How to Boost revenues in First-Price Auctions? The
Magic of Disclosing Only Winning Bids from Past
Auctions
Philippe Jehiel∗
Peter Katuščák†
Fabio Michelucci‡
March 10, 2017
Abstract
We present an experiment to evaluate revenue implications of two disclosure policies available to a long-run auctioneer: disclosure of all bids from past auctions and disclosure of winning bids only. Using a first-price auction with two bidders, we find that disclosing the winning
bids leads to higher bids and revenues in the long run. We propose to explain this finding by
allowing a share of bidders to be naive in that, when presented with historical winning bids,
they mistakenly best-respond to that distribution, failing to realize that winning bids are not
representative of all bids. We also develop a method to estimate the fraction of naive bidders in
a between-subject design and to relate it to the degree of bidders’ risk aversion. Our findings
challenge the predictive power of the Bayesian Nash equilibrium based on full bidder rationality, and they underline the selection of historical price information as a key market design
choice.
Keywords: auctions, feedback, selection neglect, experiment, market design.
JEL Classification: C91, C92, D44
WORK IN PROGRESS. PLEASE DO NOT DISTRIBUTE.
∗
Paris School of Economics and Department of Economics and ELSE, University College London, Campus
Jourdan, 48 Boulevard Jourdan, Building E, 2nd floor, office 25, 75014 Paris, [email protected].
†
RWTH Aachen University, School of Business and Economics, Templergraben 64, 52064 Aachen, Germany,
[email protected].
‡
CERGE-EI, a joint workplace of Charles University and the Economics Institute of the Czech Academy of
Science, Politických vězňů 7, 111 21 Prague, Czech Republic, [email protected].
1
1
Introduction
Consider an auction house or a long-term auctioneer who repeatedly sells identical or similar
items. Typically, such auction house has gathered a lot of bidding data from past auctions
and can disclose a selective subset of these data to current bidders. A familiar example is
eBay in which bidders have access to bidding data from previous auctions of identical or
similar items. This setting invites a question of what the long-term implications of different
disclosure policies are and what their impact on the expected auction revenue is.1
This paper presents an experiment on symmetric two-bidder private-value first-price auctions in which, prior to making their bidding decisions, bidders are presented with bidding
data from previous auctions. We investigate implications of two commonly used disclosure
policies: historical information (HI) on all bids (environment A) from some previous time interval, and HI on winning bids only (environment W). An important feature of our experiment
is that it involves subjects bidding in multiple auctions over time with no feedback about the
outcome of any auction they participate in before the end of the experiment.2 The reason for
not providing feedback on auction outcomes is to avoid dynamic consideration in the underlying strategic interaction. This allows us to provide insights for auctions in which bidders
participate just once.
Our main finding is that the choice of disclosure policy has a significant effect on the
distribution of bids, both in the short-run and in the long-run, after the distribution of bids has
reached a steady-state. In particular, using the canonical design with uniformly distributed
values, bids are 6-7% under W in comparison to A. As a result, revenues are also 6-7% higher
in W relative to A.
From a theoretical perspective, the documented effect of HI on bidding is incompatible
with bidders’ full rationality (assuming that the HI dataset is arbitrarily large). In fact, rational
bidders can recover an unbiased estimate of the distribution of all bids from the distribution
of winning bids (recall the symmetry among bidders). Thus, assuming bidders are fully rational, the steady state of a dynamic process in which bidders fictitiously best-respond to the
past distribution of opponent’s bids coincides with the Bayesian Nash equilibrium (BNE) of
the underlying game under either type of HI. Moreover, this holds true irrespective of the
distribution of risk attitude of the bidders.
Our findings cannot be explained by theories that examine how bidding is affected by the
type of posterior feedback on one’s own auction via anticipated regret (Engelbrecht-Wiggans
1
Note that even though eBay is a good example of an auction house with access to a large amount of historical
bidding data, our focus on first-price auctions does not make eBay a direct application for our setting. That said,
posted prices on eBay have become predominant over auctions and the insights from our work may shed light on
how adopting different disclosure policies, such as providing all posted prices or only transaction prices, would
affect revenues.
2
The focus on one-shot bidding is similar to designs of Neugebauer and Perote (2008) and Füllbrunn and
Neugebauer (2013). Unlike our paper, however, they do not provide any HI in the course of bidding.
2
1989, Filiz-Ozbay and Ozbay 2007). This is because, in the course of the experiment, subjects
do not know the bids chosen by their opponent in their own past auctions. Therefore they
have no way to directly experience the regret of having lost the auction when they could have
profitably won it with a higher bid.3
We propose to explain the difference in the two treatments by a neglect of the selection
bias in the data in W. Specifically, the distribution of winning bids is not representative of the
distribution of all bids, as winning bids are a biased sample of all bids (this bias is clearly
toward higher bids given that the distribution of winning bids first-order stochastically dominates the distribution of all bids). First, we assume that all bidders fictitiously best-respond
to the HI presented to them. Second, we postulate that some share of subjects in W (that we
later estimate using our experimental data) erroneously assume that the shown distribution
of winning bids is representative of all bids and accordingly best-respond to this distribution
as if it were the distribution of all bids.4 Specifically, we consider a model in which bidders
may differ in their risk attitudes, and we parameterize these using the constant relative risk
aversion (CRRA) specification (Cox, Smith and Walker 1982, Cox, Smith and Walker 1992).
Under such specification, the effect of naïveté as just described is equivalent to being more
risk averse. In particular, bidders in W bid more aggressively when they are subject to the
selection neglect than when they are not, hence behaving as if they were not naive, but rather
more risk averse. As a result, the overall distribution of bids in W is shifted to the right over
time in comparison to A.
Notice that our between subjects design poses a challenge for identification of the fraction
of naive bidders. The reason is that, at the individual level, a given bid in the winning bid
treatment can be interpreted as a bid of a naive, but less risk averse bidder, or as a bid of a
sophisticated (i.e., non-naive), but a more risk averse bidder. In Section 6, we develop a procedure to overcome this identification problem using aggregate-level data. In this procedure,
we first estimate the distribution of risk aversion in A based on the assumption of fictitious
bidding. We then repeat the same exercise in W, assuming that all the bidders are sophisticated. If such assumption is correct, the two estimated distributions should be statistically
indistinguishable (we draw the subjects for the two treatments randomly from the same population). To the extent they are not, the difference reveals the degree to which bidders are
naive at different levels of risk aversion. Overall, we estimate that 57% of subjects are naive.
Additionally, we outline how to identify the extent of naïveté at various levels of risk aversion,
3
By the same token, our findings cannot be explained by learning theories based on individual auction outcome experience as in Ockenfels and Selten (2005) (see the next section for a more detailed discussion of the
related literature).
4
An alternative approach in which HI could play a role involves subjects anchoring their bids on the HI in an
attempt to imitate past behavior. We note that imitation is not easily defined in this case given that subjects are
not told the values attached to the bids shown in HI. Furthermore, if by anchoring decisions on HI one means
replicating in some way the shown HI, this may lead subjects to submit bids above their valuations, which is
intuitively unappealing and is observed only extremely rarely in our data.
3
which is an under-explored and potentially important area of research. However, the resulting
estimation procedure is data-hungry and the size of our dataset does not provide the necessary
power to draw clear conclusions on this relation. Therefore our contribution in this respect is
mainly methodological.
Our work has potentially profound implications. On the theory side, it challenges the
predictive power of the BNE based on rational bidders. It suggests that equilibrium beliefs
about the distribution of actions of the opponents may be sensitive to framing of HI about
play even if the frame is irrelevant for the content of the provided information. On the market
design side, it opens new ground for HI to have a key design role as a virtually costless and
effective instrument to raise expected revenues. In particular, our findings may shed light on
why it is a relatively common practice to disclose only winning bids in a number of auctions.5
The rest of the paper is structured as follows. Section 2 surveys the related theoretical and
experimental work and positions our paper relative to this literature. Section 3 presents the
theoretical model, which serves as the basis for our analysis of the experimental data. Section
4 presents details of our experimental design. Section 5 presents findings from the experiment.
In Section 6, we develop a strategy for identification of the overall fraction of naive bidders
and also for identifying this fraction separately for various levels of risk aversion. We also
estimate both of these objects, but, given the extent of our data, we have enough power to draw
conclusions only about the overall fraction. Finally, Section 7 concludes and suggests avenues
for further research. The Appendix contains experimental instructions and a demographic
questionnaire we used at the end of the experiment.
2
Related Literature
Our work is related to different strands of literature. First, there are earlier theoretical contributions pointing out that manipulating HI from past auctions can affect bidders’ beliefs and,
by extension, their bidding decisions. Second, there is theoretical and experimental literature
that examines the effect on bidding of feedback from auctions the bidder has, or is going to,
participate in. This literature proposes either a learning approach based on outcome impulses
from past auctions or a modification of standard preferences allowing for regret anticipation
when the outcome of the current auction is realized. Finally, there are interesting but less
related papers that deal with information disclosure in connection with strategic repeated bidding effects (absent by design in our work).6
5
For example, Priceline.com, a site matching, among others, hotels with potential customers, and allowing
them to bid a price for their desired room, provides bidders with samples of winning bids from the past. See
http://www.priceline.com/promo/hotel_winning_bids.html for more information.
6
It should be mentioned that we do not refer to the vast literature documenting overbidding in experimental
first-price auctions, nor to the various explanations (e.g. risk aversion, joy of winning, level-k, quantal response
equilibrium) that have been proposed to account for this observation.
4
The first strand of literature is based on the application of the analogy-based expectation
equilibrium (ABEE) (Jehiel 2005) to auctions (Jehiel 2011).7 A feature that is common with
our approach is that bidders fictitiously best respond to the HI they are given. However, the
manipulation of HI differs. In our setting, bidders receive HI only on the auction type they
participate in, but they might receive a selected subset of that information. In Jehiel (2011),
bidders receive complete HI, but pooled over different types of auctions or different types of
bidders. Hence the effects present in this approach are different from the ones considered
in our setting. Another approach is proposed by Esponda (2008), who applies the concept
of self-confirming equilibrium (SCE) (Battigalli 1987, Fudenberg and Levine 1993, Dekel,
Fudenberg and Levine 2004) to first-price auctions in a context where bidders play repeatedly
and they learn from their own experience either being informed of the winning bid or of all
other bids (as well as being informed of their own payoff). In a SCE, beliefs are only required
to be consistent with the feedback that players obtain. Hence, by manipulating the richness
of the feedback, one might reach SCE in which beliefs are not correct and that differ from
the BNE. However, for the independent private value setting we consider here, the BNE is the
only possible steady state under either of the two types of HI we consider.
In the second strand of literature, one line of research treats feedback as an input into
learning how to adjust one’s own bidding strategy. In contrast to our approach, each bidder is
informed about the winning bid, perhaps the second highest bid and other bids, and whether
he/she has won or lost in previous auctions before bidding in the next auction. Ockenfels and
Selten (2005) and Neugebauer and Selten (2006) consider an environment in which having
lost, but learning that the winning bid was less than one’s value, generates an impulse to bid
higher in future auctions. On the other hand, having won but learning that the second highest
bid was lower than one’s own winning bid generates an impulse to bid lower in future auctions. Equilibrium bidding is then determined by an “impulse balance” between the upward
and downward impulses on bidding. The impulse to bid higher can be increased relative to the
impulse to bid lower by providing feedback on winning bids only, thereby generating higher
revenues. One way of interpreting the origin of the bidding impulses is that they are triggered
by an ex post experience of loser and winner regret, respectively. This leads to a second related
line of research. Engelbrecht-Wiggans (1989) and Filiz-Ozbay and Ozbay (2007) propose that
ex ante regret anticipation can extend the effects of regret even to a one-shot setting. The idea
is that knowing that one will learn about the winning (second highest) bid could induce anticipation of loser (winner) regret, thereby making bidders to bid more (less). Experimental
evidence from repeated bidding (Isaac and Walker 1985, Ockenfels and Selten 2005, Neugebauer and Selten 2006, Engelbrecht-Wiggans and Katok 2008) robustly documents that providing feedback on winning bids in one’s own past auctions results in higher bids and higher
7
See also Jehiel and Koessler (2008) for an application of this idea to general Bayesian games and Huck,
Jehiel and Rutter (2011) for a related experimental investigation.
5
expected revenues compared to the case when bidders only learn whether they have won or
not.8 In contrast, the evidence from one shot bidding experiments is mixed. Filiz-Ozbay and
Ozbay (2007) find that bids and revenues are higher when feedback on the winning bid is
given as opposed to only learning whether one has won or not. On the other hand, Katuščák,
Michelucci and Zajíček (2015), in an experiment with a larger sample size and employing
various experimental protocols, do not find any effect of feedback in a one-shot setting.9
Our design intentionally isolates the effect of HI from the effect of feedback whether a
subject won or lost past auctions that the literature above has considered, and it highlights the
role of selection neglect in explaining the boost of revenues observed when only winning bids
are disclosed. More work would be required to weigh the role of selection neglect vs other
models of learning as considered by Ockenfels and Selten (2005) in contexts in which bidders
act repeatedly and observe on the spot whether they win or lose.
The third strand of literature examines situations where the same set of bidders plays the
same auction repeatedly. In such settings, there are additional strategic considerations which
may interplay with the choice of feedback type, and that are absent in our setting. Bergemann
and Hörner (2010) and Cason, Kannan and Siebert (2011) consider repeated bidding by a
fixed set of bidders, with each bidder having a value (or a cost, in procurement auctions) that
remains constant in each auction repetition. In this environment, non-anonymous HI about
previous auction repetitions creates a channel through which bidders’ bids signal their values,
which has an impact on the rest of the interaction. In our setting, these considerations are
absent since bidding groups are drawn from a large pool of bidders and HI is anonymous.
Another consideration that arises when bidders play repeatedly is tacit collusion. Dufwenberg
and Gneezy (2002) consider a repeated common value auction in which each bidding group
is randomly drawn from a fixed population.10 If all bids from previous auctions are disclosed,
bidders may attempt to bid low early on in order to induce other bidders to also bid low in
future auctions. In our setting, collusion-building is unlikely to operate. First, given that the
HI is anonymous, it is virtually impossible to identify whether a group of bidders is attempting
to initiate collusion. Second, with private values, a low bid might very well be competitive if
the bidder’s (unobserved) value is low, making it hard to detect deviations. Finally, Sonsino
and Ivanova-Stenzel (2006), motivated by observing eBay, test whether subjects searching for
more HI perform better.11 This is a complementary question to the one that we address in
which the same HI is imposed on all bidders and the effect of varying the richness of HI on
bidding and revenues is analyzed instead.
8
There is weaker (mixed) evidence from studies employing repeated bidding that feedback on the second
highest bid in one’s own past auctions results in less aggressive bidding and lower expected revenues.
9
For similar null findings, see also Nagatsuka (2016) and Ratan and Wen (2016).
10
The common value is constant and publicly known.
11
The authors provide evidence that bidders who use the option to retrieve information and who search with
more effort obtain higher payoffs on average.
6
3
Theory
Consider a first-price auction with two bidders who simultaneously submit a non-negative bid.
The winner is the bidder who submits the highest bid (in case of ties, the winner is drawn at
random) and the price paid is the bid submitted by the winning bidder. Each bidder has a
two-dimensional type (later a three-dimensional type, see below). The first dimension of the
type is given by the bidder’s valuation vi . The second dimension is given by the bidder’s risk
attitude. Following Cox et al. (1982), we parameterize the degree of risk aversion (loving)
using the constant relative risk aversion (CRRA) utility function, letting u(z) = z αi , αi > 0,
where 1 − αi is the CRRA coefficient of bidder i. The bidders are assumed to be identical
ex ante in that their type distributions are identical. Moreover, we assume that the type draws
of the two bidders are independent (implying private values) and that a bidder’s value is independent of his risk attitude. In general, valuations are drawn from distribution G with a
cumulative distribution function (cdf) G(∙) with the density g(∙).12 In the experimental design,
the valuations are drawn from the uniform distribution on [0, 100]. Some of our theoretical developments also utilize this distribution (on [0, 1], in that setting). The risk attitude parameter
αi is drawn from a distribution F . We do not require that F to have a density. We therefore
allow for lumps in the distribution of risk aversion (for example, everyone having the same
CRRA coefficient). In some of our theoretical results, we assume that there is an upper bound
α < ∞ on the support of α, i.e., that there is a limit to the extent of risk loving. We will
estimate the distribution of ln(α) in Section 6.
The rules of the auction together with the assumptions on the distributions of valuations
and risk attitudes, if these are assumed to be common knowledge, define the underlying
Bayesian game. We will refer to the symmetric Bayesian Nash equilibrium (BNE) of this
game in our theoretical developments. However, rather than assuming common knowledge of
these distributions and therefore relying on the BNE as an empirical prediction, we utilize it
as a theoretical benchmark.
Instead, our main interest is to understand the effect of providing different types of historical information (HI) to bidders. In one case, referred to as the A environment, bidders
are provided with a complete data set of past bids (from identical auctions played during a
certain period in the recent past). In another case, referred to as the W environment, bidders
are provided with only a selection of the dataset, namely, only the winning bids. Since we do
not assume that the distributions of values and risk attitudes are commonly known, the provided HI is the only source of information bidders have to form their beliefs about the bidding
behavior of their opponent. We assume that bidders believe that the provided HI captures how
their opponent is going to bid. Accordingly, bidders bidding in period t fictitiously best reply
12
We refer to a distribution by the capital letter, to the corresponding cdf by this letter complemented by “(∙)”
and to the corresponding density by the corresponding lower-case letter complemented by “(∙)”.
7
to the HI from auctions in period t − 1 (all of them, not only those auctions in which they
participated).13
Specifically, let At−1 be the distribution of all bids that a bidder observes in environment
A at a generic time period t − 1. Analogously, let Wt−1 be the distribution of winning bids
in environment W at t − 1. In environment A and time period t, a bidder i of type (vi , αi )
best-responds to the distribution At−1 by picking a bid
αi
bA
t (vi , αi ) = argmax (vi − b) At−1 (b).
(1)
b
In environment W , the choice of the bidding response depends on a bidder’s ability to correctly infer how the probability of winning depends on his or her bid. For this purpose, we
assume that bidders may be of two types. First, naive bidders consider the distribution Wt−1
to be representative of all bids. Hence, a naive bidder i of type (vi , αi ) best-responds to the
distribution Wt−1 by picking a bid
αi
bN
t (vi , αi ) = argmax (vi − b) Wt−1 (b).
(2)
b
That is, naive bidders are subject to the selection neglect, and best-respond to Wt−1 as if it
were At−1 . Second, sophisticated bidders are able to infer the distribution of all bids from
the distribution of winning bids and best-respond to the former. In particular, given the cdf of
p
winning bids Wt−1 (∙), the cdf of all bids is given W̃t−1 (∙) ≡ Wt−1 (∙). Hence, a sophisticated
bidder of type (vi , αi ) best-responds to the distribution Wt−1 by picking a bid
bSt (vi , αi ) = argmax (vi − b)αi
b
p
Wt−1 (∙)
≡ argmax (vi − b)αi W̃t−1 (∙).
(3)
b
The binary naivete type enlarges the bidder type space to three dimensions. In parallel to
valuations and risk attitudes, we assume that the presence or absence of naivete is independent
across bidders. Moreover, we assume that it is also independent of valuations. However, we
allow the incidence of naivete to be correlated with the bidder’s risk attitude. In particular,
let λ(α) be the fraction of naive bidders (i.e., the probability of being naive) among bidders
whose risk attitude is given by α. While the shape of λ(∙) affects the sequence of (winning)
bid distributions in environment W , it has no impact in environment A. Moreover, if all
bidders were sophisticated in W , there would be no difference between environments A and
W (ignoring the possible effect of a small sample size). We will thus attribute the observed
difference between A and W as resulting from the fact that λ(∙) is bounded away from 0 and
13
Note that since we are mainly interested in steady state comparisons, we do not model in detail the underlying
learning process. In particular, we are not specific about how bidders would bid at an initial period for which no
previous HI is available. For instance, one could assume that a bidder’s bid is a random draw from U [0, vi ].
8
we will suggest an empirical method to estimate λ(∙) in Section6.
A simple observation is that, in environment W , a naive bidder of type (vi , αi ) never bids
less, and generically bids more, than a sophisticated bidder of the same type irrespective of
the shape of Wt−1 (∙). To see this, observe that the problem of a sophisticated bidder of type
(vi , αi ) is to maximize αi ln(vi −b)+1/2 ln Wt−1 (b) with respect to b. The analogous problem
of a naive bidder of the same (vi , αi ) type is to maximize αi ln(vi − b) + ln Wt−1 (b), which is
the same as maximizing (αi /2) ln(vi − b) + 1/2 ln Wt−1 (b). As a result, the marginal expected
utility of a bid of a naive bidder is always at least as high as, and typically strictly higher than,
the one of the sophisticated bidder. Another observation that follows from this comparison is
that a naive bidder of type (vi , αi ) bids identically to a sophisticated bidder of type (vi , αi /2),
irrespective of the shape of Wt−1 (∙).14
Given the type distribution (including the incidence of naivete) and given some initial
distribution functions A0 (∙) and W0 (∙), the equations (1)-(3) define sequences {At (∙)}t>0 and
{Wt (∙)}t>0 . Our main interest is the comparison of the long run behavior of the two dynamic
systems. Assuming convergence, this boils down to the comparison of the steady states of the
two systems, in which the distributions of bids, denoted by A∞ and W̃∞ , respectively, are the
same in all periods.15
The steady states in environment A correspond to the BNE of the corresponding Bayesian
game and these have been analyzed by Cox et al. (1982). While no close form solution is
available in general, we will refer to BN E(F ) as the corresponding BNE (assumed to be
unique), recalling that F indexes the distribution of risk attitude in the population.16
The proposition below describes how the steady state in environment W relates to the one
in environment A, indicating that the effect of naïveté corresponds to a downward shift in the
distribution of the risk attitude parameter in the sense of first order stochastic dominance.
Proposition 1. Let BN E(F ) be the steady state in environment A, where F is the distribution of the risk attitude parameter. Then the steady state in environment W with the
same distribution of the risk attitude parameter is given by BN E(F ∗ ), where F ∗ (α) =
R 2α
F (α) + α λ(z)dF (z).
Proof. This is an immediate consequence of the observation that in environment W a naive
bidder of type (vi , αi ) behaves like a sophisticated bidder of type (vi , αi /2), thereby implying
that the steady state in environment W coincides with the steady state in environment A in
14
A byproduct of the latter observation is that when observing a bid b in environment W , it is impossible to
determine whether the bidder who placed the bid is naive or sophisticated. Since subjects in our experiment did
not participate in both environments A and W , there is no way to identify whether a given subject is naive or
sophisticated from our data.
15
More work is required to analyze under what circumstances the proposed fictitious play dynamics converges.
When analyzing our data, we will check whether the distribution of bids varies less over time toward the end of
the experiment. Such finding would suggest that the convergence assumption might be legitimate.
16
The distribution of valuations G is not varied in our analysis, thereby explaining why we do not index the
BNE by G.
9
which the mass of bidders with the risk attitude parameter less than α is given by F ∗ (α). Q.
E. D.
The observation that naivete corresponds to a downward shift in the distribution of the
risk attitude parameter together with the intuition that a higher risk aversion (lower αi ) corresponds to higher bids given valuations suggests that the steady state distribution of bids in
environment W first-order stochastically dominates the analogous distribution in environment
A. It turns out that verifying this conjecture is quite difficult in general. However, the following proposition establishes the result for the case of uniformly distributed valuations (on the
interval [0, 1], without loss of generality) and an upper bound on the degree of risk loving in
the population.
Proposition 2. Suppose that valuations are distributed uniformly on [0, 1] and that there exists
α < ∞ such that F (α) = 1 (i.e., there is an upper bound α on the risk attitude parameter
in the population). Then W̃∞ first-order stochastically dominates A∞ . By extension, the
distribution of auction revenue in the steady state of environment W first-order stochastically
dominates the distribution of revenue in the steady state of environment A, implying a higher
expected revenue in the steady state of environment W .
Proof. By Proposition 1, A∞ is given by the equilibrium distribution of bids in BN E(F )
and W̃∞ is given by the equilibrium distribution of bids in BN E(F ∗ ). Begin by focusing on
A
environment A. It can be shown that the support of A∞ is an interval [0, b ] and that A∞ has
a continuous positive density on its support.17 As a result, the optimal bid of a bidder of type
First, note that A∞ (b) > 0 for any b > 0. If there existed b > 0 such that A∞ (b) = 0, then there would be
a positive mass of bidders with valuation less than b who are overbidding their valuations and risking winning
the auction with a loss. But all these bidders could avoid such fate by bidding up to their value instead, a
contradiction. Second, the previous observation implies that every bidder with a positive valuation has a positive
expected utility when bidding above 0 and below his valuation, and any such bid dominates bidding at or above
his valuation. As a result, the optimal bid of a bidder of type (vi , αi ) for any vi > 0 is strictly below vi . Third,
note that A∞ cannot have any mass points and therefore A∞ (∙) must be continuous, with A∞ (0) = 0. If there
existed a probability mass point at b, then any bidder bidding at b ((almost, if b = 0) all of whom, by the previous
argument, have a valuation higher than b) could, by nudging his bid to slightly more than b, discontinuously
improve his probability of winning while only continuously decreasing his payoff conditional on winning, hence
improving his expected utility, a contradiction. Fourth, there can be no gaps in the support of A∞ , implying that
A
A
its support is an interval [0, b ] and that A∞ (∙) is strictly increasing on [0, b ]. If there existed b1 < b2 such that
0 < A∞ (b1 ) = A∞ (b2 ) < 1 and A∞ (b2 ) < A∞ (b) for all b > b2 , then, for a sufficiently small ε > 0, anyone
bidding in the interval [b2 , b2 + ε] could improve his payoff conditional on winning by bidding b1 instead, while
decreasing his probability of winning only slightly, overall improving his expected utility, a contradiction. Fifth,
since A∞ (∙) is continuous and strictly increasing, it has a well-defined positive density a∞ (∙) almost everywhere.
A
In fact, must have a continuous density on [0, b ]. If there were a point b with an upward jump in the density, then
anyone who found it optimal to increase his bid up to within ε below b for a sufficiently small ε > 0 would find
it advantageous to increase his bid further to above b, a contradiction. If, on the other hand, there were a point b
with a downward jump in the density, then there would have to be a mass point of bids at b, a contradiction. To
see this, consider the set of types who bid in the interval (b, b + ε) for a sufficiently small ε > 0. Now consider
another set of types that is obtained from the former set by reducing each bidder’s valuation by δ > 0 large
enough so that a bid above b is no longer optimal but at the same time small enough so that a bid below b is not
optimal either. Because the latter set has a positive mass, there would have to be a mass point of bids at b.
17
10
(vi , αi ) with vi > 0 is strictly between 0 and vi (bidding 0 leads to a zero chance of winning,
bidding at or above vi leads to a zero or a negative surplus conditional on winning, with a
positive chance of winning). Focusing on this region, the marginal log-expected utility of a
bid is given by
∂
αi
a∞ (b)
ln [(vi − b)αi A∞ (b)] = −
+
.
(4)
∂b
vi − b A∞ (b)
At this point, we assume that A∞ (∙) is log-concave, implying that a∞ (∙)/A∞ (∙) is strictly
decreasing. Under this assumption, the derivative in (4) is strictly decreasing in b, positive for
b = 0 and negative as b approaches vi . Hence the optimal bid bA
∞ (vi , αi ) is implicitly given by
vi = bA
∞ (vi , αi ) + αi
A∞ [bA
∞ (vi , αi )]
.
A
a∞ [b∞ (vi , αi )]
(5)
Note that since the derivative in (4) is strictly increasing in vi , bA
∞ (vi , αi ) is strictly increasing
in vi . Moreover, since the derivative in (4) is strictly decreasing in αi , bA
∞ (vi , αi ) is strictly
decreasing in αi . That is, less risk-averse bidders bid less and more risk-averve bidders bid
more for the very same valuation.
Now consider what types of bidders would submit a particular bid b. Equation (5) implies
that it would be all types (vi , αi ) that satisfy
vi = b + αi
A∞ (b)
.
a∞ (b)
(6)
It follows that if the risk attitude parameter were unbounded, the highest risk attitude type αi
that would ever submit the bid b is given by
RA (b) ≡ (1 − b)
a∞ (b)
.
A∞ (b)
(7)
Note that since a higher bid at vi = 1 requires a lower value of αi , RA (∙) is strictly decreasing.
Given these observations, it follows that the mass of bids equal to or higher than b is
A∞ (b)
1−b−α
dF (α)
1 − A∞ (b) =
a∞ (b)
0
"
#
Z RA (b)
1
= (1 − b) F [RA (b)] − A
αdF (α)
R (b) 0
EF [αi |αi ≤ RA (b)]
A
,
= (1 − b)F [R (b)] 1 −
RA (b)
Z
RA (b)
where EF (∙) is the (conditional) expectation of αi based on the distribution F . This implies
that
EF [αi |αi ≤ RA (b)]
A
A∞ (b) = 1 − (1 − b)F [R (b)] 1 −
.
(8)
RA (b)
11
Now, as suggested by Cox et al. (1982), denote by bA the highest bid ever submitted by any
bidder with the risk attitude parameter α. This must be any bidder of the type (1, α). Therefore
bA is implicitly given by
(9)
RA (bA ) = α.
As a result, for any b ≤ bA ,
EF [αi |αi ≤ RA (b)] = EF (αi )
and
F [RA (b)] = 1,
and therefore
EF (αi )
A∞ (b) = 1 − (1 − b) 1 − A
R (b)
A∞ (b)
= b + EF (αi )
.
a∞ (b)
This is a first-order differential equation that has a closed-form solution
A∞ (b) = [1 + EF (αi )]b.
(10)
This implies, among other things, that up to the bid of bA , each bidder’s bidding strategy is
linear in valuation, in particular
bA
∞ (vi , αi ) =
vi
.
1 + αi
(11)
We could carry out the analysis in environment W analogously and, in parallel to (10),
arrive at the conclusion that, for bids b ≤ bW , where bW is the highest steady-state bid of
bidders with αi = α, defined analogously to (7) and (9),
W̃∞ (b) = [1 + EF ∗ (αi )]b.
(12)
By construction (see Proposition 1), F first-order strictly stochastically dominates F ∗ , implying that EF ∗ (αi ) < EF (αi ), and therefore
W̃∞ (b) < A∞ (b)
(13)
A
W
for all b ∈ (0, min{bA , bW }]. Now suppose that there exists b ∈ (min{bA , bW }, max{b , b })
such that W̃∞ (b) = A∞ (b). Then, because both W̃∞ (∙) and A∞ (∙) are continuously differentiable, and because of (13), it must be the case that w̃∞ (b) ≥ a∞ (b). But then, by (4) and its
12
analogue for environment W , the types of sophisticated bidders who bid more than b in environment A also do so in environment W . Moreover, there is a positive mass of naive bidders
who bid up to b in environment A, but would bid more than b in environment A if their risk
attitude parameter αi were halved, and who therefore also bid more than b in environment W .
As a result, the mass of bidders bidding more than b is higher in environment W than it is in
environment A, implying that W̃∞ (b) < A∞ (b), a contradiction. Therefore, due to (13) and
due to continuity of both W̃∞ (∙) and A∞ (∙), it must be that case that W̃∞ (b) < A∞ (b) for all
W
b ∈ (0, b ). That is, W̃∞ first-order stochastically dominates A∞ .
By extension, the cumulative distribution function of the winning bid, and hence rev2
enue, is given by W̃∞
(∙) = W∞ (∙) in environment W and by A2∞ (∙) in environment A, with
W
2
(b) < A2∞ (b) for all b ∈ (0, b ). As a result, the former distribution first-order stochastiW̃∞
cally dominates the latter distribution. Q. E. D.
The following example illustrates the comparison of the two steady states.
Example 1. Assume that vi is uniformly distributed on [0, 1] and all the bidders have the same
risk attitude parameter α. Also, let λ ≡ λ(α) ∈ (0, 1). Then in environment A,
A∞ (b) = (1 + α) b,
with the support of A∞ being [0, 1/(1 + α)]. The steady-state bidding function is given by
bA
∞ (vi ) =
In environment W ,
W̃∞ (b) =
vi
.
1+α
λα
1+α−
2
b
for b ∈ [0, 1/(1 + α)]. For b > 1/(1 + α), W̃∞ (∙) is characterized by the differential equation
w̃∞ (b) =
with the initial condition
W̃∞
λα
W̃ (b)
2 W̃ (b) − λb − (1 − λ)
1
1+α
=1−
λα
.
2(1 + α)
The steady-state bidding function of the sophisticated bidders is given by
bS∞ (vi ) =
vi
.
1+α
For the naive bidders, for vi ≤ (1 + α/2)/(1 + α), it is given by
bN
∞ (vi ) =
vi
.
1 + α/2
13
Figure 1: This figure illustrates Example 1. bA
∞ (∙) is the steady-state bidding function of all
S
N
the bidders in environment A. b∞ (∙) and b∞ (∙) are the steady-state bidding functions of the
sophisticated and naive bidders, respectively, in environment W . The shape of bN
∞ (∙) when its
value exceeds 1/(1 + α) is unknown. We draw it as a concave function based on simulations
in Van Boening et al. (1998).
For vi > (1 + α/2)/(1 + α), it is (implicitly) given by
−1
bN
∞ (vi ) = W̃∞ (λvi + 1 − λ).
Figure 1 plots the steady-state bidding functions for both types of bidders in Example 1.
4
Experimental Design
Our experiment involves repeated two-bidder first-price auctions (FPAs) in two treatments: HI
on all bids (treatment A) and HI on winning bids (treatment W ). We first describe a representative session from the perspective of a subject receiving information about the experiment.
After this description, we discuss specific design choices and noise reduction measures.
After entering the lab and taking their seats, subjects in both treatments were given the very
same printed instructions. They also received a sheet of paper and a pencil in case they wanted
to take notes. The instructions explained (in non-technical terms) that the 24 participants in
14
the session were divided into 2 groups of 12 and that members of each group would interact
only among themselves. This interaction consisted of rounds of first-price auctions, each time
against one randomly determined opponent from the same group. The subjects were explained
the rules of the auction, the fact that they would play 11 blocks of 6 auction rounds, and that
everyone would get paid on the basis of the same 5 randomly chosen rounds at the exchange
rate of 1 point=10 CZK (Czech crowns). On top of that, each subject knew he/she would be
paid a 100 CZK show-up fee.18
The instructions informed subjects that values in block 1 were generated independently
across subjects using the same random number generator. The instructions also stated that
value generation for later blocks would be explained at the end of block 1, that there would be
no feedback on the outcome of the individual auctions until the very end of the experiment,
and that the bidding would be followed by a demographic questionnaire. Finally, a shot of a
bidding screen used in block 1 was provided.19 We then provided subjects with a quiz consisting of eight multiple-choice questions testing subjects’ understanding of the instructions.20 In
block 1, each subject was presented with six consecutive bidding screens. Each screen listed
the subject’s value realization for that round and the subject was asked to submit a bid.
Block 1 was identical in treatments A and W . After bidding in block 1 was completed, a
second set of printed instructions was distributed. This time, the instructions were treatmentspecific. In both treatments, subjects were told that their value realizations in all subsequent
blocks would be the same, including the order, as their value realizations in block 1. At
the same time, they were reminded that the matching into bidding pairs was random and
independent in each bidding round within and across blocks. In treatment A, subjects were told
that, from block 2 onwards, they would receive historical information about the 72 bids from
the auctions played in their group in the previous block. In treatment W , subjects were told
that, from block 2 onwards, they would receive historical information about the 36 winning
bids from the auctions played in their group in the previous block. Then the subjects were
presented with a detailed explanation of the instruments used to display historical information
from the previous block (see below for details) together with a shot of a bidding screen used
in blocks 2 through 11. It should be stressed that, in treatment W , we used two different
pairings of subjects: an Earnings Pairing and an Information Pairing. The former was used
for the determination of the actual payoffs, while the latter was used for the determination of
winning bids shown in the HI. The instructions were explicit about the difference between the
two pairings.21
18
At the time of the experiment, the exchange rate was approximately 27.5 CZK/EUR and 20.1 CZK/USD.
The screen contained a link to a built-in calculator that the subjects could use while bidding.
20
The quiz is presented in the Appendix. After the subjects answered all the questions, their answers were
checked by an experimenter. Incorrect answers were infrequent. In case of an incorrect answer, an explanation
was provided and a subject was given additional time to correct the answer. The experiment proceeded into block
1 only after all the subjects answered all the questions correctly.
21
We discuss the reason for using the two separate pairings below.
19
15
Subjects had the opportunity to go over the instructions at their own pace and privately
ask questions. Then treatment-specific quizzes consisting of ten multiple-choice questions
were administrated.22 Then bidding resumed for blocks 2 through 11 and after the 66 rounds
were completed, a demographic questionnaire was distributed.23 Finally, the subjects were
presented with a feedback screen showing the subject’s bid, the opponent’s bid, and the winner
for each of the 66 auctions. The computer then determined the 5 payoff-relevant auction
rounds and displayed the results of those, including the subject’s earnings in points and in
CZK, on a new screen.
In what follows, we discuss some important design aspects in greater detail.
Earnings Pairing vs. Information Pairing:
The main reason why we used two different pairings in W was to avoid subjects using
the HI to infer whether their bids in the previous block were winning on not. Such inference
could compromise the comparison with treatment A in at least two ways. First, subjects
would become aware of their round-specific payoffs. Even though, given that the experimental
earnings are determined by five randomly chosen rounds, this is not enough to infer much
about one’s final cash earnings, such payoff information could trigger an income effect on
subsequent play. Second, bidders could realize loser regret or, coined differently, winning
and losing impulses. By avoiding such confounds, we argue that any difference of bidding
between A and W can be attributed to the difference of beliefs about the distribution of the
opponent’s bids.24
HI generation and its richness:
Two reasons led us to form HI at the level of a block and to use 6 bidding rounds in a block.
First, we wanted each subject to bid for several different values given a certain realization of
HI. This allows estimating a bidding strategy for each subject given the realization of HI.
Second, we needed to form a rich enough history of past bids. With 6 rounds of bidding in a
block, a history of all bids consists of 72 bids and a history of winning bids consists of 36 bids.
The design of HI we use represents a good compromise between achieving these objectives
and not overloading subjects with too many auctions.25
We chose to include one’s own (winning) bid in the HI. The main reason for doing so
(rather than excluding own (winning) bid from the HI shown to the subject) is that in this
22
The quizzes from both treatments are presented in the Appendix. Again, when checking on the quiz answers,
we observed very good understanding of the instructions.
23
Because of the necessity to collect all the bidding information for the previous block, bidding moved to the
next block only when all the subjects in a given group had completed bidding in the previous block.
24
An alternative way to exclude effects related to feedback from own past auctions would be to exclude the own
auction outcomes from HI provided to a subject. However, this has the undesirable effect of different subjects
being exposed to different subsets of HI (see the discussion below for an elaboration).
25
We use 11 blocks because our extensive pilot experiment testing had shown that the distribution of bidding
function slopes stabilizes around blocks 9 and 10 and there might sometimes be a last-block effect in terms of
subjects bidding slightly less.
16
way all subjects receive the same HI. This facilitates comparison of bidding strategies across
subjects. Another reason is that in field applications, auctioneers would most likely provide
aggregate HI, without filtering out previous bids of a given bidder.
Elimination of repeated bidding effects:
In our experiment, subjects bid repeatedly rather than just once. We use two measures to
avoid repeated bidding effects. First, in each bidding round, a subject’s opponent is chosen
randomly and anonymously from among the other 11 group members using a new, independent draw. Second, subjects do not receive any feedback on the outcome of the auctions they
have played until the very end of the experiment. An alternative way of precluding potential
repeated bidding effects would be to use cohorts of one-shot bidders, each having access to HI
on bidding of the previous cohort. However, such design would be very demanding in terms
of the number of subjects and therefore exceedingly costly.26
Display of HI to subjects:
In terms of displaying HI to subjects, the main trade-off is between presenting something
easy to understand on the one hand, and providing sufficiently detailed information about the
distribution of (winning) bids on the other hand. We achieve both objectives by presenting
the HI in two formats. In order to provide an overall assessment of what the (winning) bid
distribution in the previous block looks like, the subjects are provided with a histogram with
10 bars, each representing the percentage of all (winning) bids placed within the bins [0, 10],
(10, 20],...,(80, 90] and (90, ∞). We use a histogram since it should be familiar to most subjects from textbooks, magazines, newspapers or introductory statistical classes. In order to
allow subjects to recover arbitrarily fine details of the distribution, we provide HI also in a
second format. By inputting numbers into two boxes, subjects can recover an exact percentage of (winning) bids between the two numbers. Moreover, entering only one number in the
left (right) box, they can obtain an exact percentage of (winning) bids at and above (below)
that number.
Lack of information about distribution of opponents’ values:
Recall that while each subject’s value is drawn from the uniform distribution on [0, 100],
the shape of this distribution is not common knowledge. This departure from the more standard practice of announcing to subjects the distribution from which the values are drawn is
motivated by two considerations. First, we want to stay close to the theoretical model introduced in the previous section in which bidders can only use HI on bids to form their beliefs.
Providing information about the distribution of values interferes with this objective by giving
subjects an additional source of information for forming their beliefs. Second, in the majority
26
The reason is that to form a rich-enough HI, one would need many subjects in each cohort. Also, it takes at
least several cohorts for bidding behavior to stabilize. On top of that, each sequence of cohorts observing HI on
the previous cohort would form one statistically independent cluster of data. To gain statistical power, one would
need at least several such clusters, hence further increasing the required number of subjects.
17
of field applications, bidders do not have information about the distribution of opponents’ valuations. If desired, such distribution would typically have to be inferred from other sources,
predominantly from HI on bids in past auctions.
Subjects’ payments:
Subject payments are determined by 5 randomly chosen rounds out of the 66 bidding
rounds. This is a reasonable compromise between minimizing hedging effects on the one
hand and providing sufficiently strong incentives and reducing luck-driven variance of subject
earnings on the other hand.
4.1
Noise Reduction Measures
Our analysis is based on comparing slopes of bidding strategies at subject-block level across
blocks and across the two treatments (see the next section). We take a number of measures to
minimize the noise involved in such comparisons.
First, estimation of bidding strategy slopes might be subject to a large degree of noise if
all 6 value realizations of a subject in a block happen to be clustered around the same low
or medium value. To avoid this, these six values, rather than being drawn independently
from the uniform distribution on [0, 100], are drawn from uniform distributions on [0, 100/6],
(100/6, 200/6], ..., (500/6, 100], respectively, and then randomly scrambled. This way the
realized values provide an even coverage of the support of the value distribution for each
subject in each block, while preserving the form of the compound distribution (uniform on
[0, 100]). Of course, the values of a single subject are not independent across the six rounds.
However, independence of generation across subjects and random scrambling of order imply
that knowing one’s value in a given round does not help to predict the values of the opponent
in that or any other round.27 Furthermore, for each subject, it is indeed true that both his and
his opponent’s value in a given round are independently drawn from the uniform distribution
on [0, 100].
Second, the variation in bidding behavior from one block to another should not be attributable to changes in value realizations or changes in the formation of bidding pairs (in
treatment W ). We take two steps to achieve this. For each subject, we repeat the value realizations and their ordering in each block. Moreover, in each block, we also repeat the pairing of
subjects for the purpose of constructing the HI on winning bids in treatment W (Information
Pairing).28
27
Subjects are explicitly informed about this fact in the instructions.
The former measure gives another layer of importance to the even coverage of the support of the value
distribution. Since value realizations repeat block after block, the even value generation avoids a scenario in
which a subject could face unluckily low value realizations round after round, block after bloc. Such subject
could easily end up frustrated due to having little chances of winning (with a big surplus), which could introduce
noise into the data.
28
18
Third, any differences in bidding behavior between the two treatments should be attributable
to the different HI provided rather than changes in value realizations. For this reason, we use
the very same value realizations, subject-by-subject and round-by-round, in all bidding groups
in both treatments. That is, in any session, a subject sitting at a terminal notionally labelled
i ∈ {1, .., 12} in treatment A has the same value realizations in each round of a block, and
these are identical to the analogous realizations of1 as a subject sitting at a terminal notionally
labelled i in treatment W .
Fourth, in order to limit noise originating from different timing of sessions for the two
treatments, each session consists of 24 subjects split into two independent groups of 12 subjects, one in each treatment. Of the 24 subjects in each session, the allocation to the two
treatments is random, subject only to gender balancing (see below).
Fifth, we also want to avoid potential noise from different gender composition of subjects
in the two treatments. This is because previous literature (Chen, Katuščák and Ozdenoren
2013, Pearson and Schipper 2013) has documented that there are gender differences in bidding
in first-price auctions with independent private values, with women typically bidding more
than men. To achieve this, for each session, we recruited the same number of men and women
by means of separate recruitment campaigns. Due to a random no-show-up, this resulted in
an approximately gender-even pool of subjects coming to the lab. In all but two sessions, we
were able to utilize 12 men and 12 women.29 Of these, 6 men and 6 women were assigned to
each treatment. Moreover, men (women) were directed to occupy the terminals with the same
notional numbers in the two treatments. This means that men (women) had the same value
realizations in the two treatments.30
4.2
Logistics and Subjects
Altogether, we ran 10 laboratory sessions of 24 subjects each, utilizing 240 subjects in total. In each session, half of the subjects participated in treatment A and the other half in
treatment W . That is, we have data on 10 independent bidding groups of 12 subjects (120
subjects in total) in each treatment. In terms of gender composition, we utilized 119 male and
121 female subjects. All the sessions were conducted between December 2013 and February 2014 at the Laboratory of Experimental Economics at the University of Economics in
Prague.31 The experiment was conducted using a computerized interface programmed in zTree (Fischbacher 2007). Subjects were recruited using the Online Recruitment System for
Economic Experiments (Greiner 2015) from our subject database. Most of our subjects were
29
One session utilized 10 men and 14 women and one session utilized 13 men and 11 women.
In the session with 10 men and 14 women, 5 men and 7 women were allocated to each treatment. In
particular, in each of the two treatments, one woman took a terminal usually allocated to men. In the session with
13 men and 11 women, 7 men and 5 women we allocated to treatment A and 6 men and 6 women were allocated
to treatment W . In the former, one man took a terminal usually allocated to women.
31
See http://www.vse-lee.cz/eng/about-lee/about-us.
30
19
students from the University of Economics in Prague. A minority were students of other
universities in Prague. According to the demographic questionnaire, at the time of the experiment, 45 percent of the subjects did not hold any degree, 46 percent held a bachelor’s degree
and 9 percent held a master’s degree. Regarding the field of study, 6 percent had a mathematics or statistics major, 7 percent had a science, engineering or medicine major, 67 percent had
an economics or business major, 8 percent had a social science major other than economics
or business, and 12 percent had a humanities or some other major. Almost 99 percent of our
subjects were between 18 and 28 years old, with three subjects being older (up to 39). Also,
47 percent of the subjects claimed to have had some experience with online auctions, 3 percent
with offline auctions and 10 percent claimed experience with both types of auctions. Subjects
were paid in cash in Czech crowns (CZK) at the end of their session. Each session lasted
approximately 2 hours with an average earning of 421 CZK.32
5
Results
This section presents our empirical findings. Subsection 5.1 discusses “steady-state” treatment effects on bidding behavior. Subsection 5.2 analyzes "steady-state” treatment effects on
average auction revenues. When presenting the results, we pay special attention to blocks 1,
10 and 11. Block 1 is treatment-free, so we can check whether the two subject groups differ
due to non-treatment reasons. We take blocks 10 and 11 to approximate the steady-state bidding behavior under the two feedback types. We consider block 10 alongside with block 11
because behavior in block 11 may be affected by last-period effects, whereas behavior in block
10 should be less so.33 When considering statistical significance, unless noted otherwise, we
employ two-sided tests at 95% confidence level. In block 1 comparisons, standard errors are
adjusted for clustering at subject level. For all other comparisons, standard errors are adjusted
for clustering at bidding group level.
5.1
Bidding
As the initial step of the analysis, we estimate the slope of the bidding function for each individual subject in each block using OLS. This measure is a summary statistic of the behavior
of a subject in a particular block. We assume that the bidding function is linear and has a
zero-intercept. With vijt denoting the value and bijt denoting the corresponding bid of subject
32
For a comparison, an hourly wage that students could earn at the time of the experiment in research assistant
or manual jobs typically ranged from 75 to 100 CZK.
33
We observed some indication of a last block effect in some pilot experiments.
20
Figure 2: Average Bidding Function Slopes by Treatment and Block
.82
Slope
.8
.78
.76
.74
1
2
3
4
5
6
Block
All Bids
7
8
9
10
11
Winning Bids
i in round j of block t, the estimate of slope for subject i in block t is given by
[ it =
slope
P6
6
X
j=1 vijt bijt
=
P6
2
j=1 vijt
j=1
2
vijt
P6
k=1
2
vikt
!
bijt
.
vijt
(14)
That is, the estimated slope is a square-value-weighted average of the six individual bid/value
ratios in a given block. We exclude 9 subjects from the subsequent analysis due to slopes of
their bidding strategies systematically exceeding 1.
Figure 2 plots the average of the bidding function slopes by treatment and block. This
figure captures the overall findings. In block 1, which is treatment-free, behavior in the two
treatments should differ only by noise. Indeed, the average slopes are very close in the two
treatments at 0.751 (with the standard error of 0.012) in treatment A and 0.737 (0.017) in
treatment W . Notice that such values are higher than in previous experiments (for instance,
Katuščák et al. (2015), who use the same estimation procedure, find an average bid value ratio
of 0.69). This could be due to the fact that in the current experiment we do not provide any
information on the distribution of values, which implies a more uncertain (and ambiguous)
environment for the subjects in the initial block. In the following blocks, a discrepancy between the two treatments arises. Subjects in W start bidding more than their counterparts in
A. In the final blocks 10 and 11, the average slope in A is 0.764 (0.010) and 0.768 (0.011),
respectively, changed little from block 1 (t-test p-values of 0.116 and 0.055, respectively). On
the other hand, the corresponding average slopes in W are 0.821 (0.016) and 0.812 (0.018),
respectively. In case of W , this constitutes a statistically significant increase over the average
slope in block 1 (t-test p-values of 0.000 in both cases). In proportional terms, bids in W in
21
Figure 3: Effect of Treatment (W − A) on the Average Bidding Function Slopes by Block
.1
.08
Slope Difference
.06
.04
.02
0
-.02
-.04
1
2
3
4
5
6
Block
Treatment Difference
7
8
9
10
11
95% Conf. Interval
blocks 10 and 11 are 7.5 percent and 5.7 percent higher, respectively, than bids in in the same
blocks of A.
Figure 3 plots the estimate of the treatment effect (W minus A) on the average slope by
block, together with its 95% confidence interval. The estimated treatment effect is positive
in blocks 2 through 11 and it is statistically significant in blocks 4, 5 and 9 through 11. By
the final blocks 10 and 11, the treatment effect reaches 0.057 (0.016) and 0.045 (0.018), respectively. Moreover, block-by-block, we compute the average bidding function slope in each
bidding group (10 in each treatment) and we perform the Mann-Whitney ranksum test for the
equality of distributions of the average slopes.34 Starting from block 4 and with the exception
of blocks 6 and 7, we reject the null hypothesis in favor of the distribution under W first-order
stochastically dominating the one under A.
Figure 2 also shows that subjects in W bid slightly less on average that subjects in A do in
block 1. This suggests that the former might bid less than the latter do if they were subjected
to treatment A instead. In order to see what the treatment effect looks like after removing
this non-treatment-related bidding difference, Figure 4 plots the difference in differences in
average bidding strategy slopes between block i and block 1. By construction, the differencein-differences is zero in block 1. In all the remaining blocks, it is positive and statistically
significant. This suggests that the lack of statistical significance for the treatment effect in
some blocks visible in Figure 2 might indeed be due to the heterogeneity of subjects across
the two treatments.
To picture the overall impact of the treatment on bidding behavior, Figure 5 presents kernel
34
Note that subjects do not interact across the individual bidding groups, so each bidding group presents a
statistically independent observation.
22
Figure 4: Effect of Treatment (W − A) on the Average Bidding Function Slopes by Block
after Subtracting the Treatment Effect Estimate in Block 1
.1
Slope D-in-D to Block 1
.08
.06
.04
.02
0
1
2
3
4
5
6
Block
Treatment D-in-D to Block 1
7
8
9
10
11
95% Confidence Interval
estimates of the probability density function (pdf) of the bidding function slopes as well as its
empirical cumulative distribution function (cdf) by the two treatments for blocks 1, 10 and
11. There is very little observable difference between the distributions under A and W in
block 1 (Kolmogorov-Smirnov test p-value of 0.645). In contrast to that, in blocks 10 and
11, with an exception of a few slope realizations below 0.5, the distribution under W firstorder stochastically dominates the distribution under A. Indeed, the Kolmogorov-Smirnov
test rejects the null hypothesis of no difference between the two distributions in all blocks
from block 3 onward (with the p-value being 0.001 or less from block 6 onward).
Put together, these findings show that in the “steady state”, as approximated by blocks 10
and 11, W robustly and significantly shifts the distribution of bidding strategy slopes to the
right relative to A. That is, given the very same values, subjects tend to bid more under W
than they do under A. This finding supports the corresponding theoretical prediction that we
laid out in Section 3.
5.2
Average Revenues
Having discussed individual bidding behavior, we now switch our attention to auction revenues. Given a set of two-bidder auctions, we define the average revenue as the average of the
winning bids in these auctions.
In each block, there are many possible ways of matching subjects into pairs within a bidding group. Just in any single round, there are 11 × 9 × 7 × 5 × 3 × 1 = 10, 395 unique ways
of matching subjects into pairs. Moreover, since the order of value presentation is randomized
23
24
Pdf
Cdf
.5
.6 .7
Slope
.8
.9
1
.4
.5
.6 .7
Slope
.8
.9
1
0
0
.4
.6
.8
1
.2
.3
Cdf, Block 1
.2
.4
.6
.8
1
0
.4
0
.3
1
2
3
4
5
1
2
3
4
Pdf, Block 1
Cdf
5
Pdf
.3
.3
.6
.7
Slope
.8
.5
.6 .7
Slope
.8
Cdf, Block 10
.5
All Bids
.4
.4
Pdf, Block 10
1
1
Winning Bids
.9
.9
0
.2
.4
.6
.8
1
0
1
2
3
4
5
.3
.3
.4
.4
.6 .7
Slope
.8
.5
.6 .7
Slope
.8
Cdf, Block 11
.5
Pdf, Block 11
Figure 5: Distribution of Bidding Function Slopes by Treatment in Blocks 1, 10 and 11
Pdf
Cdf
.9
.9
1
1
25
Winning Bids
40
All Bids
60
80
0
20
40
All Bids
80
45-degree line
60
Average Revenue, Block 10
Realizations
0
20
0
0
20
40
60
80
20
40
60
80
Average Revenue, Block 1
Winning Bids
0
20
40
60
80
0
20
40
All Bids
60
Average Revenue, Block 11
Figure 6: Average revenues across the Treatments in Blocks 1, 10 and 11
Winning Bids
80
across different rounds within a block and subjects see bidding feedback only after each block
(rather than round), one should also consider matches across different rounds of a given block.
To estimate the treatment effect on average revenue in a given block, we generate 1,000 bootstrap draws from the data on values and bids in this block. Each draw is generated as follows.
At the level of a bidding group, we first randomly (with the uniform distribution) draw one of
the 6 rounds, separately and independently for each subject. Next, we randomly pair the 12
members of the bidding group into pairs. Any pattern of pairing is equally likely. We then
use the very same pattern of selected rounds and pairs in each bidding group. Finally, we use
the values and bids from the chosen rounds and the pairing pattern to determine the auction
winners and average revenues separately in each treatment.35 In each block, each bootstrap
draw hence generates a matched pair of average revenues, one for each treatment.
The resulting bootstrap data for blocks 1, 10 and 11 is plotted in Figure 6. The scatterplot
for block 1 is symmetric around the diagonal. Precisely, the fraction of bootstrap draws for
which the average revenue in W exceeds the one in A plus one half of the fraction of cases
in which the two average revenues are equal, a measure we call “fraction revenue W > A”,
is equal to 0.394. In the absence of any difference between the two treatments in block 1, we
would expect this measure to be close to 0.5. The fact that the measure is below 0.5 is due to
the fact that subjects in W appear to bid a bit less in block 1 than subjects in A do (see Figure
2).36 In contrast to that, the vast majority of points in the scatterplots for blocks 10 and 11 lie
above the diagonal. Precisely, fraction revenue W > A is equal to 0.894 in block 10 and 0.855
in block 11. Hence, the average revenue appears to be systematically higher in W than in A.
To capture the magnitude of the difference, the average of the ratio of average revenue under
W to that under A is 1.068 and 1.058 in blocks 10 and 11, respectively. Also, regressing the
average revenue in W on that in A and imposing a zero intercept gives a coefficient of 1.067
and 1.056 in blocks 10 and 11, respectively.37 Hence, on average, the steady-state average
revenue is about 6 percent higher in W than it is in A. For a comparison, the analogous
measures for block 1 are 0.985 (average revenue ratio) and 0.974 (revenue regression slope).
We also conduct formal tests to examine these informal conclusions. The key is to obtain
distributions of the various measures reported in the previous two paragraphs under the null
hypothesis of no treatment effect.38 To do that, we employ a “super-bootstrap” procedure that
works as follows. In each super-bootstrap draw, we first randomly allocate the data from the
20 bidding groups into two groups of 10 and pretend that the first set of 10 groups corresponds
to A and the second set of 10 groups corresponds to W . Then, based on this artificial treatment
35
In case a bootstrap draw involves one of the 9 subjects who has been excluded due to overbidding, their
matched data is excluded from computing the averages.
36
See below for a formal statistical test of the null hypothesis that the fraction is equal to 0.5.
37
Note that the regression coefficient is simply a square A-revenues-weighted average of the 1,000 revenue
ratios. The reasoning is analogous to the one following equation 14.
38
Note that using t-tests based on conventional standard errors for averages and regression coefficients is
inappropriate since the 1,000 bootstrap draws are not statistically independent.
26
Table 1: Results of the “Super-bootstrap” Tests
Block 1
Block 10
Block 11
Fraction revenues
0.394
(0.260)
0.894
(0.000)
0.855
(0.000)
Average revenues ratio
0.985
(0.284)
1.068
(0.002)
1.058
(0.010)
Revenues regression slope
0.974
(0.286)
1.067
(0.004)
1.056
(0.016)
Note: P-values for two-sided tests of the null hypothesis are in parentheses.
assignment, we obtain the 1,000 bootstrap draws and compute the average revenue for each
bootstrap draw using the same procedure as described above for the original bootstrap draw.
Next, we compute all three statistics of interest (fraction revenue W > A, average revenue
ratio, revenue regression slope). We repeat the whole procedure 1,000 times, thus obtaining
1,000 super-bootstrap realizations of the three statistics under the null hypothesis. We then
examine where in these distributions the original statistic realizations are located and compute
p-values for two-sided tests of the respective null hypotheses. The null hypotheses are that
that fraction revenues W > A is equal to 0.5, the average revenues ratio is equal to 1 and the
revenues regression slope is equal to 1. The results are reported in Table 1. In block 1, we do
not reject the null hypothesis for any of the three measures. To the contrary, we do reject the
null hypothesis for all three measures in blocks 10 and 11.
Overall, these findings document that in the “steady state”, as approximated by blocks 10
and 11, W significantly increases the average revenue realizations in comparison to A. That
is, given the very same values, the auctioneer realizes a higher auction revenue on average.
These findings support the theoretical hypothesis outlined in Section 3.
6
Identification and Estimation of the Fraction of Naive Bidders
In this section, we first present a strategy for identifying the fraction λ of naive bidders overall
and for each level of risk aversion. We retain the indexation structure introduced in Section
5. That is, i indexes bidders and, implicitly, also bidding groups (with there being I bidders
in total), j ∈ {1, .., 6} indexes bidding rounds within a block and t indexes blocks (with there
being T blocks in total). Identification is a question of what objects of interest can be unambiguously recovered from a dataset with infinitely many observations. In the present context,
27
the term “infinitely” refers to two dimensions. First, it refers to each bidder bidding in infinitely many blocks (that is, T → ∞). As a result, assuming noisy bidding responses based
on maximizing expected CRRA utility, each bidder’s CRRA coefficient can be precisely recovered. Second, it refers to there being infinitely many bidding groups and therefore bidders
(that is, I → ∞). As a result, the distribution of the CRRA coefficient in the population can
be precisely recovered.
We then present an estimation procedure based on this strategy and its results. It turns
out that our dataset provides a sufficient power to have a reasonably precise estimate of the
overall fraction of naive bidders. However, the dataset is too small for precisely estimating the
fraction of naifs for different risk aversion levels.
6.1
Identification
Recall from Section 3 that, in W , a naive bidder bids indistinguishably from a sophisticated
bidder with half-as-high parameter α. Therefore it is impossible to use data from W only to
identify the incidence of naivete. When combining the data from both treatments, identification would be straightforward in a within-subject data. But our design is between-subject.
Hence the main challenge of the identification strategy is to outline how λ, and its dependence
on α (denoted by λ(∙)), can be recovered from such data.
Using the notation from Section 3, let κi ≡ −ln(αi ). That is, κi is a inverse monotone
transformation of αi , with κi = 0 corresponding to risk neutrality, κi > 0 corresponding to
risk aversion, and κi < 0 corresponding to risk loving. For future reference, we call κi a
bidder’s “risk aversion” and denote by H(∙) the cumulative distribution function of this risk
aversion parameter. The reason for employing this transformation is that it turns out to be
more convenient to work with empirically than αi .
To allow for deviations from fully optimizing behavior, we assume that instead of choosing
the peak of his expected utility as assumed in Section 3, a bidder might deviate to a different
bid. In the spirit of quantal response equilibrium (McKelvey and Palfrey 1995), we add a noise
element to the risk aversion parameter. Specifically, the realization of the noise-inclusive risk
aversion of bidder i in round j of block t is given by κi + εijt , where εijt is an i.i.d. noise
term with E(εijt ) = 0 and V ar(εijt ) = σi2 . Noiseless optimization corresponds to the special
case of σi2 = 0. For σi2 > 0, the bidder typically perceives the marginal expected utility of
the bid to be either higher (εijt > 0) or lower (εijt < 0) than it really is and hence bids more
or less, respectively, than his noiseless optimum. We are not making any assumption on the
functional form of the distribution of εijt , nor on the magnitude of σi2 . In principle, both can
be recovered from the data, but this is not necessary for our purpose.
28
Under this modification, in treatment A, a bidder chooses his bid according to
bijt = argmax(vij − b)e
−(κi +εijt )
Ai,t−1 (b).
(15)
b∈(0,vij )
Here, Ai,t−1 (∙) is the empirical cumulative distribution function of bids in block t−1 in the bidding group to which bidder i belongs. Recall that value realizations repeat block-by-block, so
values are not subscripted by t. In order to proceed, we need to take the first-order condition.
However, given that Ai,t−1 (∙) is discrete, it does not have a well-defined density with respect to
the Lebesgue measure. To resolve this problem, we assume that bidders approximate Ai,t−1 (∙)
by its continuously differentiable smoothed version. Given that bidders are primarily shown
a histogram based on Ai,t−1 (∙) rather than the cdf itself, we actually assume that they first approximate ai,t−1 (∙), the density based on Ai,t−1 (∙), by means of a smooth function âi,t−1 (∙), and
then obtain the approximation Âi,t−1 (∙) of Ai,t−1 (∙) by integrating the density estimate. There
are many ways of how to approximate the density. We assume that the density approximation
√
is given by the kernel estimate based on Epanechnikov kernel (K(z) = 0.75(1 − 0.2z 2 )/ 5 if
√
|z| < 5 and K(z) = 0 otherwise) with the “oversmoothed” bandwidth (h ' 1.1325σn−1/5 ,
with σ being the standard deviation in the dataset) and bid support restriction to [0, 100].39 To
obtain the approximation of the cdf, we integrate the density using the trapezoid method based
on evaluating the density on the grid {0, 0.01, 0.02, ..., 99.99, 100}.
By applying a natural logarithm transformation of the objective function and taking the
first-order condition, the optimal bid bijt is characterized by
"
κi = −ln (vij − bijt )
âi,t−1 (bijt )
Âi,t−1 (bijt )
#
− εijt .
(16)
#)
(17)
By taking the expectation, if follows that
κi = −E
(
"
ln (vij − bijt )
âi,t−1 (bijt )
Âi,t−1 (bijt )
.
This equation shows how risk aversion of an individual bidder in A can be identified from his
value and bid data. Repeating this identification for each bidder, we can identify H(∙).
In treatment W , we can identify risk aversion of a particular bidder analogously if we
assume that this bidder is naive or, alternatively, that he is sophisticated. A naive bidder
simply best-responds to Wi,t−1 (∙), the empirical cumulative distribution function of winning
bids in block t − 1 in the bidding group to which bidder i belongs. In parallel to (17), using
39
See Jann (2007) and Salgado-Ugarte, Shimizu and Taniuchi (1995) for further technical details. The estimation was implemented using the command kdens in Stata 14.
29
analogous density and cdf approximations, it follows that
κi = −E
(
"
ln (vij − bijt )
ŵi,t−1 (bijt )
Ŵi,t−1 (bijt )
#)
.
(18)
We assume that a sophisticated bidder first carries out the same approximations
as a naive
q
c̃
(∙). The
bidder, and then recovers the approximation of the cdf of bids by W
Ŵ
i,t−1 (∙) ≡
q i,t−1 b̃ i,t−1 (b) = ŵi,t−1 (b)/ 2 Ŵi,t−1 (b) . In
bidder then computes the corresponding density by w
parallel to (17), it then follows that
(
"
#)
b̃ i,t−1 (bijt )
w
κi = −E ln (vij − bijt )
c̃
W
i,t−1 (bijt )
#)
( "
ŵi,t−1 (bijt )
+ ln(2).
= −E ln (vij − bijt )
Ŵi,t−1 (bijt )
(19)
Repeating the identification inherent in (19) and (18) for each bidder, we can identify HN (∙)
and HS (∙), the cumulative distribution functions of risk aversion in W assuming universal
naivete and assuming universal sophistication, respectively.
Comparison of (19) and (18) reveals that being naive is, in terms of bidding, observationally equivalent to being sophisticated and having risk aversion increased by ln(2). This is just
an equivalent restatement of the observation that we have made in Section 3 and that underlies Proposition 1. Vice versa, being sophisticated is observationally equivalent to being naive
and having risk aversion lowered by ln(2). As a side remark, this implies, as already mentioned, that between-subject data on values and bids cannot be used to distinguish between
sophistication and naïveté at the individual level. It then follows that, for any κ,
HS (κ) = HN [κ − ln(2)].
(20)
This implies that the two distributions are horizontal shifts of each other. Because of this, from
now on we will only work with HS (∙).
Identification of λ (and λ(∙)) is based on comparing H(∙) with HS (∙). The key identification assumption is that the distribution of risk aversion in the population of bidders is the same
in both treatments. This is justified by the random assignment of subjects to the two treatments. Under this assumption, consider who the bidders are whose implied risk aversion in
W , when assuming sophistication, is lower than or equal to κ. First, it is all the sophisticated
bidders whose risk aversion is less than or equal to κ. Second, it is all the naive bidders whose
30
risk aversion is less than or equal to κ − ln(2). Hence, for any κ,
HS (κ) =
Z
κ
−∞
[1 − λ(y)]dH(y) +
= H(κ) − Λ(κ),
where
Λ(κ) ≡
Z
Z
κ−ln(2)
λ(y)dH(y)
−∞
(21)
κ
λ(y)dH(y).
(22)
κ−ln(2)
Since we have already discussed how to identify H(∙) and HS (∙). As a result, using (21), we
can identify Λ(∙) by
Λ(κ) = H(κ) − HS (κ)
(23)
for any κ.
The next issue is how to recover λ(∙) from Λ(∙). As the next example illustrates, this is
impossible without additional assumptions:
Example: Let H(∙) have a density h(∙) that is strictly positive on (−∞, ∞). Suppose that
given Λ(∙), λ1 (∙) satisfies (22) for all κ. Now consider λ2 (∙), which is a perturbation of λ1 (∙)
given by λ2 (κ) ≡ λ1 (κ) + α(κ)/h(κ), where α(κ) alternates between −1 on intervals ((n −
1) ln(2)/2, n ln(2)/2] for odd n and 1 on intervals ((n − 1) ln(2)/2, n ln(2)/2] for even n. By
construction, λ2 (∙) 6= λ1 (∙), but it also satisfies (22).
One could make a number of alternative assumptions in an attempt to pin down the shape of
λ(∙). Any such assumption is in some way arbitrary, however.40 For this reason, we prefer not
to make such assumption. Instead, we focus on what objects of interest can be identified from
40
For example, one way to proceed would be to make an assumption that the support of risk aversion is
bounded from below (i.e., there is an upper bound on risk loving), but not from above (i.e., there is no bound
on risk aversion), and that H(∙) has a strictly positive density h(∙) on its support. Then, by assumption, there
exists κ > −∞ such that H(κ) = 0 and h(κ) > 0 for all κ > κ. It then follows from (22) that, for any
κ ∈ (κ, κ + ln(2)),
Z
Λ(κ) =
κ
λ(y)dH(y),
κ
implying that Λ0 (κ) = λ(κ)h(κ), and therefore
λ(κ) = Λ0 (κ)/h(κ).
(24)
For any κ ≥ κ + ln(2), (22) implies that
Λ0 (κ) = λ(κ)h(κ) − λ[κ − ln(2)]h[κ − ln(2)],
giving
λ(κ) =
Λ0 (κ) + λ[κ − ln(2)]h[κ − ln(2)]
.
h(λ)
(25)
Conditional on the value of κ, (24) identifies λ(∙) on (κ, κ + ln(2)), and (25) then recursively identifies λ(∙) on
[κ + ln(2), ∞). To operationalize this identification procedure, however, one needs to specify the value of κ ex
ante. Any such assumed value is arbitrary. Moreover, it is not even clear why there would be an upper bound on
risk loving.
31
Λ(∙). In particular, for any κ in the support of H, we can identify a locally-smoothed version of
λ(∙) given by the risk-aversion weighted average of λ(∙) in the interval (κ − ln 2/2, κ + ln 2/2].
Denoting the smoothed fraction λ∗ (∙), we have that
λ∗ (κ) ≡
R κ+ln(2)/2
κ−ln(2)/2
λ(y)dH(y)
H[κ + ln(2)/2] − H[κ − ln(2)/2]
Λ(κ + ln 2/2)
=
H[κ + ln(2)/2] − H[κ − ln(2)/2]
H(κ + ln 2/2) − HS (κ + ln 2/2)
=
H[κ + ln(2)/2] − H[κ − ln(2)/2]
(26)
The last equality gives an explicit formula for identifying λ∗ (∙) from H(∙) and HS (∙), whose
identification we have discussed above.
Given λ∗ (∙), there are multiple ways of identifying the overall fraction of naifs λ. The
latter is always given by a risk-aversion weighted average of λ∗ (∙) evaluated on a grid with a
step size of ln 2. Formally,
λ=
X
z∈Z
λ∗ (κ0 + z ln 2){H[κ0 + (z + 0.5) ln(2)] − H[κ0 + (z − 0.5) ln(2)]},
(27)
where Z is the set of integers and κ0 ∈ (− ln 2/2, ln2/2]. The multiplicity comes from
different picks of the anchor κ0 , but each pick results in the same value of λ.
6.2
Estimation
Our estimates of individual risk aversion coefficients in A and in W assuming sophistication
are based on sample analogues of (17) and (19), respectively. That is, the estimate of risk
aversion of bidder i in A is given by
"
#
11
6
âi,t−1 (bijt )
1 XX
ln (vij − bijt )
κ̂i = −
.
60 t=2 j=1
Âi,t−1 (bijt )
(28)
Analogously, the estimate of risk aversion of bidder i in W , assuming that he/she is sophisticated, is given by
#
"
11
6
ŵi,t−1 (bijt )
1 XX
+ ln(2).
ln (vij − bijt )
κ̂i = −
60 t=2 j=1
Ŵi,t−1 (bijt )
(29)
Given the empirical distributions of κ̂i in the two treatments, we estimate thedensity functions of κ in each treatment using kernel estimators with the same settings, except for the
boundary conditions, as we use to estimate densities of bids in the previous subsection. Both
32
density estimates are evaluated on the grid {zs : z ∈ {−105, −99, ..., 524, 525}} with the step
size of s ≡ (ln 2/2)/35 (approximately 0.01). We then obtain the estimates of H(∙) and HS (∙)
by integrating the respective density estimates using the trapezoid method. The resulting density estimates, their underlying histograms, and Ĥ(∙) and ĤS (∙) are displayed in Figure 7.
One can observe a clear shift in the two distributions in the sense of first order stochastic dominance. This simply reflects the assumption of universal sophistication in W used to obtain
ĤS (∙).
Our estimate of λ∗ (∙) is based on the sample analogue of (26) and is, for any κ on the
sub-grid {ks : k ∈ {−70, −64, ..., 489, 490}}, given by
Ĥ(κ + ln 2/2) − ĤS (κ + ln 2/2)
λb∗ (κ) =
.
Ĥ[κ + ln(2)/2] − Ĥ[κ − ln(2)/2]
(30)
Finally, our estimate of the overall fraction of naifs in the population is based on the sample
analogue of (27). To operationalize this estimator, we need to choose a bounded grid with
the step size of ln 2 that covers virtually all of the empirical support of κ. We use the grid
{z ln 2 : z ∈ {−1, 0, .., 7}} (this corresponds to choosing κ0 = 0). Our estimate is hence
given by
λ̂ =
X
z∈{−1,0,..,7}
λb∗ (z ln 2){Ĥ[(z + 0.5) ln(2)] − Ĥ[(z − 0.5) ln(2)]}.
(31)
To evaluate the precision of our estimates of λ and λ∗ (∙), we conduct a bootstrap analysis. In particular, using the dataset of κ̂, we repeatedly draw (with replacement) samples of
size equal to our dataset, preserving the stratification by treatment and clustering by bidding
group.41 Altogether, we draw 1,000 bootstrap samples and for each sample we compute λ̂ and
λb∗ (∙). The relevant confidence intervals for the baseline estimates of λ̂ and λb∗ (∙) are then given
by the 2.5th and 97.5th percentile of the distribution of the bootstrap estimates (point-wise by
κ in case of λb∗ (∙)).
The average fraction of naive bidders is estimated to be 57%. There is, however, variation
in the extent of naïveté over the support of κ. Figure 8 plots the estimate of λ∗ (∙). Bidders who
are close to risk-neutral (κ close to 0) appear to be more sophisticated. Naïveté then increases
with risk aversion until about κ = 1, where it reaches more than 70% of the population,
and decreases with risk aversion afterwards. However, as can be seen by looking at the 95%
confidence interval, the estimate of this relation is overall quite noisy. In particular, we cannot
reject the hypothesis that λ∗ (∙) is flat. Hence, the limits of our data do not allow us to draw
a clear conclusion on the shape of the relation between naïveté and risk aversion. We hope
that our methodology can prove useful in future work trying to address how naivete relates to
41
That is, each bootstrap sample has as many observations in each treatment as the original sample. Also, each
time a particular subject is drawn into the sample, so are all the other subjects from the same bidding group.
33
Figure 7: Estimated distributions of risk aversion in A and in W assuming universal sophistication
34
Figure 8: Estimated Fraction of Naive Bidders as a function of κ
1.5
1.25
1
.75
.5
.25
0
-.25
-.5
-.75
-1
0
.5
1
1.5
2
κ
2.5
λ bar (shifted)
3
3.5
4
95% CI
various dimensions of heterogeneity in bidder preferences.
7
Conclusion and Discussion
We compare two disclosure rules of historical information (HI) in first-price auctions: all
bids and winning bids only. Under the standard theory of Bayesian Nash Equilibrium with
fully rational bidders, the choice among the two disclosure policies should be irrelevant in the
steady state. Instead, our experimental test, based on auctions with two bidders, shows that
actual bidding behavior differs across the two settings. In particular, HI on winning bids leads
to higher bids and revenues in the long-run. To explain this finding, we propose that, when
faced with HI on winning bids, a fraction of bidders mistakenly best-respond to the historical
distribution of winning bids rather than to the historical distribution of all bids. On the theory
side, our work challenges the prediction that historical market information should not matter
for equilibrium outcomes as long as it is rich enough for forming correct beliefs. On the market
design side, it suggests that, in terms of revenues, a long term auctioneer should disclose
only the past winning bids rather than all bids. Even though our experimental test focuses
on auctions with two-bidders, the market design implications should extend to settings with
higher numbers of bidders and also to other market institutions in which the determination of
beliefs about the behavior of the opponent(s) plays a crucial role.
A corollary implication that arises from our results is that the “overbidding puzzle” ob35
served in many experiments on first-price auctions might to some extent be an artifact of bidders being given HI on winning bids only from previous experimental rounds. Even though we
do not believe that this is the sole reason for “overbidding", our result underlines the importance of careful consideration of information feedback rules for interpretation of experimental
results.
It is an interesting question how our theoretical reasoning extends to auctions with more
than two bidders. In first-price auctions, a sufficient statistic for the formulation of a bidder’s
best response is the distribution of the highest competing bid (DHCB). In the case of two
bidders, under HI on winning bids, formulation of best response requires conversion of HI
into DHCB, and this is where bounded rationality comes in to play a role. In contrast, the
information that bidders get under HI on all bids is, in the steady state, exactly the DHCB. That
is, the bidders are given exactly the distribution they need to best-respond to in the Bayesian
Nash Equilibrium, without necessity to perform any further calculations. As a result, the twobidder case provides a natural benchmark for investigating the role of bounded rationality
since its role should be limited to only one type of HI. In contrast, in auctions with n > 2
bidders, the bidders are, in the steady state, not given the DHCB under either of the two
HI types. This opens up the possibility that, in the process of converting the provided HI
into the DHCB, bounded rationality might play a role under both HI types. The neglect of
selection bias theory that we propose still implies that naivete affects bidding only under HI
on winning bids, though. This is because even though the naive bidders fail to understand
that bids and winning bids are different objects, one could assume that they otherwise perform
all the necessary steps, including computation of order statistic distributions, to determine the
DHCB. In particular, naive bidders, when presented with HI on all bids, A(∙), form correct
beliefs about the DHCB, An−1 (∙). When presented with HI on winning bids, however, they
1
use W (∙) in place of W n (∙) to form beliefs about the DHCB. After computing the order
n−1
statistic distribution, they believe that the DHCB is given by W n−1 (∙) rather than W n (∙).
An alternative type of mistake that bidders might make is to confuse the HI they are provided
with for the DHCB. This type of mistake would imply that, under HI on all bids, naive bidders
best-respond to A(∙), whereas under HI on winning bids, they best-respond to W (∙). As a
result, in comparison to their best response under correct beliefs, naive bidders end up bidding
less under HI on all bids and more under HI on winning bids. The predictions of the neglect
of selection bias theory and this alternative theory are indistinguishable in auctions with two
bidders, while they differ in auctions with more than two bidders.42
42
Under the neglect of selection bias theory, the revenue difference between the two HI types is entirely driven
by incorrect beliefs about the DHCB of naive bidders under HI on winning bids, and their resulting overbidding
relative to the best response. Moreover, this overbidding grows larger with the number of bidders. Under the
alternative theory, under HI on winning bids, the overbidding compared to the best response diminishes with the
number of bidders. On the other hand, underbidding compared to the best response under HI on all bids increases
with the number of bidders.
36
The discussion above hints at the variety of biases that might be at play in more complex
environments than the one we look at. It also implies that there is a vast scope for future
research in order to pin down more precisely the biases bidders are subject to in various market
environments in response to different types of historical market information.
8
Acknowledgements
Special thanks go to Tomáš Miklánek for his excellent job with programming and organizing
the experiment. This study was supported by the Grant Agency of the Czech Republic (GAČR
402/11/1726). All opinions expressed are those of the authors and have not been endorsed by
GAČR, Paris School of Economics, University College London, RWTH Aachen University
or CERGE-EI. We would like to thank seminar participants at RWTH Aachen University,
Aarhus University, University of Arkansas, Higher School of Economics in Moscow, University of Technology in Sydney and participants at numerous conferences for their feedback and
suggestions that helped to improve the paper.
37
References
Battigalli, Pierpaolo, “Comportamento Razionale ed Equilibrio nei Giochi e Nelle
Situazioni Sociali,” Dissertation, Bocconi University, Milano 1987.
Bergemann, Dirk and Johannes Hörner, “Should Auctions be Transparent?,” 2010.
COWLES FOUNDATION DISCUSSION PAPER NO. 1764.
Boening, Mark V. Van, Stephen J. Rassenti, and Vernon L. Smith, “Numerical
Computation of Equilibrium Bid Functions in a First-Price Auction with Heterogeneous
Risk Attitudes,” Experimental Economics, 1998, 1, 147–159.
Cason, Timothy, Karthik Kannan, and Ralph Siebert, “An Experimental Study of
Information Revelation Policies in Sequential Auctions,” Management Science, 2011, 57
(4), 667–688.
Chen, Yan, Peter Katuščák, and Emre Ozdenoren, “Why Can’t a Woman Bid More Like a
Man?,” Games and Economic Behavior, January 2013, 77 (1), 181–213.
Cox, James C., Vernon Smith, and James M. Walker, “Auction Market Theory of
Heterogenous Bidders,” Economics Letters, 1982, 9, 319–325.
,
, and
, “Theory and Misbehavior of First-Price Auctions: Comment,” The
American Economic Review, December 1992, 82 (5), 1392–1412.
Dekel, Eddie, Drew Fudenberg, and David K. Levine, “Learning to Play Bayesian
Games,” Games and Economic Behavior, February 2004, 46 (2), 282–303.
Dufwenberg, Martin and Uri Gneezy, “Information disclosure in auctions: an experiment,”
Journal of Economic Behavior and Organization, 2002, 48, 431–444.
Engelbrecht-Wiggans, Richard, “The Effect of Regret on Optimal Bidding in Auctions,”
Management Science, 1989, 35 (6), 685–692.
and Elena Katok, “Regret and Feedback Information in First-Price Sealed-Bid
Auctions,” Management Science, April 2008, 54 (4), 808–819.
Esponda, Ignacio, “Information Feedback in First Price Auctions,” RAND Journal of
Economics, Summer 2008, 39, 491–508.
Filiz-Ozbay, Emel and Erkut Y. Ozbay, “Auction with Anticipated Regret: Theory and
Experiments,” American Economic Review, April 2007, 54 (4), 1407–1418.
Fischbacher, Urs, “z-Tree: Zurich Toolbox for Ready-made Economic Experiments,”
Experimental Economics, 2007, 10 (2), 171–178.
Fudenberg, Drew and David K. Levine, “Self-Confirming Equilibrium,” Econometrica,
May 1993, 61 (3), 523–545.
Füllbrunn, Sascha and Tibor Neugebauer, “Varying the number of bidders in the
first-price sealed-bid auction: experimental evidence for the one-shot game,” Theory and
Decision, 2013, 75 (3), 421–447.
Greiner, Ben, “Subject Pool Recruitment Procedures: Organizing Experiments with
38
ORSEE,” Journal of the Economic Science Association, July 2015, 1 (1), 114–125.
Huck, Steffen, Philippe Jehiel, and Tom Rutter, “Feedback spillover and analogy-based
expectations: A multi-game experiment,” Games and Economic Behavior, 2011, 71 (2),
351–365.
Isaac, Mark R. and James M. Walker, “Information and Conspiracy in Sealed Bid
Auctions,” Journal of Economic Behavior and Organization, 1985, 6, 139–159.
Jann, Ben, “Univariate kernel density estimation,” 2007. Downloaded from
http://fmwww.bc.edu/RePEc/bocode/k/kdens.pdf on Feb. 17, 2017.
Jehiel, Philippe, “Analogy-based expectation equilibrium,” Journal of Economic Theory,
2005, 123, 81–104.
, “Manipulative Auction Design,” Theoretical Economics, 2011, 6 (2), 185–217.
and Frédéric Koessler, “Revisiting games of incomplete information with
analogy-based expectations,” Games and Economic Behavior, 2008, 62 (2), 533–557.
Katuščák, Peter, Fabio Michelucci, and Miroslav Zajíček, “Does Feedback Really Matter
in One-Shot First-Price Auctions?,” Journal of Economic Behavior and Organization,
2015, Forthcoming.
McKelvey, Richard D. and Thomas R. Palfrey, “Quantal Response Equilibria in Normal
Form Games,” Games and Economic Behavior, 1995, 10, 6–38.
Nagatsuka, Masao, “The Effect of Information Feedback on Bidding Behavior:
Experimental Study on Regret Theory in Auctions,” Journal of Behavioural Economics
Finance Entrepreneurship Accounting and Transport, 2016, 4 (2), 18–23.
Neugebauer, Tibor and Javier Perote, “Bidding ‘as if’ risk neutral in experimental first
price auctions without information feedback,” Experimental Economics, 2008, 11,
190–202.
and Reinhard Selten, “Individual behavior of first-price auctions: The importance of
information feedback in computerized experimental markets,” Games And Economic
Behavior, 2006, 54, 183–204.
Ockenfels, Axel and Reinhard Selten, “Impulse Balance Equilibrium and Feedback in First
Price Auctions,” Games And Economic Behavior, 2005, 51 (1), 155–170.
Pearson, Matthew and Burkhard C. Schipper, “Menstrual Cycle and Competitive
Bidding,” Games and Economic Behavior, March 2013, 78 (2), 1–20.
Ratan, Anmol and Yuanji Wen, “Does regret matter in first-price auctions?,” Economics
Letters, 2016, 143, 114–117.
Salgado-Ugarte, I. H., M. Shimizu, and T. Taniuchi, “Practical rules for bandwidth
selection in univariate density estimation,” Stata Technical Bulletin, 1995, 26, 23–31.
Sonsino, Doron and Radosveta Ivanova-Stenzel, “Experimental internet auctions with
random information retrieval,” Experimental Economics, 2006, 9 (4), 323–341.
39