# Dirac plus Nambu Monopoles in the Standard Model

George Lazaridesa, Qaisar Shafib, Tanmay Vachaspatic aSchool of Electrical and
Computer Engineering, Faculty of Engineering, Aristotle University of
Thessaloniki, Thessaloniki 54124, Greece
bBartol Research Institute, Department of Physics and Astronomy, University of
Delaware, Newark, DE 19716, USA
cPhysics Department, Arizona State University, Tempe, AZ 85287, USA.

###### Abstract

We show how in the standard electroweak model three $SU(2)_{L}$ Nambu
monopoles, each carrying electromagnetic (EM) and Z- magnetic fluxes, can
merge (through Z-strings) with a single $U(1)_{Y}$ Dirac monopole to yield a
composite monopole that only carries EM magnetic flux. Compatibility with the
Dirac quantization condition requires this composite monopole to carry six
quanta ($12\pi/e$) of magnetic charge, independent of the electroweak mixing
angle $\theta_{w}$. The Dirac monopole is not regular at the origin and the
energy of the composite monopole is therefore divergent. We discuss how this
problem is cured by embedding $U(1)_{Y}$ in a grand unified group such as
$SU(5)$. A second composite configuration with only one Nambu monopole and a
colored $U(1)_{Y}$ Dirac monopole that has minimal EM charge of $4\pi/e$ is
also described. Finally, there exists a configuration with an EM charge of
$8\pi/e$ as well as screened color magnetic charge.

##  I Introduction

It is widely recognized that the Standard Model (SM) based on the gauge
symmetry $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}/Z_{3}\times Z_{2}$ does not
contain topologically stable finite energy monopoles. This stems from the fact
that the vacuum manifold in the SM is a three-sphere and does not have a non-
trivial second homotopy group that is required for the existence of
topological finite energy monopoles. Note that this result is unaffected if
additional scalar and/or matter fields are introduced in the SM. Nevertheless,
as we show, interesting composite topological structures are present in the SM
and its extensions such as grand unified theories (GUTs).

It was recognized sometime ago Nambu:1977ag ; Vachaspati:1992fi ;
Achucarro:1999it  that a ‘t Hooft-Polyakov type HOOFT1974276 ; Polyakov:1974ek
monopole configuration can be realized within the $SU(2)_{L}$ sector of the
SM. This is achieved by utilizing a Higgs doublet field configuration which
effectively behaves like a scalar $SU(2)_{L}$ triplet with hypercharge $Y=0$.
This yields a monopole configuration that carries one unit of $T_{L}^{3}$
magnetic charge, where $T_{L}^{3}={\rm diag}(1,-1)$ denotes the third
generator of $SU(2)_{L}$. In other words, the monopole emits both
electromagnetic (EM) and Z magnetic flux, where the latter appears in a flux
tube (or Z string). The EM and Z magnetic flux carried by the monopole are
found to be $(4\pi/e)\sin^{2}\theta_{w}$ and
$(4\pi/e)\sin\theta_{w}\cos\theta_{w}$ respectively where $\theta_{w}$ is the
electroweak mixing angle with $\sin^{2}\theta_{w}=0.23$. This monopole cannot
be purely electromagnetic in nature with the Z flux screened within its core
(size $\sim M_{Z}^{-1}$) because it would violate Gauss’s law pertaining to
the magnetic component of $U(1)_{Y}$.

A recent paper PhysRevD.103.095021  shows that the Nambu monopole
configurations routinely arise in GUTs, with the simplest example provided by
SU(5). A new feature here is the presence of an elementary scalar $SU(2)_{L}$
triplet with $Y=0$ that acquires an induced vacuum expectation value (VEV)
from the electroweak breaking. In addition, the quantized value of the Z flux
is also predicted by exploiting the well-known prediction of
$\sin^{2}\theta_{w}=3/8$.

Motivated by these considerations, it seems appropriate to enquire whether a
purely electromagnetic monopole configuration can be realized in the SM.
Clearly, this cannot happen without involving the $U(1)_{Y}$ sector, in
contrast to the claim in Refs. Hung:2020vuo ; Ellis:2020bpy . Ignoring the
singular nature of a Dirac monopole dirac  for a moment, it turns out that a
purely electromagnetic monopole can be realized by combining via Z strings a
single Y monopole of appropriate magnetic charge with three Nambu monopoles.
This composite monopole structure carries a magnetic charge of ($12\pi/e$) in
order to satisfy the Dirac quantization conditions in the presence of quarks
and charged leptons. All dependence on $\sin^{2}\theta_{w}$ disappears in
arriving at this conclusion. We show how the embedding of $U(1)_{Y}$ in a
grand unified framework such as SU(5) resolves the singular nature of the
above composite monopole. As expected, the monopole turns out to be superheavy
with mass comparable to the GUT scale. Two additional examples are provided of
configurations that carry EM charges of $4\pi/e$ and $8\pi/e$ as well as
screened color magnetic charge.

##  II Combining Nambu and Dirac Monopoles in the Standard Model

We are admitting the possibility of a $U(1)_{Y}$ Dirac monopole in the SM.
From the Dirac quantization condition such a monopole carries magnetic
hypercharge

$m_{Y}=\frac{12\pi}{g^{\prime}}n,\ \ n\in{\cal Z},$ (1)

where $g^{\prime}$ is the hypercharge gauge coupling constant, and the left-
handed quark doublet $(u,d)_{L}$ has hypercharge $1/6$. Next consider a Nambu
monopole in the symmetry breaking $SU(2)_{L}\to U(1)_{L}$. Such a monopole has
weak magnetic charge given by the condition

$m_{L}=\frac{2\pi}{g/2}n^{\prime}=\frac{4\pi}{g}n^{\prime},\ \
n^{\prime}\in{\cal Z},$ (2)

where $g$ is the $SU(2)_{L}$ coupling constant. We can work in the unitary
gauge for the $SU(2)_{L}$ monopole such that the magnetic flux from the
monopole consists only of $W^{3}_{\mu}$ gauge field, where $W^{a}_{\mu}$
denote the usual $SU(2)_{L}$ gauge fields.

We now wish to put the Nambu $SU(2)_{L}$ and Dirac $U(1)_{Y}$ monopoles
together and re-express the charges in terms of the post-electroweak symmetry
breaking gauge fields: $Z_{\mu}$ and $A_{\mu}$. The relation between
$(W^{3}_{\mu},Y_{\mu})$ and $(Z_{\mu},A_{\mu})$ is given by

$\displaystyle W^{3}_{\mu}=\cos\theta_{w}Z_{\mu}+\sin\theta_{w}A_{\mu},$ (3)
$\displaystyle Y_{\mu}=\cos\theta_{w}A_{\mu}-\sin\theta_{w}Z_{\mu}.$ (4)

Therefore, the $Z_{\mu}$ and $A_{\mu}$ magnetic charges on $n^{\prime}$
$SU(2)_{L}$ monopoles are

$m_{LZ}=\frac{4\pi n^{\prime}}{g}\cos\theta_{w},\ \ m_{LA}=\frac{4\pi
n^{\prime}}{g}\sin\theta_{w},$ (5)

and on $n$ $U(1)_{Y}$ monopoles are

$m_{YZ}=-\frac{12\pi n}{g^{\prime}}\sin\theta_{w},\ \ m_{YA}=\frac{12\pi
n}{g^{\prime}}\cos\theta_{w}.$ (6)

The net $Z_{\mu}$ and $A_{\mu}$ magnetic charges on a conglomerate of $n$
$U(1)_{Y}$ and $n^{\prime}$ $SU(2)_{L}$ monopoles are

$\displaystyle m_{Y+L,Z}$ $\displaystyle=$ $\displaystyle\frac{4\pi
n^{\prime}}{g}\cos\theta_{w}-\frac{12\pi n}{g^{\prime}}\sin\theta_{w},$ (7)
$\displaystyle m_{Y+L,A}$ $\displaystyle=$ $\displaystyle\frac{4\pi
n^{\prime}}{g}\sin\theta_{w}+\frac{12\pi n}{g^{\prime}}\cos\theta_{w}.$ (8)

This configuration should not have any net $Z$ magnetic charge because the $Z$
magnetic fields are confined once the electroweak symmetry is broken. Any net
$Z$ flux would form a string that would confine the monopole conglomerate to
an anti-conglomerate. Thus, we require

$\frac{4\pi n^{\prime}}{g}\cos\theta_{w}-\frac{12\pi
n}{g^{\prime}}\sin\theta_{w}=0.$ (9)

Together with the relations

$\cos\theta_{w}=\frac{g}{\sqrt{g^{2}+g^{\prime 2}}},\ \
\sin\theta_{w}=\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime 2}}},$ (10)

the constraint in (9) gives

$n^{\prime}=3n,$ (11)

and so the conglomerate should contain three times as many Nambu monopoles as
the Dirac Y-monopoles.

Next let us consider the EM magnetic charge on the conglomerate. From (8),
taking into account (10) and (11), we get

$m_{Y+L,A}=\frac{12\pi}{e}n$ (12)

where $e=gg^{\prime}/\sqrt{g^{2}+g^{\prime 2}}$ is the electromagnetic
coupling constant. Therefore the minimal EM charge of the conglomerate is
$6\times 2\pi/e$ and this result is independent of $\sin^{2}\theta_{w}$.

In Fig. 1 we sketch the minimal allowed monopole conglomerate. From (5), we
see that the minimal ($n^{\prime}=1$) magnetic flux along the Z-string is
$4\pi/g_{z}$, where $g_{z}=g/c_{w}$ is the gauge coupling associated with the
Z gauge boson with $c_{w}=\cos\theta_{w}$. This string corresponds to a $4\pi$
rotation about the corresponding generator $T^{3}_{L}/2-s^{2}_{w}Q$ and
therefore it carries magnetic flux $T^{3}_{L}-2s^{2}_{w}Q$, where
$s_{w}=\sin\theta_{w}$. The minimal Nambu monopole also carries EM magnetic
charge $(4\pi/g)s_{w}=(4\pi/e)s^{2}_{w}$ corresponding to magnetic flux
$2Qs^{2}_{w}$.

Figure 1:  A $U(1)_{Y}$ Dirac monopole (red) with minimal EM magnetic flux
$6Qc_{w}^{2}$, where $c_{w}=\cos\theta_{w}$, gets attached to three Nambu
$SU(2)_{L}$ monopoles (blue) each with EM magnetic flux $2Qs_{w}^{2}$, where
$s_{w}=\sin\theta_{w}$. The Z-component of the $SU(2)_{L}$ flux of the Nambu
monopoles is confined in strings with flux $T^{3}_{L}-2s^{2}_{w}Q$ that
connect them to the $U(1)_{Y}$ Dirac monopole. The conglomerate yields an
electromagnetic monopole that carries six quanta ($12\pi/e$) of EM magnetic
charge.

In the next section we show how a monopole configuration with EM charge
$4\pi/e$ is realized by merging a single Nambu monopole with a colored
$U(1)_{Y}$ monopole.

##  III Nambu and $U(1)_{Y}$ Monopoles in $SU(5)$

The embedding of the SM in a grand unified framework has important
ramifications, especially as far as our current discussion is concerned. For
instance, in minimal SU(5) the Nambu monopole contains an inner core due to
the presence of a heavy $SU(2)_{L}$ scalar triplet field that acquires an
induced VEV through mixing with the SM Higgs doublet. The Nambu monopole
carries $(3/4)(2\pi/e)$ unit of Dirac magnetic charge, and the Z magnetic flux
is also predicted [6]. The composite electromagnetic monopole discussed above
is now realized by considering a $U(1)_{Y}$ monopole with winding number six,
which appears from the symmetry breaking $SU(5)\xrightarrow[]{}SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}/Z_{3}\times Z_{2}$. At this stage the monopole has a
core size of order $M_{GUT}^{-1}$ and mass around $10\;M_{GUT}$. Clearly, the
singularity encountered with such a monopole in the SM setting has now
disappeared.

Following electroweak breaking the monopole carries an appropriate combination
of EM and Z magnetic fluxes, with the latter in the form of Z-strings. In
order to retain a finite energy configuration this $U(1)_{Y}$ monopole in
$SU(5)$ must merge with three Nambu monopoles to yield a composite structure
that carries six quanta of Dirac magnetic charge (Fig. 1). Note that in the
absence of quarks just two quanta $(4\pi/e)$ would have been compatible with
the Dirac quantization condition. Indeed, in the $SU(5)$ breaking to
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}/Z_{3}\times Z_{2}$ the fundamental
topologically stable monopole carries $SU(3)_{c}$, $SU(2)_{L}$ and $U(1)_{Y}$
magnetic charges. The subsequent breaking of the SM symmetry to
$SU(3)_{c}\times U(1)_{em}/Z_{3}$ yields a monopole carrying a single quantum
$(2\pi/e)$ of Dirac magnetic charge as well as screened color magnetic charge.
Quark confinement in this case is required in order for the Dirac quantization
condition to be satisfied beyond the screening radius
$\sim\Lambda_{QCD}^{-1}$.

Finally, let us note that a purely electromagnetic monopole carrying three
quanta of Dirac magnetic charge and accompanied by no other field exists in
SU(5), and a classical solution has been shown in Ref. RUBAKOV1983240 .

##  IV Colored Dirac Monopoles

The smallest homotopically non-trivial loop in the SM gauge group
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}/Z_{3}\times Z_{2}$ consists of a
$2\pi$ rotation about $Y$ – for the left-handed quark doublet $Y=1/6$ –
accompanied by a $\pi$ rotation about $T^{3}_{L}={\rm diag}(1,-1)$ and a
$2\pi/3$ rotation about $T^{8}_{c}={\rm diag}(1,1,-2)$. The minimal singular
Dirac monopole then has magnetic flux $Y+T^{3}_{L}/2+T^{8}_{c}/3$. The
magnetic flux of the doubly charged Dirac monopole can be written as
$2Y+2T^{8}_{c}/3$ since the loop in $SU(2)_{L}$ becomes homotopically trivial
and can be removed. After electroweak symmetry breaking, the colored Dirac
monopole can absorb the Z-flux emerging from the Nambu monopole (Fig. 2) (see
also Ref. Preskill:1992bf ). The total EM plus color flux of the conglomerate
is $2Q+2T^{8}_{c}/3$ with the color part screened for distances larger than
the QCD scale.

Figure 2:  A doubly charged colored Dirac monopole (pink) and a Nambu monopole
(blue) connected by a Z-string with magnetic flux $T^{3}_{L}-2s^{2}_{w}Q$. The
total flux of this configuration is $2Q+2T^{8}_{c}/3$ of which the colored
part will be screened due to QCD effects.

##  V Structure and stability

Figs. 1 and 2 show the purely EM monopoles in terms of their constituent Dirac
and Nambu monopoles. However, we do not expect these diagrams to realistically
show classical solutions of the field theory. For example, the colored
monopole of Fig. 2 is more likely to be a structureless, point-like
$SU(3)\times U(1)_{Y}/Z_{3}$ Dirac monopole surrounded by a cloud of
electroweak fields that make up the Nambu monopole. Other solutions have been
proposed in Ref. Cho:1996qd  though these solutions do not correspond to the
correctly quantized monopoles depicted in Figs. 1 and 2 – the conglomerate in
Fig. 1 has three (not one) Nambu monopoles, and the conglomerate in Fig. 2 has
net color charge.

If the Dirac monopoles are regularized as $SU(5)$ monopoles, Fig. 2
corresponds to the winding 2 monopole resulting from the symmetry breaking to
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}/Z_{3}\times Z_{2}$, while Fig. 1
corresponds to the winding 6 grand unified monopole. The odd winding $SU(5)$
monopoles carry $SU(2)_{L}$ magnetic charge and once electroweak symmetry
breaks, two odd winding monopoles combine to result in an $SU(2)_{L}$ neutral
monopole Ng:2008mp . Further, it has been shown that the $SU(5)$ monopole with
winding $n=5$ is unstable to fragmentation into $n=2+3$ monopoles, and
monopoles with winding $n\geq 7$ are unstable to fragmentation into
$n=6+(n-6)$ monopoles Vachaspati:1995yp ; Liu:1996ea . Then the only other
relevant winding to consider in the $SU(5)$ regularization is $n=4$. This
monopole will connect with two Nambu monopoles to form a conglomerate that
carries color and EM magnetic charges but the color magnetic field will get
screened to give a pure EM monopole with flux $4Q$ (EM charge $8\pi/e$) at
large distances.

##  VI Summary

We have explored how purely EM monopole configurations may arise in the SM. We
do not expect such configurations to have well defined energy because of the
singular nature of the Dirac monopole associated with the $U(1)_{Y}$ sector of
the electroweak model. Barring this, we identified a configuration consisting
of a single $U(1)_{Y}$ Dirac monopole which is linked via Z strings to three
Nambu monopoles. This composite monopole carries six units ($12\pi/e$) of EM
magnetic charge in order to be compatible with the Dirac quantization
condition in the presence of quarks. Although the Y and Nambu monopoles both
carry $\theta_{w}$ dependent EM and Z magnetic charges, the end result leaves
$\theta_{w}$ undetermined in the $SU(2)_{L}\times U(1)_{Y}/Z_{2}$ framework.
We also show how this singular composite monopole gets regularized and obtains
a finite, albeit superheavy mass, from embedding the SM in a GUT. In the
second example a colored $U(1)_{Y}$ monopole merges with a single Nambu
monopole to yield a composite topological structure carrying an EM magnetic
charge of $4\pi/e$ as well as screened color magnetic charge. A third example
consists of two Nambu monopoles and a single colored Dirac $U(1)_{Y}$ monopole
and has EM magnetic charge $8\pi/e$.

Before concluding let us note that models such as $SU(4)_{c}\times
SU(2)_{L}\times SU(2)_{R}$ Pati:1974yy  and $SU(3)_{c}\times SU(3)_{L}\times
SU(3)_{R}$ predict electric charge quantization Lazarides:2019xai ;
Lazarides:2021tua , but gauge coupling unification in these models is not
automatic. In this case the masses of the composite monopoles can lie in the
multi-TeV range. Finally, it is worth stressing that these two models also
predict the existence of topologically stable magnetic monopoles that
respectively carry two and three Dirac quanta of magnetic charge, and which
may be accessible at high energy colliders.

Acknowledgments. We thank Amit Tiwari for drawing the figures and P.Q. Hung
and Nick Mavromatos for discussions. This work is supported by the Hellenic
Foundation for Research and Innovation (H.F.R.I.) under the “First Call for
H.F.R.I. Research Projects to support Faculty Members and Researchers and the
procurement of high-cost research equipment grant” (Project Number: 2251).
Q.S. is supported in part by the DOE Grant DE-SC-001380. T.V. is supported by
the U.S. Department of Energy, Office of High Energy Physics, under Award DE-
SC0019470 at ASU.

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# Demand Analysis and Customized Product Offering Design on E-Commerce
Platform

Dipankar Das
Assistant Professor, Goa Institute of Management
Email ID:dipankar3das<EMAIL_ADDRESS>
###### Abstract

It can be observed that the purchasing decision of an individual consumer in
an electronic marketplace is determined by a set of factors, such as personal
characteristics of the consumer, product pricing, minimum price-quantity
combination offered, decision-making space, and underlying motivation of the
consumer. These factors are combined to form a consumer’s choice problem
domain, which plays a pivotal role in the product offering. In this study, we
attempt to focus on how the products? Offered can be customized by
incorporating the quantity and pack size of the products along with the
factors above to form a more extensive domain for examining the combined
effects of all of these factors on demand. Accordingly, the demand function is
defined by a novel method invoking the extended domain of choice problem in
the electronic marketplace. Consequently, the predictable uncertainty
associated with the consumer’s demand function may disappear, increase the
likelihood of earning optimum revenue through customized combinations of the
components of the extended domain of choice problem, and improve the
understanding of the fluctuations in consumer demand. Finally, we propose a
generalized price response function with standard properties applicable to
E-Commerce.

Keyword: Price Response Function, Demand Function, E-Commerce, Platform,
Information, Choice Set, Non-convexity, Fuzzy Set, Supply Chain and Revenue
Maximization.

##  1 Introduction

It is rather common in the E-commerce marketplace that the decision to
purchase an item/product by any customer depends on the customer’s
characteristics, the price of the product, offered minimum price-quantity
combinations, decision-making space, and intention of the customer. These
factors are combined to form a proposed option set for the customers (which
may be considered a domain or decision set). From this domain a customer
chooses the most preferred bundle/set ((refer to Figure 1)111In Figure 1; =
INR. $1$ USD $\approx 83.04$ INR Feb 11 at 10:03 PM GMT+5:30 .). Based on the
merged combination of customers request and its revenue/profit margin, a
particular platform creates a choice set for the buyers and displays it for
selection.
Supposing a potential customer desires to buy a product from a well-known and
widely used e-commerce platform, Amazon, that customer starts searching for
the desired products by setting some filtering criteria. In Amazon’s web
browser, these criteria are sequentially organized as follows: prime new
stores, department, customer ratings, brand, price range (maximum and
minimum), deals, category of items, membership service, specialty, cuisine,
discount, availability, etc. When purchasing an item/product, any customer
selects the best option for every criterion and then waits for the platform’s
reaction by returning the results. The respective platform reads the selected
options based on different criteria and then provides customers with a long
list of the items/products that meet their requirements.
Any deviation of choice set offered by the platform from the choice set
selected by the customer de facto leads to problems in the identification of
the individuals as the customers of the products (Rusmevichientong et al.,
(2006); Aggarwal et al., (2004); Calvete et al., (2019); Domínguez et al.,
(2021)). Therefore, the aggregate customer demand becomes indeterminate, since
this mismatch leads to the shifting of the customers to a different domain
where they can accomplish their goal. Under this circumstance, it is hard for
any e-commerce platform to forecast the outcome without determining the degree
of individual and aggregate utility.
Through the earlier mentioned mismatch, any platform can earn supplementary
revenueDomínguez et al., (2021). Therefore, a platform has two fundamental
objectives: to earn revenue due to this mismatch and to retain the customer.
To accomplish these objectives, it is necessary to comprehend the customer’s
demand function. The platform could prepare customized offers by manipulating
the domain and knowing the demand function. Usually, a platform chooses a
strategy excluding the domain of the consumer, and based on the rationality of
the customer, unfavorable outcomes in any platform shift them to a different
platform. Each platform caters to different consumer incentives based on their
purchase history. As a result, this generates a specific reference point.
Accordingly, different platforms meet the objectives of different customers.
To understand the mapping of choice from one domain to another, it is
essential to perceive the various types of domain/decision spaces, the
respective platform’s choice set, and its mapping behavior. This understanding
enables a platform to analyze and predict the customer’s preference pattern
and set profit maximizing and envy-free pricesFernandes et al., (2016);
Anshelevich et al., (2017). Furthermore, it is important here to say that
price is one of the components of revenue, and by using this instrument, any
firm can maximize its revenue through the balance in supply and demand
appropriatelyRoth et al., (2020).
Against this backdrop, there are twofold objectives of this study. Firstly,
this study attempts to identify the determinants of the customers’ demand for
e-commerce platforms. Accordingly, we offer a novel method for determining the
demand function of a single sale posting on an e-commerce platform,
incorporating many customer characteristics. This will enable the e-commerce
platforms to maximize their objectives and satisfy customers. This research
describes the topological behavior of the decision set/domain during the first
phase. It proposes a new method for calculating the demand function, which
depends on the abovementioned factors. In the second phase, the study outlines
the customer’s criteria for switching from one domain to another and the
revenue maximization strategy. The proposed method has been extended from the
previous work explained in the revenue-maximizing strategyDas, 2022b ; Das,
2022a .
The second objective is to propose some solutions for the decision-makers,
which develop a trade-off between the maximization of sales revenue and
retainment of the customers for a particular platform. For the analysis of
this study, we mainly focus on the commodities that an e-commerce platform can
customize in terms of quantity and pack size. For instance, FMCG products are
fruits, vegetables, sugar, dry fruits, cereals, grains, and wheat.
To the best of our knowledge, none of the existing studies proposes a domain’s
complex topological behavior and a novel approach for determining the
customer’s demand to avoid the mismatch problem. Thus, this study’s expected
findings will be important for e-commerce platforms in managing their profit
margins and satisfying their customers. Moreover, creating customized offers
by anticipating demand is any platform’s primary source of additional revenue,
reducing fluctuations. We demonstrate that if a consumer’s motivation is
known, the information presented on an offer may be utilized to ascertain the
consumer’s perceived demand. Consequently, the proposed strategy can aid in
selecting the optimal domain pair and the lowest offered price-quantity pair
while maintaining the exogeneity of motives and network offers. This may
prevent consumers from switching to a different e-commerce platform.
Moreover, a study shows that online retailers exhibit higher price dispersion
than their offline counterpartsAparicio et al., (2023).

Answering these questions will allow the platform to analyze and predict the
agent/preference consumer’s pattern and set profit-maximizing, envy-free
prices (Fernandes et al., (2016)).

Figure 1: Pricing of a product by Amazon India.

The rest of this paper is structured as follows. In section two, we discuss
related literature. The third section describes the model that offers a novel
approach to determine the customer demand to overcome the mismatch problem. In
section four, we explain the price response function and its properties.
Section five discusses the forms and properties of the revenue function.
Section six concludes, and section seven discusses the limitations and future
scope of the current work.

##  2 Related Literature

Recently, revenue management (RM) has emerged as one of the most successful
areas of operations research applications. The decisions regarding demand
management and the processes and systems required to make such decisions are
the primary emphasis of RM. The objective is to maximize revenue by managing
supply and demand through price fluctuations. According to Talluri et al.,
(2004), the more customer heterogeneity in a supply chain, the greater the
possibility to exploit this heterogeneity strategically and tactically to
maximize profits.
The Price of every item/product in a multi-product pricing model is determined
by its position in the list of itemsRoth et al., (2020). The output of any
platform might not reflect the domain of consumers preference. This leads to
unsatisfactory outcomes Rusmevichientong et al., (2006); Aggarwal et al.,
(2004); Calvete et al., (2019); Domínguez et al., (2021). The finding of
another study reveals that online retailers exhibit higher price dispersion
than that of their offline counterparts Aparicio et al., (2023).
The e-commerce platform already occupies a significant market share despite
consumers not minimizing their overall spending on online purchases. This
reveals the substantial preference of the consumers in buying from the
platform even though they fail to reduce their expenditure from these
purchases Das and Jadhav, (2022)
In addition, greater variation in demand over time due to seasonality,
disruptions, or other variables leads to an increase in the difficulty of
managing the demand decisions. Consequently, the most essential element of RM
is forecasting demand in digital or online markets. According to Foster and
Vohra Foster and Vohra, 1999a , decisions on the risks, benefits, and
consequences that are the outcome of the actions initiated by the decision-
maker must be taken before the actual course of events in any organization.
However, in most e-commerce platforms (as seen in our example), the decision
maker’s objective may need to be more well-defined, resulting in a matching
problem.
In the online decision problem, the decision must be made before knowing which
state of the world will be obtained. In that case, the loss incurred in each
period does not depend on decisions taken in earlier periods. This Regret in
the On-line Decision Problem has been studied in Foster and Vohra Foster and
Vohra, 1999b . The overall goal of pricing is to maximize the discounted
profit over a finite sequence of pricing periods.
The selection of a profitable price requires knowledge of the demand curve.
This knowledge can only be obtained by observation of the demand at different
prices in different periods. A model stating that the price variations
observed in a market can, in part, be explained by rational learning behavior
by firms (Lobo and Boyd, (2003)). A revenue management problem has been
addressed where the decision is to set the dynamic pricing without knowing the
demand function over a finite sales horizon to maximize expected revenues,
given an initial inventory (Besbes and Zeevi, (2009); Farias and Van Roy,
(2010).
Consumer preference heterogeneity and search frictions have been studied by
Morozov et al. Morozov et al., (2021). Besides traditional methods like
interviews and focus groups to identify customer needs for marketing strategy
and product development, user-generated content is a promising alternative for
identifying customer needs (Timoshenko and Hauser, (2019)).
Dzyabura and Hauser (Dzyabura and Hauser, (2019) propose the product
recommendation using consumers preference weights learning has been proposed.
In the specific context of revenue management, previous studies should have
considered certain assumptions regarding the unpredictability of demand in a
marketplace such as an e-commerce platform. As a direct consequence, following
the same pricing mechanism strategy may not maximize overall revenue when
there is a mismatch due to consumer and seller domain preferences. Our
research addresses this gap by examining a set of consumer parameters that may
be utilized to accurately derive the demand function and modify the price
depending on the anticipated value of the suggested demand function. In
particular, our paper proposes that revenue may be maximized to varying
degrees by making the domain non-convex. This approach allows managers to
arrive at a strategic decision while reducing the amount of ambiguity
associated with the consumer demand function.

##  3 Model

The model section has been divided into the proposed demand function and the
price response function with properties. Next, we introduce the following
definitions.

###### Definition 1

Minimum Price-Quantity Pair Offered: On the digital market, the minimum
quantity or purchase value of a commodity bundle must be ordered to get the
commodity with or without discount offers222In our example, the ”Minimum
Price-Quantity Pair Offered” is 108.68 for 415g. Customers are not permitted
to place orders lower than this pair.

###### Definition 2

The domain is the decision space on which the agent makes decisions.

###### Definition 3

Decision Space: This is the platform’s offering of (price, quantity) pairs.
This set has the properties of being both convex and non-convex. The domain
will be convex if the commodity prices are linear and non-convex if the
commodity prices are non-linear333In our example, this set is (108.68, 415g),
(99.00, 415g (Pack of 1)), (291.00, 415g (Pack of 2)),(495.00, 415g (Pack of
4)) and a combo offer of three commodities together at 926.36. As a result,
this offer set is non-convex.

###### Definition 4

Consumer Motive: This is a consumer’s revealed preference when comparing two
offers or commodity alternatives. In a pair-wise comparison, this could be the
preferred option. Each consumer belongs to a market segment and is defined by
a set of partial preferences, represented by a directed acyclic graph (DAG)
with products as nodes. This is also used in another context in this article.
In general, it is a measure of the preference (or degree of importance) of an
item in a previous transaction compared to others.

###  3.1 Derivation of the Demand Function and Analysis

Let a platform prepare and offer different types of choice sets/domains before
a customer fulfills the above three definitions viz. (i)Minimum Price-Quantity
Pair Offered, (ii)Domain/Decision-space/Set of all Price-Quantity Pairs
Offered for Sale, and (iii)Consumer Motive. And under each criterion, a set of
finite possibilities is there. Hence, the offer may be prepared and posted by
altering the above three criteria. Let the customer want to buy two
commodities at a time $X_{1}\&X_{2}$, i.e., $X_{i};i=1,2$. Each commodity can
be offered in a finite number of ways called choice sets. Combining these two
commodities, three possible domains could be offered. These are (i)Convex
Domain Case with Linear prices of both $X_{1}$ and $X_{2}$ ,(ii) Convex Domain
Case with Linear $X_{1}$ Price and Non-Linear $X_{2}$ Price,(iii)Non-Convex
Domain Case with Non-Linear Prices of both $X_{1}$ and $X_{2}$. These are
treated as $\mathcal{D}=\\{D_{1},D_{2},D_{3}\\}$ or
$\mathcal{D}\triangleq\\{D_{j}|j=1,2,3\\}$. This is not a problem of ranking
these domains to maximize revenue; rather, it is a choice of a revenue-
maximizing domain subject to conditional purchase probability and demand
uncertainty. These three domains are made of two commodities and lie in the
$j^{th}$ domain $D_{j}$. I assume that the probability of domain $j$ being
satisfactory is independent of everything else and denote it by $\lambda_{j}$,
i.e., conditional on viewing the two products on domain $D_{j}$, the
probability that she (customer) would purchase on this domain $D_{j}$. Hence,
the first task done here is to derive the demand function of each commodity
$X_{1}\&X_{2}$ under each domain $D_{j}$ using the above four factors
affecting the demand functions and after that using this demand function, and
the purchase probability revenue maximization conditions and strategy have
been discussed.
The derivation of the perceived demand function and the customized offer
function have been derived in three ways. The assumption is that the consumer
buys two commodities $X_{1}\&X_{2}$ to maintain the simplicity. The three
cases are (i) Convex Domain Case with Linear prices of both $X_{1}$ and
$X_{2}$, (ii) Convex Case with Linear $X_{1}$ price, and Non-Linear $X_{2}$
Price and (iii) Non-Convex Domain Case with Non-Linear prices of both $X_{1}$
and $X_{2}$.

###  3.2 Membership Function and Two-Way Consistency Condition

The present model is based on a suitable choice of Membership Functions in
understanding consumer behavior. The consumer motive is captured here by the
membership function. Among numerous forms of fuzzy membership functions, three
are commonly used: linear, parabolic, or reversed parabolic, as shown in
Figure 2, and these are explained in Fan et al., (2019). In Figure 2, curve
$a$ is the linear fuzzy membership function, representing the reasonable
decision-maker. Curve $b$ is the reversed parabolic function for the
conservative decision-maker, and curve $c$ is the explanatory function for the
adventurous decision-maker. Regarding neoclassical economic theory, linear
fuzzy membership represents risk-neutral behavior, reversed parabolic function
describes risk-averse behavior, and parabolic function represents risk-taking
behavior.

Figure 2: Fuzzy membership Functions of Different Forms

When solving standard economic problems, the linear fuzzy membership function
shaped like curve $a$ is adopted, mainly to simplify the mathematical
description. Although the forms of fuzzy membership functions are different,
the basic ideas are consistent. The linear membership function is generally
expressed as:

$\phi_{f_{i}}(x)=\dfrac{f_{i}(x)-f_{i}^{0}}{f_{i}^{1}-f_{i}^{0}}$

and

$\displaystyle\phi_{f_{i}}(x)=0$ if $\displaystyle f_{i}(x)\leq f_{i}^{0}$ (1)
$\displaystyle\phi_{f_{i}}(x)=1$ if $\displaystyle f_{i}(x)\geq f_{i}^{1}$

In Eq. 1, $f_{i}^{0}$ is at an unacceptable level, whereas $f_{i}^{1}$ is at a
fully desirable level; the fuzzy membership function $\phi_{f_{i}}(x)$
reflects the membership degree of function $f(x)$ between $f_{i}^{0}$ and
$f_{i}^{1}$. The membership degree can be considered an actual number from the
interval $[0,1]$.

##### Two-way consistency of the choice problem.

The present study is based on the condition of rationality and is due to Das,
2022b . There exists a two-way consistency of the choice problem. This means
the choice would be the same if the decision-maker moved from a large set to a
small one and from a small one to a large one. This happens provided the
decision-maker can interpret the information correctly for each set.
The above definition indirectly says that the decision-maker is deciding with
respect to a reference. It can be achieved in both ways, from a large or a
small set, if the reference point is in both sets.
This rationality condition is supported by the situation given by ArrowArrow,
(1959). It says that if some elements are chosen out of the set, say $S_{2}$
and then the range of alternatives is narrowed to $S_{1}$ but still contain
some previously chosen elements, no previously unchosen piece becomes chosen,
and no one selected part becomes unchosen. Let the set of alternatives are in
$\textbf{S}=\\{x_{1},...,x_{n}\\}$. Therefore, let any choice problem
$C:S_{i}\rightarrow\textbf{S}$ where, $S_{i}\subseteq 2^{n}$. $2^{n}$ is the
set of all subsets. The preference relation on S is reflexive, transitive, and
asymmetric, and the relation is $R=\textit{"better than or equal
to"}i.e.\succcurlyeq$. The choice problem is any subset $S_{i}$, and the
choice function is $C_{\succcurlyeq}(S_{i})\rightarrow\textbf{S}$. Let there
are two choice problems; $S_{1}\&S_{2}\subseteq\textbf{S}$. This two-way
consistency can be found in the degree of preference being constant in two
sets. From Proposition 1 in Das, 2022b , If the agent maintains a constant
preference in two-choice sets, the choice problem will follow a two-way
consistency.
let $S_{1}$ and $S_{2}$ are two subsets from $S$. Then, the following two
Equations are true.

$\forall S_{1}\subseteq
S_{2}\subseteq\textbf{S}\text{if}C_{\succcurlyeq}(S_{2})\in
S_{1}\text{then}C_{\succcurlyeq}(S_{1})=C_{\succcurlyeq}(S_{2})$ $\forall
S_{1}\subseteq S_{2}\subseteq\textbf{S}\text{if}C_{\succcurlyeq}(S_{1})\in
S_{2}\text{then}C_{\succcurlyeq}(S_{2})=C_{\succcurlyeq}(S_{1})$

Moreover, information in the pattern $S_{1}$ is equal to $S_{2}$. This means,

$[C_{\succcurlyeq}(S_{1}|H)]=[C_{\succcurlyeq}(S_{2}|H)]=[x_{i}|H]$

This means the agent selects the object $x_{i}\in S$ when there were only $n$
objects and appear sequentially in ascending order in pattern $S_{1}$and also
selects the same thing $x_{i}\in S$ when $n$ and appear sequentially in
descending order in pattern $S_{2}$. In other words, the agent could correctly
measure the information index/degree of preference for the object $x_{i}\in S$
as it would happen in pattern $S_{2}$. If these two choices are the same, the
agent would be two-way consistent. Here, $H$ is the degree of preference, and
it is the same in both subsets $S_{1}\&S_{2}\subseteq S$, for example.

####  3.2.1 Framework

The model is based on some important variables. These are,
$X_{1},X_{2}\in\mathbb{R}_{+}$. Lowercase (viz. $x_{1}\&x_{2}$) indicates the
quantity in real numbers. In the present section, the demand for $X_{1}$ and
$X_{2}$ of a consumer and in aggregate (industry) has been derived
theoretically without knowing the individual consumers’ utility functions.
Therefore, the domain of the decision making is $Z(=X_{1}\times
X_{2})\subseteq\mathbb{R}_{+}^{2}$. The price per unit of $X_{1}$ is $p_{1}$
and the price per unit of $X_{2}$ is $p_{2}$. The total expenditure for the
two variables is $m$. A degree of preference towards the variable $x_{1}$ is
denoted by a membership function $\mu\in\mathbb{R}_{+}$. Such that
$\mu:X_{1}\rightarrow[0,1]$. A degree of preference towards the variable
$x_{2}$ is denoted by a membership function $\phi\in\mathbb{R}_{+}$. Such that
$\phi:X_{2}\rightarrow[0,1]$. $x_{1min}$ is the minimum of $X_{1}$ required to
maintain the minimum quality of $X_{2}$. $x_{2min}$ is the minimum $X_{2}$
required to maintain the quality of the goods and services in the market.
$x_{1max}\&x_{2max}$ are the variables that indicate the maximum consumption
can be achieved under $C$ for $X_{1}$ and $X_{2}$, respectively. The framework
and the analysis here are based on the robust microeconomics framework given
by Anwar Shaikh Shaikh, (2016). First, the demand function of $X_{1}$ and
$X_{2}$ for an individual consumer will be derived. After that, the aggregate
demand function will be derived.

####  3.2.2 Individual Demand Functions of $X_{1}$ and $X_{2}$

For any price vector $p$,the input vector $z$ and the expenditure $m$, the set
of attainable alternatives is: $B(p,m)=\\{z\in
Z\subseteq\mathbb{R}_{+}^{2}|pz\leqslant m\\}$. A subset of $Z$ having the
form $B(p,m)$ is called the budget set. One can obtain another available set
by modifying the price $p$ and the expenditure $m$. Hence, $\mathcal{B}$ is
treated as the family of the budget sets. Therefore, the choice/demand
function is $f:\mathcal{B}\rightarrow Z$.
The derivation of the demand functions is divided into two parts, in an
(i)Convex domain and an (ii) Non-Convex domain. The price will be linear if
the market has high competition. This case would be treated as a convex case.
Moreover, if the $X_{2}$ product/service producers compete imperfectly and set
the price non-linearly, this case would be treated as a non-convex case. The
derivation of the individual demand function is based on the two-way
consistency condition, as stated above. That considers that the consumer would
maintain the previous proportion or the degree of preference towards a
commodity. The fundamental intuition behind this model is that customers will
continue to purchase the same product unless a trigger event forces them to
consider other products. This may be termed as the motive of the consumer in
our empirical measurement. The detailed derivation of the proposed method can
be found in Das, 2022b ; Das, 2022a . Hence, the demand functions can be
derived in simple language by solving the following Equations individually for
the two commodities $X_{1}\&X_{2}$ respectively.

$\mu=\dfrac{(x_{1}-x_{1min})}{(x_{1max}-x_{1min})}$

so that $0\leqslant\mu\leqslant 1$. Here, I assume for the time being that
both $x_{1min}$ and $\mu$ are independent of prices. Then, for each $\mu$, we
can derive the corresponding per capita consumption demand for $x_{1}$. The
logic is also the same for $X_{2}$.

$\phi=\dfrac{(x_{2}-x_{2min})}{(x_{2max}-x_{2min})}$

The analysis in the next three subsections assumes that $m$=1000.

####  3.2.3 Convex Domain Case with Linear prices of $X_{1}\&X_{2}$
Commodities

Figure 3: Convex Domain Analysis

Figure 3 depicts the Convex Domain Analysis case. Let Figure 4 be any offer by
an online retail platform of fruits and vegetables. This is the situation
where in the consideration set from Figure 4, commodity [1] and [10] are
treated $x_{1}\&x_{2}$ respectively.

Figure 4: Set of Ten Different Types of Products offered by BigBasket along
with the Consideration Set-One of the Consumer

$x_{1}=(1-\mu)x_{1min}+\mu(\dfrac{m}{p_{1}})-\mu(\dfrac{p_{2}}{p_{1}})x_{2min}$
(2)
$x_{2}=(1-\phi)x_{2min}+\phi(\dfrac{m}{p_{2}})-\phi(\dfrac{p_{1}}{p_{2}})x_{1min}$
(3)

The two Equations 2 & 3 are the demand equations of $X_{1}$ and $X_{2}$
respectively444The derivations are in the appendix. It is apparent from
Equations 2 & 3 that for each good, the quantity demanded responds negatively
to a rise in its price at any given income (Figure 5 in A and E). Yet, it
requires no specific model of consumer behavior.
From Figure 4, the consideration set consists of two commodities, and we
consider the following values for the parameters:
$p_{1}=35;p_{2}=38.49;x_{1min}=2pcs;x_{2min}=200g,m=1000$. The parameters
$\mu,\phi$ are unknown; we have taken three different values to measure the
mismatch. These are $\mu,\phi=\\{0.10,0.20,0.30,0.50,0.70,0.80,0.9\\}$ as the
least preferred to highly preferred ranges. The commodity purchase ranges are
$x_{1}=\\{2,4,6,8,...\\}\&x_{2}=\\{200,400,600,800,...\\}$ respectively and
are in $\mathbb{Z}_{+}$. But to derive the perceived demand function, we need
to assume that $x_{1},x_{2}\in\mathbb{R}_{+}$. This means the domain is linear
i.e. $0.175x_{1}+0.19x_{2}=1000$ assuming $x_{1},x_{2}\in\mathbb{R}_{+}$.
Here, $p_{1}=0.175/\text{per g}\&p_{2}=0.19/g$. Hence, assuming the minimum
order quantity as $1pcs$ for $x_{1}$ and $1g$ for $x_{2}$, the consumer’s
motive as given by $\mu\&\phi$, linear or convex domain the two demand
functions can be derived as below. Here, we also assume that 2pcs =200g.

$x_{1}=(1-\mu)+\mu(\dfrac{1000}{p_{1}})-\mu(\dfrac{0.19}{p_{1}})$ (4)
$x_{2}=(1-\phi)+\phi(\dfrac{1000}{p_{2}})-\phi(\dfrac{0.175}{p_{2}})$ (5)

And if considering the minimum order of $x_{1}\&x_{2}$, the demand functions
are given below.

$x_{1}=(1-\mu)200+\mu(\dfrac{1000}{p_{1}})-\mu(\dfrac{0.19}{p_{1}})200$ (6)
$x_{2}=(1-\phi)200+\phi(\dfrac{1000}{p_{2}})-\phi(\dfrac{0.175}{p_{2}})200$
(7)

On the other hand, the offered pair set or the supply schedule is given by
$\\{(35,200g),(70,400g),(105,600g),\\\ ...\\}$ for $x_{1}$ and
$\\{(38.9,200g),(76.98,400g),(115.47,600g),...\\}$ for $x_{2}$. Using these
data pairs, the supply equation can be derived as
$p_{1s}=0.175x_{1}\&p_{2s}=0.1904x_{2}+0.8199999999999932$ for $x_{1}\&x_{2}$
respectively.
Remarks 1 For all values of $\mu\&\phi$ and $p_{1}\&p_{2}$ the demands of
$x_{1}\&x_{2}$ are higher in presence of minimum order constraints in
Equations 6 & 7 than in Equations 4 & 5 respectively. As a result, the revenue
will be higher in the presence of minimum order constraints. This minimum
order pair is a part of the domain. Hence, the equilibrium price will be more
to maximize the revenue as proposed in Equation 2 & 3.
Numerical Results In the following simulated result Figure 5, we have shown
that the equilibrium price is higher in the revenue functions of commodity
$x_{1}\&x_{2}$ in the presence of the minimum purchase constraints. On the
other panel, we have shown that demand and supply are equal at that
equilibrium price. There are two Panels. In Panel 1, the analysis for
commodity $x_{1}$ has been done, and in Panel 2, for commodity $x_{2}$. In
both cases, the results are the same. Diagrams A and E show two curves for
$x_{1}\&x_{2}$ respectively. Blue curves are based on Equations 4 & 5. While
green curves are based on 6 & 7. The curves are drawn assuming
$\mu\&\phi=\\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9\\}$. The demands curve is a
high degree of $\mu\&\phi$ higher. Moreover, demand increases and shifts above
in the presence of minimum order constraints. This can be seen that the green
curves are above the blue curve. The other Diagrams are drawn by changing the
quadrant. Combining B and C, D was drawn, and F, G, and H were combined. The
red line is the platform’s supply line. The equilibrium quantity will be
determined by considering the aggregate demand not depicted here. The platform
is the price taker. So, the Diagram shows that the equilibrium quantity will
be meager because the demand curves show a very high degree of elasticity.
This is because the offered commodities are necessary goods. Here, in the
aggregate demand functions, the parameters $\mu\&\phi$ are nothing but the
maximum of the individual firm’s preferences for $X_{1}$ and $X_{2}$.
Aggregate Demand Functions. Here, in the aggregate demand function555The
derivations are in the appendix, the parameters $\mu\&\phi$ are nothing but
the maximum of the individual customer’s preferences for $X_{1}$ and $X_{2}$.

$\sum_{i=1}^{N}x_{1_{i}}=(1-\mu_{k})\sum_{i=1}^{N}x_{1min_{i}}+(\dfrac{\mu_{k}}{p_{1}})\sum_{i=1}^{N}m_{i}-\mu_{k}(\dfrac{p_{2}}{p_{1}})\sum_{i=1}^{N}x_{2min_{i}}$
(8)
$\sum_{i=1}^{N}x_{2_{i}}=(1-\phi_{k})\sum_{i=1}^{N}x_{2min_{i}}+(\dfrac{\phi_{k}}{p_{2}})\sum_{i=1}^{N}m_{i}-\phi_{k}(\dfrac{p_{1}}{p_{2}})\sum_{i=1}^{N}x_{1min_{i}}$
(9)

The industry faces the same prices, viz. $p_{1}\&p_{2}$, but different budget
i.e. $C_{i}$.Moreover, $x_{1min_{i}},x_{2min_{i}}$ are also different for the
different customers.The aggregate $\mu$ is
$\mu_{k}=\\{\mu_{1}\cup\mu_{2}\cup...\cup\mu_{N}\\}=\max_{i=1}^{N}\mu_{i}$ and
the aggregate $\phi$ is
$\phi_{k}=\\{\phi_{1}\cup\phi_{2}\cup...\cup\phi_{N}\\}=\max_{i=1}^{N}\phi_{i}$
.The same thing is true for $(1-\mu_{k})=\max_{i=1}^{N}(1-\mu)_{i}$ and
$(1-\phi_{k})=\max_{i=1}^{N}(1-\phi)_{i}$. Because,
$\mu+(1-\mu)=1\&\phi+(1-\phi)=1$. Moreover,
$\mu,\phi\in[0,1]\subseteq\mathbb{R}_{+}$.

Figure 5: Numerical Analysis of Convex Domain Case with Linear $X_{1}$ price
and Linear $X_{2}$ Price

The platform will achieve maximum revenue because of the high volume of
aggregate minimum orders and the high degree of aggregate preferences
$\mu_{k}\&\phi_{k}$.

####  3.2.4 Convex Case with Linear $X_{1}$ price and Non-Linear $X_{2}$ Price

Figure 6: Convex Case with linear $X_{1}$ price and Non-Linear $X_{2}$
Price:(i) Figure 7: Convex Case with linear $X_{1}$ price and Non-Linear
$X_{2}$ Price:(ii)

Two possibilities are depicted in 6 & 7. Derivations are given in the
appendix.

Figure 8: Set of Ten Different Types of Products offered by BigBasket along
with the Consideration Set-Two of the Consumer

Let another offer by any platform is represented in Figure 8. This is the
situation where in the consideration set from Figure 8, [1] and [4] are
treated $x_{1}\&x_{2}$ respectively. The offered price for $x_{2}$ follows a
step function and is nonlinear (see Figure 11). As a result, the domain is
non-convex. For a non-convex domain, the solution will be either at the corner
or in the interior of the feasible region based on the constraints. As a
result, the revenue will be more than in the previous case. In this situation,
the price of commodity $x_{1}$ is linear, but the price of commodity $x_{2}$
is non-linear so that the solution will be in the interiorDas, 2022b ; Das,
2022a . Numerical parameters are the same for $x_{1}$ but different for
$x_{2}$. $p_{2}^{1}=0.2$ for $x_{2min}=100g$ but after $100g$ the next
quantity is more than $200$, i.e., $250$. As a result, the gap between the
perceived and the offered demand curve will be more than in the previous case.
Now, if the $p_{2}^{1}=0.1$ for $x_{2min}=200g$, then the domain would be non-
convex, and the following demand functions would persist. Let the vector
$z=(x_{1},x_{2})$ for two variables and,$z\in
Z\subseteq\mathbb{R}_{+}^{2}$.There are two budget sets that the firm is
facing.These are,
$B^{1}=\\{z_{i}\in\mathbb{R}_{+}^{2}:\sum_{i=1}^{2}p_{i}^{1}x_{i}\leq
m^{1}\\}\&~B^{2}=\\{z_{i}\in\mathbb{R}_{+}^{2}:\sum_{i=1}^{2}p_{i}^{2}x_{i}\leq
m^{2};\text{if},x_{2}>\bar{x}_{2}\\}$. Moreover,$m^{2}>m^{1},p_{1}^{1}\neq
p_{1}^{2}\&p_{2}^{1}<p_{2}^{2}$, because if you buy more of $x_{2}$, you have
to pay more and as a result, the relative price of $x_{1}$ will also change.
Here, $\bar{x}_{i}\text{for},i=1,2$ are any constant real numbers of
$x_{1}\&x_{2}$.The different prices for these constraints create the price
non-linearly666The derivations are in the appendix.

$x_{1}=\dfrac{\mu_{i}}{p_{1}^{1}}[(m^{1}-p_{2}^{1}x_{2min})-p_{1}^{1}x_{1min}]+x_{1min}$
(10)
$x_{2}=\dfrac{\phi_{i}}{p_{2}^{1}}[(m^{1}-p_{1}^{1}x_{1min})-p_{2}^{1}x_{2min}]+x_{2min}$
(11)

Remarks 2 The price of commodity $x_{2}$ is higher in the higher quantity buy,
so the solution will be at the lowest quantity. This is why
$p_{2}^{2},m^{2},p_{1}^{2}$ are not there in the above two equations. This can
be checked from Figure 8 that the consumer would buy 100g at $0.20$. It is
also clear from Equations 10 & 11777The derivations are in the appendix that
whatever the price, there is a minimum demand for each commodity.
Aggregate Demand Functions.Here, is the aggregate demand functions for $X_{1}$
and $X_{2}$.888The derivations are in the appendix

$\sum_{i=1}^{N}x_{1_{i}}=\mu_{k}[p_{1}^{1}\sum_{i=1}^{N}m_{i}^{1}-\dfrac{p_{2}^{2}p_{1}^{1}\sum_{i=1}^{N}m_{i}^{1}}{m^{2}}x_{2min_{k}}]+\mu_{k-1}[\dfrac{\sum_{i=1}^{N}m_{i}^{1}}{p_{1}^{1}}-\dfrac{p_{2}^{1}}{p_{1}^{1}}\sum_{i=1}^{N-1}x_{2min_{i\neq
k}}]+(1-\mu_{k})\sum_{i=1}^{N}x_{1min_{i}}$

and

$\sum_{i=1}^{N}x_{2_{i}}=\phi_{k}[\dfrac{\sum_{i=1}^{N}m^{2}_{i}}{p_{2}^{2}}-\dfrac{\sum_{i=1}^{N}m_{i}^{2}}{p_{1}^{1}p_{2}^{2}\sum_{i=1}^{N}m_{i}^{1}}x_{1min_{k}}]+\phi_{k-1}[\dfrac{\sum_{i=1}^{N}m_{i}^{1}}{p_{2}^{1}}-\dfrac{p_{1}^{1}}{p_{2}^{1}}\sum_{i=1}^{N-1}x_{1min_{i\neq
k}}]+(1-\phi_{k})\sum_{i=1}^{N}x_{2min_{i}}$

Here, $\sum_{i=1}^{N-1}\mu_{i\neq k}=\cup_{i=1\neq k}^{N-1}(\mu_{i\neq
k})=\max[\mu_{1,}...,\mu_{N-1}]=\mu_{k-1}$
and $\sum_{i=1}^{N-1}\phi_{i\neq k}=\cup_{i=1\neq k}^{N-1}(\phi_{i\neq
k})=\max[\phi_{1},...,\phi_{N-1}]=\phi_{k-1}$ for an order
$\mu_{1}<...<\mu_{k-1}<\mu_{k}$
and $\phi_{1}<...<\phi_{k-1}<\phi_{k},[\mu_{k}+\sum_{i=1}^{N-1}\mu_{i\neq
k}]=\max[[\mu_{k}]+\max_{i=1}^{N-1}\mu_{i\neq k}]=\mu_{k}$ and
$[\phi_{k}+\sum_{i=1}^{N-1}\phi_{i\neq
k}]=\max[[\phi_{k}]+\max_{i=1}^{N-1}\phi_{i\neq
k}]=\phi_{k}$.Moreover,$x_{1min_{k}}\&x_{2min_{k}}$ are the minimum fixed
demand for the firm $k$. This case has proposed Proposition 1.

###### Proposition 1

If the price of $X_{2}$ in the market is in the non-linearly set whereas
$X_{1}$ is linearly priced, then it suggests that the suppliers of $X_{2}$
would enjoy some degree of monopoly power. On the other hand, the $X_{1}$
would enjoy a linear price. The firms who can use $X_{2}$ at a higher cost
would use it, and the rest would prefer $X_{1}$. Hence, there would be
multiple equilibria in the market. As a result, $X_{1}$ and $X_{2}$would act
as complementary to each other.

##### Proof:

The proof is in the appendix. $\square$

####  3.2.5 Non-Convex Domain Case with Non-Linear $X_{1}$ price and Non-
Linear $X_{2}$ Price

Figure 9: Non-Convex Domain Analysis

Non-Convex Domain Case has been depicted in 9.

Figure 10: Set of Ten Different Types of Products offered by BigBasket along
with the Consideration Set-Three of the Consumer Figure 11: The Non-Linear
Price Offer

This is the situation where in the consideration set from Figure 11 commodity
[1] & [11] (from Figure 11)and either [5] or [10] (from Figure 10) is treated
as $x_{1}\&x_{2}$ respectively. The offered price for $x_{1}\&x_{2}$ follows a
step function and is nonlinear. For example, in Figure 11, the price per unit
is 0.0540, and the minimum quantity bought is 250g. If the consumer buys at
least 1kg or 1000g, the per unit price will reduce to 0.05350/g. Hence, the
minimum quantity bought is 1000g or 1kg. This is a complete non-convex domain,
and the demand functions for these two commodities can be derived below. Let
the vector $z=(x_{1},x_{2})$ for two variables and,$z\in
Z\subseteq\mathbb{R}_{+}^{2}$.There are two budget sets that the firm is
facing.These are,
$B^{1}=\\{z_{i}\in\mathbb{R}_{+}^{2}:\sum_{i=1}^{2}p_{i}^{1}x_{i}\leq
m;\text{if},~x_{1}\leq\bar{x}_{1}\\}\&~B^{2}=\\{z_{i}\in\mathbb{R}_{+}^{2}:\sum_{i=1}^{2}p_{i}^{2}x_{i}\leq
m;\text{if},x_{1}>\bar{x}_{1}\&x_{2}<\bar{x}_{2}\\}$.Here,
$\bar{x}_{i}\text{for},i=1,2$ are any constant real numbers of $x_{1}\&x_{2}$.
The firm faces two budget constraints at the same time, so the domain becomes
non-convex because the price vectors
$(p_{1}^{1},p_{2}^{1})(p_{1}^{2},p_{2}^{2})$ for the two budget sets are
different and depends on the quantity purchase, i.e., $\bar{x}_{i},i=1,2$.
Assuming
$(m-\bar{m}_{1}p_{1}^{2})=p_{1},(m-\bar{m}_{2})p_{2}^{1}=p_{2}\&(m-\bar{m}_{1})(m-\bar{m}_{2})=M$.
Assuming that the minimum expenditure in $x_{1}\&x_{2}$ are constant i.e.
$p_{1}^{1}x_{1min}=\bar{m}_{1},p_{2}^{2}x_{2min}=\bar{m}_{2}$. There are two
possibilities viz. $\bar{m}_{1}\neq\bar{m}_{2}$ &
$\bar{m}_{1}=\bar{m}_{2}=\bar{m}$.The different prices for these constraints
create a non-linear price. The derived demand functions999The derivations are
in the appendix are given below.

$x_{1}=\dfrac{\mu}{p_{1}}[\dfrac{p_{1}}{p_{1}^{2}}(m-p_{2}^{2}x_{2min})-p_{1}x_{1min}]+x_{1mn}$
(12)
$x_{2}=\dfrac{\phi}{p_{2}}[\dfrac{p_{2}}{p_{2}^{1}}(m-p_{1}^{1}x_{1min})-p_{2}x_{2min}]+x_{2min}$
(13)

Using Equations 12 & 13 I derived as below.

$x_{1}^{*}=\dfrac{m-p_{2}^{1}x_{2}}{p_{1}^{1}}$ (14)
$x_{2}^{*}=\dfrac{m-p_{1}^{2}x_{1}}{p_{2}^{2}}$ (15)

Aggregate Demand Functions. The aggregate demand function101010The derivations
are in the appendix can be found by summing for $N$ number of consumers in the
industry. First, we need to convert the demand functions 12 & 13 into
aggregate terms, as below. The aggregate demand function for $X_{2}$ depends
on the degree of preference of the large customer.

$\sum_{i=1}^{N}x_{1_{i}}^{*}=\dfrac{\mu_{k}}{p_{1}}[\dfrac{p_{1}}{p_{1}^{2}}(\sum_{i=1}^{N}m_{i}-p_{2}^{2}\sum_{i=1}^{N}x_{2min_{i}})-p_{1}\sum_{i=1}^{N}x_{1min_{i}}]+\sum_{i=1}^{N}x_{1mn_{i}}$

and

$\sum_{i=1}^{N}x_{2_{i}}^{*}=\dfrac{\phi_{k}}{p_{2}}[\dfrac{p_{2}}{p_{2}^{1}}(\sum_{i=1}^{N}m_{i}-p_{1}^{1}\sum_{i=1}^{N}x_{1min_{i}})-p_{2}\sum_{i=1}^{N}x_{2min_{i}}]+\sum_{i=1}^{N}x_{2min_{i}}$

Finally, using Equations (25) and (26), the final range of the demand
equations can be found.These are;
$(\sum_{i=1}^{N}x_{1_{i}},\sum_{i=1}^{N}x_{2_{i}})\in[(\sum_{i=1}^{N}x_{1_{i}}^{**},\sum_{i=1}^{N}x_{2_{i}}^{**})],(\sum_{i=1}^{N}x_{1_{i}}^{***},\sum_{i=1}^{N}x_{2_{i}}^{***})$.
Remarks 3. The interpretation is very clear from the Equations 12, 13, 14, 15,
i.e., there is a two-way complementary relationship that exists between the
two commodities, viz. $x_{1}\&x_{2}$. Say, for example, for commodity $x_{1}$.
In the first round, the demand depends on the minimum expenditure to buy
$x_{2}$. The higher the minimum expenditure incurred, the lower the demand for
$x_{1}$. Now, if all the parameters are fulfilled and decide $x_{1}$ in 12,
the next level or the final decision will be taken in Equation 14. Again, the
demand of $x_{2}$ matters based on Equation 11. Higher is the demand of
$x_{2}$ lower is the demand of $x_{1}$. The same analysis is true for $x_{2}$.
Therefore, the higher the non-linearity of the step functions, the lower the
demand, and the consumer would not prefer that domain to buy. Rather, they
prefer to buy from a domain with a linear domain. The choice of domain and the
conditions thereof have been explained in the following section as an
application toward consumer behavior. Thus far, it is clear that the platform
will be able to earn more revenue if the domain contains step functions of
price-quantity relationship or non-linearity. Because of this nonlinearity,
the choice set becomes very small for the consumer. As a result, the consumer
needs to spend more on that domain.
Remarks 4. From the above model, it is clear that the minimum quantity
constraints create the decision space nonconvex. Hence, the proposed method
also suggests that the nonconvex domain does not guarantee that the solution
will be at the optimum point. Clearly and accurately, the consumer’s decision-
making domain is a lattice. All the parameters, viz. prices, discounts,
minimum order quantities, etc… are in integer lines. As a result, the solution
considered is in real space, and the nonconvex domain provides a set of
solutions in the interior. Now, it is necessary to know how many lattice
points there are in possible solutions. The individual demand function derived
in equations 2& 3 predicts the demand quantities in real numbers for different
minimum purchase constraints and other parameters. Hence, the second question
is to identify the lattice commodity bundle nearest to the real one concerning
the demand function defined in actual variables. The above questions can be
solved in a process as follows. The estimated/projected demand vector in real
numbers can be determined from the individual demand function. After that, a
set of available lattice points will be determined. In the end, the nearest
lattice points need to be derived and match the same.

##### Remarks 5

The revenue is dependent on the choice of the products offered, respected
prices, the minimum order constraints, the motive of the customer as
represented by the degree of preference for a product in different slabs of
each offer, etc… So far, I have considered that the customer motive viz,
$\mu,\phi$ are identical in both linear and non-linear offers. This means that
it was not dependent on slabs. For the sake of a clear understanding of the
price response function, hereafter, the motive of the customer is different in
the nonlinear offer for each slab so that the expected revenue can be
calculated for a fixed and random attention span. This means each customer may
have a different preference for various slabs of a given offer. Let us try to
understand the using Figure 7. To buy Fresho Beans, there are three slabs in
the left panel: viz. (1 kg, @ 53.50),(500g, @27),(250g, @13.50). So far, it
was assumed that the degree of preference, say $\mu$, to derive the demand
function of this commodity is constant in these three slabs. To generalize,
$\mu$ is different for these three slabs.

##  4 The Price Response Function and its Properties

This section explains the properties of the generalized price response
function, as given by the following equation for commodity $x_{i}$:

$x_{i}(p_{i}\mid\mu^{k},x_{i,min}^{k},m,p_{j}^{k},x_{j,min}^{k};\forall
j,k=1,...,N)=(1-\mu^{k})x_{i,min}^{k}+\mu^{k}(\dfrac{m}{p_{i}^{k}})-\mu^{k}(\dfrac{p_{j}^{k}}{p_{i}^{k}})x_{j,min}^{k}$
(16)

In economic textbooks, the price-response function is analogous to the market
demand function. There is, however, a significant distinction. The price-
response function describes the demand for a single seller’s goods as a
function of the price given by that seller. This contrasts the concept of a
market demand curve, which outlines how a whole market will react to price
changes. This distinction is significant because various firms competing in
the same market confront different price-response functions Phillips, (2021);
Pinchuk, (2006). Eq.16 gives the general demand function of commodity
$x_{i}\forall i=1,...,N$ and for all possible slabs $k=1,...,N$ and for other
commodity $x_{j}\forall j=1,...,N$. If the platform offers in the non-convex
domain, then for each slab, the price response function can also be calculated
using Eq.16 instead of calculating the overall response for that domain.
Because each slab offer is linear, nonlinearity arises when considering a set
of linear offers for all slabs. For example, in Figure 7, to buy Fresho Beans,
there are three slabs; this is considered the set of offers viz. $D_{1}$= (1
kg, @ 53.50),(500g, @27),(250g, @13.50). The set $D_{1}$ is a nonlinear offer,
but individual elements or individual offers in that set are linear. We can
easily derive Eq.16 by taking any other set like this for another commodity.
Let the offer set for other commodity is $D_{2}$= (1 kg, @ 55),(500g,
@20),(250g, @12). Then for each ordered pair of the set $D_{1}\times
D_{2}$=[{(1 kg, @ 53.50)(1 kg, @ 55)},{(500g, @27)(500g, @20)},{(250g,
@13.50)(250g, @12)}] Equation 16 can be calculated, and those will be the
price response functions of the two commodities under each pair of the domain
$D_{1}\times D_{2}$. This will increase the predictability and the responses
due to price, number of slabs, minimum order constraints, consumer motive, and
budget.

##### Non-negativity

The demand function
$x_{i}(p_{i}\mid\mu^{k},x_{i,min}^{k},m,p_{j},x_{j,min}^{k};\forall
j,k=1,...,N)$ is greater than or equal to zero for all prices i.e.
$x_{i}(p_{i})\geq 0\forall p_{i}$.

##### Downward Sloping

There is a negative relationship exists between $x_{i}(p_{i})\&p_{i}$ and its
has been shown in Fig. 5 and the equation of negatively sloped curve is
$x^{\prime}(p_{i,min}^{k})=-[\dfrac{m\mu}{p^{2}}-\dfrac{p_{j}^{k}x_{j,min}^{k}\mu}{p^{2}}]$
for $[\dfrac{m\mu}{p^{2}}-\dfrac{p_{j}^{k}x_{j,min}^{k}\mu}{p^{2}}]>0$.

##### Continuity

It will have no gaps or jumps. The price response function proposed here has a
small number of discrete steps because these are for only one customer. As the
number of customers increases, the price response function will approach a
more continuous function. While price-response functions are not truly
continuous because prices of the discrete nature of customer demand and
minimum order constraints and price can only occur at discrete intervals, they
are often treated as continuous for mathematical convenience.

##### Hazard Rate

Hazard Rate of a price response function at a particular price is equal to
minus one times the slope of the price response function at that price divided
by the demand; i.e., $h(p)=-\dfrac{x^{\prime}(p)}{x(p)}$. This means the
hazard rate is positive because the slope is negative. And the hazard rate is
given by
$h(p)=-\dfrac{\mu(-m+p_{j}^{k}x_{j,min}^{k})}{(-m\mu+p_{i}^{k}x_{i,min}^{k}(\mu-1)+p_{j}^{k}x_{j,min}^{k}\mu)^{2}}$.

##### Price Elasticity

The most common measure of the sensitivity of demand to price is price
elasticity, defined as the ratio of the percentage change in demand to the
percentage change in price. Formally, it can be written as
$\epsilon(p_{1},p_{2})=-\dfrac{100\\{[x(p_{2})-x(p_{1})]/x(p_{1})\\}}{100\\{(p_{2}-p_{1})/p_{1}\\}}$
where $\epsilon(p_{1},p_{2})$ is the elasticity of a price change from $p_{1}$
to $p_{2}$. However, the point elasticity at $p$ is written as
$\epsilon(p)=-\dfrac{x^{\prime}(p_{1})p_{1}}{x(p_{1})}$. In other words, the
point elasticity equals $1$ times the slope of the demand curve times the
price divided by demand. And the Price Elasticity is given by
$\epsilon(p)=-\dfrac{p_{i}^{k}\mu(-m+p_{j}^{k}x_{j,min}^{k})}{(-m\mu+p_{i}^{k}x_{i,min}^{k}(\mu-1)+p_{j}^{k}x_{j,min}^{k}\mu)^{2}}$.

##### Price Response and Willingness to Pay

The willingness-to-pay approach assumes that each potential customer has a
maximum willingness to pay (sometimes called a reservation price) for a
product or service. A customer will purchase if and only if the price is less
than her maximum willingness to pay. It can derive the willingness-to-pay
distribution from the price-response function using111111Assuming that the
price-response function is finite. $w(p)=-\dfrac{x^{\prime}(p)}{x(0)}$. The
demand at price $p=0$ is infinite and the limit value is also infinite i.e.
$\lim_{p_{i}^{k}\to\ 0}x(p_{i}^{k})=\infty\textit{sign}(m\mu-
p_{j}^{k}x_{j,min}^{k}\mu)$. This means if $\textit{sign}(m\mu-
p_{j}^{k}x_{j,min}^{k}\mu)$ becomes zero, then it will be undefined.
Otherwise, it will be positive infinity for positive signs and negative
infinity for negative signs. Therefore the willingness to pay at $p_{i}^{k}=0$
is $0$. The willingness to pay exists at any positive limit value. Here, I
have considered three possible cases.
(i) At $p_{i}^{k}=0.01$ the willingness to pay is
$\dfrac{\frac{m\mu}{p^{2}}-\frac{p_{j}^{k}x_{j,min}^{k}\mu}{p^{2}}}{10.0m\mu-10.0p_{j}^{k}x_{j,min}^{k}\mu-1.0x_{i,min}^{k}x_{j,min}^{k}+1.0x_{i,min}^{k}}$
(ii) At $p_{i}^{k}=0.001$ the willingness to pay is
$\dfrac{\frac{m\mu}{p^{2}}-\frac{p_{j}^{k}x_{j,min}^{k}\mu}{p^{2}}}{100.0m\mu-100.0p_{j}^{k}x_{j,min}^{k}\mu-1.0x_{i,min}^{k}x_{j,min}^{k}+1.0x_{i,min}^{k}}$
(iii) At $p_{i}^{k}=0.001$ the willingness to pay is
$\dfrac{\frac{m\mu}{p^{2}}-\frac{p_{j}^{k}x_{j,min}^{k}\mu}{p^{2}}}{100.0m\mu-100.0p_{j}^{k}x_{j,min}^{k}\mu-1.0x_{i,min}^{k}x_{j,min}^{k}+1.0x_{i,min}^{k}}$
Therefore, it is clear that the willingness to pay is an increasing function
with respect to price, given the other factors affecting the willingness to
pay. Given the other factors, the willingness to pay can be increased by
increasing the prices. This is so the customer has selected the particular
slab and wants to stay put. Once the preferred slab and the degree of
preference of the product in question are known, the pricing will be the
revenue maximizer.

##### Purchase probability.

I assume no offers in each slab have identical characteristics (the
combination of price-quantity and conditional purchase probability). Given
that position $k$ is within a customer attention span, her purchase
probability of the product at slab $k$ is

$\prod_{j=1}^{k-1}(1-\lambda_{j}).\lambda_{k}$

where, $\lambda_{k}$ is the probability that the customer will select a slab
$k$ given the prices and other factors affecting the demand. In other words,
the customer purchases a product at the slab $k$ if it is satisfactory while
all products displayed earlier slabs are not. $(1-\lambda_{k})$ is the effect
that a product at slab $j$ exerts on the products displayed later. This is the
probability that the product at the $j^{th}$ slab has not been preferred over
$k$.

##  5 Revenue Function

The revenue function depends on the parameters viz. membership function,
minimum order constraints of the commodity and the related commodities, prices
af of the commodity and the related commodities, and the choice of slabs by
the customer. Moreover, consumers prefer to select the convex domain over the
noon convex domain, which can be represented by the probability elements
attached to it. In this case, the revenue can be maximized by changing other
factors, keeping the commodity’s price exogenous. The platform controls the
demand function using a choice-set policy. The platform’s goal is to choose
prices $p_{i}$ to maximize the expected revenue from a potential consumer.
From the earlier introduction, we see that the total expected revenue
$E[x_{i}(p_{i})]=R(x_{i})$ from a consumer with an individual demand function
for each slab and with an attention span of $y$ can be expressed as

$\mathcal{R}(k,y,x_{i})\triangleq\sum_{k=1}^{\min(y,N)}\left[\prod_{j=1}^{k-1}(1-\lambda_{j})\right]\lambda_{k}.x_{i}(p_{i}^{k}).p_{i}^{k}$
(17)

##### Remarks 6

The Eq. 17 is the expected revenue function and depends on the random choice
of the slab of a given customized commodity offer. The predicted revenue
function is affected by the addition and removal of slabs. This has been
explained by explaining the creation of a non-convex domain to earn more
revenue and push the customer to buy more, given the price in each slab.

###### Proposition 2

The expected revenue function is affected by the addition and removal of
slabs. The higher the number of slabs, the higher the convexity, and the lower
the expected revenue. On the other hand, the lower the number of slabs, the
higher the non-convexity and the higher the expected revenue.

##### Proof:

The proof is in the appendix. $\square$

##### Remarks 7

From Proposition 2, the expected revenue depends on the slabs’ choice. Because
the higher number of slabs will be supported by the lower prices, and on the
other hand, if the customer selects more elevated slabs for the lower prices,
then the factor $\left[\prod_{j=1}^{k-1}(1-\lambda_{j})\right]\lambda_{k}$
will be lower to get the expected value. This will again reduce the expected
revenue. To maximize the expected revenue, there should be more than one slab
to create the non-convex domain. As a result, the consumer will buy more
quantity. On the other hand, if the slabs are increased, the customer will
tend to buy less at a lower price. The slabs’ choice depends on the
willingness to pay and the intention to buy more. This section generalizes the
price-response function as defined in Eq.(16) with respect to different risk
behavior.

##  6 Conclusions and the Managerial Implications

This article suggests a straightforward mechanism for slab pricing on an
e-commerce platform. The article’s main idea is that customers will be
considered homogeneous if the quantity is provided in a compact set, which is
closed and bounded. However, diverse customers can be produced if the same
quantity is provided by making the domain non-convex. The study above
indicates that income increases with increasing non-convexity.
This means that in the case of any two commodities, as explained here, the
revenue will be higher in terms of higher quantity sold. This is the case
where Non-Convex Domain Case with Non-Linear $X_{1}$ Price and Non-Linear
$X_{2}$ Price; the less revenue will be earned in the case Convex Domain Case
with Linear $X_{1}$ Price and Non-Linear $X_{2}$ Price, and finally the case
Convex Domain Case with Linear $X_{1}$ Price and Linear $X_{2}$ Price.
The platform must choose the offering domain, figure out how many slabs to
offer, and then calculate the equilibrium price for that domain to maximize
income. The convex domain scenario will be more elevated, and the degree of
nonlinearity will be lower the more slabs there are. Conversely, the greater
the flexibility to apply non-linear prices, the fewer slabs there are. The
revenue will drop as the number of slabs rises. Therefore, this article
explains that choosing two slabs is the best option. However, the predicted
revenue will drop if there is a more than two-slab increase. Consequently, the
study suggests a novel method for figuring out non-linear prices by choosing
the right amount of slabs, prices, minimum order restrictions, etc. A
marketing manager has to remember that the least amount of money needed for
matching must be spent when choosing the domain and spending to optimize
revenue. If not, the offer would still be available, but the customer,
assuming the platform employs the same method for choosing the domain and
pricing, would go to another one where the matching price is the lowest.

##  7 Limitation

The present article proposes a model of creating slabs by setting non-linear
prices in each slab using a proposed elegant model. The present article does
not consider the $n$-commodity case. Therefore, in future research may be
developed for $n$ commodities. Conflict of Interest: we certify that we have
no affiliations with or involvement in any organization or entity with any
financial interest (such as honorarium, educational grants, participation in
speakers bureaus, membership, employment, consultancies, stock ownership, or
other equity interest; and expert testimony or patent-licensing arrangements),
or non-financial interest (such as personal or professional relationships,
affiliations, knowledge or beliefs) in the subject matter or materials
discussed in this manuscript.
Conflict of Interest
I certify that we have no affiliations with or involvement in any organization
or entity with any financial interest (such as honorarium, educational grants,
participation in speakers Bureaus, membership, employment, consultancies,
stock ownership, or other equity interest, and expert testimony or patent-
licensing arrangements), or non-financial interest (such as personal or
professional relationships, affiliations, knowledge, or beliefs) in the
subject matter or materials discussed in this manuscript.
Data Availability Statements
I do not analyze or generate datasets because our work proceeds with a
theoretical and mathematical approach.
Appendix
The proofs are available on demand. To access the proofs, you are to request
the author via email at<EMAIL_ADDRESS>.

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