# Interictal intracranial EEG for predicting surgical success: the importance
of space and time

Yujiang Wang  Nishant Sinha  Gabrielle M Schroeder  Sriharsha Ramaraju  Andrew
W McEvoy  Anna Miserocchi  Jane de Tisi  Fahmida A Chowdhury  Beate Diehl
John S Duncan  Peter N Taylor  School of Computing Science, Newcastle
University, UK  Institute of Neuroscience, Newcastle University, UK  Institute
of Neurology, Queen Square, London WC1N 3BG, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Predicting post-operative seizure freedom using functional correlation
networks derived from interictal intracranial EEG has shown some success.
However, there are important challenges to consider. 1: electrodes physically
closer to each other naturally tend to be more correlated causing a spatial
bias. 2: implantation location and number of electrodes differ between
patients, making cross-subject comparisons difficult. 3: functional
correlation networks can vary over time but are currently assumed as static.

In this study we address these three substantial challenges using intracranial
EEG data from 55 patients with intractable focal epilepsy. Patients
additionally underwent preoperative MR imaging, intra-operative CT, and post-
operative MRI allowing accurate localisation of electrodes and delineation of
removed tissue.

We show that normalising for spatial proximity between nearby electrodes
improves prediction of post-surgery seizure outcomes. Moreover, patients with
more extensive electrode coverage were more likely to have their outcome
predicted correctly (ROC-AUC $>$0.9, p$<<$0.05), but not necessarily more
likely to have a better outcome. Finally, our predictions are robust
regardless of the time segment.

Future studies should account for the spatial proximity of electrodes in
functional network construction to improve prediction of post-surgical seizure
outcomes. Greater coverage of both removed and spared tissue allows for
predictions with higher accuracy.

###### keywords:

Epilepsy , surgery , network , intracranial EEG , surgical outcome prediction

††journal: arXiv

##  1 Introduction

Surgery is an effective treatment for epilepsy, with over half of patients
achieving outcomes of postoperative seizure freedom (de Tisi et al., 2011).
For patients not seizure free after surgery, a possible explanation is the
incomplete removal of the epileptogenic zone, defined as the area of cortex
that is indispensable for seizure generation (Rosenow and Luders, 2001). More
recently, the concept of the epileptogenic ‘network’ has emerged, recognising
multiple brain regions and connections between them to be responsible for
generating seizures (Bartolomei et al., 2017; Spencer, 2002). Identification
of the epileptogenic network in each patient is extremely challenging, as
removal of multiple brain regions and connections between them may lead to
seizure freedom. Several recent studies have highlighted the potential network
properties that are specific to the epileptogenic network and used various
network properties to predict post-operative outcome (Richardson, 2012; Englot
et al., 2015; Munsell et al., 2015; Morgan et al., 2017; Taylor et al., 2018).

Functional networks inferred using intracranial EEG (iEEG) have received
considerable attention in this context. These functional networks measure iEEG
signal similarity as connection strength between iEEG channels. Indeed,
studies using iEEG-derived networks have demonstrated their value for
predicting patient outcomes when using ictal (Wilke et al., 2011; Burns et
al., 2014; Goodfellow et al., 2016; Park and Madsen, 2018; Yang et al., 2018)
and inter-ictal data (Palmigiano et al., 2012; Sinha et al., 2014;
Zweiphenning et al., 2016; Sinha et al., 2017; Tomlinson et al., 2017; Shah et
al., 2019).

The potential of using (only) interictal data is particularly attractive in a
clinical setting (Korzeniewska et al., 2014; Shah et al., 2019), but despite
promising results from previous studies, several challenges remain. First,
electrodes that are physically closer are also more likely to be highly
correlated (Lagarde et al., 2018; Betzel et al., 2019). This means that
functional networks derived from iEEG are always dependent on the spatial
layout of electrodes, which differs from patient to patient. Second, the
individualised configuration of intracranial electrodes also means that
different number of electrodes are sampling tissue that is ultimately removed
or spared by surgery. This makes comparison of network properties across
patients difficult. Third, although iEEG functional networks fluctuate over
time (Kuhnert et al., 2010; Geier et al., 2015), it is not currently known if
these fluctuations affect their use to predict surgical outcome. To enable the
translational use of interictal functional iEEG networks, it is crucial to
answer these questions.

##  2 Materials & Methods

###  2.1 Patients and interictal EEG and MRI preprocessing

This retrospective study analysed data from 55 patients with refractory focal
epilepsy from the National Hospital for Neurology and Neurosurgery (NHNN) who
had intracranial EEG (iEEG) followed by resection and clinical follow-up of at
least 12 months. iEEG data was acquired using a mixture of grid, strip, and
SEEG recordings. The iEEG data was anonymised and exported, then analysed
under the approval of the Newcastle University Ethics Committee (2225/2017).
Patient metadata is shown in supplementary material S1

The iEEG data analysed consisted of continuous one-hour segments of interictal
EEG that were at least two hours away from seizures. Artefactual channels were
removed by visual inspection, and all remaining channels were subsequently re-
referenced to common average. Each channel was then notch filtered at 50 Hz
(iir filter with Q factor 50, 4th order zero phase lag) and bandpass filtered
(Butterworth 4th order zero phase lag) between 1-70 Hz.

To delineate the iEEG electrodes that overlapped with the subsequently
surgically removed tissue, we first mapped the spatial position of the iEEG
electrodes to the space of the pre-operative T1w MRI using intraoperative MRI
and CT in a semi-automated fashion (Hamilton et al., 2017). We also manually
delineated the surgically removed tissue in space of the pre-operative MRI
using rigid body registration of the post-operative T1w MRI to the pre-
operative T1w MRI (Taylor et al., 2018). Any iEEG electrode contact that was
within the 5mm distance of surgically removed tissue was deemed as a “removed
electrode”. All others were marked as “spared electrodes”. For one patient
(IDP 851), a post-operative MRI was unavailable and the surgery report from
the clinical team was therefore used to identify the resected electrodes. The
procedure is summarised schematically in Fig 1A.

###  2.2 Functional networks derivation and network quantification

Functional brain networks were derived from the one-hour iEEG segments. We
applied Pearson correlation to 2 second sliding windows (without overlap) of
the broad band (1-70 Hz) iEEG data and subsequently averaged the correlation
matrices over all windows to obtain one functional network (average
correlation) matrix per subject.

To quantify the network properties of each node we use the node strength of
the network, which is a measure of the total level of correlation of a node.
It has been suggested that this quantity is indicative of epileptogenic
tissue, and we show in supplementary material S3 that it recapitulates a
quantity derived from a dynamical model of epileptogenic tissue we previously
suggested (Sinha et al., 2017). To quantify if the node strength of removed
electrodes differed from the node strength of the spared electrodes, we used
the area under the receiver-operator curve (AUC), which is equivalent to the
normalised non-parametric Mann-Whitney U statistic. We chose this measure
because it is based only on the rank order of the node strength, and thus
robust to outliers and non-normal distributions in node strength. In the
following we will term this measure $D_{RS}$ which stands for the
Distinguishability of the Removed node strength vs. the Spared node strength
and is a single value per patient. This procedure is summarised schematically
in Fig 1B.

Figure 1: Summary of the processing pipeline. A CT and post-operative T1
weighted MRI scans were linearly registered to the pre-operative T1 weighted
MRI scan. Electrodes, shown in blue, were localised from the registered CT and
marked with respect to the cortical surface. A mask was additionally manually
drawn (shown in red and projected to the cortical surface for visualisation)
to cover those areas removed by surgery. Electrodes located within 5 mm of the
volumetric surgery mask were then identified as ‘removed’ (shown in red) and
all others as identified as ‘spared’ (shown in black). The Euclidean distance
between each electrode is also calculated in millimetres. B Two second non-
overlapping correlation matrices are computed from the EEG time series, and
their mean (temporal average) matrix calculated. By summing the rows of the
temporal average matrix, the node strength was calculated. The difference in
node strengths for removed and spared electrodes was then computed as the
$D_{RS}$ measure, with 1 indicating perfect separation of the removed vs.
spared tissue and 0.5 indicating no separation. C To derive a spatially
normalised temporal average matrix we applied a spatial regression which was
pre-calculated based on spared electrodes from good outcome patients. After
applying the regression, we retained the residuals as the spatially normalised
temporal average matrix, which allows to calculate a normalised node strength
and $D_{RS}$ value. The pipeline was applied to each patient.

###  2.3 Spatial normalisation

The fact that electrodes that are closer together in space are more likely to
be correlated introduces a bias in the functional networks that depends on
each subject’s implantation. We therefore applied a spatial normalisation to
reduce this bias. We used spared electrodes from good outcome (ILAE class 1)
patients only - which represents signals from non-epileptogenic tissue - as a
baseline to establish how correlation coefficients change as a function of
spatial (Euclidean) distance. Once this baseline function was found, all
correlation coefficients in the functional network matrices were normalised by
computing the residuals to this baseline function. The spatially-normalised
correlation coefficients now indicate the extent of correlation between two
electrodes that is beyond what is expected due to their spatial proximity to
each other. This procedure is summarised schematically in Fig 1C.

###  2.4 Spatial coverage

Due to clinical need, the spatial coverage of the implanted electrodes differs
from patient to patient. Thus, the spatial sampling of the surgically
removed/spared tissue differs in terms of total number of electrodes and
amount of tissue removed. To account for these differences, we obtained the
number of removed electrode contacts and the number of spared electrode
contacts in each patient. We then successively excluded subjects from our
analysis based on the minimum number of removed and spared electrode contacts
($n_{x}$) to observe the effect of spatial coverage on our results. For
example, at $n_{x}=1$, all 55 subjects are included. At $n_{x}=20$, only
subjects who have at least 20 removed electrode contacts and at least 20
spared electrode contacts are retained for analysis, which in our case
consists of 27 subjects. In our analysis we scan the $n_{x}$ parameter from 1
to 40 and report the results for each value. The rationale behind this measure
is that larger $n_{x}$ values can be interpreted as providing a better network
representation and sampling of both removed and spared networks. It then
follows that if we have captured the network better, our discrimination
between outcome groups should improve.

###  2.5 Temporal variability

A longstanding open question in the field is if the temporal variability of
the interictal functional networks (Kuhnert et al., 2010; Geier et al., 2015)
affect its ability to delineate epileptogenic tissue.

We first addressed the question of time scale by dividing the one-hour iEEG
segment into several smaller non-overlapping segments (segments of length 4
seconds, 10 seconds, 20 seconds, 40 seconds, 1 minute, 3 minutes, 6 minutes,
and 10 minutes) and measuring their performance when repeating the same
analysis. Specifically, for each segment, we apply a 2s non-overlapping
windows to create an average functional network matrix for this segment. For
example, the 40 second segment functional network matrix is created from 20
windows, whereas the 10 second segment is generated from only 5 windows.

We next investigated the performance of two other separate one-hour segments
from the same subject (at least 2 hours away from seizures, and at least 4
hours away from the other iEEG segments). In some patients, it was not
possible to find such a second or third one-hour segment, and thus the number
of patients decreases slightly (53 subjects had a second segment, and 51
subjects had a third segment).

All analyses were performed independently on each segment; all the steps
including spatial regression onwards were performed for each segment without
knowledge of the other segments.

###  2.6 Statistical analysis of relationship to surgical outcome

To investigate if $D_{RS}$ contains useful information to explain post-
surgical outcomes, we tested if $D_{RS}$ is different between good (ILAE class
1) and poor outcome (ILAE class 2 and above) patients. We measured the area
under the receiver-operator curve (AUC) as a main indicator, where AUC equal
to 1 shows that $D_{RS}$ can fully distinguish good and poor outcome patients.
Conversely, an AUC of 0.5 indicates that $D_{RS}$ cannot distinguish between
outcome groups. We also tested for the statistical significance of the AUC by
performing the ranksum test between the good and poor outcome patients for the
$D_{RS}$ measure. We obtained 95% confidence intervals of the AUC based on a
logit transformation (Qin and Hotilovac, 2008). Supplementary material S2
shows equivalent results using a cross validated AUC.

###  2.7 Data availability

We will make all functional network matrices and analysis code underlying all
result figures available upon acceptance of the manuscript.

##  3 Results

In this study, we compare network properties (in particular node strength) of
interictal iEEG functional networks between the surgically removed and spared
tissue to predict surgical outcome (seizure freedom) in individual patients.
We will specifically address the following three questions: (1) Does spatial
normalisation of functional networks increase the ability to distinguish
between outcome groups? (2) Does increased coverage of removed and spared
tissue lead to increased distinction between outcome groups? (3) Does the
choice of timescale or time point affect the ability to distinguish between
outcome groups?

###  3.1 Spatial normalisation of interictal functional networks improves
distinction between outcome groups

We first investigate if node strength computed from raw, spatially un-
normalised networks discriminates between outcome groups. Figure 2A,B (upper
panels) shows the node strength computed for two example patients which are
then used to calculate the $D_{RS}$ value. This single $D_{RS}$ value measures
the difference in node strength between removed and spared electrodes in an
individual. Figure 2C shows the $D_{RS}$ value for all 55 patients in our
study and we find no substantial difference in $D_{RS}$ value between outcome
groups (Fig. 2D).

Figure 2: Spatial normalisation improves the discrimination between outcome
groups. A,B Node strength before and after spatial normalisation for two
example patients. C ILAE outcome groups show similar $D_{RS}$ values, meaning
that the difference in node strength between removed and spared nodes does not
explain outcome. Each dot is an individual patient. D Receiver operating curve
for the data presented in panel C shows poor discrimination between outcome
groups. E After spatial normalisation patient groups show significant
differences in their $D_{RS}$ values, which can discriminate outcome groups
with a ROC AUC of 0.7 (F).

Given that electrodes which are more spatially proximal are more likely to
have higher functional connectivity, we next sought to determine if
normalising for this effect increases discrimination between groups using the
$D_{RS}$ measure. To this end we used a null model for spatial normalisation
which accounts for the spatial proximity of electrodes (see Methods). The
normalisation subsequently impacts the node strength (c.f. Fig. 2A,B upper and
lower panels), and in our data it improves the distinction between outcome
groups (compare Fig 2C,D with E,F). For our remaining analysis we will
therefore use spatially normalised functional networks.

###  3.2 Increased coverage of removed and spared tissue improves distinction
between outcome groups

Spatially undersampling networks can directly lead to changes in the estimated
network properties (Conrad et al., 2019), and thus we investigated the impact
of spatial sampling on our ability to distinguish outcome groups. Fig. 3A
shows two example patients: one has a large number of electrode contacts
located in both removed and spared tissue, but the other one only has eight
electrode contacts located in the spared tissue. Therefore, these two patients
are not directly comparable in terms of the network properties of their spared
tissue. To account for this issue, we successively excluded patients with a
low $n_{x}$, which is their minimum number of electrodes in both spared and
removed tissue. As we increase coverage of removed and spared tissue ($n_{x}$)
the distinction between outcome groups in terms of AUC values becomes clearer
(Fig. 3B). For example, at $n_{x}=20$ (we only use the subset of patients that
have at least 20 electrodes spared and removed tissue), 27 patients remain for
the analysis, and we find outcome class is distinguishable with an AUC=0.91
(Fig. 3C). Note also that the proportion of good vs. poor outcome patients
does not change substantially over $n_{x}$ (Fig. 3B bottom panel orange line).
Cross-validated AUCs follow a similar trend of increasing AUC for greater
coverage (supplementary information S2).

Figure 3: Greater coverage of spared and removed networks leads to greater
discrimination between outcome groups. A Removed (red) and spared (black)
electrodes are shown for two example patients. Patient 985 has greater
sampling of spared tissue (27 electrodes) than patient 865 (9 electrodes). B
Top: Scanning AUC (blue line) over $n_{x}$, where at each $n_{x}$ value, only
patients with at least $n_{x}$ electrodes in removed and spared tissue are
included in the analysis. Shaded blue area indicates the 95% confidence
interval for the AUC. For high $n_{x}$ too few subjects remained for analysis
to obtain a confidence interval. Bottom: Showing the number of patients
included (blue line) and the percentage of good outcome (orange line) patients
for each $n_{x}$ value. C At a value of $n_{x}=20$, $D_{RS}$ values between
outcome groups, and ROC curve are shown.

###  3.3 Interictal functional networks fluctuate over time, but these
fluctuations do not affect the distinction between outcome groups

For practical applications using interictal iEEG functional networks to
delineate epileptogenic tissue, it is important to understand if and how our
results change if different underlying data are used. Specifically, we
investigate if networks generated from segments of different duration and from
different time points affect our main findings. We systematically scanned
segments of shorter durations and measured their ability to distinguish
outcome groups in our cohort (Fig. 4A,B). Typically, a 10 second segment does
not perform significantly worse than a 1 hour segment over all $n_{x}$. For
all segments of all lengths, their AUCs lie largely within the 95% confidence
interval of the AUC for the one-hour segment (Fig. 4A,B). We do however note
that the AUC varies more from segment to segment for shorter segments
($<=$10s), indicating that consistency of results may drop for short segments.

Figure 4: Consistent results for networks sampled at different times. A Left:
AUCs for 20 separate ten second segments are shown as individual thin blue
lines. AUC of the 1h segment (thick blue line) and 95% confidence intervals
(shaded blue area) are reproduced from figure 3B for reference. Right: same
for 20 separate one minute segments. B AUC for segments of different length
shown for $n_{x}=1$ and $n_{x}=20$. The horizontal blue line (shaded area)
indicates the AUC (confidence interval) for the one-hour segment for
reference. Thick horizontal blue bars indicate the mean AUC of all segments of
a particular length. C Node strength for two example patients for three
separate one-hour time segments. Time segment 1 panels are identical to those
in the lower panels of figure 2A,B. D Thick blue line and shaded area are from
the first one-hour segment and reproduced from figure 3B for reference. Purple
and green lines indicate the AUC for the second and third one-hour segments,
respectively. Note that all functional networks of all segments were derived
by averaging over correlation matrices from 2s non-overlapping windows in this
figure. I.e. the window size to obtain correlations of the time series stayed
constant in all segments.

To test if a different one-hour time segment would change our AUCs
substantially, we repeated the same analysis for two different one-hour time
segments that are at least four hours away from any other segment in each
patient. Fig. 4C shows the node strength of two example subject over all three
one-hour segments, and although there are some variations, the gross spatial
pattern remains stable. This consistency is also reflected in the AUCs
distinguishing outcome groups (Fig. 4D), where the two new segments lie within
the confidence interval of the original segment.

##  4 Discussion

Interictal iEEG network-based approaches to predict seizure freedom after
surgery and to identify epileptogenic tissue have attracted interest in recent
years. However, major issues regarding spatial bias, incomplete coverage, and
temporal stability have remained relatively unexplored. Our study makes three
key contributions in this regard. First, we found that spatial normalisation
substantially increases the discrimination between outcome groups. Second, we
found that increased coverage of removed and spared networks leads to greater
distinction between outcome groups, but not necessarily better outcomes.
Third, we show that our results are consistent for a wide range of time scales
from minutes up to hours. Our work confirms that interictal iEEG network
analysis holds value for predicting seizure freedom after surgery.

Spatial normalisation of functional networks is a natural way to measure
signal similarity relative to a baseline. The need to establish baselines
derived from “healthy” tissue for iEEG has also been recognised by others
(Frauscher et al., 2018; Betzel et al., 2019). Besides spatial information,
white matter pathways and shared gene expression between regions also explain
functional relationships (Betzel et al., 2019). In future, these and other
variables could also be included to normalise functional networks in iEEG to
enhance the detection of the pathological aspects.

Intracranial EEG only samples specific subnetworks in the brain, which can
vary widely between patients. It is clear that such subnetworks do not
necessarily have the same properties as the whole-brain network. Recent
analyses demonstrated that even leaving out one node from iEEG functional
networks can dramatically change their network properties (Conrad et al.,
2019). The implication is that the property of the epileptogenic
tissue/network may change depending on the subnetwork sampled, which may
explain some conflicting results in the literature (Zijlmans et al., 2019).
Thus, the restricted spatial sampling inherent in iEEG is a natural limitation
in the context of functional networks, and we showed that it directly impacts
upon how informative the functional networks are for distinguishing outcome
groups. Other studies using different patient cohorts (with different
implantation strategies) may therefore achieve better or worse AUCs as a
direct consequence of the coverage of the patients in the study. Finally, it
is important to note that in our study, increased coverage (higher $n_{x}$)
leads to increased discrimination between outcome groups, but does not
necessarily lead to better outcomes. Future studies combining scalp EEG or
other recording modalities with iEEG may reveal how the iEEG subnetwork can be
related to whole-brain networks to improve localisation.

Temporal fluctuations of iEEG functional networks are well-studied during
epileptic seizures and the pre-ictal periods (Burns et al., 2014; Campo et
al., 2018; Schroeder et al., 2019); however, the interictal iEEG functional
networks are often treated as static (Palmigiano et al., 2012; Sinha et al.,
2017; Shah et al., 2019) and in contrast to the ictal networks, it is
suggested that interictal networks are stable over time (Kramer et al., 2011;
Chapeton et al., 2017). Rather than determining stability or fluctuations of
interictal functional networks as such, we asked the simpler question of
whether the time scale or time point matters for discriminating between
outcome groups. In our cohort, the time scale and time point did not
dramatically impact our results. However, this result should not be directly
interpreted as evidence for stability of the interictal functional networks.
Future work should investigate what aspects of interictal iEEG are
variable/static, which may also highlight the causal link between interictal
EEG and epileptogenic tissue.

Taken together, our results support the use of interictal intracranial EEG
networks for predicting surgical outcome and provide considerations and
practical solutions for its clinical use. Future studies should investigate
the generalisability of the approach across multiple clinical sites, and
assess the combined use with noninvasive whole-brain approaches, such as high
density scalp EEG.

##  5 Acknowledgements

We thank Gerold Baier, Louis Lemieux, Richard Rosch, and members of the CNNP
lab (www.cnnp-lab.com) for discussions on the analysis and manuscript. We
thank Catherine Scott, Roman Rodionov, and Sjoerd Vos for helping with data
organisation. B.D. receives support from the NIH—National Institute of
Neurological Disorders and Stroke U01-NS090407 (The Center for SUDEP Research)
and Epilepsy Research UK. P.N.T. and Y.W. gratefully acknowledge funding from
Wellcome Trust (208940/Z/17/Z and 210109/Z/18/Z). The authors declare no
conflict of interest. The funders played no role in the design of the study.

##  6 Author contributions

Y.W. and P.N.T conceived the idea, developed the methods, and wrote the code.
B.D. oversaw clinical acquisition and annotation of the NHNN patient EEG data.
J.D. oversaw clinical acquisition and annotation of the UCLH patient MRI data.
JdT. oversaw clinical annotation of the NHNN patient metadata. F.A.C.
contributed to clinical acquisition of UCLH patient EEG data. Y.W. and P.N.T
organised the data and performed the analysis. Y.W. and P.N.T. drafted the
manuscript and produced all figures. All authors participated in critically
reviewing and revising the manuscript.

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# Istanbul Flower Auction: The Need for Speed

Isa Hafalir, Donglai Luo, Cong Tao University of Technology Sydney. Email:
<EMAIL_ADDRESS>of Technology Sydney. Email:
<EMAIL_ADDRESS>of Technology Sydney. Email:
<EMAIL_ADDRESS>
###### Abstract

We examine the unique format of the Istanbul Flower Auction and compare it to
traditional Dutch and English auctions, emphasizing the need to auction large
volumes rapidly. In a model with time costs, we study how this auction format,
which cleverly combines Dutch and English auction mechanisms, manages time
costs by dynamically adapting to initial bidding behaviors. Our numerical
analysis considers specific time cost functions and reveals the high
performance of the Istanbul Flower Auction in comparison to standard auction
formats, in terms of both auctioneer and bidder utilities. This work
highlights the critical role of auction design in improving social welfare,
particularly in scenarios demanding the quick sale of numerous lots.

JEL classification code: C72, D44

Keywords: auction theory, time costs

##  1 Introduction

The study of auction mechanisms has long been a subject of interest in the
field of economics, given their extensive application in various markets for
the sale of goods and services (Milgrom,, 1989; Klemperer,, 1999). Auctions
are particularly vital in markets where goods need to be sold quickly and
efficiently, such as those for perishable items that are auctioned off in
numerous lots within a predetermined period. Fresh produce auctions, which
deal with the sale of perishable goods like fruits, vegetables, fish, and
flowers, are critical for ensuring these items reach consumers in a timely
manner (Cassady,, 1967). These markets are characterized by their need for
rapid sales to facilitate transactions between a large number of sellers and
buyers within very tight time frames.

Among the various auction formats, the Dutch auction has been widely adopted
in markets dealing with perishable goods, praised for its speed and
efficiency. In this format, the auctioneer sets a high starting price, which
is progressively lowered until a bid is made, ensuring a swift sale of the
item at hand. However, the Istanbul Flower Auction introduces a nuanced
approach to the auction mechanism by incorporating elements of both the Dutch
and English auction formats, adapting the mechanism based on the initial
responses of the bidders. Specifically, given a starting price $s$, Istanbul
Flower Auction operates as a Dutch auction if no one bids at $s$, and it
operates as an English auction (among the initial bidders) if at least one
bidder bids at $s$. This hybrid model is innovative in its flexibility,
potentially offering a superior solution to the unique challenges faced in
perishable goods markets.

In this paper, we study a comparative analysis of the Istanbul Flower Auction
against the backdrop of traditional Dutch and English auctions. By focusing on
the perishable goods market, particularly the sale of flowers in Istanbul, we
explore how different auction formats can impact the outcomes and auctioneer’s
and bidders’ utilities. The theoretical framework centers on the utility for
both auctioneers and bidders, taking into account the time cost as a function
of the total duration of the auction. Through this lens, we study the Istanbul
Flower Auction, Dutch Auction and English Auction in terms of the utilities of
the auctioneer and the bidders in perishable goods markets.

We study how the expected utility for the auctioneer in Istanbul Flower
Auctions is affected by various factors, including the auction’s starting
price and the time cost functions for both the auctioneer and the bidders. We
establish, especially in scenarios where any party incurs time costs, that the
Istanbul Flower Auction offers superior utility for the auctioneer compared to
traditional Dutch and English auctions (Proposition 2). This advantage is
pronounced when the auctioneer values time highly, and the bidding strategy of
the bidders is either non-increasing or invariant with respect to the starting
price. We also explore how bidders with time costs can gain more from the
Istanbul Flower Auction compared to Dutch and English auctions, emphasizing
the auction’s ability to adapt its starting price for optimized expected
utility (Proposition 3). Our numerical analysis supports the theoretical
insights, showing that the Istanbul Flower Auction format significantly
benefits both auctioneers and bidders, particularly under conditions of high
time costs. We find that the auctioneer’s preference for the Istanbul format
largely depends on her own time costs, with other auction characteristics
playing a lesser role.

Our study is motivated by the premise that the choice of auction format can
significantly affect market outcomes, especially in sectors where time is of
the essence. By examining the Istanbul Flower Auction, we seek to contribute
to the broader discourse on auction theory and practice, offering insights
that could inform the design and selection of auction formats in various
contexts. The findings of this research may have implications for auctioneers
and market organizers worldwide, suggesting that flexibility in auction design
could be key to optimizing sales processes, especially in markets where a
large number of items need to be sold in very tight time frames.111These
auctions are not only used for selling fresh produce. Used car auctions, which
are used by dealerships or leasing companies, can sell hundreds to thousands
of vehicles in a single event. Similarly, large wine auctions can involve the
sale of thousands of bottles or cases, including rare and vintage wines, over
the course of a single event.

The rest of the paper is organized as follows. The introduction (Section 1)
ends with a review of the related literature. Section 2 introduces our model
and solves for the equilibrium of Istanbul Flower Auctions. Section 3 provides
theoretical results on the payoff comparison of Istanbul Flower Auctions
against the backdrop of Dutch and English Auctions. Section 4 provides our
numerical results. Section 5 concludes. We provide all proofs in the Appendix.

###  1.1 Brief review of the literature

There is a long strand of literature on the descending Dutch and the ascending
English auctions, exploring various analogs and extensions derived from the
standard models. Comprehensive overviews of these studies can be found in
Klemperer, (2004) and Milgrom, (2004). Our contribution to the literature is
studying a unique hybrid auction format that has been implemented in Turkish
flower markets for decades.

Hybrid auctions typically allow for both ascending and descending prices
throughout the auction process, with many variants in practice. For example,
eBay provides a selling mechanism that provides bidders with a “buy-it-now”
option simultaneously with the English auction process. Mathews, (2004) shows
that under such circumstances, impatient sellers would gain extra revenue by
setting an attractive “buy-it-now” price. Furthermore, Azevedo et al., (2020)
consider a descending buyout price and attributed its advantage over standard
formats to information acquisition costs. In another example, Katok and Roth,
(2004) study multi-unit Dutch auctions for homogeneous goods with divisible
lots where the price can drop and then rise, and explained it by synergy
effects between parts of the lot.

Recent studies have recognized the importance of auction speed in the design
of mechanisms for real-world markets. Banks et al., (2003) document the trade-
off between efficiency and speed in the Federal Communication Commission
spectrum auctions, where speed and revenue can be enhanced through the
improved design proposed by Kwasnica et al., (2005). Andersson and Erlanson,
(2013) numerically show that a hybrid Vickrey-English-Dutch algorithm is
faster than the Vickrey-English or Vickrey-Dutch auctions. The role of speed
in auctions has also been highlighted in experimental studies, where
participants’ impatience was linked to their enjoyment of participation (Cox
et al.,, 1983) or designation costs (Katok and Kwasnica,, 2008). However, none
of these models formally incorporates the time cost of auction participants
into the optimization problem, except for Hafalir et al., (2023).

A similar hybrid auction format is used in various fish markets (including
Honolulu and Sydney fish markets,) for which a comprehensive analytical and
experimental study on how the time costs of the participants affect the
auction speed and revenue is provided in Hafalir et al., (2023). Unlike the
fish auctions in Honolulu and Sydney, in Istanbul Flower Auctions the
ascending or descending price is determined by the initial decision of bidders
rather than occurring as serial stages. More specifically, the main difference
between the Honolulu-Sydney fish auction and the Istanbul flower auction is
that in the former auction, the price can start going up after a bid in the
Dutch phase (whereas this is not allowed in the latter auction.)

##  2 Model

We consider a single-item auction with $n$ bidders in which the total time
lapsed in the auction is important for both the auctioneer and the bidders.
The private values of bidders are independently and identically distributed
according to a twice differentiable cumulative distribution function
$F(\cdot)$ over $[0,1]$ with a corresponding density function $f(\cdot)$. As
the clock ticks, the time cost is modeled as a decreasing and differentiable
discounting multiplier $c_{A}(t)$ for the auctioneer and $c_{B}(t)$ for the
bidders.

More specifically, if the auction ends when the bidder with value $v$ wins at
a price $p$ after $t$ units of time have passed, the auctioneer’s utility is

$U_{A}=p\cdot c_{A}(t)$

and the winning bidder’s utility is

$U_{B}=(v-p)\cdot c_{B}(t)$

where the time cost function of the auctioneer $c_{A}(t)$ and that of bidders
$c_{B}(t)$ are strictly decreasing functions of time $t$ (when time costs
exist.) More specifically, for $i=A,B$, we have
$c_{i}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, where $c_{i}(0)=1$; moreover,
$c_{i}^{\prime}(t)<0$ except when $c_{i}(t)=1$ for all $t\geq 0$. We refer to
$c_{i}(t)=1$ as the “no time cost” case for the auctioneer ($i=A$) and the
bidders ($i=B$), respectively.

Bidders who lose the auction would get a utility of $0.$ We refer to the gains
of the auctioneer as utility as opposed to revenue because the auctioneer’s
utility is not necessarily equal to the revenue in the presence of time costs.

We also define several auxiliary functions which will be useful later. Let us
denote the distribution of the highest value of $n-1$ random variables
independently distributed according to $F(\cdot)$ by a cumulative distribution
function $G(\cdot)$ with a corresponding density function $g(\cdot)$, i.e.
$G(v)=F^{n-1}(v)$. Let us denote the joint density function of the highest
value equals to $v$ and the second highest value equals to $x$ among $n$
random variables identically and independently distributed according to
$F(\cdot)$ by $h(v,x)$, we have $h(v,x)=n(n-1)f(v)f(x)F^{n-2}(x)$.

###  2.1 Istanbul Flower Auctions

The Istanbul flower auction proceeds as follows.222For a more detailed
description, see Öz and Çalışkan, (2010). It begins with a starting price
announced by the auctioneer and bidders’ decisions to _bid_ or _wait_ at the
starting price. Then, the auction turns into one of the following two formats
conditional on the number of bidders bidding at the starting price:

  1. (1)

If no one bids at the starting price, the auction then operates as a Dutch
auction: the price starts going down until someone bids (or the auction ends
with no winner when the price drops to its minimum.)

  2. (2)

If at least one bids at the starting price, the auction then operates as an
English auction for those initial bidders only: the price starts going up
until the second last of the initial bidders leaves the auction (the rise of
the price stops immediately at the starting price if there is only one bidder
who bids at the starting price).

Bidders simultaneously make decisions on whether to bid at the starting price.
We assume that decisions are made instantly, without the implied time cost.
Moreover, in equilibrium, the decision to bid also reflects the bidder’s
preference for the auction format in a consistent way such that she will not
join the other format.

We begin our analysis with the bidder’s behavior. Consider a starting price
$s\in[0,1]$ and bidders implement symmetric increasing bidding strategies.
They need to decide on (i) whether to bid at the starting price and (ii) at
which price to bid or leave when they are active bidders in the subsequent
Dutch or English auction.

We first discuss those bidders who choose to wait. Intuitively, it is always
sub-optimal for a bidder with item value $v<s$ to bid at the beginning,
because she can never acquire a positive payoff in the subsequent English
auction. Therefore, as we restrict our attention to the monotonic increasing
bidding strategy, we focus on the case featuring a cutoff $p(s)\in[0,s]$, such
that the bidder only bids at the starting price if her private value is
greater than $p(s)$ and wait otherwise.

In the subsequent Dutch auction, given the starting price $s$, let us denote
the symmetric equilibrium bid function by $b(v,s)$. Since this Dutch auction
starts from $s$ and not at $1,$ it could be possible that there is a cluster
of bids at $s$: i.e., we may have $b(v,s)=s$ for all $s\in[\lambda(s),p(s)]$
for some $\lambda\in(0,p(s))$. We assume ties break evenly for those bidders
potentially choosing to cluster their bids at $s$. Lemma 1 shows that, in
equilibrium, there will be no cluster bids at $s$, and the value cutoff $p(s)$
is determined by the starting price $s$ via the Dutch bidding function when
$p(s)<1$.

###### Lemma 1

In equilibrium, the probability that a tie occurs in the Dutch auction is
zero. Moreover, the unique solution to the cutoff value $p(s)$ as a function
of the starting price $s$ is given by $b(p(s),s)=s$ when $p(s)<1$.

A necessary condition for $b(v,s)$ to be a symmetric equilibrium strategy is
that the bidder maximizes her payoff at her own value $v$ instead of any
alternative $v^{\prime}$. For $v,v^{\prime}\leq p(s)$, it is the Dutch auction
problem (in the presence of time costs):

$v=\mathop{\arg\max}_{v^{\prime}}G(v^{\prime})(v-b(v^{\prime},s))c_{B}(s-b(v^{\prime},s))$

which relates to the following ordinary differential equation derived from the
first-order condition

$\frac{\partial b(v,s)}{\partial
v}=\frac{g(v)}{G(v)}\frac{(v-b(v,s))c_{B}(s-b(v,s))}{(v-b(v,s))c_{B}^{\prime}(s-b(v,s))+c_{B}(s-b(v,s))}.$
(1)

Indeed, the Dutch bidding strategy $b(v,s)$ is defined by the differential
equation 1 and the initial condition $b(0)=0$.

Now, consider a bidder with value $v$ who bids to initiate the English
auction, i.e., $v\geq p(s)$. She will remain in the auction and compete with
other bidders (if any) until the price reaches her value333Even with the
presence of time costs, since time costs are in the multiplicative form, it is
easy to see that the weakly dominant strategy would be to remain in the
auction till the price reaches the bidder’s value.. Her expected utility is
given by 444The superscript $FE$ corresponds to the flower auction in the
English auction phase. Similarly, $FD$: flower auction in Dutch auction phase,
$D$: Dutch auction, $E$: English auction, $F$: Flower auction.

$EU_{B}^{FE}(v,s)=(v-s)G(p(s))+\int_{p(s)}^{v}c_{B}(x-s)(v-x)dG(x)$

where the first term is her expected utility from winning the item at the
starting price when no other bidder is competing with her, and the second term
is her expected utility from winning the item at the highest price among other
competitors.

As such, when an Istanbul Flower Auction starts at $s$, the ex-ante expected
utility for bidders is given by

$EU_{B}^{F}(s)=\int_{0}^{p(s)}c_{B}(s-b(v,s))(v-b(v,s))G(v)dF(v)\\\
+\int_{p(s)}^{1}\left[(v-s)G(p(s))+\int_{p(s)}^{v}c_{B}(x-s)(v-x)dG(x)\right]dF(v)$

The resulting utility for the auctioneer is given by

$EU_{A}^{F}(s)=\int_{0}^{p(s)}c_{A}(s-b(v,s))b(v,s)dF^{n}(v)\\\
+\int_{p(s)}^{1}\left(\int_{0}^{p(s)}sh(v,x)dx+\int_{p(s)}^{v}c_{A}(x-s)xh(v,x)dx\right)dv$

Now, the optimization problem for the auctioneer is to select the best initial
price $s^{*}$ in order to optimize her utility. that is

$s^{*}=\mathop{\arg\max}_{s}EU_{A}^{F}(s)$

We can argue that the equilibrium that we consider for the Istanbul Flower
Auction is efficient. The reasons are given by: (i) the Dutch phase bidding
function is assumed to be increasing in the bidders’ item values, and (ii)
each bidder remains in the auction until the price reaches her item value
during the English phase. The following remark notes the efficiency of the
Istanbul Flower Auction.

###### Remark 1

In equilibrium, Istanbul Flower Auction is efficient, in the sense that the
item is allocated to the bidder with the highest item value.

Note that, in this setup, efficiency in the sense of “highest-valued bidder
wins the item” does not imply social welfare maximization, since time costs
also enter the bidders’ and the auctioneer’s utilities.

####  2.1.1 Dutch auctions and English auctions

Istanbul flower auction inherently contains elements of both English and Dutch
auctions and can be explicitly converted to either format by setting the
initial price at $s=0$ or $s=1$. When $s=1$, as in the Dutch auctions, the
price continuously descends from $1$ until the first bidder stops the clock
and takes it.

In this case, the expected utility for the auctioneer is given by

$EU_{A}^{D}=EU_{A}^{F}(1)=\int_{0}^{1}c_{A}(1-b(v,1))b(v,1)dF^{n}(v).$

The ex-ante expected utility for bidders is given by

$EU_{B}^{D}=EU_{B}^{F}(1)=\int_{0}^{1}c_{B}(1-b(v,1))(v-b(v,1))G(v)dv.$

The English auction is also incorporated into the Istanbul Flower Auction
framework by setting the starting price at $s=0$. In this case, the ex-ante
expected utilities for the auctioneer and the bidders are given by

$\displaystyle
EU_{A}^{E}=EU_{A}^{F}(0)=\int_{0}^{1}\int_{0}^{v}c_{A}(x)xh(v,x)dxdv,$
$\displaystyle
EU_{A}^{E}=EU_{A}^{F}(0)=\int_{0}^{1}\int_{0}^{v}c_{B}(x)(v-x)dG(x)dF(v).$

Hence, the Istanbul flower auction can be viewed as a hybrid mechanism that is
conducted as either an English or a Dutch auction. The initial price
calibrates the occurrence of each format. In the following example, we show
that the variability of the initial price at the Istanbul flower auction
renders it a utility superiority of the auctioneer over Dutch and English
auctions.

###  2.2 An illustrative example

###### Example 1

Consider a 2-bidder Istanbul Flower Auction where both the auctioneer and the
bidders share the same exponential time cost function
$c_{A}(t)=c_{B}(t)=e^{-t}$. The private value of each bidder is independently
and uniformly distributed on the unit interval $[0,1]$. The auctioneer sets
the starting price $s$.

The cutoff value $p(s)$ is determined by equating the expected utility of the
cutoff bidder in the Dutch phase and the English phase. If the auction starts
as a Dutch auction, the Dutch bidding strategy is determined by the
differential equation below with the initial condition $b(0,s)=0$:

$\dfrac{\partial b(v,s)}{\partial
v}=\frac{v-b(v,s)}{v}\frac{1}{1-(v-b(v,s))}.$

The expected utility of the cutoff bidder with value $p$ in the Dutch phase is
given by

$EU_{B}^{FD}(p,s)=e^{-(s-b(p,s)}(p-b(p,s))p.$

If the auction starts as an English auction, the expected utility of the
cutoff bidder is given by

$EU_{B}^{FE}(v,s)=(p-s)p.$

We solve $p$ from $EU_{B}^{FD}(p,s)=EU_{B}^{FE}(p,s)$ numerically and observe
that $p(s)$ increases from $0$ to $1$ in $(0,0.592)$ and fixes at $1$ in
$(0.592,1)$.

The expected utility of the auctioneer is given by

$EU_{A}^{F}(s)=\int_{0}^{p}2e^{s-b(v)}vb(v)dv-2(p^{2}+p+1)e^{s-p}-2(p-1)ps+6e^{s-1}.$

By plugging in the $p(s)$ function calculated above, we can numerically find
that the auctioneer’s utility is maximized at the starting price
$s^{*}=0.498$. The cutoff value at the optimal starting price is
$p(s^{*})=0.861$, which means that the bidder with item value $v\in[0,0.861]$
will wait for the Dutch phase, while the bidder with item value
$v\in(0.861,1]$ will bid to start the English phase at the beginning of the
auction.

Figure 1 compares the revenue performance of the flower auction with the Dutch
auction ($s=1$) and the English auction ($s=0$). The auctioneer’s utility is
higher than the English auction as long as she sets the Istanbul Flower
Auction differently. Although the Istanbul Flower Auction outperforms the
Dutch auction in most cases, it is possible for the auctioneer to get a worse
result by setting a sufficiently low starting price; in this example, it is
$s<0.043$. The expected utility for the auctioneer at the optimal starting
price is $0.340$, which is the peak of the utility curve. It outperforms the
Dutch utility of $0.216$ by $57\%$, and outperforms the English utility of
$0.207$ by $64\%$. This also outlines the importance of the auctioneer’s
decision on the starting price.

Figure 1: Example: Revenue Comparison

##  3 Payoff Comparison

In this section, we compare the payoffs of the auctioneer and bidders in the
Istanbul flower auction, Dutch auction, and English auction formats.

###  3.1 Utility for the auctioneer

The first-order derivative of the expected utility for the auctioneer in
Istanbul flower auctions is given by

$\begin{split}\frac{dEU_{A}^{F}(s)}{ds}=&\quad\frac{dp(s)}{ds}[c_{A}(s-b(p(s),s))b(p(s),s)-s]nF^{n-1}(p(s))f(p(s))\\\
&+\frac{dp(s)}{ds}[s-c_{A}(p(s)-s)p(s)]n(n-1)f(p(s))F^{n-2}(p(s))[1-F(p(s))]\\\
&+nF^{n-1}(p(s))[1-F(p(s))]\\\
&-\int_{p(s)}^{1}\int_{p(s)}^{v}c^{\prime}_{A}(x-s)xh(v,x)dxdv\\\
&+\int_{0}^{p(s)}\left[c^{\prime}_{A}(s-b(v,s))(1-\frac{\partial
b(v,s)}{\partial s})b(v,s)+c_{A}(s-b(v,s))\frac{\partial b(v,s)}{\partial
s}\right]dF^{n}(v).\end{split}$ (2)

It can be verified that this derivative is zero for all $s\in[0,1]$ when both
the auctioneer and the bidders have no time cost. This is the classic revenue
equivalence result (Myerson,, 1981; Riley and Samuelson,, 1981). We document
it in the following proposition.

###### Proposition 1

Istanbul flower auctions satisfy the Revenue Equivalence Principle when both
the auctioneer and bidders have no time costs, i.e.,
$EU_{A}^{F}(s)=\int_{0}^{1}\left[v-\frac{1-F(v)}{f(v)}\right]dF^{n}(v)$ when
$c_{A}(t)=1$ and $c_{B}(t)=1$.

However, when either the auctioneer or the bidders incur time costs, the
revenue equivalence principle no longer holds. In this more interesting case,
we provide comparative static analyses based on the local properties of the
auctioneer utility function in our time-costly setting.

Recall that $EU_{A}^{D}=EU_{A}^{F}(1)$, it is sufficient to claim that the
Istanbul Flower Auction is a better choice for the auctioneer than the Dutch
auction, by showing that the expected utility for the auctioneer is decreasing
at $s=1$ (or the derivative of the expected utility at $s=1$ is negative),
because then it must reach higher utility for some interior $s$. Similarly,
the Istanbul Flower Auction is better than the English auction if the expected
utility for the auctioneer is increasing at $s=0$ (or the derivative of the
expected utility at $s=1$ is positive), because $EU_{A}^{E}=EU_{A}^{F}(0)$.
Hence, by establishing the sign of the first-order derivative of the expected
utility for the auctioneer with respect to the starting price in stylized
scenarios, the revenue comparison follows naturally.

It is evident from the expression of the first-order derivative that the
properties of the terms $\frac{\partial b(v,s)}{\partial s}$ and
$\frac{dp(s)}{ds}$ are crucial in determining its sign.

We first note that $\frac{\partial b(v,s)}{\partial s}$ has an upper bound:

###### Lemma 2

$\frac{\partial b(v,s)}{\partial s}<1$.

Intuitively, the variation of the starting price is only partly reflected in
the variation of the Dutch bidding strategy.

Then, we describe how the Dutch-English cutoff value $p(s)$ in Istanbul Flower
Auctions changes with the starting price $s$ set by the auctioneer:

###### Lemma 3

There exists a $\tilde{s}\in(0,1)$ such that

  1. (a)

$p(s)<1$ and $\frac{dp(s)}{ds}>0$ when $s\in[0,\tilde{s})$.

  2. (b)

$p(s)=1$ and $\frac{dp(s)}{ds}=0$when $s\in[\tilde{s},1]$.

The cutoff value is always increasing with the starting price, as long as
Istanbul Flower Auctions can indeed proceed into either the Dutch or English
phase. After the starting price exceeds a certain value, the Dutch-English
cut-off is fixed at $1$. That is, all bidders will wait for the Dutch phase in
Istanbul Flower Auctions, so it simply becomes Dutch auctions starting at
price $s$.

This lemma also helps us conceptualize the “triviality” of the equilibrium in
the Istanbul Flower Auction: (i) the equilibrium is non-trivial when
$s\in(0,\tilde{s})$ in that bidders differ in their decisions between waiting
for the Dutch phase and bidding for the English phase at the beginning of the
auction, while (ii) the equilibrium is trivial when $s=0$ where all bidders
bid in the English phase or $s\in\cup[\tilde{s},1]$ where all bidders bid at
the Dutch phase.

So far, we have only established the upper bound of $1$ for $\frac{\partial
b(v,s)}{\partial s}$, which is insufficient for further analysis. Further
insights into how the Dutch phase bidding function reacts to changes in the
starting price can be obtained by imposing mild regularities on the bidder
time cost function:

###### Assumption 1

For any $t\in[0,1]$, the time cost function for bidders $c_{B}(t)$ satisfies
one of the following conditions:

  1. (a)

$\frac{d}{dt}\left[\frac{c^{\prime}_{B}(t)}{c_{B}(t)}\right]<0$, e.g. linear
cost $c_{B}(t)=1-\mu t$.

  2. (b)

$\frac{d}{dt}\left[\frac{c^{\prime}_{B}(t)}{c_{B}(t)}\right]=0.$, e.g.
exponential cost $c_{B}(t)=e^{-\mu t}$.

  3. (c)

$\frac{d}{dt}\left[\frac{c^{\prime}_{B}(t)}{c_{B}(t)}\right]>0.$, e.g. the
hyperbolic cost $c_{B}(t)=\frac{1}{1+\mu t}$.

This assumption requires the relative decreasing rate of the cost function to
change monotonically. Coincidentally, the three cases includes three commonly
used time cost (discounting) functions, linear, exponential, and hyperbolic,
respectively.

The relationship between the sign of $\frac{\partial b(v,s)}{\partial s}$ and
the assumed class of time cost functions is given by:

###### Lemma 4

In equilibrium, the bidder’s Dutch phase bidding function $b(v,s)$ in the
Istanbul Flower Auction is (a) weakly increasing, (b) invariant, and (c)
weakly decreasing in the starting price $s$, for each corresponding class of
time cost functions in Assumption 1.

To illustrate, we build numerical results for the three representative
functions. The lemma 4 states that the Dutch phase bidding function will
either shift upward (see Figure 2(a)), stay the same (see Figure 2(b)), or
shift downward (see Figure 2(c)) when the starting price increases, depending
on whether the cost functions are linear, exponential, or hyperbolic.

(a) Linear cost

(b) Exponential cost

(c) Hyperbolic cost

Figure 2: Lemma 4

With the assumption and lemmas introduced previously in this section, the
revenue comparison between the Istanbul Flower Auctions and the Dutch or
English auctions can be established in almost all scenarios. We now state our
primary analytical finding on the general advantage of Istanbul Flower
Auctions.

###### Proposition 2

When at least one party incurs time cost, the comparison between the Istanbul
Flower Auction and other standard auction formats from the perspective of the
auctioneer can be established in the following scenarios:

  1. (a)

Under Assumption 1(b) or 1(c), when the auctioneer has time cost, the Istanbul
Flower Auction with starting price $s^{*}$ is strictly better than the Dutch
auction for the auctioneer. In this case, the equilibrium is non-trivial,
i.e., $EU_{A}^{F}(s^{*})>EU_{A}^{D}$ and $s^{*}\in(0,\tilde{s})$ when
$c^{\prime}_{A}(t)<0$ and $\frac{\partial b(v,s)}{\partial s}\leq 0$.

  2. (b)

Under Assumption 1(a) or 1(b), when the auctioneer has no time cost, the
Istanbul Flower Auction with starting price $s^{*}$ gives the same expected
utility to the auctioneer as the Dutch auction. In this case, the equilibrium
is trivial, i.e., $EU_{A}^{F}(s^{*})=EU_{A}^{D}$ and $s^{*}\in[\tilde{s},1]$
when $c_{A}(t)=1$ and $\frac{\partial b(v,s)}{\partial s}\geq 0$.

  3. (c)

Under Assumption 1(c), when the auctioneer has no time cost, the Istanbul
Flower Auction with starting price $s^{*}$ is strictly better than the Dutch
auction for the auctioneer, i.e., $EU_{A}^{F}(s^{*})>EU_{A}^{D}$ when
$c_{A}(t)=1$ and $\frac{\partial b(v,s)}{\partial s}<0$.

  4. (d)

Under Assumption 1(a) or 1(b) or 1(c), the Istanbul Flower Auction with
starting price $s^{*}$ is strictly better than the English auction for the
auctioneer, i.e., $EU_{A}^{F}(s^{*})>EU_{A}^{E}$.

This proposition demonstrates the general superiority of Istanbul Flower
Auctions compared to the two conventional auction formats: Dutch and English.
The “strictly better” result for the comparison with English auctions is
established for all time-costly scenarios, where the additional assumption on
the time cost function is redundant. The “strictly better” or “indifferent”
result for the comparison with Dutch auctions depends on whether the
auctioneer incurs a time cost and on how the cost function is specified by
Assumption 1.

Istanbul Flower Auction outperforms the Dutch auctions when the auctioneer has
time cost and the Dutch phase bidding strategy of the bidder is non-increasing
in the starting price. Specifically, the auctioneer with time cost strictly
prefers Istanbul Flower Auctions when bidders face exponential, hyperbolic, or
no time cost (see Figure 3).

(a) Exponential cost

(b) Hyperbolic cost

Figure 3: A numerical example for Proposition 2 with auctioneer with time cost

When the auctioneer has no time cost, the Istanbul flower auction might appear
as a revenue equivalent but more complicated choice compared to the Dutch
auction (see Figure 4(a) and 4(c)), though it strongly depends on the assumed
shape of the cost function (see the counterexample in Figure 4(b)). By
contrasting “no time cost” scenario in Figure 4(a) with the costly scenario in
Figure 3(a), we observe a close connection between the auctioneer’s time cost
and her preference for Istanbul Flower Auctions. Moreover, it is worth noting
that Dutch auctions outperform English auctions when only bidders incur time
costs.

(a) Exponential cost

(b) Linear cost

(c) Hyperbolic cost

Figure 4: A numerical example for Proposition 2 with the auctioneer with no
time cost

We summarise the auctioneer’s preference for the three auction formats in
Table 1.

Table 1: Auctioneer’s preference for auction formats

| Assumption on bidder time cost
---|---
Auctioneer time cost | 1(b) no cost | 1(a) | 1(b) costly | 1(c)
No cost | $\textbf{IFA}=\textbf{DA}=\textbf{EA}$ | $\textbf{IFA}=\textbf{DA}>\text{EA}$ | $\textbf{IFA}=\textbf{DA}>\text{EA}$ | $\boxed{\textbf{IFA}}>\text{DA}>\text{EA}$
Costly |  $\boxed{\textbf{IFA}}>\text{DA}$ $\text{IFA}\geq\text{DA}$ | $\boxed{\textbf{IFA}}>\text{DA}$ | $\boxed{\textbf{IFA}}>\text{DA}$ |
$\boxed{\textbf{IFA}}>\text{EA}$ | $\text{IFA}>\text{EA}$ | $\boxed{\textbf{IFA}}>\text{EA}$ | $\boxed{\textbf{IFA}}>\text{EA}$

  * *

IFA — optimal Istanbul Flower auctions, DA — Dutch auctions, EA — English
auctions. The most preferred auction format is marked by bold text. The non-
trivial IFA is marked by boxed text.

In addition, note that two functional forms, the exponential cost and the
hyperbolic cost, are typically employed to model the time cost (or impatience)
in the literature. They will be the primary focus of our numerical analysis in
the next section. Therefore, we restate our findings in the main proposition
for these two scenarios below.

###### Remark 2

The English auction is always strictly suboptimal for the auctioneer, except
for the no time cost case for the auctioneer and the bidders.

###### Remark 3

Under exponential time cost, Istanbul Flower Auctions are weakly better than
Dutch auctions and strictly better than English auctions for the auctioneer
when at least one party incurs time cost. The indifference can only occur when
the auctioneer has no time cost.

###### Remark 4

Under hyperbolic time cost, Istanbul Flower Auctions are strictly better than
Dutch auctions and English auctions for the auctioneer when at least one party
incurs time cost.

###  3.2 Expected utilities for the bidders

Analogously, we can compare the ex-ante expected utility for bidders by
studying its first-order derivative at $s=0$ and $s=1$. Although the
optimality of the starting price is usually not aligned between the auctioneer
and the bidders,555In the optimal Istanbul flower auction, the auctioneer
would choose the starting price to maximize her own utility, not the bidders’
expected utilities. we demonstrate that bidders with time costs can benefit
from a certain starting price in $(0,1)$; that is, they prefer some
specifications of Istanbul Flower Auctions over the Dutch and English
auctions.

###### Proposition 3

We have:

  1. (a)

When bidders have time costs, there exists some starting price $\hat{s_{1}}<1$
such that Istanbul flower auction is strictly better than Dutch auctions for
the bidders, and this equilibrium is non-trivial, i.e., there exists
$\hat{s_{1}}<\tilde{s}$ such that $EU_{B}^{F}(\hat{s_{1}})>EU_{B}^{D}$ when
$c^{\prime}_{B}(t)<0$.

  2. (b)

When bidders have time costs, there exists some starting price $\hat{s_{2}}<1$
such that Istanbul flower auction is strictly better than English auctions for
the bidders, i.e., there exists $\hat{s_{2}}>0$ such that
$EU_{B}^{F}(\hat{s_{2}})>EU_{B}^{E}$ when $c^{\prime}_{B}(t)<0$.

We verify stronger results regarding the bidder’s preference of Istanbul
Flower Auctions over Dutch auctions numerically in the next section, for the
two functional formats of the time cost that we focus on.666In the next
section, we calculate the expected utilities of the bidders for the starting
price that is chosen to maximize the auctioneers expected utility. In fact,
the utility function of the bidders with respect to the starting price $s$
always follows the same pattern of increasing and then decreasing. We know
that $s=0$ is equivalent to the English auction and $s=1$ is equivalent to the
Dutch auction; therefore, the Istanbul Flower Auction is the preferred choice
for the bidders in these scenarios.

##  4 Numerical Analysis

We conduct a numerical analysis to evaluate the performance of Istanbul Flower
Auctions and Dutch auctions, focusing on two specific time cost
functions—exponential cost and hyperbolic cost, varying time costs of the
auctioneer and the bidders. Our findings indicate that opting for the Istanbul
Flower Auction format is more advantageous for both the auctioneer and the
bidders, and it becomes significantly more favorable when both the auctioneer
and the bidders have time costs. Furthermore, the main factor driving the
preference for Istanbul Flower Auctions for the auctioneer (bidders) stems
from the auctioneer’s (bidders’) own time costs, showing minimal influence
from other auction characteristics. Lastly, we observe that Istanbul flower
auctions’ relative advantage over Dutch auctions does not quickly fade when we
have more bidders.

(a) Exponential time cost

(b) Hyperbolic time cost

Figure 5: Expected auctioneer utility differences of Istanbul Flower Auctions
and Dutch auctions Figure 6: FA auctioneer benefit under high time cost

(a) Exponential time cost

(b) Hyperbolic time cost

Figure 7: Expected bidder utility differences of Istanbul Flower Auctions and
Dutch auctions Figure 8: FA bidder benefit under high time cost

###  4.1 Utility for the auctioneer

As illustrated in Figure 5, Istanbul Flower Auctions is strictly better than
Dutch auctions in terms of the expected utility for the auctioneer, except for
the auctioneer with no time cost (who is indifferent between the two auction
formats.) The relative difference between Istanbul Flower Auctions and Dutch
auctions becomes significantly larger when the time cost of the auctioneer is
higher.

When there is more competition in the auction, i.e., more bidders, we have the
common observation that the relative advantage of Istanbul Flower Auctions
diminishes, since more competition leads to quicker bids, and therefore
mitigates the effect of time cost. However, the relative advantage of Istanbul
Flower Auctions does not quickly attenuate to near zero within a practical
range of bidder numbers, as long as the auctioneer time cost is high. For
example (see Figure 6), when the time cost parameters are set to $0.95$, which
means that a unit of value depreciates almost half after a unit of time
elapsed under hyperbolic cost and depreciates even more under exponential
cost, the auctioneer utility is about $50\%$ higher in 2-bidder auctions and
is still at least $10\%$ higher in 10-bidder auctions when compared to Dutch
auctions.

The impact of the bidder time cost on the difference in utility for the
auctioneer is minimal. If we keep the auctioneer time cost constant, the range
of variation in the auctioneer utility difference due to different bidder time
costs is always less than $25\%$ of the maximum difference within the
parameter range $\mu\in[0,1]$ that we are examining. The profit that the
auctioneer gains from Istanbul Flower Auctions consistently decreases as the
bidders’ costs increase under exponential cost. However, under hyperbolic
cost, the auctioneer’s benefit may either decrease or first decrease and then
increase.

These observations suggest that the auctioneer’s time cost significantly
influences their preference for Istanbul Flower Auctions. If the auctioneer
values the speed of the auctions, they are inclined to choose Istanbul Flower
Auctions over Dutch auctions, irrespective of the number or type of bidders
present.

###  4.2 Utility for bidders

Although the initial price set by the auctioneer may not be ideal for the
bidders, Istanbul Flower Auctions are consistently more advantageous than
Dutch auctions in terms of the expected utility for bidders in all scenarios
depicted in Figure 7, except for bidders with no time costs, who are
indifferent between the two auction formats. The relative advantage of
Istanbul Flower Auctions significantly increases when the time cost of bidders
is higher.

When varying the number of bidders, we observe that the relative advantage of
Istanbul Flower Auctions becomes smaller in a more competitive market. It is
important to highlight that this benefit remains significant within a
practical range of bidders, provided that the cost of time for bidders is
substantial. For example, using the time cost scenario from the preceding
subsection, the bidder’s benefit exceeds $40\%$ in 2-bidder auctions and
remains above $10\%$ even in 10-bidder auctions when switching from the Dutch
auction format to the Istanbul Flower Auction format.

The time cost of the auctioneer does not play an important role in the bidder
utility difference. When fixing the bidder time cost, the range of variation
in the bidder utility difference due to different auctioneer time costs is
always within $25\%$ of the maximum difference within the parameter range
$\mu\in[0,1]$. Moreover, the difference in bidder utility consistently becomes
smaller as the time cost of the auctioneer increases under either time
discounting function we consider.

Similar to our findings on the auctioneer utility difference, these
observations suggest that the time cost of the bidder herself is the primary
factor that influences her preference for Istanbul Flower Auctions, regardless
of the number of bidders or the time cost of the auctioneer.

##  5 Conclusion

Our analysis reveals that the Istanbul Flower Auction format, with its
innovative approach to combining Dutch and English auction elements, offers
distinct advantages in achieving higher utilities for both auctioneers and
bidders. This format proves particularly beneficial in markets characterized
by high time sensitivity, such as perishable goods auctions. The flexibility
to switch between auction types based on initial bidding activity allows for
more efficient price discovery and less time costs. It can lead to higher
utility for all parties involved, especially under conditions of time cost.
The numerical findings suggest that the auctioneer’s or bidder’s preference
for the Istanbul Flower Auction largely depends on their own time costs. This
insight aligns with real-world scenarios where sellers need to sell goods
swiftly. Their urgency justifies the use of Istanbul Flower Auction regardless
of the buyers’ perspectives. Moreover, the auction is beneficial for all
parties: it does not disadvantage bidders even when they incur no cost by
participating, supporting the adoption of Istanbul Flower Auction in the
Istanbul flower market.

These findings suggest that adopting flexible auction formats like the
Istanbul Flower Auction could enhance outcomes for all the parties involved in
various auction-based markets, particularly those where the speed of the
auction is important due to the large number of items needing to be sold in a
fixed amount of time.

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## Appendix: Omitted Proofs

Proof of Lemma 1. In contrast to the Dutch auction setting where the price
continuously declines from $1$ and we can solve for a strictly increasing
bidding function; the bidding function here may be capped by the starting
price $s$ when the bidder’s private value exceeds certain cutoff
$\lambda(s)\geq s$, i.e., Therefore, the bidding function for the subsequent
Dutch auction is given by

$\beta(v,s)=\begin{cases}b(v,s)&v\leq\lambda(s)\\\ s&\lambda(s)\leq v\leq
p(s)\end{cases}$

and the bidder’s expected utility is given by

$EU_{B}^{FD}(v,s)=\begin{cases}c_{B}(s-b(v,s))(v-b(v,s))G(v)&v\leq\lambda(s)\\\
\sum\limits_{k=0}^{n-1}\binom{n-1}{k}\frac{v-s}{k+1}F^{n-1-k}(\lambda(s))[F(p(s))-F(\lambda(s))]^{k}&\lambda(s)\leq
v\leq p(s)\end{cases}$

where $b(\cdot,s)\rightarrow[0,s]$ is a strictly increasing and differentiable
function with $b(0)=0$ given the starting price $s$.

Step 1 We prove $\lambda(s)=p(s)$ in two cases, depending on whether the
subsequent auction is possible to be an English auction.

Let’s start from the case $p(s)<1$, where both Dutch and English can be the
subsequent auction format. Suppose $\lambda(s)<p(s)$, a bidder with value
$v=p(s)$ should be indifferent between initially not bidding or bidding at the
starting price $s$ in equilibrium, that is

$EU_{B}^{FD}(p(s),s)=EU_{B}^{FE}(p(s),s).$

We know that

$EU_{B}^{FE}(p(s),s)=(p(s)-s)G(p(s))$

and

$\displaystyle EU_{B}^{FD}(p(s),s)$
$\displaystyle=\sum\limits_{k=0}^{n-1}\binom{n-1}{k}\frac{p(s)-s}{k+1}F^{n-1-k}(\lambda(s))[F(p(s))-F(\lambda(s))]^{k}$
$\displaystyle=\frac{p(s)-s}{n}\frac{F^{n}(p(s))-F^{n}(\lambda(s))}{F(p(s))-F(\lambda(s))}$

Let’s define an auxiliary function

$k(x)=\frac{1-x^{n}}{n(1-x)}=\frac{1+x+\cdots+x^{n-1}}{n}.$

It is obvious that $k^{\prime}(x)>0$ for $x\in(0,1)$, and $k(1)=1$. Then
notice that $\frac{F(\lambda(s))}{F(p(s))}\in(F(\lambda(s)),1)$ and

$k\left(\frac{F(\lambda(s))}{F(p(s))}\right)=\frac{EU_{B}^{FD}(p(s),s)}{EU_{B}^{FE}(p(s),s)},$

we conclude that there is no equilibrium such that $\lambda(s)<p(s)$ when
$p(s)<1$.

Now let’s look into the case $p(s)=1$, where Istanbul flower auctions become
Dutch auctions starting at the given price $s$. Suppose $\lambda(s)<p(s)$, any
Dutch bidder whose bid is capped at $s$ will deviate to bidding at the
beginning, because she will win the item at the price $s$ for sure in the
English auction instead of possibly competing with other bidders in the Dutch
auction.

Step 2 Given that $\lambda(s)=p(s)$ in equilibrium, when $p(s)<1$, use the
indifference condition at $v=p(s)$ again, we have

$\displaystyle EU_{B}^{FD}(p(s),s)$ $\displaystyle=EU_{B}^{FE}(p(s),s)$
$\displaystyle c_{B}(s-b(p(s),s))(p(s)-b(p(s),s))G(p(s))$
$\displaystyle=(p(s)-s)G(p(s))$ $\displaystyle
c_{B}(s-b(p(s),s))(p(s)-b(p(s),s))$ $\displaystyle=p(s)-s$

By definition of the time cost function $c_{B}(0)=1$, so $b(p(s),s)=s$ is a
solution. Let’s define an auxiliary function

$l(p)=p-s-c_{B}(s-b(p,s))(p-b(p,s)).$

Compute its first-order derivative and reformat it with the ordinary
differential equation derived for $b(\cdot,s)$, we have

$\displaystyle l^{\prime}(p)$
$\displaystyle=1-\left[-c_{B}^{\prime}(s-b(p,s))\frac{\partial
b(p,s)}{\partial p}(p-b(p,s))+c_{B}(s-b(p,s))(1-\frac{\partial
b(p,s)}{\partial p})\right]$
$\displaystyle=1-c_{B}(s-b(p,s))\left[1-\frac{g(p)}{G(p)}(p-b(p,s))\right]$
$\displaystyle>0$

because $c_{B}(\cdot)\in[0,1]$ and $p\geq b(p,s)$. Therefore, $l(p)$ is
strictly increasing in $p$, and the solution to $l(p)=0$ is unique, which is
implicitly given by $b(p,s)=s$.

When $p(s)=1$, then we do not necessarily have $b(p,s)=s$, but the following
must be true $b(p,s)\leq s.$

$\square$

Proof of Lemma 2.

Starting from the Dutch bidding strategy in equation 1, we take the partial
derivative with respect to $s$ on both sides, we get:

$\displaystyle\frac{\partial^{2}b(v;s)}{\partial s\partial v}=$
$\displaystyle\frac{g(v)}{G(v)}\big{(}\frac{-\frac{\partial b}{\partial
s}}{(v-b(v;s))\frac{c^{\prime}_{B}(s-b(v;s))}{c_{B}(s-b(v;s))}+1}+$
$\displaystyle(v-b(v))(-\frac{-\frac{\partial b}{\partial
s}\frac{c^{\prime}_{B}(s-b(v;s))}{c_{B}(s-b(v;s))}+(v-b(v;s))\cdot\frac{\partial\frac{c^{\prime}_{B}(s-b(v;s))}{c_{B}(s-b(v;s))}}{\partial
s}}{[(v-b(v;s))\frac{c^{\prime}_{B}(s-b(v;s))}{c_{B}(s-b(v;s))}+1]^{2}})\big{)}$

To simplify, let
$\gamma(v;s):=\frac{c^{\prime}_{B}(s-b(v;s))}{c_{B}(s-b(v;s))}$, the above
equation turns into:

$\frac{\partial^{2}b(v;s)}{\partial s\partial
v}=\frac{g(v)}{G(v)}\big{(}\frac{-\frac{\partial b}{\partial
s}}{(v-b(v))\gamma(v;s)+1}+(v-b(v))(-\frac{-\frac{\partial b}{\partial
s}\gamma(v;s)+(v-b(v))\cdot\frac{\partial\gamma(v;s)}{\partial
s}}{[(v-b(v))\gamma(v;s)+1]^{2}})\big{)}$

and

$\frac{\partial\gamma(v;s)}{\partial s}=(1-\frac{\partial b}{\partial
s})\frac{d\gamma}{d(s-b(v))}$

Then,

$\frac{\partial^{2}b(v;s)}{\partial s\partial
v}=\frac{g(v)}{G(v)}\frac{-\frac{\partial b}{\partial
s}-(v-b(v))^{2}(1-\frac{\partial b}{\partial
s})\frac{d\gamma}{d(s-b(v))}}{[(v-b(v))\gamma(v;s)+1]^{2}}$

Note $\frac{\partial b}{\partial s}$ is a function of $v$, assuming $b(v;s)$
is well-behaved such that we could interchange the order of partial
derivative, then $\frac{\partial b}{\partial s}$ is determined by a
differential equation:

$\frac{d\frac{\partial b(v;s)}{\partial
s}}{dv}=\frac{g(v)}{G(v)}\frac{-\frac{\partial b}{\partial
s}-(v-b(v))^{2}(1-\frac{\partial b}{\partial
s})\frac{d\gamma}{d(s-b(v))}}{[(v-b(v))\gamma(v;s)+1]^{2}}$

Since we focus on the class of symmetric increasing bidding function, we have
$b(0)\equiv 0$ as the bidder with the lowest type has no interest in either
overbidding or underbidding. It follows that $\frac{\partial b(v;s)}{\partial
s}|_{v=0}=0$ is the initial condition of the differential equation. It is
obvious when $\frac{\partial b(v)}{\partial s}\rightarrow 1$ at certain point,
$\frac{d\frac{\partial b(v;s)}{\partial s}}{dv}<0$ at that point. Therefore,
the curve $\frac{\partial b}{\partial s}(v)$ could never reach the border of
$\frac{\partial b(v)}{\partial s}=1$ when the initial condition starts under
the line.

$\square$

Proof of Lemma 3.

The starting price $s$ functions as a cap on the Dutch phase bids. By Lemma 2,
we know that the Dutch phase bidding curve will never move upward faster than
the cap $s$. It is also evident that $b(1,s)<1$. Therefore, when increasing
$s$ from $0$ to $1$, it departs upwards from the Dutch phase bidding curve and
finally becomes non-restrictive to the Dutch phase bidding strategy after
reaching some interim $\tilde{s}$. By Lemma 1, we can solve $\tilde{s}$ from
$b(1,\tilde{s})=\tilde{s}$, and use the property of the $l(p)$ function
defined in the proof of the lemma to prove that $p(s)\equiv 1$ for all
$s\in[\tilde{s},1]$.

Then, let us consider $s\in[0,\tilde{s})$ where the Dutch phase bids of high-
value bidders are capped. By Lemma 1 we know that

$b(p(s),s)=s.$

Taking the derivative with respect to $s$ on the above equation, we have

$\frac{\partial b(v,s)}{\partial v}\frac{dp(s)}{ds}+\frac{\partial
b(v,s)}{\partial s}=1.$

We are looking for a strictly increasing Dutch bidding strategy, which means
$\frac{\partial b(v,s)}{\partial v}>0$. By Lemma 2, we have $\frac{\partial
b(v,s)}{\partial s}<1$. Therefore, we know that

$\frac{dp(s)}{ds}>0.$

$\square$

Proof of Lemma 4.

We first investigate the properties of the Dutch phase bidding function at the
intersection point if two bidding functions with different $s$ possibly
intersect with each other. Let us write the Dutch ODE in the following form:

$\frac{G(v)}{g(v)}=\frac{v-b(v,s)}{\dfrac{\partial b(v,s)}{\partial
v}}\frac{1}{(v-b(v,s))\dfrac{c_{B}^{\prime}(s-b(v,s))}{c_{B}(s-b(v,s))}+1}.$

Recall that we focus on such a Dutch phase bidding strategy that is strictly
increasing in the item value $v$. Therefore, the second denominator must be
positive for any $v$ and $s$ in equilibrium. Consider two starting prices
$s_{1}$ and $s_{2}\geq s_{1}$. Suppose there exists a $v_{0}\in(0,1]$ such
that $b(v,s_{1})=b(v,s_{2})=b_{0}$, then we have

$\frac{v_{0}-b_{0}}{\dfrac{\partial b(v_{0},s_{1})}{\partial
v}}\frac{1}{(v_{0}-b_{0})\dfrac{c_{B}^{\prime}(s_{1}-b_{0})}{c_{B}(s_{1}-b_{0})}+1}=\frac{v_{0}-b_{0}}{\dfrac{\partial
b(v_{0},s_{2})}{\partial
v}}\frac{1}{(v_{0}-b_{0})\dfrac{c_{B}^{\prime}(s_{2}-b_{0})}{c_{B}(s_{2}-b_{0})}+1}.$

Then, the comparison between $\frac{\partial b(v_{0},s_{1})}{\partial v}$ and
$\frac{\partial b(v_{0},s_{2})}{\partial v}$ is determined by the relationship
between $\frac{c_{B}^{\prime}(s_{1}-b_{0})}{c_{B}(s_{1}-b_{0})}$ and
$\frac{c_{B}^{\prime}(s_{2}-b_{0})}{c_{B}(s_{2}-b_{0})}$. We also know that
$s_{1}<s_{2}$, so we have the following result:

  1. 1.

$\frac{\partial b(v_{0},s_{1})}{\partial v}<\frac{\partial
b(v_{0},s_{2})}{\partial v}$ under condition (a).

  2. 2.

$\frac{\partial b(v_{0},s_{1})}{\partial v}=\frac{\partial
b(v_{0},s_{2})}{\partial v}$ under condition (b).

  3. 3.

$\frac{\partial b(v_{0},s_{1})}{\partial v}>\frac{\partial
b(v_{0},s_{2})}{\partial v}$ under condition (c).

Then we investigate the properties of the Dutch phase bidding function at the
left end $0$. Apply Euler’s method to the ODE with the initial condition
$b(0,s)=0$, for $h\rightarrow 0$ we have

$\displaystyle b(h,s)\approx b(0,s)+h\frac{\partial b(0,s)}{\partial v}=0$
$\displaystyle b(2h,s)\approx b(h,s)+h\frac{\partial b(h,s)}{\partial
v}\approx\frac{g(h)}{G(h)}\frac{1}{\dfrac{c_{B}^{\prime}(s)}{c_{B}(s)}+\dfrac{1}{h}}$

and finally have

$\frac{\partial b(h,s)}{\partial
v}\approx\frac{g(h)}{G(h)}\frac{1}{h\dfrac{c_{B}^{\prime}(s)}{c_{B}(s)}+1}.$

Taking the limit we get $\frac{\partial b(0,s)}{\partial v}$. Similarly, the
comparison between $\frac{\partial b(0,s_{1})}{\partial v}$ and
$\frac{\partial b(0,s_{2})}{\partial v}$ is determined by the relationship
between $\frac{c_{B}^{\prime}(s_{1})}{c_{B}(s_{1})}$ and
$\frac{c_{B}^{\prime}(s_{2})}{c_{B}(s_{2})}$. Given that $s_{1}<s_{2}$, we
have the following result:

  1. 1.

$\frac{\partial b(0,s_{1})}{\partial v}<\frac{\partial b(0,s_{2})}{\partial
v}$ under condition (a).

  2. 2.

$\frac{\partial b(0,s_{1})}{\partial v}=\frac{\partial b(0,s_{2})}{\partial
v}$ under condition (b).

  3. 3.

$\frac{\partial b(0,s_{1})}{\partial v}>\frac{\partial b(0,s_{2})}{\partial
v}$ under condition (c).

Finally, use the ODE initial condition $b(0,s_{1})=b(0,s_{2})=0$ together with
the above results, we conclude that:

  1. 1.

$b(v,s_{1})\leq b(v,s_{2})$ for any $v\in(0,1]$, or equivalently
$\frac{\partial b(v,s)}{\partial s}\geq 0$, under condition (a).

  2. 2.

$b(v,s_{1})=b(v,s_{2})$ for any $v\in(0,1]$, or equivalently $\frac{\partial
b(v,s)}{\partial s}=0$, under condition (b).

  3. 3.

$b(v,s_{1})\geq b(v,s_{2})$ for any $v\in(0,1]$, or equivalently
$\frac{\partial b(v,s)}{\partial s}\leq 0$, under condition (c).

Note that we only have a weak monotonicity here because we cannot exclude the
possibility that the two bidding functions are tangent to each other at some
point.

$\square$

Proof of Proposition 2.

(a) By Lemma 1 and Lemma 3, we either have

$s\in[0,\tilde{s}),p(s)<1,\frac{dp(s)}{ds}>0,b(p(s),s)=s$

or have

$s\in(\tilde{s},1],p(s)=1,\frac{dp(s)}{ds}=0,b(p(s),s)<s.$

Specifically, $p(s)$ is not differentiable at $\tilde{s}$: the left-sided
limit belongs to the first case, and the right-sided limit belongs to the
second case. Plugging these into equation 2, the first-order derivative of the
auctioneer utility for $s\in[\tilde{s},1)$ is given by

$\frac{dEU_{A}^{F}(s)}{ds}=\int_{0}^{1}\left[c^{\prime}_{A}(s-b(v,s))(1-\frac{\partial
b(v,s)}{\partial s})b(v,s)+c_{A}(s-b(v,s))\frac{\partial b(v,s)}{\partial
s}\right]dF^{n}(v).$

Given that the auctioneer has a time cost, that is, $c^{\prime}_{A}(t)<0$, and
our assumption that $\frac{\partial b(v,s)}{\partial s}\leq 0$, we finally
have

$\frac{dEU_{A}^{F}(s)}{ds}<0,s\in[\tilde{s},1).$

Therefore, we must have $EU_{A}^{F}(s^{*})>EU_{A}^{F}(s)\geq
EU_{A}^{F}(1)=EU_{A}^{D}$ for any $s\in[\tilde{s},1]$. It is obvious that
$s^{*}<\tilde{s}$, and $s^{*}>0$ is proved in (d), so the optimal Istanbul
Flower Auction is non-trivial.

(b) Again, by Lemma 1 and Lemma 3, we can compute that for the auctioneer with
no time cost $c_{A}(t)\equiv 1$, when $s\in(0,\tilde{s})$, the first and
fourth terms on the right-hand side of equation 2 are zeros, the sum of the
second and third terms is positive because we can prove that
$\frac{dp(s)}{ds}(p-s)<\frac{F(p(s))}{(n-1)f(p(s))}$ by taking the derivative
of $s$ on both sides of the equation
$b(p(s),s)=p(s)-\int_{0}^{p(s)}\frac{G(y)c_{B}(s-b(y,s))}{G(p)}dy=s$, and the
last term is simplified to

$\int_{0}^{p(s)}\frac{\partial b(v,s)}{\partial s}dF^{n}(v).$

Given our assumption that $\frac{\partial b(v,s)}{\partial s}\geq 0$, we
finally have

$\frac{dEU_{A}^{F}(s)}{ds}>0,s\in(0,\tilde{s}).$

When $s\in(\tilde{s},1)$, the only difference is that the first four terms are
all zeros, and finally we have

$\frac{dEU_{A}^{F}(s)}{ds}\geq 0,s\in(\tilde{s},1).$

Therefore, we must have
$EU_{A}^{D}=EU_{A}^{F}(1)=EU_{A}^{F}(s^{*})>EU_{A}^{F}(s)$ for any
$s\in[0,\tilde{s})$. It is obvious that $s^{*}\geq\tilde{s}$, which means that
the optimal Istanbul Flower Auction is trivial.

(c) Again, by Lemma 1 and Lemma 3, we can compute that for the auctioneer with
no time cost $c_{A}(t)\equiv 1$, when $s\in(\tilde{s},1)$, the first four
terms on the right-hand side of equation 2 are all zeros, and the last term
becomes

$\int_{0}^{p(s)}\frac{\partial b(v,s)}{\partial s}dF^{n}(v).$

Given our assumption that $\frac{\partial b(v,s)}{\partial s}<0$, we finally
have

$\frac{dEU_{A}^{F}(s)}{ds}<0,s\in(\tilde{s},1).$

Therefore, we must have $EU_{A}^{F}(s^{*})>EU_{A}^{F}(s)\geq
EU_{A}^{F}(1)=EU_{A}^{D}$ for any $s\in(\tilde{s},1]$.

(d) When the auctioneer has time cost, that is, $c^{\prime}_{A}(t)<0$, using
equation 2 we can compute that

$\frac{dEU_{A}^{F}(0)}{ds}=-\int_{0}^{1}c^{\prime}_{A}(x,s)xh(v,x)dxdv<0.$

Therefore, we must have

$EU_{A}^{F}(s^{*})>EU_{A}^{F}(0)=EU_{A}^{E}.$

When the auctioneer has no time cost, that is, $c_{A}(t)\equiv 1$, we can
write the Dutch phase bidding function as

$b(v,s)=v-\int_{0}^{v}\frac{G(y)c_{B}(s-b(y,s)}{G(v)c_{B}(s-b(v,s)}dy.$

Let us denote the special case of no bidder time cost by subscript $0$. We
have

$b_{0}(v,s)=v-\int_{0}^{v}\frac{G(y)}{G(v)}dy<b(v,s).$

Then, for the Dutch auction, we have

$EU_{A}^{D}=EU_{A}^{F}(1)=\int_{0}^{1}b(v,1)dF^{n}(v)>\int_{0}^{1}b_{0}(v,1)dF^{n}(v)=EU_{A0}^{D}.$

For the English auction, since the winner will always pay the second highest
item value and the auctioneer has no time cost, the expected utility of the
auctioneer is always the same, which is given by

$EU_{A}^{E}=EU_{A}^{F}(0)=\int_{0}^{1}\int_{0}^{v}xh(v,x)dxdv=EU_{A0}^{E}.$

By the Revenue Equivalence Principle, we have

$EU_{A0}^{E}=EU_{A0}^{D}=\int_{0}^{1}\left[v-\frac{1-F(v)}{f(v}\right]dF^{n}(v).$

Therefore, we must have

$EU_{A}^{F}(s^{*})\geq EU_{A}^{F}(1)>EU_{A}^{F}(0)=EU_{A}^{E}.$

$\square$

Proof of Proposition 3. (a) For $s\in(\tilde{s},1)$, we can compute that

$\begin{split}\frac{dEU_{B}^{F}(s)}{ds}=\int_{0}^{1}\left[c^{\prime}_{B}(s-b(v,s))(1-\frac{\partial
b(v,s)}{\partial s})(v-b(v,s))\right.\\\ \left.-c_{B}(s-b(v,s))\frac{\partial
b(v,s)}{\partial s}\right]G(v)dF(v).\end{split}$

Using the assumption together with Lemma 2, we know that $\frac{\partial
b(v,s)}{\partial s}\in[0,1)$. Then, for bidders with time costs, we finally
have

$\frac{dEU_{B}^{F}(1)}{ds}<0,s\in(\tilde{s},1).$

Therefore, there exists $\hat{s_{1}}<\tilde{s}$ such that
$EU_{B}^{F}(\hat{s_{1}})>EU_{B}^{F}(1)=EU_{B}^{D}$.

(b) We can compute that

$\frac{dEU_{B}^{F}(0)}{ds}=-\int_{0}^{1}\int_{0}^{v}c^{\prime}_{B}(x)(v-x)dG(x)dF(v).$

Then, for bidders with time costs, we always have

$\frac{dEU_{B}^{F}(0)}{ds}>0.$

Therefore, there exists $\hat{s_{2}}>0$ such that
$EU_{B}^{F}(\hat{s_{2}})>EU_{B}^{F}(0)=EU_{B}^{E}$. $\square$