# Towards Effective Resource Procurement in MEC: a Resource Re-selling
Framework

Marie Siew, Shikhar Sharma, Kun Guo, Desmond Cai, Wanli Wen, Carlee Joe-Wong,
Tony Q.S. Quek  Marie Siew, Shikhar Sharma and Carlee Joe-Wong are with the
Electrical and Computer Engineering Department, Carnegie Mellon University,
Pittsburgh, PA 15213.
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Kun Guo is with the National Mobile Communications
Research Laboratory, Southeast University, Nanjing 210096, China, and also
with the School of Communications and Electronics Engineering, East China
Normal University, Shanghai 200241, China.
E-mail<EMAIL_ADDRESS>Desmond Cai. E-mail<EMAIL_ADDRESS>Wanli
Wen is with the School of Microelectronics and Communication Engineering,
Chongqing University, Chongqing 400044, China.
E-mail<EMAIL_ADDRESS>T. Q. S. Quek is with the Singapore
University of Technology and Design, Singapore 487372, and also with the
Yonsei Frontier Lab, Yonsei University, South Korea.
E-mail<EMAIL_ADDRESS>Corresponding author: Tony Q.S. Quek.

###### Abstract

On-demand and resource reservation pricing models, widely used in cloud
computing, are currently used in Multi-Access Edge Computing (MEC).
Nevertheless the edge’s resources are distributed and each server has lower
capacity. If too much resources were reserved in advance, on-demand users may
not get their jobs served on time, jeopardizing MEC’s latency benefits.
Concurrently, reservation plan users may possess un-used quota. Therefore, we
propose a sharing platform where reservation plan users can re-sell unused
resource quota to on-demand users. To investigate the mobile network
operator’s (MNO’s) incentive of allowing re-selling, we formulate a 3-stage
non-cooperative Stackelberg Game and characterize the optimal strategies of
buyers and re-sellers. We show that users’ actions give rise to 4 different
outcomes at equilibrium, dependent on the prices and supply levels of the
sharing and on-demand pools. Based on the 4 possible outcomes, we characterise
the MNO’s optimal prices for on-demand users. Numerical results show that
having both pools gives the MNO an optimal revenue when the on-demand pool’s
supply is low, and unexpectedly, when the MNO’s commission is low. We develop
an interactive prototype, and show that users’ decision distributions in
studies on our prototype are similar to that of our decision model.

###### Index Terms:

Edge computing, Network economics, Sharing Platforms, Game Theory

##  1 Introduction

Edge computing enables a wide variety of low latency and computationally
intensive services on mobile and other resource constrained devices (e.g
Internet of Things (IoT) devices). These low latency services include video
analytics, real-time analytics, virtual and augmented reality (VR/ AR), and
connected vehicle decision making. Edge computing brings the power of cloud
computing to the network edge, with servers placed at edge access points e.g.
base stations or wifi access points [1]. Users and device owners can offload
computationally intensive tasks to the nearby edge servers, and receive them
within latency requirements [2], due to the close proximity of the edge
servers. The wide-area-network (WAN) delay of cloud computing will be
avoided[3].

As computing resources are limited at the edge, researchers have been actively
investigating the computation offloading and resource allocation problem, [2,
4, 5, 6, 7]. Besides the technical challenges of computation offloading,
resource pricing solutions to optimize revenue [8, 9, 10, 11, 12, 13, 14, 15,
16] can serve as complementary mechanisms to control user demands and ensure
good quality-of-service despite limited available resources, while maintaining
the network operator’s incentive. Currently, Edge computing providers have
been adopting the traditional cloud computing pricing models [17, 18, 19]: the
On-Demand pricing model (OPM) and the Resource Reservation model (RRM), e.g.
in AWS Local Zones [20] and Juniper [21]. These two models cater to different
types of users. In the On-Demand pricing model, service users pay for
computing resources, as and when they need it, and do not make any long term
commitment [22]. In MEC, this is suitable for individual mobile users using,
for example, virtual reality or mobile gaming services once, or a couple of
times, not across consistent timeslots across the span of months; for firms
during special one-off events involving augmented reality displays or machine
learning platforms. In contrast, for the Resource reservation model, service
users reserve computing resource instances in advance, across multiple
timeslots, (e.g. for a year), for a discounted price [23]. This caters to
users (or firms) who use computing resources in bulk, at periodic and pre-
determined timings. These may include IoT vendors with periodic data analytic
requirements for their IoT networks or firms behind public augmented reality
setups, with a constant need of computing capabilities.

Besides maximizing revenue, will such plans be able to meet user demand? While
edge computing is advantageous latency-wise through the proximity to end
users, unlike cloud data centers with thousands of servers [24], the
distributed nature of MEC involves small-scale data centers placed at the
network edge [2, 1]. The server capacity at edge nodes may be limited due to
physical constraints [7]. Therefore, the difficulty from jointly using the On-
Demand pricing and Resource Reservation pricing models in MEC is that after
the resource units are reserved in advance by the Resource Reservation users,
there might be fewer resource units left for on-demand users. Consequently,
on-demand users may experience a longer service latency, which may jeopardize
the utility of MEC since services which use edge computing are often latency
sensitive [2, 3]. Cloud providers complement the on-demand and resource
reservation models and monetize excess demand through spot instances, leasing
the unused capacity at data centers to users at a discounted price [25]. We
are unable to consider spot pricing in MEC because unlike the cloud, there is
a lack of spare capacity at the edge. Furthermore, raising the price of
resource reservations to control the demand, may not suffice as users with
long-term usage needs may continue to prefer fixed upfront expenses compared
to on-demand expenses that may fluctuate and may be hard to predict.

Therefore, to improve the QoS of users, we propose a sharing platform, titled
the Sharing Quota Model (SQM), Here, reservation plan users can re-sell their
excess un-utilized resource quota via the platform to on-demand users, while
the MNO collects a commission. This could benefit all parties. The on-demand
users can attain a higher QoS (lower delay) and a higher probability of
getting their jobs served, the reservation plan users will obtain extra income
and the MNO collects a commission. Higher customer satisfaction also benefits
the MNO in the long run. It is unclear, however, whether the MNO always has an
incentive to offer such a platform, as the SQM may discourage on-demand users
from purchasing capacity directly from the MNO. To best integrate the sharing
platform with the on-demand pricing and resource reservation models, the
following research questions arise:

a) Given the sharing platform, would on-demand users still buy from the on-
demand platform? What would the aggregate equilibrium behavior look like?

b) Given the users’ decisions, it is not clear if the MNO has an incentive to
provide this platform: will it lead to sub-optimal revenue? How should it set
prices such that revenue is optimized, whilst minimizing the unmet demand of
the on-demand users?

c) How can we design the platform? Is it more practical if resource quota re-
selling is automatically or manually decided?

While there have been resource sharing (or trading) platforms in networks,
e.g. mobile data market trading platforms [26, 27, 28], and crowdsourced
community wireless networks [29], their market structures are different.
Specifically, the mobile data market users buy plans which have the same
structure, differentiated by usage levels. In crowdsourced community wireless
network model formulations, given prices and membership choices of other
users, users decide their network access time. This differs from our market
structure, where there are two types of plans (on-demand vs reservation),
determined by service application type and usage patterns. Our work also
differs from other MEC resource allocation or revenue maximization work, as it
uniquely considers how the on demand and resource reservation model can be
improved in MEC. Therefore, to answer the above questions and investigate the
feasibility of the sharing platform, we perform game theoretic analysis on the
three-sided market, formulating a three- stage non-cooperative Stackelberg
Game amongst the MNO, re-sellers (reservation plan users) and buyers (on-
demand users). Our analysis characterises the decision making and conflicting
incentives of the three parties, with the MNO setting the on-demand pool and
sharing pool’s prices to optimize revenue, the on-demand users choosing the
pool to purchase from, aiming to maximize their payoff functions, and the
reservation users also aiming to maximize their payoff functions.

Our contributions are summarized as follows:

  * •

With a direct application of on-demand and resource reservation pricing plans
in MEC, on-demand users face potentially limited resource availability at the
edge, impacting their QoS, while resource reservation plan users might have
reserved excess un-utilized resource quota. Therefore we propose a novel model
integrating a sharing platform, for the re-selling of unused resource quota
from reservation plan users to on-demand users.

  * •

To investigate the MNO’s incentive on having a sharing platform and to
optimize the MNO’s prices for revenue maximization, we will use a 3-stage
Stackelberg Game formulation to characterise how the MNO, buyers and re-
sellers interact at equilibrium. In this sequential game, the MNO selects
prices to optimize its revenue. Following which, the re-sellers (reservation
plan users) choose whether or not to sell, forming the sharing pool. Viewing
the prices and relative supplies, the buyers choose which pool (or not) to buy
from. We incorporate two platform design options - automated re-selling of
unused quota, and manual decision making for re-sellers.

  * •

Using backwards induction, we firstly characterise the buyer’s optimal
strategies for the above two platform design options (Proposition 1). Our
analysis shows that the equilibrium behavior amongst the buyers and re-sellers
possesses a 4 region structure (Proposition 2, with each region corresponding
to whether or not the on-demand and sharing pools coexist). The likelihood of
each of the 4 regions happening depend on the relative magnitudes of prices
and supply levels of the two pools. Based on the 4 region structure, we
optimize the on-demand and sharing prices (which also impacts the sharing
supply), to optimize the MNO’s revenue, and provide insights on the optimal
price (Theorems 1 and 2, Corollary 1).

  * •

We provide numerical results, which show how the system behaves at the optimal
point. Our results show that for different willingness to pay and usage
distributions, having both the sharing and on-demand pool gives the MNO an
optimal revenue when the on-demand pool’s supply $q_{o}$ is low, and when
$\delta$, the platform’s commission is low, but not below a certain threshold.
It would be expected that a high commission would give the platform a higher
sharing revenue, nevertheless our results show that it incurred a trade-off in
dissuading reservation plan users from re-selling.

  * •

We created a prototype, an interactive website, to conduct preliminary user
studies on user decision making in our resource re-selling framework.Our
analysis shows that there is no statistically significant evidence that the
empirical distribution function of the data from the user study and the
empirical distribution function of the data from the model’s predictions (Eq.
1, Prop. 1-2) are distinct.

The rest of this paper is organized as follows. In Section 2, we introduce the
related works. In Section 3, we describe the system model. Following which, we
formulate the Stackelberg Game in Section 4. In Section 5, we present the game
analysis and insights. We present numerical results in Section 6, present our
prototype and preliminary users studies in Section 7, discuss future areas of
investigation in Section 8, and finally conclude this paper in Section 9.

##  2 Related Works

In this work, we propose a novel system model incorporating a resource re-
selling platform, to improve the integration of the on-demand and resource
reservation pricing models in MEC, given the limited resources at the edge. We
also discuss how our model differs from other platforms in networked systems.

Edge Computing resource allocation: Researchers have been actively optimizing
resource allocation in MEC, balancing the user’s delay requirements with
minimizing the energy consumption, under various scenarios [4, 5, 6, 7, 30].
Another line of work is collaborative edge computing to maintain the QoS in
light of user mobility. This involves strategies like VM or service migration
across base stations [31, 32, 33, 14] and user association optimization [34].

Edge Computing revenue maximization: These works involve optimizing revenue
while maintaining the quality of service for users (delay, job service rate,
etc) [8, 9, 10, 12, 13, 11]. For example, [8] used the Fisher market model to
model the MEC market. [9] jointly optimized over the resource prices and the
devices’ budget allocation strategies. [10] modelled a dynamic game between
the edge resource owners, infrastructure providers, and service users. While
our model follows the user payoff model of [10], we model a distinct scenario,
involving a novel hybrid sharing, on-demand and resource reservation
framework. Thus, unlike [10] we must further account for re-seller incentives,
leading to distinct results. In our prior work [35], we proposed a model where
users constantly change whether to be an ’owner’ or ’renter’ of excess
resource units, along with dynamic pricing mechanisms. We did not investigate
the MNO’s incentive of having a sharing platform. To the best of our
knowledge, there has not been work on how to improve the integration of the
on-demand and the resource reservation pricing models in MEC

Cloud Computing pricing: Maximizing revenue in cloud computing by improving on
the on-demand and reserved instance models was studied in [19, 36, 37, 38].
These works focused on spot instances, which sell the spare capacity at data
centers, at discounted prices [25]. For example, [19] proposed a truthful
auction for spot instances. [36] proposed a dynamic pricing policy. [37]
maximized revenue through capacity segmentation. Unlike in cloud computing,
each edge server has limited resources, making the spot instances model not
feasible in edge networks.

Other Platforms in Networked Systems: The mobile data market has seen the
existence of platforms (e.g. China Mobile Hong Kong’s 2nd exChange Market)
which provide a secondary data market, allowing users to buy and sell data
from each other [26, 27]. These works [26, 27, 28] perform a game theoretic
analysis to characterise the equilibrium and optimal decisions under various
scenarios, as well as propose alternative platform designs [39]. This platform
has a different market structure from the edge computing market \- here all
the users buy resources under the same type of scheme (monthly data plans with
usage quotas) and are differentiated by usage levels. In the edge computing
market, users have vastly different kinds of applications and usage patterns,
which suit different usage schemes: some users have consistent usage patterns
(e.g. IoT vendors) and suit the resource reservation model, and some users
have more sporadic, one-off usage patterns (e.g. individual mobile users using
applications like virtual reality) and are suited to the on-demand model.

##  3 The Resource Procurement Model

We consider an edge MNO that provides edge computing service to $N$ users. The
MNO has 2 resource procurement schemes for users to choose from, namely the
Resource Reservation Model (RRM) and On-Demand Pricing Model (OPM). These
models, which are commonly used as cloud pricing and procurement schemes [17,
18, 19], are currently being used in edge networks [20, 21]. Under OPM, the
users purchase credits to offload their jobs at the edge servers ”on the go”,
as and when they need it. They pay $p_{o}$ per unit of resource consumed (CPU
cycles), to the MNO. Customers of on-demand pricing could include individual
mobile users, e.g. users of applications like AR/VR or mobile gaming, or firms
who are using AR or machine learning applications for one-off special events
or displays. These users’ demand for computing is sporadic, not following a
consistent usage pattern across timeslots.

The RRM is catered to customers who have consistent computing requirements
across different timeslots. The RRM allows them to reserve (i.e. commit to)
computing resource in advance across multiple timeslots, e.g. for a year, at a
discounted price per timeslot of $p_{r}$. Customers of the reservation plans
include users who use computing resources in bulk, at periodic timings, and
more frequently in general. For example, these may include IoT vendors with
constant sensing and data analysis requirements for their large IoT network,
autonomous driving firms, entities behind long-term public augmented reality
setups.

The user has chosen between the resource procurement schemes OPM and RRM based
on its service application’s usage requirements: sporadic, one-off vs
consistent, frequent and long-term. The details of the user usage model will
be in covered the following subsections. The computing resources at the edge
server are first provisioned to the reserved users, with the remaining
resources forming the on-demand pool.

Nevertheless, unlike cloud resource allocation where data centers have huge
amount of computing resources, the server capacity at edge nodes is more
limited [2, 1], and the data processing rate is less powerful [40]. Hence,
after resources are reserved in advance by RRM users, there might be less
resources left for on-demand users. At the same time, RRM users may find that
for specific timeslots, they have excess computing resource quota for CPU
cycles which they do not require. In this work we propose and explore the
impact of adding a resource sharing scheme, the Sharing Quota Model (SQM). In
this integrated scheme (see Figure 1), RRM users act as re-sellers, selling
their excess unused resource quota at a price of $p_{r}$ per unit of resource
to on-demand users (buyers). The resold resources form the sharing pool and
increase the total supply available to on-demand users (buyers). The MNO acts
as a middleman and takes a commission of $\delta$ from the transaction. We
investigate the impact that adding SQM has on the profits of the MNO, and
hence whether this scheme is feasible.

Figure 1: The integration of the resource-quota sharing platform with the On-
Demand and Resource Reservation models.

Re-sellers: The re-sellers are users who have previously reserved resources
from RRM, and who have excess resources available for sale. Under the proposed
sharing scheme SQM, they have 2 choices $s=\\{sell,no\\}$ in each timeslot.
They can choose whether to sell their excess computing resource quota via the
MNO’s platform to on-demand users (buyers), or not. Without loss of
generality, we consider the payoff per unit resource quota for seller $i$,
which is expressed as:

$\pi_{i}^{s}(g_{i})=\begin{cases}(1-\delta)p_{r}-g_{i},&\text{if}\ s=sell\\\
0,&\text{if}\ s=no.\end{cases}$ (1)

Recall that $p_{r}$ is the per unit price set by the MNO for the transaction,
and define $\delta$ as the commission level which the MNO takes from the
transaction. We let $g_{i}$ represent the inconvenience cost of selling for
seller $i$ (factoring in uncertainty of usage, and the inconvenience of
checking the app to decide to share). In this paper, we consider per unit
resources in the payoff functions of both buyers and re-sellers. Buyers and
re-sellers with multiple units are considered as multiple buyers or re-
sellers. Note that if we had considered users buying/re-selling distinct
quantities of resources, with the resource usage level of users following
$U[0,1]$, the impact would be that the supply and revenue functions are
multiplied by a constant factor.

The inconvenience cost depends on the extent to which each re-seller is
willing and able to expend time, energy and resources to periodically check
the app, factor in the uncertainty of personal usage, and decide whether or
not to sell the excess resource quota. This differs across different vendors
and owners, and differs across applications. For example, it would be easier
for an IoT vendor to dedicate resources to check the app, as comapared to an
individual MEC user. Some individual MEC users are more likely to check than
others as they are more interested in earning extra income. Besides this ,it
is more difficult for users of some applications to make the decision to re-
sell, as their usage for that particular timeslot is yet to be known, and more
unpredictable. $g_{i}$ is a random variable which follows the uniform $U[0,1]$
distribution. It captures the varying levels of personal costs across the
users. While the distribution has an impact on the way revenue is computed, we
perform simulations using both uniform and beta (which has a gaussian-like
structure) distributions in Section 6, showing similar results across
distributions.

As we can see from Eq. (1), a re-seller’s decision is impacted by the price it
receives $p_{r}$, the MNO’s commission level $\delta$, and the inconvenience
cost $g_{i}$. Re-seller $i$ will share/sell when its profit $(1-\delta)p_{r}$
is greater than its cost $g_{i}$. If the inconvenience cost is too high, it
will not. To obtain the total proportion of re-sellers willing to sell/share,
we integrate over the users whose inconvenience cost $g_{i}$ is less than or
equals to $(1-\delta)p_{r}$. As $g_{i}$ follows the uniform $U[0,1]$
distribution, the proportion of re-sellers willing to share/sell will be:

$\begin{split}prop&=\int^{(1-\delta)p_{r}}_{0}dg_{i}\\\
&=(1-\delta)p_{r}.\end{split}$ (2)

The quality of the sharing pool, $q_{s}$, indicates the total supply of
resource quota available for buyers, and therefore is dependent on the
proportion of re-sellers who are willing to share their unused quota,
$(1-\delta)p_{r}$. It can be expressed as111Function $q_{s}(p_{r})$ may have
various forms in realistic MEC systems, but our analysis permits any
functional form. We will discuss a concrete form for numerical results in
Section 6.:

$q_{s}(p_{r})=f((1-\delta)p_{r}),$ (3)

with $q_{s}(p_{r})$ being a strictly increasing, twice differentiable and
concave function. This supply level has an impact on the user experienced
delay.

Buyers: The buyers are on-demand users who purchase quota to offload their
jobs at the edge servers on the go, i.e. not in advance. These buyers send
their computing jobs to the edge server, with different willingness to pay
(represented by $u_{i}$ for user $i$). The willingness to pay is the buyer’s
utility of job computation at the edge server. For example, some jobs are more
urgent and considered more of a necessity than others (e.g. public AR/VR
displays, autonomous vehicle computation), and therefore the buyers will still
be willing to pay despite the price $p_{r}$ increasing. On the other hand,
some jobs are considered less urgent, or are not viewed as necessities from
the buyer’s perspective, resulting in a lower willingness to pay: given a high
price of resource quota $p_{r}$, the buyer is more likely to choose not to
purchase the edge computing service. We let $u_{i}$ follow the uniform
$U[0,1]$ distribution, to represent how the willingness to pay differs across
users. Note that the distribution has no impact on the payoff maximizing
strategies of the buyers, or the equilibrium outcome of user decisions in
light of fixed prices. While it has an impact on the way revenue (Eq. (6)) is
computed, we perform simulations using both uniform and beta distributions in
Section 6 which show similar trends and effects across distributions.

Under SQM, these on-demand users now have 3 options $b\in\\{\text{on-demand
pool, sharing pool, {\color[rgb]{0,0,0}none}}\\}$.222For simplification, we
use $b=o$, $b=s$, and $b=n$ to represent $b=\text{on-demand pool}$,
$b=\text{sharing pool}$, and $b=\text{{\color[rgb]{0,0,0}none}}$,
respectively. Note that, $b=\text{none}$ means not buying from both the on-
demand pool and sharing pool. Therefore the payoff of buyer $i$ is:

$\pi_{i}^{b}(u_{i})=\begin{cases}u_{i}q_{o}-p_{o},&\text{if}\ b=\text{on-
demand pool}\\\ u_{i}{\color[rgb]{0,0,0}q_{s}(p_{r})}-p_{r},&\text{if}\
b=\text{sharing pool}\\\ 0,&\text{if}\ b=\text{not buying},\end{cases}$ (4)

where $u_{i}$ is buyer $i$’s willingness to pay for a unit resource quota, as
described earlier and used in [10]. The payoff when users choose the on-demand
and sharing pool respectively is their willingness to pay (satisfaction)
$u_{i}$ multiplied by the quality/supply of the pool ($q_{o}$ or $q_{s}$)
minus the payment ($p_{o}$ or $p_{r}$). The quality/supply ($q_{o}$ or
$q_{s}$) indicates the relative supply of resources available at the on-demand
and sharing pool respectively. The quality at the on-demand pool, $q_{o}$, is
the amount of supply of computing resources (in CPU cycles) at the on-demand
pool, after the resource reservation users have reserved their portion.333Note
that in traditional centralised cloud computing, the supply at data centers
(i.e. the $q_{o}$ equivalent) would be much larger in magnitude, resulting in
less of a need for a sharing pool, according to Eq. (4). As mentioned earlier,
the quality at the sharing pool, $q_{s}(p_{r})$, is a function of the
proportion of re-sellers who are willing to re-sell. The higher the supply at
each pool, the higher the likelihood the buyer is able to get its job served,
hence improving its QoS in terms of delay experienced. Therefore the buyer’s
payoff $\pi_{i}^{b}(u_{i})$ is an increasing function of the quality
$q_{s}(p_{r})$ or $q_{o}$. As different buyers have different willingness to
pay $u_{i}$, different buyers will make different choices, based on which
choice maximizes their individual payoff. In the next few sections, we
investigate the equilibrium behavior of the users, and the optimal decisions
for the MNO to make based on that. We summarize the key notations used
throughout this paper in Table I.

TABLE I: Summary of key Notations Notation | Definition
---|---
$u_{i}$ | Willingness to pay of buyer $i$.
$g_{i}$ | The inconvenience cost of re-seller $i$.
$\delta$ | The MNO’s commission percentage.
$p_{r}$ | Price per-unit of re-sold quota at the sharing platform.
$p_{o}$ | Price per-unit resource at the on-demand pool.
$q_{o}$ | Supply, and hence quality at the on-demand pool.
$q_{s}(p_{r})$ | Supply and hence quality at the sharing platform.
$\pi_{i}^{s}(g_{i})$ | Payoff of re-seller $i$.
$\pi_{i}^{b}(u_{i})$ | Payoff of buyer $i$.
$R(p_{r},p_{o})$ | Revenue of the MNO.

##  4 Problem Formulation

The MNO’s objective is to optimize its revenue, when integrating the sharing
scheme SQM with the existing pricing models. The MNO’s decisions will be made
based on the aggregate behavior of behavior of buyers and re-sellers at
equilibrium. Its Revenue Maximization Problem (RMP) is:

$\textbf{RMP}:\underset{p_{o},p_{r}}{\text{max}}\ R(p_{r},p_{o})$ (5)

where

$\begin{split}R(p_{r},p_{o})=&N\int^{1}_{0}p_{o}\mathbb{1}_{\\{\pi^{b=o}_{i}(u_{i})>max(\pi^{b=s}_{i}(u_{i}),0)\\}}du_{i}\\\
&+N\int^{1}_{0}p_{r}\delta\mathbb{1}_{\\{\pi^{b=s}_{i}(u_{i})>max(\pi^{b=o}_{i}(u_{i}),0)\\}}du_{i}.\end{split}$
(6)

In Eq. (6), $N$ is the number of users, and $p_{o}$ and $p_{r}$ are the price
per unit resource at the on-demand pool and the sharing pool, respectively.
Hence, the first term on the right hand side is the MNO’s revenue attained
from the on-demand pool, where
$\mathbb{1}_{\\{\pi^{b=o}_{i}(u_{i})>max(\pi^{b=s}_{i}(u_{i}),0)\\}}$ is the
indicator function indicating if buyer $i$ gains a higher payoff from choosing
the on-demand pool, over the sharing pool and not buying at all. The second
term is the MNO’s revenue from the sharing pool, where
$\mathbb{1}_{\\{\pi^{b=s}_{i}(u_{i})>max(\pi^{b=o}_{i}(u_{i}),0)\\}}$ is the
indicator function indicating if buyer $i$ gains a higher payoff from choosing
the sharing pool, over the on-demand pool and not buying at all.

To analyse the strategic behaviours and the dynamic interaction amongst the
three parties and to obtain equilibrium insights, we formulate a non-
cooperative Stackelberg Game [41] (Fig. 2).  In Stage 1, the MNO (the leader)
sets the on-demand and sharing pool prices $p_{o}$ and $p_{r}$, to maximize
its revenue $R(p_{r},p_{o})$, as in Eq. (7). In Stage 2A, given $p_{r}$ and
$\delta$, the re-sellers (followers) individually decide $s$, whether they
would like to share or not. A re-seller’s decision depends on which choice
maximizes their individual payoff $\pi_{i}^{s}$, following Eq. (8). The
collective decision of all re-sellers would result in a supply/quality of
$q_{s}(p_{r})$ for the sharing pool, which would influence the decisions of
the buyers in Stage 2B. In Stage 2B, given the prices $p_{r}$ and $p_{o}$, and
the supply/qualities $q_{s}(p_{r})$ and $q_{o}$, the buyers (followers)
individually decide $b$, which pool they would buy from. A buyer’s decision
depends on which choice maximizes their individual payoff $\pi_{i}^{b}$, as
presented in Eq. (9). The decisions of the followers (the re-sellers and
buyers) would in turn affect the revenue of the leader MNO.

Figure 2: Game formulation: Non-cooperative game between the MNO, re-sellers
and buyers.

$Stage\ 1:\text{MNO (leader):
}(p_{r}^{*},p_{o}^{*})=\underset{p_{r},p_{o}}{\text{argmax}}R(p_{r},p_{o})$
(7) $Stage\ 2A:\text{Re-sellers:}\
s^{*}=\underset{s\in\\{s,n\\}}{\text{argmax}}\ \pi_{i}^{s}(g_{i})\\\ $ (8)
$Stage\ 2B:\text{Buyers :}\ b^{*}=\underset{b\in\\{o,s,n\\}}{\text{argmax}}\
\pi_{i}^{b}(u_{i})$ (9)

##  5 Game Analysis

In this section, we solve the Stackleberg Game (Eqs. (7), (8) and (9)),
characterizing and presenting insights on the equilibrium interaction amongst
the MNO, buyers, and re-sellers, and characterizing the MNO’s optimal prices.
We analyze two distinct platform designs which the MNO could implement for the
re-selling portion of the sharing platform. Firstly, in  Section 5.1, we
explore the option where the MNO implements automatic re-selling of unused
resource quota on the part of users. While this may not be as feasible in
practice as it involves all users knowing upfront whether or not they have
excess quota, nevertheless, this portion of the study gives insights on buyer
decisions and the aggregate outcome at equilibrium in light of MNO prices, by
isolating the MNO-buyer interaction. In Section 5.2 we explore the option
where the MNO implements manual decision making for re-sellers: re-sellers
weigh their personal inconvenience cost $g_{i}$ when deciding whether or not
to re-sell quota in the sharing pool, according to Eq. (1).

###  5.1 Case 1: Automatic Re-Selling of Unused Quota

In this subsection we analyze the scenario where the MNO implements automatic
re-selling of unused resource quota on the part of re-sellers.

As there is no decision making on the part of re-sellers, we have
$q_{s}(p_{r})$ as a constant $q_{s}$. The resulting game between the MNO and
the buyers is:

$Stage\ 1:\text{MNO decides}\
(p_{r}^{*},p_{o}^{*})=\underset{p_{r},p_{o}}{\text{argmax}}\ R(p_{r},p_{o})$
(10) $Stage\ 2:\text{Buyers decide}\
b^{*}=\underset{b\in\\{o,s,n\\}}{\text{argmax}}\ \pi_{i}^{b}(u_{i})$ (11)

####  5.1.1 The buyer’s optimal strategies

We solve this game by backward induction [41], solving Stage $2$ first, and
using the equilibrium outcome of Stage $2$ to solve Stage $1$ and obtain the
MNO’s optimal prices.

Buyers try to maximize their payoffs $\pi_{i}^{b}$ by choosing whether they
will buy computing resource quota from the on-demand pool, the sharing pool,
or not buy at all, given the prices $p_{o}$, $p_{r}$, and the relative supply
$q_{o}$ and $q_{s}$. Proposition 1 shows which strategies  buyer $i$ will
take, given  its willingness to pay $u_{i}$.

######  Proposition 1.

For given values of prices $(p_{o},p_{r})$ and quality/supply $(q_{o},q_{s})$,
the payoff maximizing strategy for buyer $i$ will be:

a) if $q_{s}>q_{o},$

$b^{*}=\begin{cases}\text{sharing\ pool}&\text{if}\
u_{i}>\max(\frac{p_{r}-p_{o}}{q_{s}-q_{o}},\frac{p_{r}}{q_{s}})\\\ \text{on-
demand\ pool}&\text{if}\
\frac{p_{o}}{q_{o}}<u_{i}\leq\frac{p_{o}-p_{r}}{q_{o}-{\color[rgb]{0,0,0}q_{s}}}\\\
\text{none}&\text{otherwise}\end{cases}$ (12)

b) if $q_{o}>q_{s},$

$b^{*}=\begin{cases}\text{sharing\ pool}&\text{if}\
\frac{p_{r}}{q_{s}}<u_{i}\leq\frac{p_{r}-p_{o}}{q_{s}-q_{o}}\\\ \text{on-
demand\ pool}&\text{if}\
u_{i}>\max(\frac{p_{o}-p_{r}}{q_{o}-q_{s}},\frac{p_{o}}{q_{o}})\\\
\text{none}&\text{otherwise}\end{cases}$ (13)

c) if $q_{o}=q_{s},$

$b^{*}=\begin{cases}\text{sharing\ pool}&\text{if}\
p_{o}>{\color[rgb]{0,0,0}p_{r}}\ \&\
u_{i}>\frac{{\color[rgb]{0,0,0}p_{r}}}{q_{s}}\\\ \text{on-demand\
pool}&\text{if}\ {\color[rgb]{0,0,0}p_{r}}>p_{o}\ \&\
u_{i}>\frac{p_{o}}{q_{o}}\\\ \text{none}&\text{otherwise}\end{cases}$ (14)

Proof: We compare $\pi_{i}^{o}(\mu_{i})$, $\pi_{i}^{s}(\mu_{i})$, and $0$, and
find the conditions under which each strategy maximizes buyer $i$’s payoff.
When $\pi_{i}^{o}(\mu_{i})$ is the largest, we have $b=\text{on-demand pool}$.
When $\pi_{i}^{s}(\mu_{i})$ is the largest, we have $b=\text{sharing pool}$.
Otherwise, $b=\text{none}$. ∎

(a) Both on-demand and sharing.

(b) Only On-demand.

(c) Only Sharing.

Figure 3: Illustrations of buyers’ payoff, and therefore the decision they
make, at different willingness to pay, with $q_{s}>q_{o}$.

Note that Prop. 1 is independent of the distribution of $u_{i}$, users’
willingness to pay. Fig. 3 provides a geometric description of Proposition 1
(Prop. 1a in particular). For example, if $q_{s}>q_{o}$ (corresponding to
Prop. 1a), and when $u_{i}$ satisfies
$\frac{p_{o}}{q_{o}}<u_{i}\leq\frac{p_{r}-p_{o}}{q_{s}-q_{o}}$, we can see
from Fig. 3a that the buyer’s payoff from the on-demand pool would be positive
and higher than that from the sharing pool. Hence, the buyer with $u_{i}$
satisfying $\frac{p_{o}}{q_{o}}<u_{i}\leq\frac{p_{r}-p_{o}}{q_{s}-q_{o}}$
would choose the on-demand pool. Proposition 1 shows that buyers with higher
willingness to pay $u_{i}$ will choose the option with higher quality/supply
($q_{o}$ or $q_{s}$ which is the gradient of all the payoff graphs in Fig. 3),
as we can see in all three graphs. For an example,  we look at the case where
$q_{s}>q_{o}$ in Fig. 3. Users with higher $u_{i}$ choose the sharing pool
when $p_{r}-p_{o}\leq 0$ (Fig. 3c), and also when
$0\leq\frac{p_{r}-p_{o}}{q_{s}-q_{o}}\leq 1$ (Fig. 3a). Nevertheless, we see
that when the price $p_{r}$ for the sharing pool is too high such that
$\frac{p_{r}-p_{o}}{q_{s}-q_{o}}>1$ (Fig. 3b, where
$\frac{p_{r}-p_{o}}{q_{s}-q_{o}}$ is the intersection of the two graphs), none
of the users will choose the sharing pool even with $q_{s}>q_{0}$.

In the special case where $q_{s}=q_{o}$ (Proposition 1c), the qualities/supply
of the two options are the same. Geometrically, this means that the gradients
of the two payoffs will be the same. If the on-demand pool has a higher price
($p_{o}>p_{r}$), the buyers will choose the sharing pool, and vice versa. When
the buyers’s willingness to pay is low
($u_{i}<\min(\frac{p_{r}}{q_{s}},\frac{p_{o}}{q_{o}})$), the buyer will choose
neither.

####  5.1.2 The MNO’s price selection strategy

Following the buyers’ optimal selection strategies (Proposition 1), and
integrating over the collective behavior of all of the buyers, there will be 4
regions at equilibrium, with boundary conditions as functions of $q_{o}$,
$q_{s}$, $p_{o}$ and $p_{r}$. These 4 regions correspond to the 4 scenarios of
whether users will choose the sharing and/or on-demand option, or choose not
to buy at all. In Proposition 2, we characterize these 4 regions.

######  Proposition 2.

For given values of prices ($p_{r},p_{o}$) and quality/supply ($q_{s},q_{o}$),
the equilibrium will satisfy exactly one of the following 4 conditions:

$\begin{split}\text{R1}:\
\\{(p_{r},p_{o})|p_{r}<\min(\frac{q_{s}}{q_{o}},1)p_{o}+(q_{s}-q_{o})^{+},\\\
p_{r}>\max(\frac{q_{s}}{q_{o}},1)p_{o}-(q_{o}-q_{s})^{+}\\}\end{split}$ (15)
$\text{R2}:\ \\{(p_{r},p_{o})|p_{o}<q_{o},\
p_{r}\geq\min(\frac{q_{s}}{q_{o}},1)p_{o}+(q_{s}-q_{o})^{+}\\}$ (16)
$\text{R3}:\ \\{(p_{r},p_{o})|p_{r}<q_{s},\
p_{r}\leq\max(\frac{q_{s}}{q_{o}},1)p_{o}-(q_{o}-q_{s})^{+}\\}$ (17)
$\text{R4}:\ \\{(p_{r},p_{o})|p_{r}\geq q_{s},\ p_{o}\geq q_{o}\\},$ (18)

where $(a)^{+}=a$ if $a\geq 0$, and $0$ otherwise. In R1, buyers buy from both
pools  or neither, like in Fig. 3a. In R2, buyers buy from the on-demand pool
or from neither, as in Fig. 3b. In R3, buyers buy from the sharing pool or
from neither, as in Fig. 3c. In R4, buyers will buy from neither pool.

Proof: The boundary conditions of the four regions are deduced through Fig. 3.
∎

Prop. 2 is independent of the distribution of $u_{i}$, users’ willingness to
pay. This  proposition indicates how the relative magnitudes of the prices
$p_{r}$, $p_{o}$, and supply $q_{s}$ and $q_{o}$ impact the equilibrium
outcome. Specifically, when $(p_{r},p_{o},q_{s},q_{o})$ satisfies the boundary
conditions of one of the regions, the collective behavior of buyers (who
behave according to Proposition 1), will result in the equilibrium outcome
being in that region. For example, when the prices $p_{o}$ and $p_{r}$ satisfy
the conditions $p_{o}<q_{o},\
p_{r}\geq\min(\frac{q_{s}}{q_{o}},1)p_{o}+(q_{s}-q_{o})^{+}$, it will not be
incentive compatible for any buyers to choose the sharing pool (regardless of
$u_{i}$), and the outcome is R2. The special case when $q_{s}=q_{o}$ in
Proposition 1 also fits this 4 region framework. If
$1<\min(\frac{p_{r}}{q_{s}},\frac{p_{o}}{q_{o}})$, the users will choose not
to buy and the resulting scenario will be R4. In the alternative where
$1<\min(\frac{p_{r}}{q_{s}},\frac{p_{o}}{q_{o}})$ does not hold, if the on-
demand price is higher ($p_{o}>p_{r}$) the users will choose the sharing pool
and we end up in R3, and vice versa for R2. This 4 region structure is unique
with respect to the results from work on other sharing platforms in networked
systems (mobile data re-selling [26, 28] and crowdsourced wireless community
networks [29]).

With the 4 disjoint regions involving different pools being purchased from
(Proposition 2), the revenue function $R(p_{o},p_{r})$ can be simplified as
follows:

$\begin{split}&R^{R1}(p_{o},p_{r})=\\\
&\begin{cases}&Np_{o}(\frac{p_{r}-p_{o}}{q_{s}-q_{o}}-\frac{p_{o}}{q_{o}})+Np_{r}\delta(1-\frac{p_{r}-p_{o}}{q_{s}-q_{o}}),\quad\text{if}\
q_{s}\geq q_{o}\\\
&Np_{r}\delta(\frac{p_{o}-p_{r}}{q_{o}-q_{s}}-\frac{p_{r}}{q_{s}})+Np_{o}(1-\frac{p_{o}-p_{r}}{q_{o}-q_{s}}),\quad\text{if}\
q_{o}>q_{s}\end{cases}\end{split}$ (19)
$R^{R2}(p_{o},p_{r})=Np_{o}(1-\frac{p_{o}}{q_{o}})$ (20)
$R^{R3}(p_{o},p_{r})=Np_{r}\delta(1-\frac{p_{r}}{q_{s}})$ (21)
$R^{R4}(p_{o},p_{r})=0$ (22)

In R1, buyers are willing to buy from both the on-demand and sharing pool. The
two terms in the objective $R^{R1}(p_{o},p_{r})$ correspond to the revenue
obtained from the two pools. In R1 with $q_{s}\geq q_{0}$,
$(\frac{p_{r}-p_{o}}{q_{s}-q_{o}}-\frac{p_{o}}{q_{o}})$ and
$1-\frac{p_{r}-p_{o}}{q_{s}-q_{0}}$ are the proportion of the buyers who
choose the on-demand and sharing pool respectively. In R4, buyers find it
optimal to not purchase from either pool, and hence the MNO’s revenue is $0$.

Therefore, to optimize the revenue maximization problem, we will split the
problem into four subproblems as follows:

$\underset{p_{o},p_{r}}{\text{max}}\ {\color[rgb]{0,0,0}R^{R1}(p_{o},p_{r})}\
\text{s.t.}\ (\ref{eq:R1regioncond})$ (23)
$\underset{p_{o},p_{r}}{\text{max}}\ {\color[rgb]{0,0,0}R^{R2}(p_{0},p_{r})}\
\text{s.t.}\ (\ref{eq:R2regioncond})$ (24)
$\underset{p_{o},p_{r}}{\text{max}}\ {\color[rgb]{0,0,0}R^{R3}(p_{0},p_{r})}\
\text{s.t.}\ (\ref{eq:R3regioncond})$ (25)
$\underset{p_{o},p_{r}}{\text{max}}\ {\color[rgb]{0,0,0}R^{R4}(p_{0},p_{r})\
\text{s.t.}\ (\ref{eq:R4regioncond})}$ (26)

For subproblem $R^{R1}(p_{o},p_{r})$, we further split it into 2 subproblems
$R^{R1a}(p_{o},p_{r})$ and $R^{R1b}(p_{o},p_{r})$ corresponding to the 2 cases
where $q_{s}\geq q_{o}$ and $q_{o}>q_{s}$. In the following, we give Theorem 1
to characterise the optimal prices $p_{o}$ and $p_{r}$.

######  Theorem 1.

When buyers make decisions according to Proposition 1, given a fixed
$(q_{o},q_{s})$, and if the condition $(\delta+1)^{2}q_{s}<4\delta q_{o}$
holds, the MNO’s optimal prices will be as follows.

  * •

To maximize $R^{R1a}(p_{0},p_{r})$ in R1, we have:

If Eq. (15) is satisfied

$\begin{split}&(p_{o,1a}^{*},p_{r,1a}^{*})=\Bigg{(}\frac{\delta(1+\delta)q_{o}(q_{s}-q_{o})}{4q_{s}\delta-(1+\delta)^{2}q_{o}},\\\
&\qquad\qquad\qquad\quad\frac{q_{s}-q_{o}}{2}+\frac{(1+\delta)^{2}q_{o}(q_{s}-q_{o})}{2(4q_{s}\delta-(1+\delta)^{2}q_{o})}\Bigg{)},\end{split}$

Else,

$(p_{o,1a}^{*},p_{r,1a}^{*})\\!=\\!\text{argmax}_{\\{(\frac{q_{o}}{2},q_{s}-\frac{q_{o}}{2}),(\frac{q_{o}}{2},\frac{q_{s}}{2})\\}}R^{R1a}(p_{o},p_{r}).$

  * •

To maximize $R^{R1b}(p_{o},p_{r})$ in R1, we have:

If Eq. (15) is satisfied,

$(p_{o,1b}^{*},p_{r,1b}^{*})\\!=\\!\\!(\frac{2\delta
q_{o}(q_{o}-q_{s})}{4\delta
q_{o}-(\delta+1)^{2}q_{s}},\frac{q_{s}(\delta+1)(q_{o}-q_{s})}{4\delta
q_{o}-(\delta+1)^{2}q_{s}}),$

Else,

$(p_{o,1b}^{*},p_{r,1b}^{*})=\text{argmax}_{\\{(q_{o}-\frac{q_{s}}{2},\frac{q_{s}}{2}),(\frac{q_{o}}{2},\frac{q_{s}}{2})\\}}R^{R1b}(p_{o},p_{r}).$

  * •

To maximize $R^{R2}(p_{o},p_{r})$, we have
$(p_{o,2}^{*},p_{r,2}^{*})=(\frac{q_{o}}{2},a)$.

  * •

To maximize $R^{R3}(p_{o},p_{r})$, we have
$(p_{o,3}^{*},p_{r,3}^{*})=(b,\frac{q_{s}}{2})$.

Note that, in the latter two cases, $a$ is any price $p_{r}$ which satisfies
the boundary conditions of R2 (Eq. (16)), and $b$ is any price $p_{o}$ which
satisfies the boundary conditions of R3 (Eq. (17)).

Proof: See Appendix For each of the subproblems, we obtain the Hessian matrix
of the objective function. The Hessians of R1a, R2 and R3 are all negative
semidefinite (having eigenvalues which $\leq 0$), and therefore these
subproblems have concave objectives. As the constraints are linear with
respect to $p_{o}$ and $p_{r}$, these subproblems are convex. For the
subproblem R1b, the Hessian is negative semidefinite under the condition
$(\delta+1)^{2}q_{s}<4\delta q_{o}$. With convexity, we can use the KKT
conditions to find their optimal prices. ∎

In Theorem 1, $(\delta+1)^{2}q_{s}<4\delta q_{o}$ is the condition under which
the Hessian of $R1b$’s objective has negative eigenvalues, and hence the
condition under which subproblem $R1b$ is convex. We use the KKT conditions to
solve the problem. For subproblem $R1a$, the gradients are $0$ at
$p^{*}_{o,1a}=\frac{2\delta q_{o}(q_{o}-q_{s})}{4\delta
q_{o}-(\delta+1)^{2}q_{s}}$ and
$p^{*}_{r,1a}=\frac{q_{s}(\delta+1)(q_{o}-q_{s})}{4\delta
q_{o}-(\delta+1)^{2}q_{s}}$. If this point is within the boundary conditions,
it is the optimal point. If it is not, the optimal point would be at one of
the boundaries (with regions $R2$ or $R3$), specifically being
$(\frac{q_{o}}{2},q_{s}-\frac{q_{o}}{2})$ (boundary with $R2$) or
$(\frac{q_{o}}{2},\frac{q_{s}}{2})$ (boundary with $R3$). Likewise for
subproblem $R1b$. For subproblems $R2$ and $R3$, the optimal solutions of
$p_{o,2}^{*}=\frac{q_{o}}{2}$ and $p_{r,3}^{*}=\frac{q_{s}}{2}$ hold at the
boundary as well. These optimal prices agree with intuition, that as the
quality/supply (likelihood of buyer getting service) $q_{o}$ (or $q_{s}$)
increases, the MNO can increase the price to increase its revenue.

###  5.2 Case 2: Re-sellers weigh inconvenience cost when deciding to re-sell
quota

In this subsection, we analyse an alternative platform design in which the MNO
implements manual decision making for re-sellers: re-sellers weigh their
inconvenience cost $g_{i}$ when deciding whether or not to re-sell quota in
the sharing pool, according to Eq. (1). As mentioned in Section 3, the
inconvenience cost $g_{i}$ factors in the re-seller’s uncertainty of usage
during the timeslot, and the re-seller’s inconvenience, willingness and
ability towards checking the app to decide whether or not to re-sell. Here
$g_{i}$ follows the uniform $U[0,1]$ distribution as it captures the varying
levels of inconvenience across users. In Section 6, we perform simulations
using both uniform and beta (which has a gaussian-like structure)
distributions, showing similar results across distributions.

Under this scenario, we solve the Stackelberg Game (Eq. (7), (8) and (9)).
Firstly, in Stage 1 the MNO decides the prices $p_{r}$ and $p_{o}$. Following
which, in Stage $2A$ the re-sellers decide whether they want to sell their
excess resource quota, based on what maximizes their payoff (Eq. (1)),
resulting in a quality/supply of $q_{s}(p_{r})$. Finally, in Stage $2B$, the
buyers would make a decision, aiming to maximize their individual payoffs (Eq.
(4)), given the values of $(p_{r},p_{o},q_{s}(p_{r}),q_{o})$. We solve the
Stackelberg Game by backward induction, solving Stage $2$ first, and using the
equilibrium outcome of Stage $2$ to solve Stage $1$ and obtain the MNO’s
optimal prices.

####  5.2.1 The re-seller’s optimal strategy

The following Proposition shows the re-seller’s optimal strategy.

######  Proposition 3.

For given $p_{r}$, re-seller $i$ with inconvenience cost $g_{i}$’s optimal
strategy would be:

$s^{*}=\begin{cases}&sell,\quad\text{if}\ (1-\delta)p_{r}>g_{i},\\\
&no,\quad\text{otherwise}\end{cases}$ (27)

Collectively, by Eq. (2) this results in a proportion $(1-\delta)p_{r}$ of re-
sellers being willing to rent out their unused resources and therefore
following Eq. (3), the total quality/supply of the sharing pool will be

$q_{s}(p_{r})\sim f((1-\delta)p_{r}),$ (28)

which is a function of price $p_{r}$.

####  5.2.2 The buyer’s optimal strategy

Given $(q_{s}(p_{r}),q_{o})$ and $(p_{r},p_{o})$, a buyer’s strategy would
still follow Proposition 1 in Section 5.1, with the substitution of $q_{s}$ by
$q_{s}(p_{r})$. Hence, their collective strategies would form distinct regions
R1 to R4 at equilibrium, where the four regions indicate whether or not the
sharing and/or on-demand pools are feasible. This is according to Proposition
2 in Section 5.1. Once again, the only difference is that $q_{s}$ is replaced
by $q_{s}(p_{r})$.

####  5.2.3 The MNO’s optimal price

Given the strategies of the re-sellers and buyers in Stages $2A$ and $2B$,
corresponding to the regions in Proposition 2, the MNO’s revenue maximizing
problem $R(p_{o},p_{r})$ can be split into the following subproblems.

$R^{R1a}_{\text{C2}}(p_{o},p_{r})$:

$\displaystyle\underset{p_{r},p_{o}}{\text{max}}$ $\displaystyle
Np_{o}(\frac{p_{r}-p_{o}}{q_{s}(p_{r})-q_{o}}-\frac{p_{o}}{q_{o}})+Np_{r}\delta(1-\frac{p_{r}-p_{o}}{q_{s}(p_{r})-q_{o}})$
(29) s.t. $\displaystyle(\ref{eq:R1regioncond}),\;q_{s}(p_{r})>q_{o}$

$R^{R1b}_{\text{C2}}(p_{o},p_{r})$:

$\displaystyle\underset{p_{r},p_{o}}{\text{max}}$ $\displaystyle
Np_{r}\delta(\frac{p_{o}-p_{r}}{q_{o}-q_{s}(p_{r})}-\frac{p_{r}}{q_{s}(p_{r})})+Np_{o}(1-\frac{p_{o}-p_{r}}{q_{o}-q_{s}(p_{r})})$
(30) s.t. $\displaystyle(\ref{eq:R1regioncond}),\;q_{o}>q_{s}(p_{r})$
$\displaystyle
R^{R2a}_{\text{C2}}(p_{o},p_{r}):\quad\underset{p_{r},p_{o}}{\text{max}}$
$\displaystyle Np_{o}(1-\frac{p_{o}}{q_{o}})$ (31) s.t.
$\displaystyle(\ref{eq:R2regioncond}),\;q_{s}(p_{r})>q_{o},$ $\displaystyle
R^{R2b}_{\text{C2}}(p_{o},p_{r}):\quad\underset{p_{r},p_{o}}{\text{max}}$
$\displaystyle Np_{o}(1-\frac{p_{o}}{q_{o}})$ (32) s.t.
$\displaystyle(\ref{eq:R2regioncond}),\;q_{o}>q_{s}(p_{r}),$ $\displaystyle
R^{R3a}_{\text{C2}}(p_{o},p_{r}):\quad\underset{p_{r},p_{o}}{\text{max}}$
$\displaystyle Np_{r}\delta(1-\frac{p_{r}}{q_{s}(p_{r})})$ (33) s.t.
$\displaystyle(\ref{eq:R3regioncond}),\>q_{s}(p_{r})>q_{o},$ $\displaystyle
R^{R3b}_{\text{C2}}(p_{o},p_{r}):\quad\underset{p_{r},p_{o}}{\text{max}}$
$\displaystyle Np_{r}\delta(1-\frac{p_{r}}{q_{s}(p_{r})})$ (34) s.t.
$\displaystyle(\ref{eq:R3regioncond}),\;q_{o}>q_{s}(p_{r}),$

and $R^{R4}_{\text{C2}}(p_{o},p_{r})$ is a constant of $0$, corresponding to
R4 in Proposition 2 (where users choose neither the on-demand nor sharing
pool). Other than R4, the other regions are considered with two subproblems
which correspond to the disjoint cases $q_{s}(p_{r})\geq q_{o}$ and
$q_{o}>q_{s}(p_{r})$. This allows us to replace the terms involving $min()$
and $()^{+}$. For example, $R^{R1a}_{\text{C2}}(p_{o},p_{r})$ and
$R^{R1b}_{\text{C2}}(p_{o},p_{r})$ both correspond to the situation when we
are in R1 (where at equilibrium, buyers choose both to buy from the on-demand
pool and the sharing pool). They differ depending on whether $q_{s}(p_{r})\geq
q_{o}$ or not.

The subproblems are not jointly convex with respect to $p_{o}$ and $p_{r}$.
Hence, with $p_{r}$ and $q_{s}(p_{r})$ given, we optimize with respect to
$p_{o}$ to attain insights on the MNO’s optimal price in R1 and R2. The MNO’s
revenues in R3 and R4 are not related to $p_{o}$ (since no users buy from the
on-demand pool) and are constant with $p_{r}$ and $q_{s}(p_{r})$ given.

######  Theorem 2.

Given $(p_{r},q_{s}(p_{r}))$, the MNO’s optimal on-demand price $p_{o}^{*}$ at
each region is given by:

  * •

In R1 with $q_{s}(p_{r})\geq q_{o}$,

$p_{o,1a}^{*}=\frac{q_{o}p_{r}(1+\delta)}{2q_{s}(p_{r},\delta)}\text{ or
}\text{argmax}_{\\{B_{1},B_{2}\\}}{\color[rgb]{0,0,0}R^{R1a}_{\text{C2}}(p_{o},p_{r})}$

  * •

In R1 with $q_{s}(p_{r})<q_{o}$,

$\begin{split}&p_{o,1b}^{*}=\frac{p_{r}(\delta+1)+q_{o}-q_{s}(p_{r},\delta)}{2}\\\
&\text{ or
}\text{argmax}_{\\{B_{1},B_{2}\\}}{\color[rgb]{0,0,0}R^{R1b}_{\text{C2}}(p_{o},p_{r})}\end{split}$

  * •

In R2 with $q_{s}(p_{r})\geq q_{o}$, $p_{o,2a}^{*}=\frac{q_{o}}{2}\text{ or
}B_{1}$

  * •

In R2 with $q_{s}(p_{r})<q_{o}$, $p_{o,2b}^{*}=\frac{q_{o}}{2}\text{ or
}B_{1}$

where $B_{1}=p_{r}-q_{s}(p_{r})+q_{o}$ and
$B_{2}=\frac{p_{r}q_{o}}{q_{s}(p_{r})}$ are points on the region boundaries.

Proof: For subproblem $R^{R1a}_{\text{C2}}$, the second derivative of the
objective function with respect to $p_{o}$ is
$\frac{-2Nq_{s}(p_{r})}{(q_{s}(p_{r})-q_{o})q_{o}}<0$ since $q_{s}(p_{r})\geq
q_{o}$ and $q_{o}>0$, indicating that the objective function is concave. As
the constraints involve linear functions of $p_{o}$, the maximization problem
is convex with respect to $p_{o}$. Therefore, we convert the constraints to
non-strict inequalities and use the KKT conditions to solve
$R^{R1a}_{\text{C2}}$, with respect to $p_{o}$. The Lagrangian is defined as
$L(p_{o},\lambda_{1},\lambda_{2})=Np_{o}(\frac{p_{r}-p_{o}}{q_{s}(p_{r})-q_{o}}-\frac{p_{o}}{q_{o}})+Np_{r}\delta(1-\frac{p_{r}-p_{o}}{q_{s}(p_{r})-q_{o}})+\lambda_{1}(p_{r}-p_{o}-q_{s}(p_{r})+q_{o})+\lambda_{2}(q_{a}(p_{r})/q_{o}p_{o}-p_{r})$,
with $\lambda_{1}$ and $\lambda_{2}$ being the dual variables corresponding to
the constraints. If the dual variables $\lambda_{1}$ (or $\lambda_{2}$) were
greater than $0$, the optimal point would lie on the boundary between region
R1 and R2 (or R3 respectively) If the dual variables are $0$, the optimal
point will be the point at which the gradient of the Lagrangian is $0$. We
solve the rest of the subproblems similarly. ∎

Theorem 2 gives us insights on how the MNO should set the on-demand price
$p_{o}$, given that the other parameters $p_{r}$ and $q_{s}(p_{r})$ are fixed.
In R1, when users buy from both the sharing and on-demand pool, the price
$p_{o}$ at which the gradient of the revenue is $0$ is
$\frac{q_{o}p_{r}(1+\delta)}{2q_{s}(p_{r})}$ for R1a and
$\frac{p_{r}(\delta+1)+q_{o}-q_{s}(p_{r})}{2}$ for R1b. We seperately analyse
the cases when $\frac{q_{o}p_{r}(1+\delta)}{2q_{s}(p_{r})}$ (the point at
which the gradient of the revenue is $0$) lies within the region’s boundaries,
and when it does not. Case 1: If
$p_{o}^{*}=\frac{q_{o}p_{r}(1+\delta)}{2q_{s}(p_{r})}$ lies within R1a’s
boundaries
($p_{r}<p_{o}+q_{s}(p_{r})-q_{o},p_{r}>\frac{q_{s}(p_{r})}{q_{o}}p_{o}$), it
is the optimal price, as the problem is convex with respect to $p_{o}$.
Likewise for $\frac{p_{r}(\delta+1)+q_{o}-q_{s}(p_{r})}{2}$ in R1b. These
functions indicate that the optimal price $p_{o}^{*}$ increases with respect
to the on-demand expected quality (supply) $q_{o}$ and platform commission
$\delta$. As the on-demand supply $q_{o}$ increases, the increase in on-demand
price $p_{o}$ would not deter users away from the on-demand pool. Likewise,
when the MNO’s commission $\delta$ increases, re-sellers will be deterred from
selling their unused resource quota, resulting in a smaller supply at the
sharing-pool. With a potentially lower payoff at the sharing pool, the
increase in the on-demand price $p_{o}$ would not deter users away from the
on-demand pool. At the same time, as the sharing supply $q_{s}(p_{r})$
increases, the sharing pool is seen as more attractive to users (due to the
potentially higher individual payoff). In light of this, the optimal on-demand
price $p_{o}^{*}$ has to decrease. Case 2: If
$\frac{q_{o}p_{r}(1+\delta)}{2q_{s}(p_{r})}$ (and
$\frac{p_{r}(\delta+1)+q_{o}-q_{s}(p_{r})}{2}$) do not lie within R1a’s
(respectively R1b’s) boundaries, the optimal points $p_{o,1a}^{*}$ (and
$p_{o,1b}^{*}$) will lie on the region’s boundaries, since the problems are
convex. Specifically, the optimal prices will be
$\text{argmax}_{\\{B_{1},B_{2}\\}}R^{R1a}_{\text{C2}}(p_{o},p_{r})$ (and
$\text{argmax}_{\\{B_{1},B_{2}\\}}R^{R1b}_{\text{C2}}(p_{o},p_{r})$).

In R2, where buyers only choose the on-demand pool, the price $p_{o}$ at which
the gradient of the revenue is $0$ is $\frac{q_{o}}{2}$. We seperately analyse
the cases when $\frac{q_{o}}{2}$ (the point at which the gradient of the
revenue is $0$) lies within the region’s boundaries, and when it does not.
Case 1: If $p_{o}^{*}=\frac{q_{o}}{2}$ lies within R2’s boundary ($p_{r}\geq
p_{o}+(q_{s}(p_{r})-q_{o})$ for $R^{R2a}_{\text{C2}}(p_{o},p_{r})$ and
$p_{r}\geq\frac{q_{s}}{q_{o}}p_{o}$ for $R^{R2b}_{\text{C2}}(p_{o},p_{r})$
respectively), it is the optimal on-demand price because the problem is convex
with respect to $p_{o}$. Therefore, the optimal on-demand price is an
increasing function of the on-demand pool’s supply $q_{o}$. This indicates
that as the supply of the on-demand pool $q_{o}$ increases, the rise in the
on-demand price will not deter users away from the on-demand pool. Case 2: If
$p_{o}^{*}=\frac{q_{o}}{2}$ does not lie within region R2’s boundaries, the
optimal point will lie on the boundary $p_{r}=B_{1}$ for
$R^{R2a}_{\text{C2}}(p_{o},p_{r})$ and $p_{r}=B_{2}$ for
$R^{R2b}_{\text{C2}}(p_{o},p_{r})$, respectively. This is because, the
optimization problems are convex with respect to $p_{o}$.

On this basis, we then focus on the joint optimization of $p_{o}$ and $p_{r}$
for the MNO’s optimal revenue, which will be the maximum of the revenues
across the four regions. According to Corollary 1, the MNO’s optimal revenue
is calculated as

$\max_{p_{r}}\quad R_{\max}(p_{r}).$ (35)

######  Corollary 1.

Given $(p_{r},q_{s}(p_{r}))$, the MNO’s optimal revenue will be

$\begin{split}&{\color[rgb]{0,0,0}R_{\max}(p_{r})}=\max\Bigg{\\{}R^{R1a}_{\text{C2}}(p_{o,1a}^{*},p_{r}),R^{R1b}_{\text{C2}}(p_{o,1b}^{*},p_{r}),\\\
&\qquad\qquad\qquad
R^{R2a}_{\text{C2}}(p_{o,2a}^{*},p_{r}),R^{R2b}_{\text{C2}}(p_{o,2b}^{*},p_{r}),R^{R3}_{\text{C2}}(p_{r}),0\Bigg{\\}},\end{split}$
(36)

where $R^{R3}_{\text{C2}}(p_{r})=Np_{r}\delta(1-\frac{p_{r}}{q_{s}(p_{r})})$
is the revenue in R3 (the region where the buyers only choose the sharing
pool), if the boundary conditions of R3 (Eq. (17)) were met. And $0$ is the
revenue in R4 (the region where the buyers choose neither the sharing pool nor
the on-demand pool), if the boundary conditions of R4 (Eq. (18)) were met.

Since the optimal on-demand price $p_{o}^{*}$ in $R_{\max}(p_{r})$ is obtained
in Theorem 2, the optimization in (35) can be addressed by using a linear
search across $p_{r}$. In Section 6, we discuss the joint optimization of
$p_{o}$ and $p_{r}$ and the resulting insights at the optimal and equilibrium
point.

##  6 Numerical Simulations

(a) The MNO’s optimal revenue and region against $q_{o}$. $\delta=0.2,a=2$,
with $u_{i},g_{i}$ following the uniform $U(0,1)$ distribution.

(b) The MNO’s optimal revenue and region against $q_{o}$. $\delta=0.2,a=2$,
with $u_{i},g_{i}$ following beta(2,2) distribution.

(c) The MNO’s optimal revenue and region against $q_{o}$. $\delta=0.2,a=2$,
with $u_{i},g_{i},$ and usage levels following the uniform $U(0,1)$
distribution.

Figure 4: When on-demand supply is low, the optimal revenue is at Region 1
(offering both a sharing and an on-demand pool). As the on-demand supply
increases, the optimal revenue transitions to Region 2 (purely on-demand).

(a) The MNO’s optimal revenue against on-demand supply $q_{o}$, $a=2$.

(b) The MNO’s optimal revenue against commission $\delta$, $a=2$.

(c) The MNO’s optimal revenue and region against $\delta$. $q_{o}=0.2,a=2$,
with $u_{i},g_{i}$ following the uniform $U(0,1)$ distribution.

(d) The MNO’s optimal revenue and region against $\delta$. $q_{o}=0.2,a=2$,
with $u_{i},g_{i}$ following the beta(2,2) distribution.

(e) The MNO’s optimal revenue and region against $\delta$. $q_{o}=0.2,a=2$,
with $u_{i},g_{i}$, and usage levels following the uniform $(0,1)$
distribution.

Figure 5: When the commission level $\delta$ is lower, the optimal revenue is
at Region 1 (offering both a sharing and an on-demand pool). As the commission
level increases, the optimal revenue transitions to Region 2 (purely on-
demand).

(a) The MNO’s revenue and region against on-demand price $p_{o}$.
$q_{o},\delta=0.2,a=2,p_{r}=0.5$, with $u_{i},g_{i}$ following the beta(2,2)
distribution.

(b) The MNO’s optimal revenue and region against on-demand supply $q_{o}$,
$\delta=0.2,a=1.5$.

(c) The individual buyer’s payoff against willingness to pay $u_{i}$,
$\delta=0.2,a=2$.

Figure 6: Further Numerical Results and Insights: The impact of the on-demand
price and normalization parameter on revenue, and how the individual buyer’s
payoff varies with the willingness to pay.

In this section we provide numerical results on the optimal and equilibrium
outcome of the Stackelberg Game (Eq. (7), (8) and (9)). These results give us
insights and show the practical implications of integrating the sharing pool
with the On-Demand Pricing and Resource Reservation model. In many of our
experiments, for each combination of $($on-demand supply $q_{o},$ commission
$\delta,$ and normalization parameter $a)$, we optimize over the prices
$(p_{o},p_{r})$, obtaining the optimal revenue, and hence optimal region. In
Section 7, we perform preliminary user studies on a prototype, to evaluate the
extent of which user decisions differ from our decision model.

Firstly, we define a specific form of $q_{s}(p_{r})$, which indicates the
relationship between the price $p_{r}$ and the resulting quality/supply
$q_{s}(p_{r})$, as follows:

$q_{s}(p_{r})=a\log(1+(1-\delta)p_{r}).$ (37)

We use a log function, which models the law of diminishing marginal returns
[42]. The relative supply/quality (i.e. likelihood of buyers getting their
jobs served) of the sharing pool increases as the proportion of re-sellers
willing to sell $(1-\delta)p_{r}$ increases. The parameter $a$ is the
normalization parameter, normalizing the function of the proportion of re-
sellers $log(1+(1-\delta)p_{r})$ willing to re-sell to a magnitude such that
the supply level $q_{s}(p_{r})$ is comparable with the supply at the on-demand
pool $q_{o}$. For our numerical simulations, we use Eq. (37) as the specific
form of $q_{s}(p_{r})$ and set the number of buyers and the number of re-
sellers ($N$) to be 50. We perform experiments with the willingness to pay
$u_{i}$ and the willingness to share $g_{i}$ following the Uniform[0,1] as
well as the Beta(2,2) distribution. Under the Uniform[0,1] distribution, the
users’ types $u_{i}$ and $g_{i}$ are evenly spread out over the range $[0,1]$.
Under the Beta(2,2) distribution, the users’ types follow an inverse parabolic
structure, with more users whose types take values towards the center of the
range $[0,1]$ and fewer users taking values towards the edge of the range
$[0,1]$, somewhat similar to a Gaussian distribution. We also model
heterogeneous user usage levels, both on the buyer side and the re-seller
side, with usage levels following the uniform[0,1] distribution.

Impact of the on-demand supply $q_{o}$ on the optimal revenue and region :
Firstly, we show how the MNO’s optimal revenue and region vary with respect to
the on-demand quality/supply $q_{o}$ in Fig. 4a, 4b and 4c where we set
$\delta=0.2$ and $a=2$. Given that we do not know what is an appropriate
normalization parameter which relates the supply function $q_{s}(p_{r})$ to
the supply of the on-demand pool $q_{o}$, we set $a=2$ first, and later
decrease the normalization parameter to $a=1.5$. To obtain the optimal revenue
and region, we optimize over the prices $(p_{o},p_{r})$, given each
combination of $($on-demand supply $q_{o},$ commission $\delta,$ and
normalization parameter $a)$. For both cases where the re-sellers’ willingness
to share and the buyers’ willingness to pay follow uniform (Fig. 4a) and beta
distribution (Fig. 4b), and when the users’ usage levels are heterogeneous and
follow the Uniform[0,1] distribution (Fig. 4c), it can be seen that for
smaller values of $q_{o}$, i.e., when the on-demand supply is lower, it is
more beneficial to supplement buyers with the sharing pool to buy quota for
job computation (Region R1). This additional income adds to the revenue of the
MNO. Nevertheless, as $q_{o}$ increases, the additional income from the
sharing pool does not outweigh the increase in revenue from the on-demand pool
$N_{\text{on-demand }}p_{o}(1-\frac{p_{o}}{q_{o}})$, hence Region R2 is
optimal.

Impact of the commission $\delta$ on the optimal revenue and region: In Fig.
5a, we plot the optimal revenue for the MNO against $q_{o}$, for different
values of commission $\delta$. Conversely, in Fig. 5b, we plot the optimal
revenue of the MNO against the commission $\delta$, for different values of
$q_{o}$. In Figs. 5c, d and e, we show the optimal region at equilibrium for
different values of $\delta$, (with $g_{i}$ and $u_{i}$ taking values from the
Uniform[0,1] and Beta(2,2) distributions, and when the users’ usage levels are
heterogeneous following Uniform[0,1] respectively). Once again, to obtain the
optimal revenue and region, we optimize over the prices $(p_{o},p_{r})$, given
each combination of $($on-demand supply $q_{o},$ commission $\delta,$ and
normalization parameter $a)$. These graphs show that when the commission
$\delta$ is smallest, it is not profitable to have sharing (hence the optimal
region is R2), because the platform does not get much revenue from sharing due
to its commission being small. As $\delta$ gets larger, it becomes profitable
to have the sharing platform (Region R1) due to the increase in commission for
the platform and hence revenue $N_{\text{on-
demand}}p_{o}(\frac{p_{r}-p_{o}}{q_{s}(p_{r})-q_{o}}-\frac{p_{o}}{q_{o}})+N_{\text{share}}p_{r}\delta(1-\frac{p_{r}-p_{o}}{q_{s}(p_{r})-q_{o}})$.
On the other hand, for larger $\delta$, the re-sellers will be less likely to
choose to re-sell their unused resource quota. This is seen in Eq. (1), as
their payoffs are inversely proportional with respect to $\delta$, and in
$q_{s}(p_{r})=a\log(1+(1-\delta)p_{r})$ (Eq. (37)) where the sharing pool’s
supply $q_{s}$ is inversely proportional to $\delta$. This results in a lower
quality/supply sharing pool, and therefore a smaller revenue which the MNO can
obtain from sharing, making Region R2 (purely on-demand) optimal at high
$\delta$. To summarize, it would have been expected that a high commission
would give the platform a higher revenue, nevertheless our results show that
it incurred a trade-off in dissuading reservation plan users from re-selling,
hence decreasing the supply and revenue.

Impact of the on-demand price $p_{o}$ on the revenue and region : In Fig. 6a,
we show how the revenue varies as the on-demand price $p_{o}$ increases, for
given values of $q_{o},\delta=0.2,a=2,p_{r}=0.5$. As the price increases, the
equilibrium region shifts from $R2$ (purely on-demand) to $R1$ (both on-demand
and sharing) as more buyers choose the sharing platform due to the on-demand
price increasing. As the on-demand price increases even further, the region
transitions to $R3$ (purely sharing) due to the on-demand price being too
high. Nevertheless, $R3$ is not optimal for the platform because it only
collects a proportion $\delta$ of the amount transacted.

Impact of the normalization parameter $a$ on the optimal revenue and region :
Given that it is not clear what is an appropriate normalization parameter
which relates the supply function $q_{s}(p_{r})$ to the supply of the on-
demand pool $q_{o}$, we explore different normalization parameters. In
contrast to Fig. 5a, we set a smaller normalization parameter $a=1.5$ for Fig.
6b. Once again, to obtain the optimal revenue and region, we optimize over the
prices $(p_{o},p_{r})$, given each combination of $($on-demand supply $q_{o},$
commission $\delta,$ and normalization parameter $a)$. With a smaller $a$, a
higher price $p_{r}$ comes with a lower quality/supply $q_{s}(p_{r})$ relative
to $q_{o}$ and therefore a lower payoff for buyers at the sharing pool. Hence,
buyers have lower incentive to buy resources from the sharing pool. That is
why region R2 (purely on-demand) gives the optimal revenue for all values of
$q_{o}$ in Fig. 6b. Nevertheless, although R1’s optimal revenue is not the
optimal over all regions under this set of parameters, users will still buy
from the sharing pool and it’s revenue is non-zero, as seen in Fig. 6b.

Analyzing individual buyer payoffs : Finally, we plot individual buyer payoffs
(Eq. (4)) at equilibrium conditions in Fig. 6c. As we can see, buyers with
lower willingness to pay $u_{i}$ choose neither pool. As their willingness to
pay (utility of the job computation) increases, their payoff increases. We can
see that for the $q_{o}=0.1$ and $q_{o}=0.2$ plots, users with high
willingness to pay choose the sharing pool, as visualised through the
different gradients.

In summary, our simulation results show that under the combination of low
$\delta$, low $q_{o}$, and large $a$, having the sharing platform (Region R1)
is optimal. Besides, it is not optimal to have Region R3 (purely sharing), due
to the MNO only being able to take a proportion $\delta$ of the revenue
$N_{\text{share }}p_{r}\delta(1-\frac{p_{r}}{q_{s}(p_{r})})$.

##  7 Prototype and Preliminary User Studies

###  7.1 Prototype Description

We created a prototype, an interactive website (see Fig. 7 for screenshots),
which is able to capture user decisions given the different parameters. The
interactive website’s url is https://shikhsh.github.io/erpMEC. In Fig. 8 we
illustrate how the real-world version of our prototype integrates with the
rest of the MEC system.

Figure 7: Screnshots of our prototype - an interactive website, which can
capture user decision making given different parameter settings, for user
studies. It’s url is https://shikhsh.github.io/erpMEC/. Figure 8: How the
real-world version of our prototype is integrated with the rest of the MEC
system.

In our prototype, when users enter the website, they will be given a choice to
select whether they are a buyer or re-seller. In the re-sellers portal, the
user is shown 1) their specific edge computing application, randomly sampled
from the list [’IoT Sensing Analytics’, ’Distributed AI’, ’Predictive
Maintence’, ’Remote Monitoring’, ’Industrial IoT’, ’Augmented Reality
Display’], 2) their remaining usage quota this time-slot, after factoring in
their planned usage level, and 3) their inconvenience cost of re-selling, in
light of potential further usage this time-slot. Note that the inconvenience
cost also includes the cost of opening the website or app. 4) The re-selling
price and platform commission, 5) the payoff equations under the different
options, and finally 6) the option buttons “re-sell” and “no”, for the user to
choose from. Parameters 1, 2, and 3 are user specific parameters, and are
uniformly randomly generated upon each user’s entrance for our prototype. When
the number of users who use our prototype in user studies is large, we can
compare the empirical distribution of user decisions to that of our model. The
parameters in 4 are fixed for all users, and can be varied across user studies
to view how users make decisions, and the aggregate outcome, under different
settings. In practice, the information shown such as the type of application
and the inconvenience cost (given potential further usage that time-slot) can
be pulled from the user’s usage history. The information on the user’s
remaining resource quota comes from the website’s back-end calculations.

In the buyers‘ portal, the user is shown 1) their specific edge computing
application, randomly sampled from the list [’IoT Sensing Analytics’, ’Mobile
Gaming’, ’Augmented Reality’, ’Data Analytics on Device’, ’Smart Home IoT
Analytics’] 2) their usage demand this time-slot, 3) their utility received,
and hence their willingness to pay, 4) the supply and prices under the sharing
pool and re-selling pool, 5) the payoff equations under the different options,
and 6) the option buttons “On-Demand Pool”, “Sharing Pool”, “No purchase”, for
the user to choose from. As per the re-seller’s portal, parameters 1, 2, and 3
are user specific parameters, and are uniformly randomly generated upon each
user’s entrance for our prototype. When the number of users who use our
prototype in user studies is large, this models the equilibrium and aggregate
outcome of heterogeneous users decision making. The parameters in 4 are fixed,
and can be varied across user studies to view how users make decisions, and
the aggregate outcome, under different settings. In practice, the information
shown such as the type of application can be pulled from the user’s usage
history. The information on the user’s remaining resource quota comes from the
website’s back-end calculations. In practice, the users’ willingness to pay
will not be shown on the website, but is inherent in users’ decision making.

Figure 9: Results of preliminary user study: The scatter points indicate the user choices given the utility and usage values they were shown on the website. The dotted lines indicate the region boundaries from the analytical model in the paper. Region | Sharing | On Demand | Neither
---|---|---|---
Analytical Model (Prop 1-2) | $\frac{pr-po}{qs-qo}<u_{i}<1$ | $\frac{po}{qo}<u_{i}<\frac{pr-po}{qs-qo}$ | $u_{i}<\frac{po}{qo}$
Percentage agreement of data and model | 77.3$\%$ | 66.7 $\%$ | 33.3 $\%$
Test Statistic (of KS test) | 0.2917 | 0.2941 | 0.2667
P-value (of KS test) | 0.2406 | 0.6487 | 0.9629
Are the data and | No significant evidence | No significant evidence | No significant evidence
model distributions distinct? |  |  |
TABLE II: Results for Buyers: Comparing data from user studies with the expected values from the model for each choice: Sharing, On-demand and Neither, through calculating the percentage agreement, and through using the Kolmogorov-Smirnov (KS) test. Our results show that there is no significant evidence of the data following a distinct distribution from our paper’s model (at significance level 0.05). Region | Re-selling | Not re-selling
---|---|---
Analytical Model (Eq 1) | $g_{i}<(1-\delta)p_{r}$ | $g_{i}>(1-\delta)p_{r}$
Percentage agreement with model (within region) | 90.9$\%$ | 52.9$\%$
Test Statistic (of KS test) | 0.4444 | 0.1824
P-value (of KS test) | 0.0975 | 0.9532
Are the data and | No significant evidence | No significant evidence
model distributions distinct? |  |
TABLE III: Results for Re-sellers: Comparing data from user studies with the
expected values from the model for each choice: Re-selling, and not re-
selling, through calculating the percentage agreement, and through using the
Kolmogorov-Smirnov (KS) test. Our results show that there is no significant
evidence of the data following a distinct distribution from our paper’s model
(at significance level 0.05).

Our prototype stores a string of information upon each user click, on our
back-end server (Fig 8). For re-sellers, the string contains [remaining usage
level for the time-slot, inconvenience cost/ uncertainty of usage $g_{i}$, re-
selling price $p_{r}$, commission level $\delta$, type of application, Y or N
indicating the user’s choice of whether or not to re-sell, the time stamp].
For buyers, the string contains [the user’s usage requirement, the utility/
willingness to pay $u_{i}$, the on-demand supply $q_{o}$, the sharing supply
$q_{s}$, the on-demand price $p_{o}$, the sharing price $p_{r}$, the user’s
choice (Sharing, On-demand, No purchase), the time stamp]. Our prototype
stores these information, without storing the identity or any other
information regarding the user. All clicks are anonymous. In practice, the
back-end server will relay the decisions of the users to the edge. Based on
this, the resource units at edge servers will be correspondingly allocated to
the different users, by the computer at the edge server (Fig 8).

###  7.2 Preliminary User Study

The goal of the user study is to verify how accurately our decision model (Eq.
1 for re-sellers, and Prop. 1-2, Fig 3a for buyers) models the way users make
decisions. We will evaluate how similar the empirical distribution function of
the data from the user study, and the empirical distribution function of the
data from the model’s predictions, are.

For this preliminary user study, we set the sharing supply available to be
slightly higher than the on-demand supply ($q_{o}=0.6,q_{s}=0.7$), and the
sharing price to be slightly higher than the on-demand price
($p_{o}=0.15,p_{r}=0.2$). For our participants, we have had 44 participants be
buyers, and 28 participants be re-sellers. Our results are in Fig. 9, Tables
III and III. Fig 9 is a scatter plot of buyers clicks, given the usage and
utility values they were shown on the website. Under our paper’s math model
(Proposition 2/ Fig 3a of paper), buyers will choose the sharing pool when
their utility value $\frac{pr-po}{qs-qo}<u_{i}<1$, will chose the on-demand
pool when $\frac{po}{qo}<u_{i}<\frac{pr-po}{qs-qo}$, and will choose neither
when $u_{i}<\frac{po}{qo}$. We can see in the scatter plot that the on-demand
and sharing region largely agrees with the paper’s model, which is
corroborated by the percentage agreement with model in Table II. The
percentage agreement is calculated as follows: For each range of utilities
$u_{i}$ corresponding to the regions [sharing, on-demand, neither] (i.e.
[$\frac{pr-po}{qs-qo}<u_{i}<1$, $\frac{po}{qo}<u_{i}<\frac{pr-po}{qs-qo}$,
$u_{i}<\frac{po}{qo}$ ]), we will calculate the percentage of data points from
our user study which agree with our model. For e.g., in the ”sharing” region,
there are altogether 22 data points, and 17 agree with our model (Prop 1-2,
and Fig. 3a), giving a percentage of $77.3\%$. We note that there is
randomness inherent in user clicks (decisions), as a math model can not fully
capture user behavior. Furthermore decisions in a prototype will have
differences from user behaviors in the actual system. Next we perform the
Kolmogorov-Smirnov (KS) test for both the buyers and re-sellers data. We
obtain the model’s data samples through using the given utility values, and
obtaining the corresponding decisions according to the model (Prop 1-2, Fig 3a
for buyers, and Eq 1 for re-sellers). With this and our data samples, we
perform a KS test to evaluate whether, for each region (Sharing, On Demand,
Neither, for buyers, Re-selling, Not Re-selling, for re-sellers) the two set
of data points (data from user studies and model) can considered to be from
the same distribution. We perform the KS test using the scipy library in
python. The KS test calculates the mean and standard deviation for each set of
data points, and obtains the KS test test statistic,

$D=\sup_{x}|F_{data}(x)-F_{model}(x)|$ (38)

which quantifies the distance between the empirical distribution functions of
the two data samples (model and user studies data). We choose a confidence
level of $95\%$. As the p-values are above the significance level of $0.05$,
there is no significant evidence that the data and model distributions are
different. Beyond our preliminary study in this work, further studies under
different parameter settings, can be made with the help of our interactive
prototype.

##  8 Future Work

For future work, we will study the long-term scenario where users may switch
between reserving resources in advance, and buying on the go through the on-
demand and sharing pools. Here, another layer would be added to the
Stackelberg game and users will make decisions based on their expected payoffs
in the long run. The expected payoff would depend on the users’ application
requirements: the probability of them needing job computation at the edge
consistently over all timeslots vs one-off and sporadically, the prices set by
the MNO and the expected supplies of the different options, which is
influenced by the aggregate decisions of other users.

Another line of investigation is to study how the sharing framework can be
implemented across multiple edge nodes, with different users being associated
with different nodes due to proximity. In light of this, the on-the-go (on-
demand and sharing) queues and the sharing supply would differ across nodes.
If users are allowed to associate with other nearby nodes, it is thus of
interest to investigate how the MNO can jointly price and allocate the
resources across nodes to the different users, to maximize revenue and
minimize the number of unmet jobs. In this new study on incentives, a
Stackelberg game formulation can be used to analyze and characterise the
interaction between the different parties, along with a matching algorithm
which matches users to resources across edge nodes, with the transmission cost
and hence distance between user and node being a factor. In the new
Stackelberg formulation, the user decision making functions will be modified
to reflect the potential supply levels at different nodes, and the likelihood
of the user attaining resources from different nodes, which is a function of
the distance.

##  9 Conclusion

In this paper we proposed a novel resource sharing platform to enhance the
usage of the on-demand pricing and resource reservation models in edge
computing. This helps to optimize the scenario where the on-demand users face
a limited supply remaining at the edge server, whilst users under the resource
reservation model have reserved excess computing resources, by allowing the
re-selling of unused resource quota. We formulated a 3-stage Stackelberg Game
to investigate the user’s optimal strategies and equilibrium behavior, as well
as whether the MNO has an incentive to provide such a resource sharing
platform. By analyzing the formulated game, we characterised the optimal
strategies of the re-sellers, buyers and MNO, and showed that there would be 4
regions (both sharing and on-demand, only on-demand, only sharing, none at
all) at equilibrium after buyers’ decisions. Based on these 4 regions, we
optimized the MNO’s revenue and analytically characterised the MNO’s optimal
on-demand prices. Our numerical results show that having a sharing pool gives
the MNO an optimal revenue when the on-demand pool’s supply is low and when
the platform’s commission is low.

## Acknowledgment

This paper is supported in part by the National Research Foundation, Singapore
and Infocomm Media Development Authority under its Future Communications
Research $\&$ Development Programme, in part by the SUTD Presidential
Postdoctoral Fellowship, under the Singapore MOE START Scheme, in part by NSF
CNS-1751075, in part by SongShan Laboratory Foundation, under Grant No.
YYJC022022007, in part by the open research fund of National Mobile
Communications Research Laboratory, Southeast University, under Grant No.
2023D04, in part by the National Natural Science Foundation of China, under
Grant No. 62301222.

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|  Marie Siew (S’19, M’21) is currently a post-doctoral researcher at the
Electrical and Computer Engineering department, Carnegie Mellon University.
She received the B.Sc. degree in Mathematical Sciences from Nanyang
Technological University, Singapore in 2016, and the Ph.D. degree from the
Singapore University of Technology and Design in 2021. Her research interests
are in the areas of network economics, edge computing, edge intelligence, and
federated learning. Dr Siew received the Best Poster Award at ACM/IEEE IPSN
2023, the SUTD Presidential Postdoctoral Fellowship, under the Singapore
Teaching and Academic Research Talent Scheme in 2021, the A*Star Computing and
Information Science Scholarship in 2019, the A*Star Undergraduate Scholarship
in 2012 from the Agency for Science, Technology and Research, and was in the
CN Yang Scholars’ Programme in NTU from 2012 to 2016.
---|---
|  Shikhar Sharma is a Masters Student in Electrical and Computer Engineering,
Carnegie Mellon University. He received his Bachelors degree in Birla
Institute of Technology and Science, Pilani in 2019. and worked as a software
development engineer at Paypal from 2019-2021.
---|---
|  Kun Guo (Member, IEEE) received the B.E. degree in Telecommunications
Engineering from Xidian University, Xi’an, China, in 2012, where she received
the Ph.D. degree in communication and information systems in 2019. From 2019
to 2021, she was a Post-Doctoral Research Fellow with the Singapore University
of Technology and Design (SUTD), Singapore. Currently, she is a Zijiang Young
Scholar with the School of Communications and Electronics Engineering at East
China Normal University, Shanghai, China. Her research interests include
wireless edge computing, caching, and intelligence.
---|---
|  Desmond Cai (S’10-M’16) received the B.Sc. degree in electrical and
computer engineering from Cornell University, Ithaca, NY, USA, in 2009 and the
Ph.D. degree in electrical engineering from the California Institute of
Technology, Pasadena, CA, USA, in 2016. He was a Research Scientist at IBM
from 2019 to 2020, and a Scientist at the Institute of High Performance
Computing from 2016 to 2019. His research interests include artificial
intelligence, optimization, network economics, and game theory. Dr. Cai
received the National Science Scholarship from the Agency for Science,
Technology, and Research in 2004, the John G. Pertsch Prize from Cornell
University in 2008, and the Sibley Prize from Cornell University in 2009.
---|---
|  Wanli Wen (S’15–M’20) received the B.S. degree from Anhui University of
Finance and Economics, Bengbu, China, in 2011, the M.S. degree in
Communication and Information Systems from Hangzhou Dianzi University,
Hangzhou, China, in 2014, and the Ph.D. degree in information and
communications engineering from Southeast University, Nanjing, in 2019. From
2019 to 2020, he was a postdoctoral research fellow at Singapore University of
Technology and Design. Since 2021, he has been an Assistant Professor in the
School of Microelectronics and Communication Engineering, Chongqing
University, China. His research interests include green communications, mobile
edge computing and caching, and federated learning.
---|---
|  Carlee Joe-Wong is the Robert E. Doherty Associate Professor of Electrical
and Computer Engineering at Carnegie Mellon University. She received her A.B.
degree (_magna cum laude_) in Mathematics, and M.A. and Ph.D. degrees in
Applied and Computational Mathematics, from Princeton University in 2011,
2013, and 2016, respectively. Her research interests lie in optimizing various
types of networked systems, including applications of machine learning and
pricing to cloud computing, mobile/wireless networks, and transportation
networks. From 2013 to 2014, she was the Director of Advanced Research at
DataMi, a startup she co-founded from her research on mobile data pricing. She
received the NSF CAREER award in 2018 and the ARO Young Investigator award in
2019.
---|---
|  Tony Q.S. Quek (S’98-M’08-SM’12-F’18) received the B.E. and M.E. degrees in
electrical and electronics engineering from the Tokyo Institute of Technology
in 1998 and 2000, respectively, and the Ph.D. degree in electrical engineering
and computer science from the Massachusetts Institute of Technology in 2008.
Currently, he is the Cheng Tsang Man Chair Professor with Singapore University
of Technology and Design (SUTD) and ST Engineering Distinguished Professor. He
also serves as the Director of the Future Communications R&D Programme, the
Head of ISTD Pillar, and the Deputy Director of the SUTD-ZJU IDEA. His current
research topics include wireless communications and networking, network
intelligence, non-terrestrial networks, open radio access network, and 6G. Dr.
Quek has been actively involved in organizing and chairing sessions, and has
served as a member of the Technical Program Committee as well as symposium
chairs in a number of international conferences. He is currently serving as an
Area Editor for the IEEE Transactions on Wireless Communications. Dr. Quek was
honored with the 2008 Philip Yeo Prize for Outstanding Achievement in
Research, the 2012 IEEE William R. Bennett Prize, the 2015 SUTD Outstanding
Education Awards – Excellence in Research, the 2016 IEEE Signal Processing
Society Young Author Best Paper Award, the 2017 CTTC Early Achievement Award,
the 2017 IEEE ComSoc AP Outstanding Paper Award, the 2020 IEEE Communications
Society Young Author Best Paper Award, the 2020 IEEE Stephen O. Rice Prize,
the 2020 Nokia Visiting Professor, and the 2022 IEEE Signal Processing Society
Best Paper Award. He is a Fellow of IEEE and a Fellow of the Academy of
Engineering Singapore.
---|---