# Social Shaping for Transactive Energy Systems

Zeinab Salehi, Yijun Chen, Ian R. Petersen, Elizabeth L. Ratnam, Guodong Shi

###### Abstract

This paper considers the problem of shaping agent utility functions in a
transactive energy system to ensure the optimal energy price at a competitive
equilibrium is always socially acceptable, that is, below a prescribed
threshold. Agents in a distributed energy system aim to maximize their
individual payoffs, as a combination of the utility of energy consumption and
the income/expenditure from energy exchange. The utility function of each
agent is parameterized by individual preference vectors, with the overall
system operating at competitive equilibriums. We show the social shaping
problem of the proposed transactive energy system is conceptually captured by
a set decision problem. The set of agent preferences that guarantees a
socially acceptable price is characterized by an implicit algebraic equation
for strictly concave and continuously differentiable utility functions. We
also present two analytical solutions where tight ranges for the coefficients
of linear-quadratic utilities and piece-wise linear utilities are established
under which optimal pricing is proven to be always socially acceptable.

##  I Introduction

Recent dramatic increases in rooftop solar, backed by energy storage
technologies including electrical vehicles and home batteries, are obfuscating
the traditional boundary between energy producer and consumer [3]. At the same
time, transactive energy systems are being designed to coordinate supply and
demand for various distributed energy sources in an electrical power network
[5], where power lines (or cables) enable electricity exchanges, and
communication channels enable cyber information exchanges. With transactive
energy systems supporting the rise of the prosumer (i.e., an electricity
producer and consumer), new ways are emerging to enable a fair [4] and
sustainable renewable energy transition [6].

In the recent literature, transactive energy systems have been designed for a
wide-range of power system applications. For example, transactive energy
systems are supporting microgrid operations [7], virtual power plant
operations [8], and the operation of bulk power systems with a proliferation
of renewable and distributed energy resources [9]. Such transactive energy
systems are primarily focused on market-based approaches for balancing
electricity supply and demand, supporting robust frequency regulation
throughout the bulk grid [10]. Typically, agents operating in transactive
energy subsystem (e.g., virtual power station) are located across a power
network and market mechanisms are in place to support individual preferences,
enabling agents to compete and collaborate with each other [11]. The aim is a
dramatic improvement in power systems operational efficiency, scalability, and
resilience. The market mechanisms used in transactive energy networks to
enable efficient and valuable energy transactions are typically drawn from
classical theories in economics and game theory [12].

In standard welfare economics theory [13] it is suggested that resources
pricing can be designed to balance supply and demand in a market. In a multi-
agent system with distributed resource allocations, agents have the autonomy
to decide their local resource consumption and exchange to optimize their
individual payoffs as a combination of local utility and income or
expenditure. In a competitive equilibrium, resource pricing is achieved when
all agents maximize their individual payoffs subject to a network-level
supply-demand balance constraint, which in turn maximizes the overall system-
level payoff [14]. More specifically, to balance network supply and demand,
resource pricing corresponds to the optimal dual variables associated with
supply-demand balance constraint, where the price also maximizes the system-
level payoff. The prospect of operating a transactive energy system as a
market via optimal pricing under a competitive equilibrium has been widely
studied in [15, 16, 17, 18, 19, 21, 20].

Early and recent studies in both the economics and engineering literature, are
primarily focused on controlling resources price volatility [23, 24, 25, 26],
rather than the resources price itself. Consequently, in practice, the
optimally computed price to balance supply and demand in a market is
potentially not socially acceptable. For example, in February 2021, the
wholesale electricity price in Texas was considered by prosumers to be
unacceptably high after widespread power outages. Prosumers that subscribed to
the wholesale price reported receiving sky-high electricity bills, which
greatly exceeded societal expectations for payment [22].

Several authors have referred to price volatility as a rapid change in the
pricing process. The Black Scholes formula [23] and Heston’s extension [24]
are the most representative models of stochastic price volatility, while
others include SABR [25] and GARCH [26]. The authors in [27] argued that
previous models did not penalize price volatility in the system-level
objective, so they proposed to modify the system-level objective to account
for price volatility, whereby constructing a dynamic game-theoretic framework
for power markets. The authors in [28] proposed a different dynamic game-
theoretic model for electricity markets, by incorporating a pricing mechanism
with the potential to reduce peak load events and the cost of providing
ancillary services. In [29], the dual version of the system-level welfare
optimization problem was considered, where an explicit penalty term on the
$L_{2}$ norm of price volatility was introduced in the system-level objective,
allowing for trade-offs between price volatility and social welfare
considerations.

In this paper, we focus on resource pricing rather than price volatility, to
support the design of socially acceptable electricity markets. We propose a
social shaping problem for a competitive equilibrium in a transactive energy
systems, aiming to bound the energy price below a socially acceptable
threshold. We focus on parameterized utility functions, where the parameters
are abstracted from the preferences of agents. We prescribe a range for the
parameters in the utility functions, to ensure resources pricing under a
competitive equilibrium is socially acceptable for all agents without creating
a mismatch in supply and demand. The idea of introducing parameterized utility
functions is informed by the concept of smart thermostat agents in the AEP
Ohio gridSMART demonstration project [30, 31]. For transactive energy systems
organized as multi-agent networks operating at competitive equilibriums, we
establish the following results.

  * •

We show the essence of the social shaping problem is a set decision problem,
where for strictly concave and continuously differentiable utility functions,
the set decision is characterized by an implicit representation of the optimal
price as a function of agent preferences.

  * •

For two representative classes of utility functions, nam-ely linear-quadratic
functions and piece-wise linear functions, the exact set of parameters that
guarantee socially acceptable pricing is established, respectively.

In our prior work [1], we introduced the concept of social shaping of agent
utility functions, which was followed by an investigation into social shaping
with linear quadratic utility functions [2]. In the current manuscript, the
results on conceptual solvability and solutions for piece-wise linear
functions are new, which are supported by the presentation of a series of new
and large-scale numerical examples.

This paper is organized as follows. In Section II, we introduce two multi-
agent transactive energy systems, which are implemented in either a
centralized or distributed transactive energy network. In Section III, we
motivate the problem of social shaping, to enable the design of socially
acceptable electricity pricing. In Section IV and Section V, we present
conceptual and analytical solutions to the social shaping problem,
respectively. Section VI presents concluding remarks.

##  II Multiagent Transactive Energy Systems

In this section, we introduce two multi-agent transactive energy systems
models, and we recall some fundamental definitions and results related to such
systems.

###  II-A Transactive Energy Systems as Multi-agent Networks

We present two simple yet representative setting where $n$ agents indexed in
the set $\mathrm{V}=\\{1,...,n\\}$, each with local energy supply and demand,
form a transactive energy system. The multi-agent network is designed to
support microgrid operations, or would support distribution grid operators
with managing ubiquitous behind-the meter renewable energy resources. For
simplicity, we assume a lossless electrical network, with extensions to
include both real and reactive network and load losses possible. When there is
no external energy resource (e.g., a microgrid operating in isolation from the
wider power grid), the $n$ agents seek to form an energy market where the
overall energy supply and demand are balanced. Such a market must incorporate
the diverse interests of individual agents, while ensuring market efficiency.

Multiagent Transactive Energy Systems (MTES). Each agent $i$ produces
electricity with a local energy resource, with $a_{i}\in\mathbb{R}^{\geq 0}$
representing _local power production_ (in kW). The overall _network generation
production_ is represented by $C:=\sum_{i=1}^{n}a_{i}$, where $C>0$ (in kW).
Each agent $i$ makes a decision on its _energy consumption load_ , denoted by
$x_{i}\in\mathbb{R}^{\geq 0}$ (in kW). Upon consuming the load $x_{i}$, agent
$i$ receives a payment (or bill) attributed to its demand preference
$f_{i}(x_{i})=h(x_{i};\theta_{i})$, where $\theta_{i}$ is the _personalized
parameter_ for the load preferences of agent $i$. Importantly, any shortfall
or surplus of energy for each agent $i$, represented by $a_{i}-x_{i}$, must be
accommodated by the transactive market. We denote the _price per unit energy_
by $\lambda\in\mathbb{R}$ (in $$\backslash$kWh), and the income (or cost) of a
transaction for agent $i$ is thereby $\lambda(a_{i}-x_{i})$ when $a_{i}>x_{i}$
(or $a_{i}<x_{i}$).

Multiagent Transactive Energy Systems with Strategic Trading (MTES-ST). We
extend the previously defined MTES, by way of supporting _strategic trading
decisions_ for each agent $i$ denoted by $e_{i}\in\mathbb{R}$. That is, the
income (or cost) of the transactions for agent $i$ is $\lambda e_{i}$, where
$e_{i}>0$ (or $e_{i}<0$). Importantly, there is an inherent constraint on
strategic trading decisions for each agent, represented by $e_{i}\leq
a_{i}-x_{i}$. Specifically, when $a_{i}>x_{i}$, there is a physical constraint
on $e_{i}$ as the amount of energy sold by agent $i$ cannot exceed
$a_{i}-x_{i}$. Furthermore, when $a_{i}<x_{i}$, there is a physical network
constraint on $e_{i}$ for agent $i$ seeking to purchase $x_{i}-a_{i}$ amount
of energy from the market.

###  II-B System-level Equilibriums

We draw on welfare theory from economics, considering an effective market
price in the context of rational agent decisions. Specifically, for both MTES
and MTES-ST, we consider the concepts of competitive equilibriums and social
welfare equilibriums to precisely quantify price effectiveness and agent
rationality.

Let $\mathbf{a}=(a_{1},...,a_{n})^{\top}\in\mathbb{R}^{n}$ be a vector
representing the _local production profile_ , or otherwise the electricity
supply available from $n$ agents. Let
$\mathbf{x}=(x_{1},...,x_{n})^{\top}\in\mathbb{R}^{n}$ be a vector
representing the _local consumption profile_ , or otherwise the electricity
demand for $n$ agents.

######  Definition 1.

For the MTES, a competitive equilibrium is the pair of (1) price denoted by
$\lambda^{\ast}\in\mathbb{R}$, and (2) local consumption profile denoted by
$\mathbf{x}^{\ast}\in\mathbb{R}^{n}$, under which the following two conditions
are satisfied.

  * (i)

The local consumption profile $\mathbf{x}^{\ast}$ maximizes each individual
agent payoff; i.e., each $x_{i}^{\ast}$ is a solution to the following
optimization problem

$\displaystyle\max_{{x}_{i}}$ $\displaystyle
h(x_{i};\theta_{i})+\lambda^{\ast}(a_{i}-x_{i})$ (1) $\displaystyle{\rm s.t.}$
$\displaystyle x_{i}\in\mathbb{R}^{\geq 0}.$

  * (ii)

The local consumption profile $\mathbf{x}^{\ast}$ balances the total energy
consumption and supply across the network; that is,

$\sum_{i=1}^{n}x_{i}^{\ast}=C.$ (2)

######  Definition 2.

Let $\mathbf{e}=(e_{1},...,e_{n})^{\top}\in\mathbb{R}^{n}$ denote a _strategic
decision profile_. A competitive equilibrium for MTES-ST is a triplet of
($\lambda^{\ast},\mathbf{x}^{\ast},\mathbf{e}^{\ast}$), under which the
following conditions are satisfied.

  * (i)

The pair $(\mathbf{x}^{\ast},\mathbf{e}^{\ast})$ maximizes the individual
payoff of each agent, i.e., $(x_{i}^{\ast},e_{i}^{\ast})$ is a solution to

$\displaystyle\max_{{x}_{i},e_{i}}$ $\displaystyle
h(x_{i};\theta_{i})+\lambda^{\ast}e_{i}$ (3) $\displaystyle{\rm s.t.}$
$\displaystyle x_{i}+e_{i}\leq a_{i},$ $\displaystyle x_{i}\in\mathbb{R}^{\geq
0},\ \ e_{i}\in\mathbb{R}.$

  * (ii)

The strategic decision profile $\mathbf{e}^{\ast}$ balances the total resource
supply and demand across the network, i.e.,

$\sum_{i=1}^{n}e_{i}^{\ast}=0.$ (4)

The aforementioned competitive equilibriums support the establishment of an
effective market with supply and demand balanced. Rational agents decisions
are support by way of maximizing their individual payoffs. From classical
welfare economic theory, competitive equilibriums guarantee Pareto optimality
in the sense that no agent can change a decision without reducing the payoff
of other agents [14, 32, 13]. Next, we consider optimality at a system-level,
in the context of a social welfare problem (i.e., in the absence of a market).

######  Definition 3.

(i) For the MTES, a social welfare equilibrium $\mathbf{x}^{\ast}$ is achieved
by way of solving the following maximization problem

$\displaystyle\max_{\mathbf{x}}$
$\displaystyle\sum_{i=1}^{n}h(x_{i};\theta_{i})$ (5) $\displaystyle{\rm s.t.}$
$\displaystyle\sum_{i=1}^{n}x_{i}=C,$ $\displaystyle x_{i}\in\mathbb{R}^{\geq
0},\,\,i\in V.$

(ii) For the MTES-ST, a social welfare equilibrium
$(\mathbf{x}^{\ast},\mathbf{e}^{\ast})$ is achieved by solving the following
maximization problem

$\displaystyle\max_{\mathbf{x},\mathbf{e}}$
$\displaystyle\sum_{i=1}^{n}h(x_{i};\theta_{i})$ (6) $\displaystyle{\rm s.t.}$
$\displaystyle\sum_{i=1}^{n}e_{i}=0,$ $\displaystyle x_{i}+e_{i}\leq a_{i},$
$\displaystyle x_{i}\in\mathbb{R}^{\geq 0},\,e_{i}\in\mathbb{R},\,\,i\in V.$

The social welfare equilibrium is consistent with the rationale of a utility-
based system planner that designs and enforces energy consumption decisions
across all agents. Such a system planner is not concerned by individual
payoffs of each specific agent, but rather, the planner aims to maximize
energy allocations across the network, even at the expense of some agents
receiving suboptimal allocations. In what follows, we investigate conditions
where both the competitive and social welfare equilibriums are equivalent.

######  Proposition 1 (as in [1]).

Suppose each $h(\cdot;\theta_{i})$ is a concave function over the domain
$\mathbb{R}^{\geq 0}$. Then for both MTES and MTES-ST, the competitive
equilibriums and the social welfare equilibriums are equivalent.

Figure 1: Centralized (left) and distributed (right) implementations of
competitive equilibriums for the proposed transactive energy systems. Each
dark blue circle represents an agent in the system; the brown circle
represents a central coordinator agent; each red line represent a
communication link. The implementations take place in three sequential phases.

Equivalence for the MTES refers to a direct mapping of the competitive
equilibrium pair ($\lambda^{\ast}$, $\mathbf{x}^{\ast}$), to the social
welfare equilibrium $\mathbf{x}^{\ast}$. Or otherwise, if $\mathbf{x}^{\star}$
is a social welfare equilibrium for the MTES, then there exists
$\lambda^{\ast}\in\mathbb{R}$ such that ($\lambda^{\ast}$,
$\mathbf{x}^{\star}$) is a competitive equilibrium. Equivalence for MTES-ST is
similarly defined, providing a guarantee that market-based agent decisions
coincide with utility-based planner decisions. More specifically, by solving
the optimization problem for the social welfare equilibrium for either MTES or
MTES-ST, the optimal dual variable is a Lagrangian multiplier associated with
the supply-demand balance constraint, which is the optimal price for the
competitive equilibrium [1].

Importantly, there is a critical difference between MTES and MTES-ST in terms
of optimal pricing. For MTES, the prices under a competitive equilibrium can
be either positive or negative. In contrast, for MTES-ST the price under a
competitive equilibrium is always non-negative. Next, we explore connections
between the two MTES and MTES-ST models.

######  Proposition 2.

Suppose each $h(\cdot;\theta_{i})$ is a concave function over the domain
$\mathbb{R}^{\geq 0}$. Assume $\lambda_{1}^{\ast}>0$ and
$\lambda_{2}^{\ast}>0$ correspond to optimal pricing signals for the
competitive equilibrium for MTES and MTES-ST, respectively. Then there holds
$\lambda_{1}^{\ast}=\lambda_{2}^{\ast}$, and the agent decisions for MTES and
MTES-ST are the same for the respective competitive equilibriums.

###### Proof.

See Appendix VII-A. ∎

###  II-C Practical Implementations of Competitive Equilibriums

In Fig. 1 we illustrate two approaches to implementing MTES or MTES-ST, each
approach including three sequential phases (i.e., P1, P2, P3).

_Approach 1: Centralized computation via an aggregator_

  * P1

The aggregator directly communicates with each agent $i\in\mathrm{V}$ to
collect their individual load preferences $\theta_{i}$ and local power
production $a_{i}$.

  * P2

The aggregator computes the overall network generation production $C$, then
the social welfare equilibirum $\mathbf{x}^{\ast}$, and identifies the
corresponding Lagrangian multiplier $\lambda^{\ast}$, which is the optimal
price.

  * P3

The aggregator directly sends $(\lambda^{\ast},\mathbf{x}^{\ast})$ to each
agent $i\in\mathrm{V}$.

_Approach 2: Distributed computation without an aggregator_

  * P1

Each agent $i\in\mathrm{V}$ selects their individual preference vector
$\theta_{i}$, and communicates ($\theta_{i},a_{i}$) to other agents.

  * P2

Each agent $i\in\mathrm{V}$ runs, for example, a distributed processor
agreement protocol [33] to receive vectors
$\bm{\theta}=(\theta_{1},\dots,\theta_{n})^{\top}$ and
$\mathbf{a}=(a_{1},\dots,a_{n})^{\top}$, and each agent computes the overall
network generation production $C$.

  * P3

Each agent $i\in\mathrm{V}$ independently solves a social welfare equilibrium
$\mathbf{x}^{\ast}$ and identifies the corresponding Lagrangian multiplier
which is the optimal price $\lambda^{\ast}$.

Each of the two implementations are underpinned by Proposition 1 that states
the competitive equilibriums are equivalent to the social welfare
equilibriums. That is, the resulting $\mathbf{x}^{\ast}$ is guaranteed to be a
competitive equilibrium. Since the social welfare equilibriums depend only on
the network generation production $C$, the processor agree protocol for the
vector $\mathbf{a}$ can be replaced by a distributed average consensus
algorithm [34], where each agent obtains the average of all $a_{i}$, or
equivalently, the network capacity $C$.

##  III The Problem of Social Shaping

In this section, we motivate and define the problem of social shaping in a
multi-agent transactive energy network. We focus on social shaping agent
preferences to support optimization-based design of electricity prices.

###  III-A Motivating Example

Here, we provide a motivating example to highlight conditions where agent
preferences significantly influence optimal pricing under competitive
equilibriums.

Example 1. Consider a MTES with four agents, where the electricity supply
available is $\mathbf{a}=(a_{1},a_{2},a_{3},a_{4})^{\top}=(5,8,7,0)^{\top}.$
Each agent $i$ has a linear-quadratic utility function of the form
$h(x_{i};\theta_{i})=-\frac{1}{2}b_{i}x_{i}^{2}+m_{i}b_{i}x_{i}$, where
$(b_{1},b_{2},b_{3},b_{4})=(2,5,3,10)$ and
$(m_{1},m_{2},m_{3},m_{4})=(6,5,6,5)$. The social welfare equilibrium is
computed by solving the optimization problem in (5), which yields
$\mathbf{x}^{\ast}=(5.12,4.65,5.41,4.82)^{\top}$. The Lagrangian multiplier
corresponding to the supply-demand balance constraint
$\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}a_{i}$ is $\lambda^{\ast}=1.765.$

Next, let $m_{4}$ take values in the interval $[5,30].$ We sample the interval
uniformly with a step-size of $1$ to obtain $26$ different values for $m_{4}.$
For each $m_{4}$, the social welfare equilibrium and optimal price are
computed. In Fig. 2, we present the optimal price $\lambda^{\ast}$ and the
optimal resource allocation $x_{4}^{\ast}$ of agent $4$ as functions of
$m_{4}$. From Fig. 2, we observe that the optimal price $\lambda^{\ast}$ at
$m_{4}=30$ exceeds the optimal price at $m_{4}=5$ by a factor of $57$.
Furthermore, we observe that the preferences of a single agent significantly
alter the electricity price, and influence the distribution of resources
allocations. $\square$

Figure 2: Top: optimal price $\lambda^{\ast}$ as functions of $m_{4}$. Bellow:
agent $4$ optimal resource allocation, $x_{4}^{\ast}$, as functions of
$m_{4}$.

###  III-B Social Shaping Problem

From Example 1, we observed that the optimal price for the MTES is in need of
social shaping at a system level. That is, we observed that one agent was able
to consume all available resources at a price that potentially prohibited
others from accessing the energy. In what follows, we consider optimal pricing
below an upper limit, where the upper limit is deemed socially acceptable by
all agents. In this way, utility functions will be restricted within a
prescribed range within our MTES framework. More importantly, socially
admissible utility functions must correspond to socially acceptable optimal
pricing. To this end, we define a social shaping problem for MTES and MTES-TS.

######  Definition 4.

(Social Shaping of Agent Preferences) Consider both MTES and MTES-ST. Let
$\Theta$ be a space of parameters, where the preference $\theta_{i}\in\Theta$
is selected by agent $i$ in the context of their utility function
$h(\cdot;\theta_{i})$ for $i\in\mathrm{V}$. Let $\lambda^{\dagger}>0$ denote
an upper energy pricing limit, which represents the threshold of socially
acceptable prices for agents $i\in\mathrm{V}$. Find the set $\Theta$, such
that agent preferences $(\theta_{1},\dots,\theta_{n})\in\Theta^{n}$ can be
selected in a way where the optimal price corresponding to the competitive
equilibriums satisfies $\lambda^{\ast}\leq\lambda^{\dagger}$.

##  IV Conceptual Social Shaping Solvability

In this section, we study the solvability of the social shaping problem at
conceptual levels.

###  IV-A MTES

Here, we introduce the social shaping problem for strictly concave and
differentiable functions. Let each $h(\cdot;\theta_{i})$ be continuously
differentiable and strictly concave. It follows from Proposition 1 that the
social welfare equilibrium and the competitive equilibrium exist and are
equivalent. Consequently, we can consider either the social welfare problem or
the competitive problem.

Let $x_{i}^{\ast}$ be a solution to (1). Since $h(\cdot;\theta_{i})$ is
continuously differentiable and strictly concave,
$h^{\prime}(\cdot;\theta_{i})$ is strictly monotone. We denote by
$l(\cdot;\theta_{i})$ the inverse function of $h^{\prime}(\cdot;\theta_{i})$.
Specifically, when $h(\cdot;\theta_{i})$ is strictly concave,
$h^{\prime}(\cdot;\theta_{i})$ is strictly monotone, and as such the inverse
must exist. As a result, we can derive

$x_{i}^{\ast}=\max\left\\{l(\lambda^{\ast};\theta_{i}),0\right\\}.$ (7)

Next, substitute (7) into the balancing equality in (2), which yields

$\sum_{i=1}^{n}\max\left\\{l(\lambda^{\ast};\theta_{i}),0\right\\}=C.$ (8)

Since $C>0$, there exists at least one agent with $x_{i}^{\ast}\neq 0$,
corresponding to $x_{i}^{\ast}=l(\cdot;\theta_{i})$. Also, as
$h^{\prime}(\cdot;\theta_{i})$ is strictly monotone, its inverse
$l(\cdot;\theta_{i})$ is also strictly monotone. Consequently, the left-hand
side of (8) is the sum of at least one strictly monotone function, implying
that the summation result is also a strictly monotone function whose inverse
exists. Fixing $\bm{\theta}=(\theta_{1},\dots,\theta_{n})$, then there holds

$\lambda^{\ast}=\hat{l}(C;\bm{\theta}),$ (9)

where $\hat{l}(\cdot;\bm{\theta})$ is the inverse of the left-hand side of
(8).

Next, we denote by $\chi_{\Theta}$ the _maximal value of the set of optimal
prices_ that support all competitive equilibriums when agent preferences are
drawn from $\bm{\theta}\in\Theta^{n}$. We define the maximal value of the set
of optimal prices by

$\displaystyle\chi_{\Theta}:=\max_{\bm{\theta}\in\Theta^{n}}\hat{l}(C;\bm{\theta}).$
(10)

From the definition of $\chi_{\Theta}$, the following result is immediately
clear indicating the social shaping problem is conceptually captured by a set
decision problem with respect to $\Theta$.

######  Theorem 1.

Consider a MTES. Suppose each $h(\cdot;\theta_{i})$ is strictly concave and
differentiable over $\mathbb{R}^{\geq 0}$. Let $\lambda^{\dagger}>0$ denote
the threshold for socially acceptable energy pricing, such that
$\lambda^{\ast}\leq\lambda^{\dagger}$. Then any set $\Theta$ satisfying
$\chi_{\Theta}\leq\lambda^{\dagger}$ solves the problem for social shaping of
agent preferences.

###  IV-B Homogenous MTES

Consider the situation where agents maintain a common preference, i.e.,
$\theta_{i}=\hat{\theta}$. For example, each agent $i$ maintains a preference
$\hat{\theta}$, which is an average of true preferences across all agents,
that is $\hat{\theta}=\frac{1}{n}\sum_{i=1}^{n}{\theta}_{i},\ i\in\mathrm{V}$,
where the set $\Theta$ is convex.

######  Theorem 2.

Consider a MTES with homogenous preferences, i.e., $\theta_{i}=\theta$ for all
$i\in\mathrm{V}$. Suppose each $h(\cdot;\theta_{i})$ is concave and
differentiable over $\mathbb{R}^{\geq 0}$. Let $\lambda^{\dagger}>0$ denote
the threshold for socially acceptable energy pricing, such that
$\lambda^{\ast}\leq\lambda^{\dagger}$. Then a solution for $\Theta$ for the
social shaping problem is given by

$\Theta=\big{\\{}\theta:h^{\prime}({C}/{n};\theta)\leq\lambda^{\dagger}\big{\\}}.$
(11)

###### Proof.

See Appendix VII-B. ∎

###  IV-C MTES-ST

In our prior work [1], we showed that $\lambda^{\ast}\geq 0$ for MTES-ST. It
follows that $\lambda^{\ast}=0$ corresponds to socially resilient pricing, as
it satisfies $\lambda^{\ast}\leq\lambda^{\dagger}$. In what follows, we
consider socially resilient pricing for MTES-ST for the case where
$\lambda^{\ast}>0$.

According to Proposition 2, when the price is positive, MTES and MTES-ST yield
the same optimal pricing $\lambda^{\ast}>0$, and agent decisions.
Consequently, we define $\chi_{\Theta}$ as (10), and introduce the following
theorem.

######  Theorem 3.

Consider a MTES-ST. Suppose each $h(\cdot;\theta_{i})$ is strictly concave and
differentiable over $\mathbb{R}^{\geq 0}$. Let $\lambda^{\dagger}>0$ denote
the threshold for socially acceptable energy pricing, such that
$\lambda^{\ast}\leq\lambda^{\dagger}$. Then any set $\Theta$ satisfying
$\chi_{\Theta}\leq\lambda^{\dagger}$ solves the problem for social shaping of
agent preferences.

##  V Analytical Social Shaping Solutions

In this section, we focus on two fundamental classes of utility functions,
i.e., linear-quadratic functions and piece-wise linear functions, and we show
the social shaping problems can be explicitly solved.

###  V-A Linear Quadratic Utility Functions

We impose the following assumption.

Assumption 1. Let
$\theta_{i}:=(b_{i},m_{i})\in\mathbb{R}^{>0}\times\mathbb{R}^{>0}$, where

$h(x_{i};\theta_{i})=-\frac{1}{2}b_{i}x_{i}^{2}+m_{i}b_{i}x_{i}.$ (12)

Fig. 3 illustrates example linear quadratic utility functions.

Figure 3: Example linear quadratic utility functions.

######  Theorem 4.

Consider a MTES with a quadratic utility function as defined in Assumption 1.
Assume $\lambda^{\dagger}$ is the upper limit of the socially acceptable
price. Suppose $(b_{\rm max},m_{\rm max})\in\mathbb{R}^{2}_{>0}$ is selected
from the following set

$\begin{gathered}\begin{aligned} \mathscr{S}_{\ast}=\left\\{m_{\rm
max}\leq\frac{C}{n},\,b_{\rm
max}\in\mathbb{R}^{>0}\right\\}\mathbin{\mathchoice{\vbox{\hbox{$\textstyle\bigcup$}}}{\vbox{\hbox{$\textstyle\bigcup$}}}{\vbox{\hbox{$\scriptstyle\bigcup$}}}{\vbox{\hbox{$\scriptscriptstyle\bigcup$}}}}\\\
\left\\{m_{\rm max}>\frac{C}{n},\,b_{\rm
max}\leq\frac{n\lambda^{\dagger}}{nm_{\rm
max}-C}\right\\}.\end{aligned}\end{gathered}$ (13)

Then $\lambda^{\ast}$ is always socially acceptable since
$\lambda^{\ast}\leq\lambda^{\dagger}$, for all utility functions satisfying
$b_{i}\leq b_{\rm max}$ and $m_{i}\leq m_{\rm max}$.

###### Proof.

See Appendix VII-C. ∎

Example 2. Consider a MTES with linear-quadratic utility functions as defined
in Assumption 1.

Let the local resource $a_{i}$ of each agent $i$, be a random number sampled
from a normal distribution $\mathcal{N}(\mu_{a},\,\sigma_{a}^{2})$ truncated
to the interval $[0,10].$ The mean value of the local resources is
$\mu_{a}=5$, and the standard deviation is $\sigma_{a}=1.25$. Overall network
generation production, $C$, is the summation of all local resources, i.e.,
$C=\sum_{i=1}^{n}a_{i}.$ Recall, the personalized parameter
$\theta_{i}=(b_{i},m_{i})$ represents the preferences of agent $i$. Let
preferences for agent $1$, $(b_{1},m_{1})$, be represented by $m_{1}$, a value
from a uniform distribution truncated to the interval
$[\frac{C}{n},\frac{100C}{n}]$, and
$b_{1}=\frac{n\lambda^{{\dagger}}}{nm_{1}-C}$. Given $(b_{1},m_{1}),$ let the
personalize parameters $\theta_{i}=(b_{i},m_{i})$ for all other agents be
sampled from two uniform distributions truncated to the interval $(0,b_{1}]$
and the interval $(0,m_{1}]$, respectively. In this manner, we have
$(b_{\max},m_{\max})=(b_{1},m_{1})$, and in what follows we verify that the
personalized parameter $\theta_{i}$ for each agent is within the range
prescribed in (13).

Consider $n=10000$ agents, and let the socially acceptable threshold
$\lambda^{{\dagger}}$ take value from $\\{20,22,24,26,28,30\\}$. For each
possible $\lambda^{{\dagger}}$, we conduct $K=1000$ numerical experiments, for
each of which we generate a parameter set
$(\mathbf{b}^{(k)},\mathbf{m}^{(k)})$, where $k=1,\dots,K$, by following the
aforementioned parameter selection process. In each experiment $k$, we input
parameters $(\mathbf{b}^{(k)},\mathbf{m}^{(k)})$ to compute the optimal price,
that is, the Lagrangian multiplier associated with the equity constraint
$\sum_{i=1}^{n}x_{i}=\sum_{i=1}a_{i}$, as in (5). In Fig. 4, we present
optimal pricing for quadratic utility functions within the range (13) under
different upper limits $\lambda^{{\dagger}}=20,22,24,26,28,30,$ each
corresponding to the respective $1000$ numerical experiments.

Next, let $\lambda^{{\dagger}}=20$ and let $n\in\\{100,1000,10000,100000\\}$.
For each possible value of $n$, we conduct $K=1000$ experiments with the
corresponding personalized parameter sets, $\theta_{i}$, obtained by applying
the aforementioned process. In Fig. 5, we present optimal pricing for
quadratic utility functions within the range (13), where $1000$ experiments
are conducted for each system scale $n=100,1000,10000,100000$.

Figure 4: Optimal pricing for quadratic utility functions within the range
(13), with upper electricity pricing limits of
$\lambda^{{\dagger}}=20,22,24,26,28,30.$ Figure 5: Optimal pricing for
quadratic utility functions within the range (13), considering multi-agent
systems of scale $n=100,1000,10000,100000$.

In Fig. 4 and Fig. 5, the middle line, and the lower and upper boundaries of
each box (interquartile range or IQR) correspond to the median, and 25/75
percentile of the $1000$ optimal prices, respectively. The lower and upper
whiskers extend maximally $1.5\times$ of IQR from 25 percentile downwards and
75 percentile upwards, respectively. The points that are located outside the
whiskers are considered data outliers.

In Fig 4, we observe that the optimal prices in the numerical experiments are
below the corresponding upper limit, i.e.,
$\lambda^{\ast}=\lambda^{{\dagger}}$, indicating the optimal pricing being
socially acceptable. In Fig. 5, we observe that the optimal prices in the
numerical experiments considering various multi-agent system scales, are lower
than the upper limit $\lambda^{{\dagger}}=20$. Our observations in both Fig. 4
and Fig. 5 correspond to, and validate Theorem 4. $\square$

###  V-B Piece-wise Linear Utility Functions

We next consider the following assumption.

Assumption 2. Let
$\theta_{i}:=(\beta_{i},\phi_{i})\in\mathbb{R}^{>0}\times\mathbb{R}^{>0}$ with

$h(x_{i};\theta_{i}):=\min\\{\beta_{i}x_{i},\phi_{i}\beta_{i}\\}.$ (14)

Fig. 6 illustrates example piece-wise linear utility functions.

Figure 6: The class of piece-wise linear functions.

######  Theorem 5.

Consider a MTES with piece-wise linear utility functions as defined in
Assumption 2. Suppose $(\beta_{\rm max},\phi_{\rm max})$ is selected from the
following set

$\begin{gathered}\begin{aligned} \mathscr{S}_{\ast}=\left\\{\phi_{\rm
max}<\frac{C}{n},\,\beta_{\rm
max}\in\mathbb{R}^{>0}\right\\}\mathbin{\mathchoice{\vbox{\hbox{$\textstyle\bigcup$}}}{\vbox{\hbox{$\textstyle\bigcup$}}}{\vbox{\hbox{$\scriptstyle\bigcup$}}}{\vbox{\hbox{$\scriptscriptstyle\bigcup$}}}}\\\
\left\\{\phi_{\rm max}\geq\frac{C}{n},\,\beta_{\rm
max}\leq\lambda^{\dagger}\right\\}.\end{aligned}\end{gathered}$ (15)

Then the resulting $\lambda^{\ast}$ is always socially acceptable since
$\lambda^{\ast}\leq\lambda^{\dagger}$, for all utility functions satisfying
$\beta_{i}\leq\beta_{\rm max}$ and $\phi_{i}\leq\phi_{\rm max}$.

###### Proof.

See Appendix VII-D. ∎

Example 3. Consider a MTES where each agent $i$ is associated with a piece-
wise linear utility function as defined in Assumption 2.

The local resources ${a_{i}}$ of each agent is obtained following the same
process described in Example 2. The network generation production, $C$, is
then, $C=\sum_{i=1}^{n}a_{i}.$ Let preferences for agent $1$,
$(\beta_{1},\phi_{1})$, be represented by $\beta_{1}=\lambda^{{\dagger}}$ and
let $\phi_{1}$ be a random number sampled from a normal distribution truncated
to the interval $[\frac{C}{n},\frac{10C}{n}]$. Let the personalize parameter
pairs for the remaining agents be sampled from two uniform distributions,
truncated to the interval $(0,\beta_{1}]$ and the interval $(0,\phi_{1}]$,
respectively. In this way, we seek to validate the design of
$(\beta_{\max},\phi_{\max})=(\beta_{1},\phi_{1})$ in the context of the range
in (15).

Let $n=10000$ denote the number of agents, and let the upper limit
$\lambda^{{\dagger}}$ take values from $\\{20,22,24,26,28,30\\}.$ For each
upper limit $\lambda^{{\dagger}}$, we carry out $K=1000$ experiments. In each
experiment $k=1,\dots,K$, a different parameter set of
$(\bm{\beta}^{(k)},\bm{\phi}^{(k)})$ is obtained upon the aforementioned
parameter generation process. For each $(\bm{\beta}^{(k)},\bm{\phi}^{(k)})$,
we solve the social welfare optimization problem (5), and the optimal dual
variable corresponding to the equity constraint
$\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}a_{i}$ is obtained as $\lambda^{\ast}$. In
Fig. 7, we present the optimal pricing for piece-wise linear utility functions
within the range (15), considering $1000$ numerical experiments for each upper
limit $\lambda^{\ast}=20,22,24,26,28,30$. From Fig. 7, we observe that the
dotted line $\lambda^{\ast}=\lambda^{{\dagger}}$, sits above all optimal
prices identified from the numerical experiments, corresponding to the optimal
price being socially acceptable.

Figure 7: Optimal pricing for piece-wise linear utility functions within the
range (15), considering upper limits $\lambda^{\ast}=20,22,24,26,28,30.$
Figure 8: Optimal pricing for piece-wise linear functions within the range
(15), for multi-agent systems of scale $n=100,1000,10000,100000$.

Next, let $\lambda^{{\dagger}}=20$ and let $n\in\\{100,1000,10000,100000\\}$.
For each possible value of $n$, we conduct $K=1000$ experiments with the
corresponding personalized parameter sets, $\theta_{i}$, obtained by applying
the aforementioned process. We then solve the optimization problem (5) and
obtain the optimal price $\lambda^{\ast}$. In Fig. 8, we present the optimal
prices for piece-wise linear functions within the range (15), considering the
respective $1000$ experiments, for each system scale
$n=100,1000,10000,100000$. In Fig. 8, we observe that all optimal prices in
the numerical experiments, considering various system scales, are also below
the upper limit $\lambda^{{\dagger}}=20$ required for social acceptance by the
agents. From Fig. 7 and Fig. 8, we observe that piece-wise linear utility
functions within the range (15), correspond to, and validate Theorem 5.
$\square$

###  V-C Further Discussions: MTES-ST

It is worth emphasizing the results in Theorem 4 and Theorem 5 for MTES
continue to be valid for MTES-ST. That is, the same set of preference
parameters will enable socially acceptable optimal pricing under competitive
equilibriums for MTES-ST. Specifically, from Proposition 2, when the price is
positive, MTES and MTES-ST are exactly the same in terms of pricing and agent
decisions. This equivalence is related to the second part of the sets proposed
in (13) and (15), i.e, $m_{\rm max}>\frac{C}{n}$ and
$\phi_{\max}\geq\frac{C}{n}$, respectively.

Regarding the first part of the proposed sets, i.e, $m_{\rm
max}\leq\frac{C}{n}$ or $\phi_{\max}<\frac{C}{n}$, Lemmas 8 and 10 in Appendix
VII-E show that under these conditions the optimal price for MTES-ST is equal
to zero, and thus socially acceptable.

##  VI Conclusions

This paper has considered the problem of shaping agent utility functions in a
transactive energy system. In the overall system design, energy supply and
demand is balanced while incorporating individual utility preferences for
agents, in a way that accommodates socially acceptable pricing. In defining
the social shaping problem, we identified a set of preference parameters to
provide guarantees that the optimal energy price is below a prescribed
threshold. Our established conceptual results indicated that the social
shaping problem is characterized by a set decision problem. We also presented
two analytical solutions in tight ranges for coefficients of linear-quadratic
utilities and piece-wise linear utilities, to further demonstrate the
application of the social shaping problem to socially acceptable electricity
pricing. Future work might include extensions of the social shaping ideas to
transactive energy systems operating dynamically, or systems that operate
under game-theoretic frameworks.

## Acknowledgements

This work was supported by the Australian Research Council under Grants
DP180101805 and DP190103615.

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##  VII Appendix

###  VII-A Proof of Proposition 2

It is assumed $h(\cdot;\theta_{i})$ is concave. Therefore, Proposition 1
states that the social welfare equilibrium and the competitive equilibrium
coincide. Consequently, either the social welfare problem or the competitive
problem can be solved. Here, we consider the competitive optimization problem
of MTES-ST in (3). The inequality constraint $x_{i}+e_{i}\leq a_{i}$ can be
written as

$x_{i}+e_{i}+s_{i}=a_{i},$ (16)

where $s_{i}\geq 0$ is the slack variable. Additionally, substituting
$e_{i}=a_{i}-x_{i}-s_{i}$ into (3) yields an equivalent form for the
optimization problem as

$\displaystyle\max_{{x}_{i},s_{i}}$ $\displaystyle
h(x_{i};\theta_{i})+\lambda^{\ast}_{2}(a_{i}-x_{i}-s_{i})$ (17) $\displaystyle
x_{i},s_{i}\in\mathbb{R}^{\geq 0}.$

Let $\lambda^{\ast}_{2}>0$. Then, the objective function in (17) is strictly
decreasing with respect to $s_{i}$. Consequently, in order for (17) to be
maximized, $s_{i}$ must be minimized, i.e., $s_{i}^{\ast}=0$. This implies
that the inequality constraint $x_{i}+e_{i}\leq a_{i}$ is active and

$x_{i}^{\ast}+e_{i}^{\ast}=a_{i}.$ (18)

Taking the summation in (18) implies

$\sum_{i=1}^{n}x_{i}^{\ast}+\sum_{i=1}^{n}e_{i}^{\ast}=C,$ (19)

where $C=\sum_{i=1}^{n}a_{i}$. Then, substituting the balancing equality
$\sum_{i=1}^{n}e_{i}^{\ast}=0$ in (4) into (19) yields

$\sum_{i=1}^{n}x_{i}^{\ast}=C.$ (20)

Furthermore, substituting $s_{i}=0$ into (17) yields

$\displaystyle\max_{{x}_{i}}$ $\displaystyle
h(x_{i};\theta_{i})+\lambda^{\ast}_{2}(a_{i}-x_{i})$ (21) $\displaystyle
x_{i}\in\mathbb{R}^{\geq 0}.$

Comparing (21) and (20) with (1) and (2) implies that the problem of MTES-ST
is equivalent to the problem of MTES. Therefore, the agent decisions for MTES
and MTES-ST are the same for the respective competitive equilibriums. A
similar analysis can be done to show $\lambda_{1}^{\ast}=\lambda_{2}^{\ast}$.

###  VII-B Proof of Theorem 2

Since $h(\cdot;\theta_{i})$ is concave, Proposition 1 holds. Therefore, either
the social welfare problem or the competitive problem can be examined. Here,
we consider the competitive optimization problem of MTES in (1). The optimal
solution $x_{i}^{\ast}$ solves the following optimization problem

$\displaystyle\max_{{x}_{i}}$ $\displaystyle
h(x_{i};\theta)+\lambda^{\ast}(a_{i}-x_{i})$ (22) $\displaystyle{\rm s.t.}$
$\displaystyle x_{i}\in\mathbb{R}^{\geq 0}.$

Let $\hat{x}_{i}$ be the value $x_{i}$ which maximizes the objective function
in the absence of any constraints. Then $\hat{x}_{i}$ is obtained when the
derivative of the objective function equals zero, i.e,

$h^{\prime}(\hat{x}_{i};\theta)=\lambda^{\ast}.$ (23)

This implies that all agents have the same $\hat{x}_{i}=\hat{x}$. Considering
the inequality constraint $x_{i}\in\mathbb{R}^{\geq 0}$ in (22), if
$\hat{x}\leq 0$ then $x_{i}^{\ast}=0$ for all agents, which contradicts the
balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$ in (2) (note that $C>0$).
Consequently, it follows $\hat{x}>0$ and $x_{i}^{\ast}=\hat{x}$, which is
positive and satisfies the inequality constraint. Additionally, from the
balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$, we yield
$x_{i}^{\ast}=\hat{x}=\frac{C}{n}$. Then, substituting $\hat{x}=\frac{C}{n}$
into (23) obtains

$h^{\prime}(C/n;\theta)=\lambda^{\ast}.$ (24)

If $\theta$ is selected in such a way that
$h^{\prime}(C/n;\theta)\leq\lambda^{\dagger}$, then we yield
$\lambda^{\ast}\leq\lambda^{\dagger}$. This leads to the set $\Theta$ in (11).

###  VII-C Proof of Theorem 4

####  VII-C1 Preliminary Lemmas

We first introduce some preliminary lemmas which are essential for the proof
Theorem 4.

######  Lemma 1.

Consider the MTES with the quadratic utility function defined in Assumption 1.
The optimal load allocation $x_{i}^{\ast}$, which is an optimal solution of
the optimization problem (1), is such that

$x_{i}^{\ast}=\max\left\\{m_{i}-\frac{\lambda^{\ast}}{b_{i}},0\right\\}.$ (25)

###### Proof.

Rearranging the optimization problem (1), $x_{i}^{\ast}$ is the solution to
the following maximization problem:

$\displaystyle\max_{{x}_{i}}$
$\displaystyle-\frac{1}{2}b_{i}x_{i}^{2}+(m_{i}b_{i}-\lambda^{\ast})x_{i}+\lambda^{\ast}a_{i}$
(26) $\displaystyle{\rm s.t.}$ $\displaystyle x_{i}\in\mathbb{R}^{\geq 0}.$

Let $\hat{x}_{i}$ be the value $x_{i}$ which maximizes the objective function
in the absence of any constraints. Then $\hat{x}_{i}$ is obtained when the
derivative of the objective function equals zero. That is,

$-b_{i}\hat{x}_{i}+(m_{i}b_{i}-\lambda^{\ast})=0,$ (27)

which implies $\hat{x}_{i}=m_{i}-\frac{\lambda^{\ast}}{b_{i}}$. Considering
the inequality constraint $x_{i}\geq 0$ in the maximization problem (26), when
$\lambda^{\ast}\leq m_{i}b_{i}$, the solution is achieved at
$x_{i}^{\ast}=\hat{x}_{i}=m_{i}-\frac{\lambda^{\ast}}{b_{i}}$ which is non-
negative and satisfies the inequality constraint. Conversely, when
$\lambda^{\ast}>m_{i}b_{i}$, then $\hat{x}_{i}$ is negative which does not
satisfy the inequality constraint, so $x_{i}^{\ast}\neq\hat{x}_{i}$. In this
case, the objective function is strictly decreasing with respect to $x_{i}$.
Consequently, in order for the objective function to be maximized, $x_{i}$
must be minimized, i.e., $x_{i}^{\ast}=0$. Therefore, when
$\lambda^{\ast}>m_{i}b_{i}$ then $x_{i}^{\ast}=0$, otherwise,
$x_{i}^{\ast}=m_{i}-\frac{\lambda^{\ast}}{b_{i}}$. ∎

######  Lemma 2.

Consider the MTES with the quadratic utility function defined in Assumption 1.
If $\sum_{i=1}^{n}m_{i}\leq C$, then $\lambda^{\ast}\leq 0$. Conversely, if
$\sum_{i=1}^{n}m_{i}>C$ then $\lambda^{\ast}>0$.

###### Proof.

(i) Consider the case $\sum_{i=1}^{n}m_{i}\leq C$. By contradiction, suppose
$\lambda^{\ast}>0$. From equation (25) we yield $x_{i}^{\ast}<m_{i}$, and
therefore, $\sum_{i=1}^{n}x_{i}^{\ast}<\sum_{i=1}^{n}m_{i}$. Since
$\sum_{i=1}^{n}m_{i}\leq C$, we obtain

$\sum_{i=1}^{n}x_{i}^{\ast}<C,$ (28)

which contradicts the balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$ in
(2). Therefore, it follows that $\lambda^{\ast}\leq 0$.

(ii) Consider the case $\sum_{i=1}^{n}m_{i}>C$. By contradiction, suppose
$\lambda^{\ast}\leq 0$. From equation (25) we obtain $x_{i}^{\ast}\geq m_{i}$,
and therefore, $\sum_{i=1}^{n}x_{i}^{\ast}\geq\sum_{i=1}^{n}m_{i}$. Since
$\sum_{i=1}^{n}m_{i}>C$, we yield

$\sum_{i=1}^{n}x_{i}^{\ast}>C,$ (29)

which contradicts the balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$ in
(2). Therefore, it follows that $\lambda^{\ast}>0$. ∎

Consider two vectors $k=(k_{1},...,k_{n})$ and
$k^{\prime}=(k^{\prime}_{1},...,k^{\prime}_{n})$, and let $k\preceq
k^{\prime}$ denote $k_{i}\leq k_{i}^{\prime}$ for all $i\in V$. Let
$m=(m_{1},...,m_{n})$ and $b=(b_{1},...,b_{n})$. Suppose $\lambda^{\ast}$ is
the optimal price associated with the pair of vectors $(m,b)$, and let
$\lambda^{\ast^{\prime}}$ be the optimal price associated with
$(m^{\prime},b^{\prime})$.

######  Lemma 3.

Consider the MTES with the quadratic utility function defined in Assumption 1.
If $\lambda^{\ast}>0$, then $m\preceq m^{\prime}$ and $b\preceq b^{\prime}$
yield $\lambda^{\ast}\leq\lambda^{\ast^{\prime}}$.

###### Proof.

Suppose $\lambda^{\ast}>0$. Substituting (25) into the balancing equality
$\sum_{i=1}^{n}x_{i}^{\ast}=C$ in (2) yields

$\sum_{i=1}^{n}\max\left\\{m_{i}-\frac{\lambda^{\ast}}{b_{i}},0\right\\}=C.$
(30)

As $m_{i}$ and $b_{i}$ increase, $\lambda^{\ast}$ must also increase so as to
compensate for the change — ensuring the balancing equality (30) holds.
Otherwise, the left-hand side of equality (30) would increase, while the
right-hand side remains constant, and so the equality would not hold. ∎

####  VII-C2 Proof of the Theorem

Now, we present the proof of Theorem 4. The quadratic utility function in
Assumption 1 is concave, so Proposition 1 holds. We investigate two cases.

Case (i) $m_{\rm max}\leq\frac{C}{n}$. In this case, $m_{i}\leq m_{\rm max}$
implies $m_{i}\leq\frac{C}{n}$ for $i\in V$, and therefore,
$\sum_{i=1}^{n}m_{i}\leq C$. Consequently, Lemma 2 implies $\lambda^{\ast}\leq
0$. Since $\lambda^{\dagger}>0$, one obtains
$\lambda^{\ast}<\lambda^{\dagger}$. Therefore, $\lambda^{\ast}$ is socially
resilient.

Case (ii) $m_{\rm max}>\frac{C}{n}$ and $b_{\rm
max}\leq\frac{n\lambda^{\dagger}}{nm_{\rm max}-C}$. If $\lambda^{\ast}\leq 0$,
then it is socially resilient. Conversely, if $\lambda^{\ast}>0$, Lemma 3
yields $\lambda^{\ast}$ is monotonically increasing with respect to $m_{i}$
and $b_{i}$, so the highest possible price $\lambda^{\ast}_{\rm max}$ is
achieved when $m_{i}=m_{\rm max}$ and $b_{i}=b_{\rm max}$ for all agents $i\in
V$. Consequently, when all agents select $m_{i}=m_{\rm max}$ and $b_{i}=b_{\rm
max}$, the balancing equality (30) results in

$n\left(m_{\rm max}-\frac{\lambda^{\ast}_{\rm max}}{b_{\rm max}}\right)=C,$
(31)

and therefore,

$\lambda^{\ast}_{\rm max}=b_{\rm max}\left(\frac{nm_{\rm max}-C}{n}\right).$
(32)

From equation (32), along with the assumption $b_{\rm
max}\leq\frac{n\lambda^{\dagger}}{nm_{\rm max}-C}$ in (13), yields
$\lambda^{\ast}_{\rm max}\leq\lambda^{\dagger}$. Since
$\lambda^{\ast}\leq\lambda^{\ast}_{\rm max}$, one obtains
$\lambda^{\ast}\leq\lambda^{\dagger}$.

Considering (i) and (ii), it follows that as long as $(b_{\rm max},m_{\rm
max})$ is constrained in the set $\mathscr{S}_{\ast}$ in (13),
$\lambda^{\ast}$ will be socially resilient.

###  VII-D Proof of Theorem 5

####  VII-D1 Preliminary Lemmas

First, we provide some lemmas which are necessary for the proof of Theorem 5.

######  Lemma 4.

Consider the MTES. If Assumption 2 holds, then $\lambda^{\ast}\geq 0$.

###### Proof.

Note that $h(x_{i};\theta_{i})$ is a non-decreasing concave function.
Therefore, according to Proposition 1 in [1], we yield $\lambda^{\ast}\geq 0$.
∎

######  Lemma 5.

Consider the MTES with the piece-wise linear utility function defined in
Assumption 2. The optimal load allocation $x_{i}^{\ast}$, which is an optimal
solution of the optimization problem (1), satisfies

$x_{i}^{*}=\left\\{{\begin{array}[]{ll}\left\\{{x:x\geq{\phi_{i}}}\right\\}&\mbox{if
$\lambda^{\ast}=0$,}\\\ {\phi_{i}}&\mbox{if $0<\lambda^{\ast}<\beta_{i}$,}\\\
\left\\{{x:0\leq x\leq{\phi_{i}}}\right\\}&\mbox{if
$\lambda^{\ast}=\beta_{i}$,}\\\ 0&\mbox{if
$\lambda^{\ast}>\beta_{i}$.}\end{array}}\right.$ (33)

###### Proof.

We investigate four cases.

Case (i) $\lambda^{\ast}=0$. In this case, the objective function in (1)
equals the utility function
$h(x_{i};\theta_{i})=\min\\{\beta_{i}x_{i},\phi_{i}\beta_{i}\\}$, shown in
Fig. 6. As can be seen, the objective function is strictly increasing in the
interval $x_{i}\in[0,\phi_{i}]$, while constant in the interval
$x_{i}\in[\phi_{i},\infty)$. Therefore, the optimal solution is achieved at
$x_{i}^{\ast}\geq\phi_{i}$.

Case (ii) $0<\lambda^{\ast}<\beta_{i}$. In this case, the objective function
is strictly increasing in the interval $x_{i}\in[0,\phi_{i}]$, while strictly
decreasing in the interval $x_{i}\in[\phi_{i},\infty)$. Therefore, the optimal
solution is achieved at $x_{i}^{\ast}=\phi_{i}$.

Case (iii) $\lambda^{\ast}=\beta_{i}$. In this case, the objective function is
constant in the interval $x_{i}\in[0,\phi_{i}]$, while strictly decreasing in
the interval $x_{i}\in[\phi_{i},\infty)$. Therefore, the optimal solution is
achieved at $0\leq x_{i}^{\ast}\leq\phi_{i}$.

Case (iv) $\lambda^{\ast}>\beta_{i}$. In this case, the objective function is
strictly decreasing in the whole interval $x_{i}\in[0,\infty)$. Therefore, the
optimal solution is achieved at $x_{i}^{\ast}=0$. ∎

######  Lemma 6.

Consider the MTES with the piece-wise linear utility function defined in
Assumption 2. If $\sum_{i=1}^{n}\phi_{i}<C$, then $\lambda^{\ast}=0$.
Conversely, if $\sum_{i=1}^{n}\phi_{i}>C$ then $\lambda^{\ast}>0$.

###### Proof.

(i) Consider $\sum_{i=1}^{n}\phi_{i}<C$. By contradiction, suppose
$\lambda^{\ast}>0$. According to (33), $\lambda^{\ast}>0$ yields
$x_{i}^{\ast}\leq\phi_{i}$ for $i\in V$, and therefore,
$\sum_{i=1}^{n}x_{i}^{\ast}\leq\sum_{i=1}^{n}\phi_{i}$. Since
$\sum_{i=1}^{n}\phi_{i}<C$, we yield

$\sum_{i=1}^{n}x_{i}^{\ast}<C,$ (34)

which contradicts the balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$ in
(2). Therefore, it follows that $\lambda^{\ast}=0$.

(ii) Consider $\sum_{i=1}^{n}\phi_{i}>C$. By contradiction, suppose
$\lambda^{\ast}=0$. According to (33), $\lambda^{\ast}=0$ yields
$x_{i}^{\ast}\geq\phi_{i}$ for $i\in V$, and therefore,
$\sum_{i=1}^{n}x_{i}^{\ast}\geq\sum_{i=1}^{n}\phi_{i}$. Since
$\sum_{i=1}^{n}\phi_{i}>C$, we yield

$\sum_{i=1}^{n}x_{i}^{\ast}>C,$ (35)

which contradicts the balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$ in
(2). Therefore, it follows that $\lambda^{\ast}>0$. ∎

####  VII-D2 Proof of the Theorem

Now, we present the proof of Theorem 5. From Fig. 6, it is obvious
$h(\cdot;\theta_{i})$ is concave. Consequently, Proposition 1 holds. Now, we
investigate two cases.

Case (i) $\phi_{\rm max}<\frac{C}{n}$. In this case, $\phi_{i}\leq\phi_{\rm
max}$ implies $\phi_{i}<\frac{C}{n}$ for $i\in V$, and therefore,
$\sum_{i=1}^{n}\phi_{i}<C$. Consequently, Lemma 6 implies $\lambda^{\ast}=0$.
Since $\lambda^{\dagger}>0$, one obtains $\lambda^{\ast}<\lambda^{\dagger}$.
Therefore, $\lambda^{\ast}$ is socially resilient.

Case (ii) $\phi_{\rm max}\geq\frac{C}{n}$ and $\beta_{\rm
max}\leq\lambda^{\dagger}$. From Lemma 4, we know that $\lambda^{\ast}\geq 0$.
If $\lambda^{\ast}=0$, then it is socially resilient. Therefore, we only study
the case under which $\lambda^{\ast}>0$.

According to (33), if $\lambda^{\ast}>\beta_{i}$ for all $i\in V$, then
$x_{i}^{\ast}=0$ for all agents. Therefore, $\sum_{i=1}^{n}x_{i}^{\ast}=0$,
which contradicts the balancing equality $\sum_{i=1}^{n}x_{i}^{\ast}=C$ in (2)
(note that $C>0$). Consequently, we obtain $\lambda^{\ast}\leq\beta_{i}$ for
at least one agent. Since $\beta_{i}\leq\beta_{max}$ and
$\beta_{max}\leq\lambda^{\dagger}$, we yield
$\lambda^{\ast}\leq\lambda^{\dagger}$.

Considering (i) and (ii), it follows that as long as $(\beta_{\rm
max},\phi_{\rm max})$ is constrained in the set $\mathscr{S}_{\ast}$ in (15),
$\lambda^{\ast}$ will be socially resilient.

###  VII-E Lemmas for MTES-ST

######  Lemma 7.

Consider the MTES-ST with the quadratic utility function defined in Assumption
1. The optimal load allocation $x_{i}^{\ast}$, which is an optimal solution of
the optimization problem (3), is obtained as (25).

###### Proof.

Expanding the optimization problem in (17) results in

$\displaystyle\max_{{x}_{i},s_{i}}$
$\displaystyle-\frac{1}{2}b_{i}x_{i}^{2}+(m_{i}b_{i}-\lambda^{\ast})x_{i}+\lambda^{\ast}a_{i}-\lambda^{\ast}s_{i}$
(36) $\displaystyle{\rm s.t.}$ $\displaystyle x_{i},s_{i}\in\mathbb{R}^{\geq
0}.$

Recall that for MTES-ST we have $\lambda^{\ast}\geq 0$ [1]. Therefore, we can
consider two cases.

Case (i) $\lambda^{\ast}=0$. In this case, (36) yields $x_{i}^{\ast}=m_{i}$.

Case (ii) $\lambda^{\ast}>0$. According to Proposition 2, both MTES-ST and
MTES have the same load allocation decisions. Therefore, based on Lemma 1 we
yield (25) is valid when $\lambda^{\ast}>0$.

Finally, considering (i) and (ii), it follows that (25) holds all the time for
MTES-ST. ∎

######  Lemma 8.

Consider the MTES-ST with the quadratic utility function defined in Assumption
1. If $\sum_{i=1}^{n}m_{i}\leq C$, then $\lambda^{\ast}=0$. Conversely, if
$\sum_{i=1}^{n}m_{i}>C$ then $\lambda^{\ast}>0$.

###### Proof.

As mentioned before, for MTES-ST we have $\lambda^{\ast}\geq 0$ [1]. We
investigate two cases.

(i) Consider $\sum_{i=1}^{n}m_{i}\leq C$. By contradiction, suppose
$\lambda^{\ast}>0$. According to (25), $\lambda^{\ast}>0$ yields
$x_{i}^{\ast}<m_{i}$ for $i\in V$, and therefore,
$\sum_{i=1}^{n}x_{i}^{\ast}<\sum_{i=1}^{n}m_{i}$. Since
$\sum_{i=1}^{n}m_{i}\leq C$, we yield $\sum_{i=1}^{n}x_{i}^{\ast}<C$, which
contradicts the equality in (20). Therefore, it follows that
$\lambda^{\ast}=0$.

(ii) Consider $\sum_{i=1}^{n}m_{i}>C$. The inequality constraint
$x_{i}+e_{i}\leq a_{i}$ in (3) yields

$\sum_{i=1}^{n}x_{i}^{\ast}\leq C.$ (37)

By contradiction, suppose $\lambda^{\ast}=0$. According to (25), we obtain
$x_{i}^{\ast}=m_{i}$, and therefore,
$\sum_{i=1}^{n}x_{i}^{\ast}=\sum_{i=1}^{n}m_{i}$. Since
$\sum_{i=1}^{n}m_{i}>C$, we yield $\sum_{i=1}^{n}x_{i}^{\ast}>C$, which
contradicts the inequality in (37). Therefore, it follows that
$\lambda^{\ast}>0$. ∎

######  Lemma 9.

Consider the MTES-ST with the piece-wise linear utility function defined in
Assumption 2. The optimal load allocation $x_{i}^{\ast}$, which is an optimal
solution of the optimization problem (3), satisfies (33).

###### Proof.

The inequality constraint $x_{i}+e_{i}\leq a_{i}$ in the optimization problem
(3) can be written as

$x_{i}+e_{i}+s_{i}=a_{i},$ (38)

where $s_{i}\geq 0$ is the slack variable. Substituting
$e_{i}=a_{i}-x_{i}-s_{i}$ into (3) yields an equivalent form for the
optimization problem as

$\displaystyle\max_{{x}_{i},s_{i}}$
$\displaystyle\min\\{\beta_{i}x_{i},\phi_{i}\beta_{i}\\}+\lambda^{\ast}(a_{i}-x_{i}-s_{i})$
(39) $\displaystyle{\rm s.t.}$ $\displaystyle x_{i},s_{i}\in\mathbb{R}^{\geq
0}.$

Recall that for MTES-ST, we have $\lambda^{\ast}\geq 0$ [1]. We investigate
two cases.

Case (i) $\lambda^{\ast}=0$. Similar to the proof of Lemma 5, case (i), the
objective function in (39) is strictly increasing in the interval
$x_{i}\in[0,\phi_{i}]$, while constant in the interval
$x_{i}\in[\phi_{i},\infty)$. Therefore, the optimal solution is achieved at
$x_{i}^{\ast}\geq\phi_{i}$.

Case (ii) $\lambda^{\ast}>0$. It is proved in Proposition 2 that for
$\lambda^{\ast}>0$, both MTES-ST and MTES yield the same results. Therefore,
this part of the proof is the same as cases (ii), (iii), and (iv) in the proof
of Lemma 5. ∎

######  Lemma 10.

Consider the MTES-ST with the piece-wise linear utility function defined in
Assumption 2. If $\sum_{i=1}^{n}\phi_{i}<C$, then $\lambda^{\ast}=0$.
Conversely, if $\sum_{i=1}^{n}\phi_{i}>C$, then $\lambda^{\ast}>0$.

###### Proof.

It is known that $\lambda^{\ast}\geq 0$ [1]. We investigate two cases.

(i) Consider $\sum_{i=1}^{n}\phi_{i}<C$. By contradiction, if
$\lambda^{\ast}>0$, then the objective function in (39) is strictly decreasing
with respect to $s_{i}$. Consequently, in order for the objective function to
be maximized, $s_{i}$ must be minimized, i.e., $s_{i}^{\ast}=0$. Following
from (38), we obtain $x_{i}^{\ast}+e_{i}^{\ast}=a_{i}$. Then, considering
$\sum_{i=1}^{n}e_{i}^{\ast}=0$ in (4), we yield

$\sum_{i=1}^{n}x_{i}^{\ast}=C,$ (40)

where $C=\sum_{i=1}^{n}a_{i}$. According to (33), when $\lambda^{\ast}>0$ then
$x_{i}^{\ast}\leq\phi_{i}$. Therefore,
$\sum_{i=1}^{n}x_{i}^{\ast}\leq\sum_{i=1}^{n}\phi_{i}$. Since
$\sum_{i=1}^{n}\phi_{i}<C$, we yield $\sum_{i=1}^{n}x_{i}^{\ast}<C$, which
contradicts the equality in (40). Therefore, it follows that
$\lambda^{\ast}=0$.

(ii) Consider $\sum_{i=1}^{n}\phi_{i}>C$. The inequality constraint
$x_{i}+e_{i}\leq a_{i}$ in (3) yields

$\sum_{i=1}^{n}x_{i}^{\ast}\leq C.$ (41)

By contradiction, suppose $\lambda^{\ast}=0$. According to (33), we obtain
$x_{i}^{\ast}\geq\phi_{i}$, and therefore,
$\sum_{i=1}^{n}x_{i}^{\ast}\geq\sum_{i=1}^{n}\phi_{i}$. Since
$\sum_{i=1}^{n}\phi_{i}>C$, we yield $\sum_{i=1}^{n}x_{i}^{\ast}>C$, which
contradicts the inequality in (41). Therefore, it follows that
$\lambda^{\ast}>0$. ∎