,α
2
1 pAsym,Inde Asym
RBSolo
bJoint,α
2
38
k
k
k
Sym
Sym
pBSym,Affi
R
b
,
b
iJoint,α iJoint,α
i ,Bi
pBSym,Inde bSym
R
iJoint,α
i
k
(46a)
+
,
(46b)
+
where we use the shorthand notations for simplicity. Accordingly, as Step 3, estimated valuations
are obtained by the slightly modified version of equations (??) under the APV assumption and
(??) under the IPV assumption as
Asym,Affi
vpNP,r,m,Joint
Asym,Inde
vpNP,r,m,Joint
9
bAsym
r,m,Joint
1
pAsym,Affi pAsym,Affi ; ηpAffi Asym
2 Pol RAsym,Affi bAsym
Joint BSolo ,BJoint
r,m,Joint , br,m,Joint ΓBSolo ,BJoint , g
2 loooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooon
piii:Joint)
bAsym
r,m,Joint
1
pAsym,Inde , gpAsym,Inde ; ηpInde .
2 Pol RAsym,Inde bAsym
Joint BSolo
r,m,Joint GBSolo
2 looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon
piv:Joint)
{
{
Construction of Quantile Points
39
(47a)
(47b)
Figure 11: Valuations and Bids: Experiment Run I
45 Degree Line
Three-Bidder Symmetric Risk-Neutral BNE
Observed Pair of Valuation and Bid, (vi=Joint , bSym
i=Joint )
18
15
Bids (in U.S. Dollars): bAsym
i=Joint
Bids (in U.S. Dollars): bSym
i=Joint
15
12
9
6
3
0
45 Degree Line
Two-Bidder Asymmetric Risk-Neutral BNE for Joint-Type Bidders
Observed Pair of Valuation and Bid, (vi=Joint , bAsym
i=Joint )
18
12
9
6
3
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Joint
15
0
18
0
(a) Joint Type: Symmetric-Auction Stage
18
15
Bids (in U.S. Dollars): bAsym
i=Solo
Bids (in U.S. Dollars): bSym
i=Solo
15
45 Degree Line
Two-Bidder Asymmetric Risk-Neutral BNE for Solo-Type Bidders
Observed Pair of Valuation and Bid, (vi=Solo , bAsym
i=Solo )
18
15
12
9
6
3
0
6
9
12
Valuations (in U.S. Dollars): vi=Joint
(b) Joint Type: Asymmetric-Auction Stage
45 Degree Line
Three-Bidder Symmetric Risk-Neutral BNE
Observed Pair of Valuation and Bid, (vi=Solo , bSym
i=Solo )
18
3
12
9
6
3
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Solo
15
0
18
0
(c) Solo Type: Symmetric-Auction Stage
3
6
9
12
Valuations (in U.S. Dollars): vi=Solo
15
18
(d) Solo Type: Asymmetric-Auction Stage
Figure 12: Valuations and Bids: Experiment Run II
45 Degree Line
Three-Bidder Symmetric Risk-Neutral BNE
Observed Pair of Valuation and Bid, (vi=Joint , bSym
i=Joint )
18
15
Bids (in U.S. Dollars): bAsym
i=Joint
Bids (in U.S. Dollars): bSym
i=Joint
15
12
9
6
3
0
45 Degree Line
Two-Bidder Asymmetric Risk-Neutral BNE for Joint-Type Bidders
Observed Pair of Valuation and Bid, (vi=Joint , bAsym
i=Joint )
18
12
9
6
3
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Joint
15
0
18
0
(a) Joint Type: Symmetric-Auction Stage
18
15
Bids (in U.S. Dollars): bAsym
i=Solo
Bids (in U.S. Dollars): bSym
i=Solo
15
45 Degree Line
Two-Bidder Asymmetric Risk-Neutral BNE for Solo-Type Bidders
Observed Pair of Valuation and Bid, (vi=Solo , bAsym
i=Solo )
18
15
12
9
6
3
0
6
9
12
Valuations (in U.S. Dollars): vi=Joint
(b) Joint Type: Asymmetric-Auction Stage
45 Degree Line
Three-Bidder Symmetric Risk-Neutral BNE
Observed Pair of Valuation and Bid, (vi=Solo , bSym
i=Solo )
18
3
12
9
6
3
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Solo
15
0
18
(c) Solo Type: Symmetric-Auction Stage
40
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Solo
15
18
(d) Solo Type: Asymmetric-Auction Stage
Figure 13: Valuations and Bids: Experiment Run III
45 Degree Line
Three-Bidder Symmetric Risk-Neutral BNE
Observed Pair of Valuation and Bid, (vi=Joint , bSym
i=Joint )
18
15
Bids (in U.S. Dollars): bAsym
i=Joint
Bids (in U.S. Dollars): bSym
i=Joint
15
12
9
6
3
0
45 Degree Line
Two-Bidder Asymmetric Risk-Neutral BNE for Joint-Type Bidders
Observed Pair of Valuation and Bid, (vi=Joint , bAsym
i=Joint )
18
12
9
6
3
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Joint
15
0
18
0
(a) Joint Type: Symmetric-Auction Stage
18
15
Bids (in U.S. Dollars): bAsym
i=Solo
Bids (in U.S. Dollars): bSym
i=Solo
15
45 Degree Line
Two-Bidder Asymmetric Risk-Neutral BNE for Solo-Type Bidders
Observed Pair of Valuation and Bid, (vi=Solo , bAsym
i=Solo )
18
15
12
9
6
3
0
6
9
12
Valuations (in U.S. Dollars): vi=Joint
(b) Joint Type: Asymmetric-Auction Stage
45 Degree Line
Three-Bidder Symmetric Risk-Neutral BNE
Observed Pair of Valuation and Bid, (vi=Solo , bSym
i=Solo )
18
3
12
9
6
3
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Solo
15
0
18
(c) Solo Type: Symmetric-Auction Stage
0
3
6
9
12
Valuations (in U.S. Dollars): vi=Solo
15
18
(d) Solo Type: Asymmetric-Auction Stage
In Summary, we observe overbidding, bidding more than risk neutral BNE, in our laboratory
auction data, as is frequently reported in the experimental auction literature. This observed overbidding is consistent with risk-averse preferences models.
Accordingly, from empirical perspective, bidders in our dataset reveal risk averse preferences, and
we estimate their risk averse preferences thorough structural estimations and by using only bid
data.
41
Figure 14: Symmetric-Auction Stage Bids: KDEs - Affiliated Value Assumption
p Sym,Affi pbSym , bSym q
(a) Experiment Run I: Γ
i i
Bi ,Bi
Sym,Affi Sym Sym
(b) Experiment Run I: gpB
i ,Bi pbi , bi q
p Sym,Affi pbSym , bSym q
(c) Experiment Run I: Γ
i i
Bi ,Bi
Sym,Affi Sym Sym
(d) Experiment Run I: gpB
i ,Bi pbi , bi q
p Sym,Affi pbSym , bSym q
(e) Experiment Run I: Γ
i i
Bi ,Bi
Sym,Affi Sym Sym
(f) Experiment Run I: gpB
i ,Bi pbi , bi q
42
Figure 15: Symmetric-Auction Stage: R Functions - Affiliated Value Assumption
32
R̂Sym,Affi
B−i ,Bi
(·, ·) Function
R̂Sym,Affi
B−i ,Bi
(·, ·) Function Values at Observed Own Bids, bSym
i
10th, 25th, 50th, 75th and 90th Quantiles of Observed Own Bids, bSym
i
28
R Function Values
24
20
16
12
8
4
0
0
3
6
9
Own Bids: bSym
i
12
15
18
Sym,Affi Sym Sym p Sym,Affi Sym,Affi
(a) Experiment Run I: RB
i ,Bi pbi , bi |ΓBi ,Bi , gpBi ,Bi q
32
R̂Sym,Affi
B−i ,Bi
(·, ·) Function
R̂Sym,Affi
B−i ,Bi
(·, ·) Function Values at Observed Own Bids, bSym
i
10th, 25th, 50th, 75th and 90th Quantiles of Observed Own Bids, bSym
i
28
R Function Values
24
20
16
12
8
4
0
0
3
6
9
Own Bids: bSym
i
12
15
18
Sym,Affi Sym Sym p Sym,Affi Sym,Affi
(b) Experiment Run II: RB
i ,Bi pbi , bi |ΓBi ,Bi , gpBi ,Bi q
32
R̂Sym,Affi
B−i ,Bi
(·, ·) Function
R̂Sym,Affi
B−i ,Bi
(·, ·) Function Values at Observed Own Bids, bSym
i
10th, 25th, 50th, 75th and 90th Quantiles of Observed Own Bids, bSym
i
28
R Function Values
24
20
16
12
8
4
0
0
3
6
9
Own Bids: bSym
i
12
15
18
Sym,Affi Sym Sym p Sym,Affi Sym,Affi
(c) Experiment Run III: RB
pb , bi |ΓBi,Bi , gpBi,Bi q
43 i ,Bi i
Figure 16: Asymmetric-Auction Stage Bids: KDEs - Affiliated Value Assumption
(a) Experiment Run I:
p Asym,Affi pbAsym , bAsym q
Γ
BSolo ,BJoint Solo
Joint
(b) Experiment Run I:
p Asym,Affi pbAsym , bAsym q
Γ
BJoint ,BSolo Joint
Solo
Note: Flipped Plot
(c) Experiment Run I:
Asym,Affi
Asym
gpB
pbAsym
Solo , bJoint q
Solo ,BJoint
(d) Experiment Run I:
p Asym,Affi pbAsym , bAsym q
Γ
Joint
BSolo ,BJoint Solo
(e) Experiment Run I:
p Asym,Affi pbAsym , bAsym q
Γ
BJoint ,BSolo Joint
Solo
Note: Flipped Plot
(f) Experiment Run I:
Asym
Asym,Affi
pbAsym
gpB
Solo , bJoint q
Solo ,BJoint
(g) Experiment Run I:
p Asym,Affi pbAsym , bAsym q
Γ
BSolo ,BJoint Solo
Joint
(h) Experiment Run I:
p Asym,Affi pbAsym , bAsym q
Γ
BJoint ,BSolo Joint
Solo
Note: Flipped Plot
(i) Experiment Run I:
Asym,Affi
Asym
gpB
pbAsym
Solo , bJoint q
Solo ,BJoint
44
Figure 17: Asymmetric-Auction Stage: R Functions - Affiliated Value Assumption
32
28
(·, ·) Function
(·, ·) Function Values at Observed Joint-Type Bids, bAsym
Joint
R̂Asym,Affi
BJ oint ,BSolo
R̂Asym,Affi
BJ oint ,BSolo
(·, ·) Function
(·, ·) Function Values at Observed Joint-Type Bids, bAsym
Solo
10th, 25th, 50th, 75th and 90th Quantiles of Observed Joint-Type Bids, bAsym
Joint
10th, 25th, 50th, 75th and 90th Quantiles of Observed Solo-Type Bids, bAsym
Solo
24
R Function Values
R̂Asym,Affi
BSolo ,BJ oint
R̂Asym,Affi
BSolo ,BJ oint
20
16
12
8
4
0
0
3
6
9
12
Asym
Asymmetric-Auction Stage Bids: bAsym
Joint or b Solo
15
18
Asym,Affi Asym Asym p Asym,Affi Asym,Affi
(a) Experiment Run I: RB
, bt
|ΓBt,Bt , gpBt,Bt qs
t ,Bt pbt
32
28
(·, ·) Function
(·, ·) Function Values at Observed Joint-Type Bids, bAsym
Joint
R̂Asym,Affi
BJ oint ,BSolo
(·, ·) Function
R̂Asym,Affi
BJ oint ,BSolo
(·, ·) Function Values at Observed Joint-Type Bids, bAsym
Solo
10th, 25th, 50th, 75th and 90th Quantiles of Observed Joint-Type Bids, bAsym
Joint
10th, 25th, 50th, 75th and 90th Quantiles of Observed Solo-Type Bids, bAsym
Solo
24
R Function Values
R̂Asym,Affi
BSolo ,BJ oint
R̂Asym,Affi
BSolo ,BJ oint
20
16
12
8
4
0
0
3
6
9
12
Asym
Asymmetric-Auction Stage Bids: bAsym
Joint or b Solo
15
18
Asym,Affi Asym Asym p Asym,Affi Asym,Affi
, bt
|ΓBt,Bt , gpBt,Bt qs
(b) Experiment Run II: RB
t ,Bt pbt
32
28
(·, ·) Function
(·, ·) Function Values at Observed Joint-Type Bids, bAsym
Joint
R̂Asym,Affi
BJ oint ,BSolo
R̂Asym,Affi
BJ oint ,BSolo
(·, ·) Function
(·, ·) Function Values at Observed Joint-Type Bids, bAsym
Solo
10th, 25th, 50th, 75th and 90th Quantiles of Observed Joint-Type Bids, bAsym
Joint
10th, 25th, 50th, 75th and 90th Quantiles of Observed Solo-Type Bids, bAsym
Solo
24
R Function Values
R̂Asym,Affi
BSolo ,BJ oint
R̂Asym,Affi
BSolo ,BJ oint
20
16
12
8
4
0
0
3
6
9
12
Asym
Asymmetric-Auction Stage Bids: bAsym
Joint or bSolo
15
18
Asym,Affi Asym Asym p Asym,Affi Asym,Affi
(c) Experiment Run III: RB
pbt , bt |ΓBt,Bt , gpBt,Bt qs
45t ,Bt
Figure 18: Symmetric-Auction Stage Bids: ECDFs & KDEs - Independent Value Assumption
1
1
Empirical CDF of Own Bids, bSym
i
ĜSym,Inde
(·) : Empirical CDF of Highest of Opponent Bids, bSym
−i
B−i
ĜSym,Inde
B−i
(·) Function Values at Observed Own Bids, bSym
i
0.8
0.8
0.7
0.7
Kernel Density Estimates
Empirical CDF Values
0.9
0.6
0.5
0.4
0.2
0.1
0.1
0
18
1
ĜSym,Inde
B−i
(·) Function Values at Observed Own Bids, bSym
i
0.8
0.7
0.7
Kernel Density Estimates
Empirical CDF Values
18
0.6
0.5
0.4
0.4
0.2
0.2
0.1
0.1
0
18
(c) Experiment Run II:
p Sym,Inde pbSym q
p Sym,Inde pq and G
G
i
Bi
Bi
(·) Values at Observed Own Bids, bSym
i
0.5
0.3
3
6
9
12
15
Symmetric-Auction Stage Bids (in U.S. Dollars): bSym
or bSym
i
−i
ĝBSym,Inde
−i
0.6
0.3
0
Kernel Density of Own Bids, bSym
i
ĝBSym,Inde
(·) : Kernel Density of Highest of Opponent Bids, bSym
−i
−i
0.9
0.8
0
3
6
9
12
15
Symmetric-Auction Stage Bids (in U.S. Dollars): bSym
or bSym
i
−i
18
(d) Experiment Run II:
Sym,Inde
gpB
pq and gpBSym,Inde
pbSym
q
i
i
i
1
1
Empirical CDF of Own Bids, bSym
i
ĜSym,Inde
(·) : Empirical CDF of Highest of Opponent Bids, bSym
−i
B−i
0.9
ĜSym,Inde
B−i
(·) Function Values at Observed Own Bids,
0.7
0.7
Kernel Density Estimates
0.8
0.6
0.5
0.4
0.4
0.2
0.2
0.1
0.1
(e) Experiment Run III:
p Sym,Inde pq and G
p Sym,Inde pbSym q
G
i
Bi
Bi
0
18
46
(·) Values at Observed Own Bids, bSym
i
0.5
0.3
3
6
9
12
15
Symmetric-Auction Stage Bids (in U.S. Dollars): bSym
or bSym
i
−i
ĝBSym,Inde
−i
0.6
0.3
0
Kernel Density of Own Bids, bSym
i
ĝBSym,Inde
(·) : Kernel Density of Highest of Opponent Bids, bSym
−i
−i
0.9
bSym
i
0.8
0
3
6
9
12
15
Symmetric-Auction Stage Bids (in U.S. Dollars): bSym
or bSym
i
−i
1
Empirical CDF of Own Bids, bSym
i
ĜSym,Inde
(·) : Empirical CDF of Highest of Opponent Bids, bSym
−i
B−i
0.9
0
0
(b) Experiment Run I:
Sym,Inde
pbSym
q
pq and gpBSym,Inde
gpB
i
i
i
(a) Experiment Run I:
p Sym,Inde pbSym q
p Sym,Inde pq and G
G
i
Bi
Bi
Empirical CDF Values
0.4
0.2
3
6
9
12
15
Symmetric-Auction Stage Bids (in U.S. Dollars): bSym
or bSym
i
−i
(·) Values at Observed Own Bids, bSym
i
0.5
0.3
0
ĝBSym,Inde
−i
0.6
0.3
0
Kernel Density of Own Bids, bSym
i
ĝBSym,Inde
(·) : Kernel Density of Highest of Opponent Bids, bSym
−i
−i
0.9
0
3
6
9
12
15
Symmetric-Auction Stage Bids (in U.S. Dollars): bSym
or bSym
i
−i
(f) Experiment Run III:
Sym,Inde
gpB
pq and gpBSym,Inde
pbSym
q
i
i
i
18
Figure 19: Symmetric-Auction Stage: R Functions - Independent Value Assumption
32
R̂Sym,Inde
B−i
(·) Function
R̂Sym,Inde
B−i
(·) Function Values at Observed Own Bids, bSym
i
10th, 25th, 50th, 75th and 90th Quantiles of Observed Own Bids, bSym
i
28
R Function Values
24
20
16
12
8
4
0
0
3
6
9
Own Bids: bSym
i
12
15
18
Sym,Inde Sym p Sym,Inde Sym,Inde
(a) Experiment Run I: RB
pbi |GBt , gpBt q
i
32
R̂Sym,Inde
B−i
(·) Function
R̂Sym,Inde
B−i
(·) Function Values at Observed Own Bids, bSym
i
10th, 25th, 50th, 75th and 90th Quantiles of Observed Own Bids, bSym
i
28
R Function Values
24
20
16
12
8
4
0
0
3
6
9
Own Bids: bSym
i
12
15
18
Sym,Inde Sym p Sym,Inde Sym,Inde
(b) Experiment Run II: RB
pbi |GBt , gpBt q
i
32
R̂Sym,Inde
B−i
(·) Function
R̂Sym,Inde
B−i
(·) Function Values at Observed Own Bids, bSym
i
10th, 25th, 50th, 75th and 90th Quantiles of Observed Own Bids, bSym
i
28
R Function Values
24
20
16
12
8
4
0
0
3
6
9
Own Bids: bSym
i
12
15
18
Sym,Inde Sym p Sym,Inde Sym,Inde
(c) Experiment Run III: RB
pbi |GBt , gpBt q
i
47
Figure 20: Asymmetric-Auction Stage Bids: ECDFs & KDEs - Independent Value Assumption
1
1
ĜAsym,Inde
BSolo
ĜAsym,Inde
BSolo
ĜAsym,Inde
BJ oint
ĜAsym,Inde
BJ oint
0.9
0.8
(·) : Empirical CDF of Solo-Type Bids, bAsym
solo
(·) Function Values at Observed Joint-Type Bids,
bAsym
Joint
0.9
(·) : Empirical CDF of Joint-Type Bids, bAsym
Joint
(·) Function Values at Observed Solo-Type Bids, bAsym
solo
0.8
Kernel Density Estimates
Empirical CDF Values
0.6
0.5
0.4
0.2
0.1
0.1
0
0
3
6
9
12
15
Asym
Asymmetric-Auction Stage Bids (in U.S. Dollars): bAsym
Joint or b Solo
18
0
3
6
9
12
15
Asym
Asymmetric-Auction Stage Bids (in U.S. Dollars): bAsym
Joint or b Solo
18
(b) Experiment Run I:
Asym,Inde
pbAsym
q
pq and gpBAsym,Inde
gpB
t
t
t
1
1
0.9
0.8
ĜAsym,Inde
BSolo
(·) : Empirical CDF of Solo-Type Bids, bAsym
solo
ĜAsym,Inde
BSolo
(·) Function Values at Observed Joint-Type Bids, bAsym
Joint
ĜAsym,Inde
BJ oint
(·) : Empirical CDF of Joint-Type Bids, bAsym
Joint
ĜAsym,Inde
BJ oint
(·) Function Values at Observed Solo-Type Bids, bAsym
solo
0.9
0.8
ĝBAsym,Inde
Solo
(·) : Kernel Density of Solo-Type Bids, bAsym
Solo
ĝBAsym,Inde
Solo
(·) Values at Observed Joint-Type Bids, bAsym
Joint
ĝBAsym,Inde
J oint
(·) : Kernel Density of Joint-Type Bids, bAsym
Joint
ĝBAsym,Inde
J oint
(·) Values at Observed Solo-Type Bids, bAsym
Solo
0.7
Kernel Density Estimates
0.7
Empirical CDF Values
(·) Values at Observed Solo-Type Bids, bAsym
Solo
0.4
0.2
(a) Experiment Run I:
p Asym,Inde pbAsym q
p Asym,Inde pq and G
G
t
Bt
Bt
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
3
6
9
12
15
Asym
Asymmetric-Auction Stage Bids (in U.S. Dollars): bAsym
Joint or b Solo
0
18
(c) Experiment Run I:
p Asym,Inde pbAsym q
p Asym,Inde pq and G
G
t
Bt
Bt
0
3
6
9
12
15
Asym
Asymmetric-Auction Stage Bids (in U.S. Dollars): bAsym
Joint or b Solo
18
(d) Experiment Run I:
Asym,Inde
gpB
pq and gpBAsym,Inde
pbAsym
q
t
t
t
1
1
ĜAsym,Inde
BSolo
ĜAsym,Inde
BSolo
ĜAsym,Inde
BJ oint
ĜAsym,Inde
BJ oint
0.9
0.8
(·) : Empirical CDF of Solo-Type Bids, bAsym
solo
(·) Function Values at Observed Joint-Type Bids,
bAsym
Joint
0.9
(·) : Empirical CDF of Joint-Type Bids, bAsym
Joint
(·) Function Values at Observed Solo-Type Bids, bAsym
solo
0.8
ĝBAsym,Inde
Solo
(·) : Kernel Density of Solo-Type Bids, bAsym
Solo
ĝBAsym,Inde
Solo
(·) Values at Observed Joint-Type Bids, bAsym
Joint
ĝBAsym,Inde
J oint
(·) : Kernel Density of Joint-Type Bids, bAsym
Joint
ĝBAsym,Inde
J oint
(·) Values at Observed Solo-Type Bids, bAsym
Solo
0.7
Kernel Density Estimates
0.7
Empirical CDF Values
(·) : Kernel Density of Joint-Type Bids, bAsym
Joint
ĝBAsym,Inde
J oint
0.5
0.3
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ĝBAsym,Inde
J oint
0.6
0.3
0
(·) : Kernel Density of Solo-Type Bids, bAsym
Solo
(·) Values at Observed Joint-Type Bids, bAsym
Joint
0.7
0.7
0
ĝBAsym,Inde
Solo
ĝBAsym,Inde
Solo
0
3
6
9
12
15
Asym
Asymmetric-Auction Stage Bids (in U.S. Dollars): bAsym
Joint or b Solo
0
18
(e) Experiment Run I:
p Asym,Inde pq and G
p Asym,Inde pbAsym q
G
t
Bt
Bt
48
0
3
6
9
12
15
Asym
Asymmetric-Auction Stage Bids (in U.S. Dollars): bAsym
Joint or b Solo
(f) Experiment Run I:
Asym,Inde
gpB
pq and gpBAsym,Inde
pbAsym
q
t
t
t
18
Figure 21: Asymmetric-Auction Stage: R Functions - Independent Value Assumption
32
28
(·) Function
(·) Function Values at Observed Joint-Type Bids, bAsym
Joint
R̂Asym,Inde
BJ oint
R̂Asym,Inde
BJ oint
(·) Function
(·) Function Values at Observed Solo-Type Bids, bAsym
Solo
10th, 25th, 50th, 75th and 90th Quantiles of Observed Joint-Type Bids, bAsym
Joint
10th, 25th, 50th, 75th and 90th Quantiles of Observed Solo-Type Bids, bAsym
Solo
24
R Function Values
R̂Asym,Inde
BSolo
R̂Asym,Inde
BSolo
20
16
12
8
4
0
0
3
6
9
12
Asym
Asymmetric-Auction Stage Bids: bAsym
Joint or b Solo
15
18
Asym,Inde Asym p Asym,Inde Asym,Inde
(a) Experiment Run I: RB
pbt |GBt
, gpBt
qs
t
32
28
(·) Function
(·) Function Values at Observed Joint-Type Bids, bAsym
Joint
R̂Asym,Inde
BJ oint
(·) Function
R̂Asym,Inde
BJ oint
(·) Function Values at Observed Solo-Type Bids, bAsym
Solo
10th, 25th, 50th, 75th and 90th Quantiles of Observed Joint-Type Bids, bAsym
Joint
10th, 25th, 50th, 75th and 90th Quantiles of Observed Solo-Type Bids, bAsym
Solo
24
R Function Values
R̂Asym,Inde
BSolo
R̂Asym,Inde
BSolo
20
16
12
8
4
0
0
3
6
9
12
Asym
Asymmetric-Auction Stage Bids: bAsym
Joint or b Solo
15
18
Asym,Inde Asym p Asym,Inde Asym,Inde
, gpBt
qs
pbt |GBt
(b) Experiment Run II: RB
t
32
28
(·) Function
(·) Function Values at Observed Joint-Type Bids, bAsym
Joint
R̂Asym,Inde
BJ oint
R̂Asym,Inde
BJ oint
(·) Function
(·) Function Values at Observed Solo-Type Bids, bAsym
Solo
10th, 25th, 50th, 75th and 90th Quantiles of Observed Joint-Type Bids, bAsym
Joint
10th, 25th, 50th, 75th and 90th Quantiles of Observed Solo-Type Bids, bAsym
Solo
24
R Function Values
R̂Asym,Inde
BSolo
R̂Asym,Inde
BSolo
20
16
12
8
4
0
0
3
6
9
12
Asym
Asymmetric-Auction Stage Bids: bAsym
Joint or bSolo
15
18
Asym,Inde Asym p Asym,Inde Asym,Inde
(c) Experiment Run III: RB
pbt |GBt
, gpBt
qs
49 t