# Social Shaping of Competitive Equilibriums
for Resilient Multi-Agent Systems

Yijun Chen, Razibul Islam, Elizabeth Ratnam, Ian R. Petersen, and Guodong Shi
Y. Chen and G. Shi are with the Australian Center for Field Robotics, The
University of Sydney, NSW, Australia. (E-mail<EMAIL_ADDRESS>guodong.shi@sydney.edu.au)R. Islam, E. Ratnam and I. R. Petersen are with the
Research School of Engineering, The Australian National University, Canberra,
Australia. (E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we study multi-agent systems with decentralized resource
allocations. Agents have local demand and resource supply, and are
interconnected through a network designed to support sharing of the local
resource; and the network has no external resource supply. It is known from
classical welfare economics theory that by pricing the flow of resource,
balance between the demand and supply is possible. Agents decide on the
consumed resource, and perhaps further the traded resource as well, to
maximize their payoffs considering both the utility of the consumption, and
the income from the trading. When the network supply and demand are balanced,
a competitive equilibrium is achieved if all agents maximize their individual
payoffs, and a social welfare equilibrium is achieved if the total agent
utilities are maximized. First, we consider multi-agent systems with static
local allocations, and prove from duality theory that under general convexity
assumptions, the competitive equilibrium and the social welfare equilibrium
exist and agree. Compared to similar results in the literature based on KKT
arguments, duality theory provides a direct way for connecting the two notions
and for a more general (e.g. nonsmooth) class of utility functions. Next, we
show that the agent utility functions can be prescribed in a family of
socially admissible functions, under which the resource price at the
competitive equilibrium is kept below a threshold. Finally, we extend the
study to dynamical multi-agent systems where agents are associated with
dynamical states from linear processes, and we prove that the dynamic the
competitive equilibrium and social welfare equilibrium continue to exist and
coincide with each other.

##  1 Introduction

Next generation technologies are leveraging the internet of things (IoT) to
support critical infrastructure systems including energy distribution and
automotive transportation, and are being organized as interconnected multi-
agent systems [6]. Such systems involve data collection, resource allocation,
and control coordination between geographically distributed subsystems. Each
subsystem, termed an ‘agent’, is an intelligent functioning unit with its own
decision, objective and preference, and remarkably, network-level goals such
as consensus, formation, and optimality can be achieved by agents interacting
with others over a network e.g., [1, 2, 3, 4, 5]. The underlying network for
multi-agent systems can be physical such as transmission lines in a power
grid, non-physical such as wireless communication channels, or a combination
of the two. The key promise of organizing subsystems into networked multi-
agent systems is a radical improvement in scalability, efficiency, and
sustainability through shared inputs and outputs, and coordinated decisions
and controls.

One important problem for multi-agent system operation is efficient resource
allocation, where demand and supply must be balanced for efficient and secure
operations at the system level. In a typical resource allocation problem,
agents have local demand and internal and external resource suppliers,
interconnected through a network that allows for transmission of the resource.
In light of classical welfare economics theory [11, 10], careful pricing of
the transmission flow potentially balances the demand against the supply
across the entire system. Agents decide on the resource consumed, and perhaps
further the resource traded, to maximize their payoffs considering both the
utility from production/consumption, and income from the trading. When network
supply and demand is balanced, a competitive equilibrium is achieved if all
agents maximize their individual payoffs; a social welfare equilibrium is
achieved if the total agent utilities are maximized [9].

The concept of resource allocation via a competitive equilibrium has been
widely applied in the smart grid literature, where consumers and suppliers of
electrical energy make trading decisions over an energy market to achieve a
competitive equilibrium [13, 14, 15, 16, 17, 18, 19]. The dynamics of power
networks can even be coupled with the market pricing dynamics [20, 21, 22, 23,
24], where energy price becomes an effective controller for quasi steady-state
grid frequency stability. We refer to [12] for a comprehensive survey on
transactive energy systems. Moreover, in climate-economy frameworks, a market
approach was also proposed as a principled way for mitigating global carbon
emissions while balancing regional interests [26, 27]. The price for carbon
emissions is calculated under the competitive equilibrium of the carbon
market, becoming a benchmark for the social cost of carbon [28]. We refer to
[29] for an excellent introduction into the dynamic integration of climate and
economy models from a feedback system perspective.

Despite the aforementioned successes in coordinating multi-agent systems via
market pricing in critical engineering and societal problems, the resilience
of the pricing mechanism is potentially a serious challenge even for
theoretically optimal equilibrium conditions. In fact, it is well known that
in welfare economics that an efficient market equilibrium may imply Pareto
optimality, while individual equity and fairness of the agents may be
discarded entirely [30]. In the context of power grids, market-driven prices
have exceeded affordable thresholds for an individual household. Specifically,
in February 2021, Texas encountered extreme cold weather conditions resulting
in a power outage disaster throughout the state, with customers on rolling
blackouts. In Texas, competitive pricing did not curtail non-critical loads to
balance the grid under the power shortage conditions [31]. Residential
customers in Texas paying wholesale electricity prices during the power
shortage event reported electricity bill shock with their bills exceeding
previous invoices by a factor of over one hundred. Moreover, in the context of
climate change mitigation, it was only possible for the Paris agreement to be
reached in 2016 after a decade-long negotiation, as the estimated social cost
of carbon was perceived as unfair among different nations [32]. It follows
that a competitive equilibrium for multi-agent systems are in need of social
shaping at the network level: only equilibriums within a prescribed range of
fairness can be accepted; only individual agent utility functions within a
prescribed range of social responsibilities can be admissible. More
importantly, the two directions of social shaping should be consistent:
socially admissible agent utility functions should always lead to socially
acceptable equilibriums.

In this paper, multi-agent systems with decentralized resource allocations are
entirely self-sustained, i.e. there is no external resource supply. First, we
consider multi-agent systems with static local allocations, and we prove that
under general convexity assumptions, the competitive equilibrium and the
social welfare equilibrium exist and agree. Our proof is based on duality
theory, as opposed to the commonly attributed KKT arguments (e.g. [17]) that
support similar results. Here duality theory provides a more direct way of
connecting the competitive equilibrium to the social welfare equilibrium, and
supports a more general (e.g. nonsmooth) class of utility functions. Next, we
investigate the case when the pricing under a competitive equilibrium is
associated with an upper bound for social acceptance. By means of constructive
analysis, we show that the agent utility function is prescribed by a family of
socially admissible quadratic functions, under which the pricing at the
competitive equilibrium is always below a threshold. Finally, we extend the
study to dynamical multi-agent systems where agents are associated with
dynamical states from linear processes, and prove that the dynamic competitive
equilibrium and the social welfare equilibrium continue to exist and coincide
in an optimal control context.

The remainder of the paper is organized as follows. In Section 2, we introduce
the multi-agent system with static decisions. In Section 3, we formulate
social shaping of competitive equilibriums. In Section 4, we formulate dynamic
pricing for resource allocation of multi-agent systems with an underlying
dynamical process. Numerical examples are presented in Section 5 and
concluding remarks are presented in Section 6.

##  2 Static Multi-Agent Systems

###  2.1 Competitive Equilibrium for Static Multi-Agent Systems

We consider a multi-agent system (MAS) with $n$ agents. The agents are indexed
in $\mathrm{V}=\\{1,\dots,n\\}$. We consider a basic MAS setup with static
agent decisions on load allocations.

MAS with Static Agent Load Decisions (MAS-SALD). Each agent $i$ holds a local
resource of $a_{i}$ units, and make a (static) decision to allocate
$x_{i}\in\mathbb{R}^{\geq 0}$ units of load for itself. The utility function
related to agent $i$ allocating $x_{i}$ amount of load is
$f_{i}(x_{i}):\mathbb{R}^{\geq 0}\mapsto\mathbb{R}$. Consequently, agent $i$
would incur an $a_{i}-x_{i}$ amount of surplus ($a_{i}>x_{i}$), or a
shortcoming ($a_{i}<x_{i}$). We assume that there is a connected network among
the $n$ agents so that they can balance the surplus and shortcomings through a
pricing mechanism. To be precise, each unit of resource across the network is
priced at $\lambda\in\mathbb{R}$. Therefore, agent $i$ will yield
$(a_{i}-x_{i})\lambda$ in income or spending.

Denoting $\mathbf{x}=(x_{1}\dots x_{n})^{\top}$ as the network resource
allocation profile, we introduce the following definitions.

######  Definition 1.

A pair of price-allocation decisions ($\lambda^{\ast}$, $\mathbf{x}^{\ast}$)
is a competitive equilibrium for the MAS-SALD if the following conditions
hold:

(i) each agent $i$ maximizes her combined payoff at ${x}^{\ast}_{i}$, i.e.,
${x}^{\ast}_{i}$ is an optimizer for the following constrained optimization
problem:

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

(ii) the total demand and supply are balanced across the network:

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

######  Definition 2.

A resource allocation profile $\mathbf{x}^{\star}$ is a social welfare
equilibrium for the MAS-SALD if it is a solution to the following optimization
problem:

$\displaystyle\max_{\mathbf{x}}$ $\displaystyle\sum_{i=1}^{n}f_{i}(x_{i})$ (3)
$\displaystyle{\rm s.t.}$
$\displaystyle\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}a_{i}$ $\displaystyle
x_{i}\in\mathbb{R}^{\geq 0};\;i\in\mathrm{V}.$

We present the following result which establishes the equivalence between a
competitive equilibrium and a social welfare equilibrium. The result is based
only on a concavity assumption for the utility functions $f_{i}$.

######  Theorem 1.

Consider the MAS-SALD. Suppose each $f_{i}(\cdot)$ is concave over the domain
$\mathbb{R}^{\geq 0}$. Then the social welfare equilibrium(s) and the
competitive equilibrium(s) coincide. To be precise, the following statements
hold.

(i) If ($\lambda^{\ast}$, $\mathbf{x}^{\ast}$) is a competitive equilibrium,
then $\mathbf{x}^{\ast}$ is a social welfare equilibrium.

(ii) If $\mathbf{x}^{\star}$ is a social welfare equilibrium, then there
exists $\lambda^{\ast}\in\mathbb{R}$ such that ($\lambda^{\ast}$,
$\mathbf{x}^{\star}$) is a competitive equilibrium.

Proof. (i) Let ($\lambda^{\ast}$, $\mathbf{x}^{\ast}$) be a competitive
equilibrium. The proof proceeds by contradiction. Suppose that
$\mathbf{x}^{\ast}$ is not a social welfare equilibrium. Then there must exist
$\bar{\mathbf{x}}^{\ast}$ such that
$\sum_{i=1}^{n}\bar{x}_{i}^{\ast}=\sum_{i=1}^{n}{x}_{i}^{\ast}=\sum_{i=1}^{n}a_{i}$,
and
$\sum_{i=1}^{n}f_{i}({x}_{i}^{\ast})<\sum_{i=1}^{n}f_{i}(\bar{x}_{i}^{\ast})$.
Consequently, there holds

$\displaystyle\sum_{i=1}^{n}\Big{(}f_{i}({x}_{i}^{\ast})+\lambda^{\ast}(a_{i}-{x}_{i}^{\ast})\Big{)}<\sum_{i=1}^{n}\Big{(}f_{i}(\bar{x}_{i}^{\ast})+\lambda^{\ast}(a_{i}-\bar{x}_{i}^{\ast})\Big{)}.$
(4)

This implies that there is at least one $m\in\mathrm{V}$ such that

$f_{m}({x}_{m}^{\ast})+\lambda^{\ast}(a_{m}-{x}_{m}^{\ast})<f_{m}(\bar{x}_{m}^{\ast})+\lambda^{\ast}(a_{m}-\bar{x}_{m}^{\ast}),$

which contradicts the fact that ($\lambda^{\ast}$, $\mathbf{x}^{\ast}$) is a
competitive equilibrium.

(ii) We pretend a proof using duality. To be consistent with the literature on
duality theory for continuous optimization, we denote $g_{i}=-f_{i}$, and
rewrite (3) as

$\displaystyle\min_{\mathbf{x}}$ $\displaystyle\sum_{i=1}^{n}g_{i}(x_{i})$ (5)
$\displaystyle{\rm s.t.}$
$\displaystyle\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}a_{i};x_{i}\in\mathbb{R}^{\geq
0},i=1,\dots,n.$

Let $\mathbf{x}^{\star}$ be a social welfare equilibrium111Note that
$\mathbf{x}^{\star}$ must be finite as the feasible set of $\mathbf{x}$ is
compact. . Then from its definition there holds
$\sum_{i=1}^{n}{x}_{i}^{\star}=\sum_{i=1}^{n}a_{i}$. Since (5) is a convex
optimization problem with a linear equality constraint, strong duality holds
[8] and we denote the optimal primal and dual costs of (5) as $p_{\ast}$ and
$d^{\ast}$, respectively.

The Lagrangian function of (5) is

$L(\mathbf{x},\lambda)=\sum_{i=1}^{n}g_{i}(x_{i})+\lambda\Big{(}\sum_{i=1}^{n}x_{i}-\sum_{i=1}^{n}a_{i}\Big{)}:(\mathbb{R}^{\geq
0})^{n}\times\mathbb{R}\mapsto\mathbb{R}.$

Then we introduce

$L^{\ast}(\lambda)=\min_{\mathbf{x}\in(\mathbb{R}^{\geq
0})^{n}}L(\mathbf{x},\lambda).$

If $\lambda^{\ast}$ is dual optimal (i.e.,
$\lambda^{\ast}\in\arg\max_{\lambda\in\mathbb{R}}L^{\ast}(\lambda)$), there
holds from strong duality [8] that

$\displaystyle d^{\ast}=L^{\ast}(\lambda^{\ast})$
$\displaystyle=\min_{\mathbf{x}\in(\mathbb{R}^{\geq
0})^{n}}L(\mathbf{x},\lambda^{\ast})$ (6) $\displaystyle\leq
L(\mathbf{x}^{\star},\lambda^{\ast})$ (7)
$\displaystyle=\sum_{i=1}^{n}g_{i}({x}_{i}^{\star})$ (8)
$\displaystyle=p^{\ast}.$ (9)

This implies the inequality from the above equation actually holds at
equality:

$\displaystyle\mathbf{x}^{\star}\in\arg\min_{\mathbf{x}\in(\mathbb{R}^{\geq
0})^{n}}L(\mathbf{x},\lambda^{\ast}).$ (10)

Note that
$L(\mathbf{x},\lambda^{\ast})=\sum_{i=1}^{n}\big{(}g_{i}(x_{i})+\lambda^{\ast}(x_{i}-a_{i})\big{)}$
implies

$\displaystyle{x}^{\star}_{i}\in\arg\max_{x_{i}\in\mathbb{R}^{\geq
0}}\big{(}f_{i}(x_{i})+\lambda^{\ast}(a_{i}-x_{i})\big{)}.$ (11)

Thus, we have proved that $(\lambda^{\ast},\mathbf{x}_{i}^{\star})$ is a
competitive equilibrium. $\square$

Clearly, in this basic multi-agent system setup, the price $\lambda^{\ast}$
associated with a competitive equilibrium could take negative values. From an
economic point of view, the resource at every agent must either be consumed or
traded, and in cases of an oversupply of resource a negative price for load
balancing would occur. From an optimization point of view, the price
$\lambda^{\ast}$ is the Lagrangian multiplier associated with an equality
constraint for a constrained optimization problem, which can take positive or
negative signs. The following result indicates that as long as one agent is
associated with a non-decreasing utility function, oversupply will not happen.

######  Proposition 1.

Consider the MAS-SALD. Suppose each $f_{i}(\cdot)$ is concave over the domain
$\mathbb{R}^{\geq 0}$. Let ($\lambda^{\ast}$, $\mathbf{x}^{\ast}$) be a
competitive equilibrium. Then $\lambda^{\ast}\geq 0$ if there exists at least
one agent $m\in\mathrm{V}$ such that $f_{m}(\cdot)$ is non-decreasing.

Proof. Let $f_{m}(\cdot)$ be non-decreasing. Assume $\lambda^{\ast}<0$. Then
$f_{m}(x_{m})+\lambda^{\ast}(a_{m}-x_{m}\big{)}$ is a strictly increasing
function with respect to $x_{m}$. Therefore, there can not be a finite
${x}^{\ast}_{m}$ such that
${x}^{\ast}_{m}\in\arg\max_{x_{m}\in\mathbb{R}^{\geq
0}}\big{(}f_{m}(x_{m})+\lambda^{\ast}(a_{m}-x_{m})\big{)}$, contradicting the
definition of the competitive equilibrium. $\square$

###  2.2 MAS with Trading Decisions

In our standing multi-agent system model, agents only decide on their
allocated load $x_{i}$, and the surplus/shortcoming $a_{i}-x_{i}$ have to go
to the network. Now we relax this restriction, and introduce the following
extended MAS.

MAS with Static Agent Load and Trading Decisions (MAS-SALTD) On top of the
MAS-SALD, each agent $i$ further makes a decision on the traded amount of
resource, denoted $e_{i}$. This $e_{i}$ is physically constrained by $x_{i}$
and $a_{i}$ in the following way

  * (i)

if $x_{i}<a_{i}$, then agent $i$ can sell, in which case $e_{i}\geq 0$ and
$e_{i}\leq a_{i}-x_{i}$;

  * (ii)

if $x_{i}\geq a_{i}$, then agent $i$ can only buy, in which case $e_{i}\leq 0$
and $e_{i}=a_{i}-x_{i}$.

Let $\lambda^{\ast}$ continue to be the price for a unit of shared resource.
Denote $\mathbf{e}=(e_{1}\dots e_{n})^{\top}$ as the vector representing the
traded resource profile across the network.

######  Definition 3.

A triplet of price-allocation-trade profile ($\lambda^{\ast}$,
$\mathbf{x}^{\ast}$, $\mathbf{e}^{\ast}$) is a competitive equilibrium for the
MAS-SALTD if the following conditions hold:

(i) Each agent $i$ maximizes her combined payoff at $(\mathbf{x}^{\ast}$,
$\mathbf{e}^{\ast})$ while meeting the physical constraint, i.e.,
$(x_{i}^{\ast}$, $e_{i}^{\ast})$ is an optimizer for the following constrained
optimization problem:

$\displaystyle\max_{{x}_{i},e_{i}}$ $\displaystyle
f_{i}(x_{i})+\lambda^{\ast}e_{i}$ (12) $\displaystyle{\rm s.t.}$
$\displaystyle x_{i}+e_{i}\leq a_{i}$ $\displaystyle x_{i}\in\mathbb{R}^{\geq
0},e_{i}\in\mathbb{R}.$

(ii) The total demand and supply are balanced across the network:

$\sum_{i=1}^{n}e_{i}^{\ast}=0.$ (13)

######  Definition 4.

A pair of resource allocation-trade profile
$(\mathbf{x}^{\star},\mathbf{e}^{\star})$ is a social welfare equilibrium for
the MAS-SALTD if it is an optimizer to the following optimization problem:

$\displaystyle\max_{\mathbf{x},\mathbf{e}}\quad$
$\displaystyle\sum_{i=1}^{n}f_{i}(x_{i})$ (14) $\displaystyle{\rm s.t.}\quad$
$\displaystyle\sum_{i=1}^{n}e_{i}=0,$ (15) $\displaystyle x_{i}+e_{i}\leq
a_{i};i\in\mathrm{V},$ (16) $\displaystyle x_{i}\in\mathbb{R}^{\geq
0},e_{i}\in\mathbb{R};i\in\mathrm{V}.$ (17)

######  Theorem 2.

Consider the MAS-SALTD. Suppose each $f_{i}(\cdot)$ is concave over the domain
$\mathbb{R}^{\geq 0}$. Then the social welfare equilibrium(s) and the
competitive equilibrium(s) continue to coincide under the shared load
decisions for the agents. To be precise, the following statements hold.

(i) If ($\lambda^{\ast},\mathbf{x}^{\ast},\mathbf{e}^{\ast}$) is a competitive
equilibrium, then $(\mathbf{x}^{\ast},\mathbf{e}^{\ast})$ is a social welfare
equilibrium.

(ii) If $(\mathbf{x}^{\star},\mathbf{e}^{\star})$ is a social welfare
equilibrium, then there exists $\lambda^{\ast}\in\mathbb{R}$ such that
($\lambda^{\ast},\mathbf{x}^{\star},\mathbf{e}^{\star}$) is a competitive
equilibrium.

Proof. (i) The proof follows the same analysis as the proof of Theorem 1(i),
where the desired connection is in place with the definitions of the
optimization goals, respectively, for the competitive equilibrium and the
social welfare equilibrium.

(ii) The key idea of the proof continues to be based on strong duality applied
to in the definition of the social welfare equilibrium, as the proof of
Theorem 1. However, now the social welfare equilibrium contains additional
inequality constraints $x_{i}+e_{i}\leq a_{i},i\in\mathrm{V}$. Inevitably they
will lead to auxiliary dual variables, in addition to the dual variable
related to the equality constraint $\sum e_{i}=0$ if we simply repeat the
proof of Theorem 1. In order to highlight the role of the dual variable
corresponding to the equality constraint, and establish it as the price in
competitive equilibrium, we need a refined treatment. To this end, we define a
set $\mathbb{X}_{i}$ for all $i\in\mathrm{V}$ in terms of the inequality
constraint in (16) as $\mathbb{X}_{i}=\\{(x_{i},e_{i})|x_{i}+e_{i}\leq
a_{i};x_{i}\in\mathbb{R}^{\geq 0};e_{i}\in\mathbb{R}\\}$. Clearly,
$\mathbb{X}_{i}$ is a polyhedral set (see Chapter 3.4.2, Duality Theory in
[7]). Denoting again $f_{i}=-g_{i}$, the problem (14)-(17) can be written as:

$\displaystyle\max$ $\displaystyle\sum_{i=1}^{n}g_{i}(x_{i})$ (18)
$\displaystyle{\rm s.t.}$
$\displaystyle(x_{i},e_{i})\in\mathbb{X}_{i},i\in\mathrm{V}$
$\displaystyle\sum_{i=1}^{n}e_{i}=0.\ $

Let $\tau$ be the Lagrange multiplier associated with constraint
$\sum_{i=1}^{n}e_{i}=0$. Subsequently, we can define the dual function where
the primal variables are in a polyhedral set as ( [7], section 5.1.6):

$L^{*}(\tau)=\sum_{i=1}^{n}L^{*}_{i}(\tau),$ (19)

where

$L^{*}_{i}(\tau)=\inf_{(x_{i},e_{i})\in\mathbb{X}_{i}}\Big{(}g_{i}(x_{i})+\tau
e_{i}\Big{)},\hskip 14.22636pti\in\mathrm{V}.$ (20)

Let $(\mathbf{x}^{\star},\mathbf{e}^{\star})$ be a social welfare equilibrium
and $\tau^{\ast}$ be the dual optimal i.e.
$\tau^{\ast}\in\operatorname*{arg\,min}\limits_{\tau\in\mathbb{R}}L^{*}(\tau)$.
Since the problem (18) is feasible and its optimal value is finite, strong
duality holds ([7], Proposition 5.2.1). This means that

$\displaystyle\sum_{i=1}^{n}g_{i}(x_{i}^{\star})$
$\displaystyle=L^{*}(\tau^{\ast})$ (21)
$\displaystyle=\sum_{i=1}^{n}\left(\inf_{(x_{i},e_{i})\in\mathbb{X}_{i}}\Big{(}g_{i}(x_{i})+\tau^{\ast}e_{i}\Big{)}\right)$
(22)
$\displaystyle\leq\sum_{i=1}^{n}g_{i}(x_{i}^{\star})+\tau^{\ast}\sum_{i=1}^{n}e_{i}^{\star}$
(23) $\displaystyle\leq\sum_{i=1}^{n}g_{i}(x_{i}^{\star}).$ (24)

Equation (21) states that the duality gap is zero, (22) comes from the
definition of the dual function, (23) follows since the minimization of
$\sum_{i=1}^{n}g_{i}(x_{i})+\tau^{\ast}\sum_{i=1}^{n}e_{i}$ over
$(x_{i},e_{i})\in\mathbb{X}_{i}$ is always less than or equal to the value at
$\sum_{i=1}^{n}g_{i}(x_{i}^{\star})+\tau^{\ast}\sum_{i=1}^{n}e_{i}^{\star}$,
(24) follows from $\sum_{i=1}^{n}e_{i}^{\star}=0$. We conclude that the two
inequalities hold with equality which implies
$(\mathbf{x}^{\star},\mathbf{e}^{\star})$ minimizes
$\sum_{i=1}^{n}g_{i}(x_{i})+\tau^{\ast}\sum_{i=1}^{n}e_{i}$ over
$(x_{i},e_{i})\in\mathbb{X}_{i}$. Therefore, there holds

$\displaystyle(\mathbf{x}^{\star},\mathbf{e}^{\star})\in\operatorname*{arg\,min}_{\begin{subarray}{c}(x_{i},e_{i})\in\mathbb{X}_{i},\\\
i\in\mathrm{V}\end{subarray}}\sum_{i=1}^{n}g_{i}(x_{i})+\tau^{\ast}\sum_{i=1}^{n}e_{i}.$
(25)

Since (25) is separable in all $i\in\mathrm{V}$, an equivalent formulation is

$\displaystyle(x_{i}^{\star},e_{i}^{\star})\in\operatorname*{arg\,min}_{(x_{i},e_{i})\in\mathbb{X}_{i}}g_{i}(x_{i})+\tau^{\ast}e_{i},\quad
i\in\mathrm{V}.$ (26)

Let us define the equilibrium price $\lambda^{\ast}$ as
$\lambda^{\ast}=-\tau^{\ast}$. It follows from (26) that
$(x_{i}^{\star},e_{i}^{\star})$ is the solution of the following optimization
problem:

$\displaystyle\max$ $\displaystyle f_{i}(x_{i})+\lambda^{\ast}e_{i}$ (27)
$\displaystyle{\rm s.t.}$ $\displaystyle x_{i}+e_{i}\leq a_{i}$ $\displaystyle
x_{i}\in\mathbb{R}^{\geq 0},e_{i}\in\mathbb{R}.\ $

Hence, we conclude that the triplet
($\lambda^{\ast},\mathbf{x}^{\star},\mathbf{e}^{\star}$) is a competitive
equilibrium. $\square$

In the presence of agent trading decisions, the price $\lambda^{\ast}$ under
any competitive equilibrium must be non-negative, as shown in the following
example.

######  Proposition 2.

Consider the MAS-SALTD. Suppose each $f_{i}(\cdot)$ is concave over the domain
$\mathbb{R}^{\geq 0}$. Let
($\lambda^{\ast},\mathbf{x}^{\ast},\mathbf{e}^{\ast}$) be a competitive
equilibrium under the agent trading decisions. Then there always holds that
$\lambda^{\ast}\geq 0$.

Proof. Assume $\lambda^{\ast}<0$. Then $f_{i}(x_{i})+\lambda^{\ast}e_{i}$ is a
strictly decreasing function with respect to $e_{i}$. Since $e_{i}$ is
unbounded below and upper bounded by $e_{i}\leq a_{i}-x_{i}$, there can not be
a finite ${e}^{\ast}_{i}$ such that
${e}^{\ast}_{i}\in\arg\max_{e_{i}\in\mathbb{R}}\big{(}f_{i}(x_{i})+\lambda^{\ast}e_{i}\big{)}$,
contradicting the definition of the competitive equilibrium. This completes
the proof. $\square$

##  3 Social Shaping for Competitive Equilibrium

Consistent with classical welfare economics theory, a competitive equilibrium,
despite being a social welfare equilibrium as well, indicates nothing about
fairness or sustainability. If the optimal pricing $\lambda^{\ast}$ is too
high, agents would option of the system, instead of participating in the self-
sustained multi-agent system. When members leave the system, the achievable
payoff for the remaining agents would go down. Therefore, the agents share a
social responsibility in shaping their utility functions so that
$\lambda^{\ast}$ is within a socially acceptable range.

###  3.1 Shaping the Competitive Equilibrium

Now we present an approach to achieve a socially acceptable competitive
equilibrium, by synthesizing a class of utility functions from which agents
can select. We make the following assumption.

Assumption 1. Each $f_{i}$ is represented by
$f_{i}(x_{i})=-\frac{1}{2}b_{i}x_{i}^{2}+k_{i}x_{i}$, where
$b_{i}\in\mathbb{R}^{\geq 0}$ and $k_{i}\in\mathbb{R}^{\geq 0}$; a utility
function $f_{i}$ is socially admissible if there hold $k_{i}\in[k_{\rm
min},k_{\rm max}]$ and $b_{i}\in[b_{\rm min},b_{\rm max}]$.

Let $\lambda^{\dagger}>0$ represent the highest pricing for $\lambda^{\ast}$
that agents can accept, and we term such a competitive equilibrium
$\lambda^{\ast}\leq\lambda^{\dagger}$ a socially resilient equilibrium. Let
$\mathbf{a}=(a_{1}\dots a_{n})^{\top}$ represent the network resource
allocation profile, and let $C:=\sum_{i=1}^{n}a_{i}$ represent the network
resource capacity. Assuming $C$ and $\mathbf{a}$ are given network
characteristics, we consider the following problem of shaping the competitive
equilibrium.

Problem. (Social Competitive Equilibrium Shaping) Consider the MAS-SALD. Find
the range for $k_{\rm min}$, $k_{\rm max}$, $b_{\rm min}$, $b_{\rm max}$ under
which there always exists a competitive equilibrium that leads to
$0\leq\lambda^{\ast}\leq\lambda^{\dagger}$, for all socially admissible
utility functions.

###  3.2 Socially Admissible Utility Functions

Denote $\mathbf{k}=(k_{1},\dots,k_{n})$ and
$\mathbf{b}=(b_{1},\dots,b_{n})^{\top}$. For two vectors
$\mathbf{l}=(l_{1},\dots,l_{n})$ and
$\mathbf{l}^{\prime}=(l_{1}^{\prime},\dots,l_{n}^{\prime})$, we write
$\mathbf{l}\preceq\mathbf{l}^{\prime}$ if there holds $l_{i}\leq
l_{i}^{\prime}$ for all $i=1,\dots,n$. In other words, $\preceq$ defines a
partial order for all vectors in $\mathbb{R}^{n}$.

Define

$\displaystyle\mathscr{S}_{\ast}:=\Big{\\{}\big{(}k_{\rm min},k_{\rm
max},b_{\rm min},b_{\rm max}\big{)}\in\mathbb{R}_{\geq 0}^{4}:\frac{nk_{\rm
min}}{b_{\rm max}}\geq C;-\frac{nk_{\rm min}}{b_{\rm max}}+\frac{nk_{\rm
max}}{b_{\rm min}}\leq C;-\frac{n\lambda^{{\dagger}}}{b_{\rm
max}}+\frac{nk_{\rm max}}{b_{\rm min}}\leq C\Big{\\}}.$ (28)

We present the following theorem.

######  Theorem 3.

Consider the MAS-SALD. Let Assumption 1 hold. The following statements hold.

(i) The competitive equilibrium is unique, and therefore, there exists a well-
defined mapping, denoted by $\mathcal{F}(\cdot,\cdot)$, that maps
$(\mathbf{k},\mathbf{b})$ to
$\lambda^{\ast}:=\mathcal{F}(\mathbf{k},\mathbf{b})$ where $\lambda^{\ast}$
belongs to the competitive equilibrium.

(ii) The competitive equilibrium is always socially resilient (ie
$\lambda^{\ast}\leq\lambda^{\dagger}$) for all socially admissible utility
functions as long as $(k_{\rm min},k_{\rm max},b_{\rm min},b_{\rm
max}\big{)}\in\mathscr{S}_{\ast}$.

(iii) Let $(k_{\rm min},k_{\rm max},b_{\rm min},b_{\rm
max}\big{)}\in\mathscr{S}_{\ast}$ be given. The mapping
$\mathcal{F}(\cdot,\cdot)$ is monotone under the partial order $\preceq$ over
$\mathbf{k}$ in the sense that

$\mathcal{F}(\mathbf{k},\mathbf{b})\leq\mathcal{F}(\mathbf{k}^{\prime},\mathbf{b})$

for all socially admissible $\mathbf{k}^{\prime}\preceq\mathbf{k}$.

Proof. (i) Under Assumption 1, all $f_{i}$ are strictly concave. Thus, the
optimization problem (3) is strictly convex, leading to a unique optimal
solution. According to Theorem 1, in a competitive equilibrium
$(\lambda^{\ast},\mathbf{x}^{\ast})$, $\mathbf{x}^{\ast}$ must be the unique
primal optimal solution for (3). While from the definition of equilibrium,
$\lambda^{\ast}$ is by definition the optimal dual variable for (3), which is
also unique.

(ii) Let $(\lambda^{\ast},\mathbf{x}^{\ast})$ be a competitive equilibrium.
Then based on the definition of competitive equilibrium, there holds
${x}_{i}^{\ast}$ is the optimal solution to

$\max_{{x}_{i}\in\mathbb{R}_{\geq
0}}-\frac{1}{2}b_{i}x_{i}^{2}+k_{i}x_{i}+\lambda^{\ast}(a_{i}-x_{i}),\quad
i=1,2,\cdots,n.$

This implies that under the condition $\lambda^{\ast}\leq k_{i}$, there holds

$\displaystyle{x}_{i}^{\ast}=\frac{k_{i}-\lambda^{\ast}}{b_{i}}.$ (29)

We substitute the above form of ${x}_{i}^{\ast}$ into the constraint
$\sum_{i=1}^{n}x_{i}=C$ to obtain

$\displaystyle\Big{(}\sum_{i=1}^{n}\frac{1}{b_{i}}\Big{)}\lambda^{\ast}=\sum_{i=1}^{n}\frac{k_{i}}{b_{i}}-C.$
(30)

We thus confirm that

$\displaystyle 0\leq\lambda^{\ast}$
$\displaystyle=\Big{(}\sum_{i=1}^{n}\frac{k_{i}}{b_{i}}-C\Big{)}/\Big{(}\sum_{i=1}^{n}\frac{1}{b_{i}}\Big{)}$
$\displaystyle\leq\Big{(}\frac{nk_{\rm max}}{b_{\rm
min}}-C\Big{)}/\Big{(}\frac{n}{b_{\rm max}}\Big{)}$ (31)

if there holds

$\frac{nk_{\rm min}}{b_{\rm max}}\leq C.$

From (3.2) we also know

$\displaystyle-\frac{nk_{\rm min}}{b_{\rm max}}+\frac{nk_{\rm max}}{b_{\rm
min}}\leq C$ (32)

guarantees $\lambda^{\ast}\leq k_{i}$. Collecting all conditions for $(k_{\rm
min},k_{\rm max},b_{\rm min},b_{\rm max}\big{)}$, we obtain that
$\lambda^{\ast}\leq\lambda^{\dagger}$ for all socially admissible utility
functions with $(k_{\rm min},k_{\rm max},b_{\rm min},b_{\rm
max}\big{)}\in\mathscr{S}_{\ast}$.

(iii) We have established that if $(k_{\rm min},k_{\rm max},b_{\rm min},b_{\rm
max}\big{)}\in\mathscr{S}_{\ast}$, then

$\displaystyle\lambda^{\ast}=\mathcal{F}(\mathbf{k},\mathbf{b})=\Big{(}\sum_{i=1}^{n}\frac{k_{i}}{b_{i}}-C\Big{)}/\Big{(}\sum_{i=1}^{n}\frac{1}{b_{i}}\Big{)}.$
(33)

It is straightforward to verify that

$\mathcal{F}(\mathbf{k},\mathbf{b})\leq\mathcal{F}(\mathbf{k}^{\prime},\mathbf{b})$

for all socially admissible $\mathbf{k}\preceq\mathbf{k}^{\prime}$. $\square$

##  4 Dynamic Multi-Agent Systems

###  4.1 MAS with Dynamic Agent Load/Trading Decisions

Here we consider the load balancing problem for dynamical multi-agent systems.

MAS with Dynamic Agent Load/Trading Decisions (MAS-DALTD). Each agent
$i\in\mathrm{V}$ is associated with a dynamical state
$\mathbf{y}_{i}(t)\in\mathbb{R}^{m}$, described by

$\displaystyle\mathbf{y}_{i}(t+1)=\mathbf{A}_{i}\mathbf{y}_{i}(t)+\mathbf{B}_{i}\mathbf{u}_{i}(t),\quad
t=0,\dots,T-1,$ (34)

where $\mathbf{u}_{i}(t)\in\mathbb{R}^{m}$ is the control input, and
$\mathbf{A}_{i}$ and $\mathbf{B}_{i}$ are real matrices with proper
dimensions. Associated with $t=0,\dots,T-1$, agent $i$ incurs a utility
function $f_{i}(\mathbf{y}_{i}(t),\mathbf{u}_{i}(t))$; the terminal utility
for agent $i$ is $\Phi_{i}(\mathbf{y}_{i}(T))$. Upon taking the control action
$\mathbf{u}_{i}(t)$, the required resource is defined by the function
$h_{i}(\mathbf{u}_{i}(t))$. Each agent can produce an $a_{i}(t)$ units of
energy at time $t$, and also makes a trading decision $e_{i}(t)$ units of
energy over the network at time $t$. Similarly,

  * (i)

if $h_{i}(\mathbf{u}_{i}(t))<a_{i}(t)$, then agent $i$ can sell, in which case
$e_{i}(t)\geq 0$ and $e_{i}(t)\leq a_{i}(t)-h_{i}(\mathbf{u}_{i}(t))$;

  * (ii)

if $h_{i}(\mathbf{u}_{i}(t))\geq a_{i}(t)$, then agent $i$ will buy, in which
case $e_{i}(t)\leq 0$ and $e_{i}(t)=a_{i}(t)-h_{i}(\mathbf{u}_{i}(t))$.

We denote $\bm{\lambda}=(\lambda_{0}\dots\lambda_{T-1})^{\top}$ as the pricing
vector through the time horizon, where $\lambda_{t}$ is the unit price for
traded energy at step $t$. Consequently, the payoff of agent $i$ throughout
$[0,T]$ is described by

$\sum_{t=0}^{T-1}\bigg{(}f_{i}(\mathbf{y}_{i}(t),\mathbf{u}_{i}(t))+\lambda_{t}e_{i}(t)\Big{)}+\Phi(\mathbf{y}_{i}(T)).$

Denote
$\mathbf{y}(t)=(\mathbf{y}_{1}(t)^{\top}\dots\mathbf{y}_{n}(t)^{\top})^{\top}$,
$\mathbf{u}(t)=(\mathbf{u}_{1}(t)^{\top}\dots\mathbf{u}_{n}(t)^{\top})^{\top}$,
and
$\mathbf{e}(t)=(\mathbf{e}_{1}(t)^{\top}\dots\mathbf{e}_{n}(t)^{\top})^{\top}$.
Further define
$\mathbf{Y}=(\mathbf{y}(0)^{\top}\dots\mathbf{y}(T)^{\top})^{\top}$,
$\mathbf{U}=(\mathbf{u}(0)^{\top}\dots\mathbf{u}(T-1)^{\top})^{\top}$ and
$\mathbf{E}=(\mathbf{e}(0)^{\top}\dots\mathbf{e}(T-1)^{\top})^{\top}$. Also
introduce
$\mathbf{U}_{i}=(\mathbf{u}_{i}(0)^{\top}\dots\mathbf{u}_{i}(T-1)^{\top})^{\top}$,
$\mathbf{E}_{i}=(\mathbf{e}_{i}(0)^{\top}\dots\mathbf{e}_{i}(T-1)^{\top})^{\top}$
and $\mathbf{a}_{i}=(a_{i}(0)^{\top}\dots a_{i}(T-1)^{\top})^{\top}$

######  Definition 5.

Let $\mathbf{y}(0)=\mathbf{y}_{0}\in\mathbb{R}^{mn}$ be given. A triple of
price-control-trading profiles
($\bm{\lambda}^{\ast},\mathbf{U}^{\ast},\mathbf{E}^{\ast}$) is a dynamic
competitive equilibrium if the following conditions hold:

(i) each agent $i$ maximizes its combined payoff under $\mathbf{U}_{i}^{\ast}$
and $\mathbf{E}^{\ast}_{i}$:

$\displaystyle\max_{\mathbf{U}_{i},\mathbf{E}_{i}}$
$\displaystyle\sum_{t=0}^{T-1}\bigg{(}f_{i}(\mathbf{y}_{i}(t),\mathbf{u}_{i}(t))+\lambda_{t}^{\ast}e_{i}(t)\Big{)}+\Phi(\mathbf{y}_{i}(T))$
(35) $\displaystyle{\rm s.t.}$
$\displaystyle\mathbf{y}_{i}(t+1)=\mathbf{A}_{i}\mathbf{y}_{i}(t)+\mathbf{B}_{i}\mathbf{u}_{i}(t),$
$\displaystyle e_{i}(t)\leq a_{i}(t)-h_{i}(\mathbf{u}_{i}(t)),t=0,\dots,T-1;$

(ii) the total demand and supply are balanced across the network for all time,
i.e., there holds

$\sum_{i=1}^{n}e_{i}(t)=0,\quad t=0,\dots,T-1.$

######  Definition 6.

Let $\mathbf{y}(0)=\mathbf{y}_{0}\in\mathbb{R}^{mn}$ be given. A pair of
control-trading profiles $(\mathbf{U}^{\star},\mathbf{E}^{\star})$ is a
dynamic social welfare equilibrium if it is a solution to the following
optimal control problem:

$\displaystyle\max_{\mathbf{U},\mathbf{E}}\quad$
$\displaystyle\sum_{i=1}^{n}\Big{(}\sum_{t=0}^{T-1}f_{i}(\mathbf{y}_{i}(t),\mathbf{u}_{i}(t))+\Phi(\mathbf{y}_{i}(T))\Big{)}$
(36) $\displaystyle{\rm s.t.}\quad$
$\displaystyle\mathbf{y}_{i}(t+1)=\mathbf{A}_{i}\mathbf{y}_{i}(t)+\mathbf{B}_{i}\mathbf{u}_{i}(t),\quad
t=0,\dots,T-1,\;i\in\mathrm{V}$ (37) $\displaystyle e_{i}(t)\leq
a_{i}(t)-h_{i}(\mathbf{u}_{i}(t)),\quad t=0,\dots,T-1,\;i\in\mathrm{V}$ (38)
$\displaystyle\sum_{i=1}^{n}e_{i}(t)=0,\quad t=0,\dots,T-1.$ (39)

###  4.2 Dynamic Competitive Equilibrium

We impose the following assumption.

Assumption 2. (i) $\Phi$ is a concave function; (ii) the $f_{i}$ are concave
functions for $i\in\mathrm{V}$; (iii) the $h_{i}$ are nonnegative convex
functions for $i\in\mathrm{V}$, and $h_{i}(\mathbf{z})<b$ defines a bounded
open set of $\mathbf{z}$ in $\mathbb{R}^{m}$ for $b>0$; (iv)
$\sum_{i=1}^{n}a_{i}(t)>0$ for all $t=1,\dots,T-1$.

We present the following result which establishes the similar connection
between the competitive equilibrium and social welfare equilibrium under this
dynamic setting.

######  Theorem 4.

Consider the MAS-DALTD with $\mathbf{y}(0)=\mathbf{y}_{0}\in\mathbb{R}^{mn}$
be given. Let Assumption 2 hold. The dynamic social welfare equilibrium(s) and
the dynamic competitive equilibrium(s) coincide and the following statements
hold.

(i) If $(\bm{\lambda}^{\ast},\mathbf{U}^{\ast},\mathbf{E}^{\ast})$ is a
dynamic competitive equilibrium, then $(\mathbf{U}^{\ast},\mathbf{E}^{\ast})$
is a dynamic social welfare equilibrium.

(ii) If $(\mathbf{U}^{\star},\mathbf{E}^{\star})$ is a dynamic social welfare
equilibrium, then there exists $\bm{\lambda}^{\ast}\in\mathbb{R}^{T}$ such
that $(\bm{\lambda}^{\ast},\mathbf{U}^{\star},\mathbf{E}^{\star})$ is a
competitive equilibrium.

Proof: (i) The proof of sufficiency follows from a similar analysis as the
proof of Theorem 1. The transition from competitive equilibrium to social
welfare equilibrium under this dynamical setting continues to be a direct
consequence of the formulations of the two underlying optimization problems.

(ii) First of all, the dynamics
$\mathbf{y}_{i}(t+1)=\mathbf{A}_{i}\mathbf{y}_{i}(t)+\mathbf{B}_{i}\mathbf{u}_{i}(t)$
with given $\mathbf{y}(0)$ ensures that any $\mathbf{y}_{i}(t)$ for
$t=1,\dots,T$ is a linear combination of $\mathbf{y}(0)$ and
$\mathbf{u}_{i}(0),\dots,\mathbf{u}_{i}(t-1)$. Therefore, we can always write
for any $i\in\mathrm{V}$ that

$\displaystyle\mathbf{y}_{i}(t)=p_{i,t}\mathbf{y}_{0}+q_{i,t}\mathbf{U}_{i},\
t=0,\dots,T$ (40)

with $p_{i,t}$ and $q_{i,t}$ being matrices with proper dimensions. As a
result, we have

$\displaystyle
f_{i}(\mathbf{y}_{i}(t),\mathbf{u}_{i}(t))=f_{i}(p_{t}\mathbf{y}_{0}+q_{t}\mathbf{U}_{i},\mathbf{u}_{i}(t)):=\tilde{f}_{i,t}(\mathbf{U}_{i}).$
(41)

In view of Assumption 2, and the fact that composition of a concave function
and an affine function continues to be concave, we conclude that
$g_{i,t}(\cdot)$ is a concave function. Similarly,

$\Phi(\mathbf{y}_{i}(t))=\Phi(p_{i,T}\mathbf{y}_{0}+q_{i,T}\mathbf{U}_{i}):=\Phi_{i}(\mathbf{U}_{i})$

where $\Phi_{i}(\cdot)$ is a concave function.

The optimization problem (36)-(39) can be equivalently rewritten as the
following convex programming:

$\displaystyle\min_{\mathbf{U},\mathbf{E}}$
$\displaystyle-\sum_{i=1}^{n}\Big{(}\sum_{t=0}^{T-1}\tilde{f}_{i,t}(\mathbf{U}_{i})+\Phi_{i}(\mathbf{U}_{i})\Big{)}$
(42) $\displaystyle{\rm s.t.}$ $\displaystyle
h_{i}(\mathbf{u}_{i}(t))+e_{i}(t)\leq a_{i}(t),\quad
t=0,\dots,T-1,\;i\in\mathrm{V}$ $\displaystyle\sum_{i=1}^{n}e_{i}(t)=0,\quad
t=0,\dots,T-1.$

Similarly, (35) can be equivalently written as convex programming

$\displaystyle\min_{\mathbf{U}_{i},\mathbf{E}_{i}}$
$\displaystyle-\sum_{t=0}^{T-1}\Big{(}\tilde{f}_{i,t}(\mathbf{U}_{i})+\lambda_{t}^{\ast}e_{i}(t)\Big{)}+\Phi_{i}(\mathbf{U}_{i})$
(43) $\displaystyle{\rm s.t.}$ $\displaystyle e_{i}(t)\leq
a_{i}(t)-h_{i}(\mathbf{u}_{i}(t)),t=0,\dots,T-1.$

Next, with Assumption 2.(iii)-(iv), we can verify that Slater’s condition
holds for (42) and (43), which guarantees strong duality in both problems [8].
Also, noting

$\displaystyle\sum_{i=1}^{n}h_{i}(\mathbf{u}_{i}(t))\leq\sum_{i=1}^{n}a_{i}(t)$
(44)

and the assumption that $h_{i}(\mathbf{z})<b$ defines a bounded open set of
$\mathbf{z}$ in $\mathbb{R}^{m}$ for $b>0$, $\mathbf{u}_{i}(t)$ takes values
in a compact set for all $i$ and $t$. Moreover, since
$h_{i}(\mathbf{u}_{i}(t))\geq 0$ holds for all $i$ and for all $t$, there
holds $e_{i}(t)\leq a_{i}(t)$. The constraint $\sum_{i=1}^{n}e_{i}(t)=0$
further ensures $e_{i}(t)\geq-\sum_{i=1}^{n}a_{i}(t)$ for all $i$ and $t$.
Thus $e_{i}(t)$ also takes values in a compact set for all $i$ and $t$. The
convex programming problem (42) leads to finite primal solution.

The Lagrangian dual function of (42) can be written as

$\displaystyle L(\mathbf{U},\mathbf{E},\bm{\lambda},\bm{\mu})$
$\displaystyle=-\sum_{i=1}^{n}\Big{(}\sum_{t=0}^{T-1}\tilde{f}_{i,t}(\mathbf{U}_{i})+\Phi_{i}(\mathbf{U}_{i})\Big{)}+\sum_{t=0}^{T-1}\sum_{i=1}^{n}\lambda_{t}e_{i}(t)+\sum_{t=0}^{T-1}\sum_{i=1}^{n}\mu_{i,t}\big{(}h_{i}(\mathbf{u}_{i}(t))+e_{i}(t)-a_{i}(t)\big{)}$
$\displaystyle=\sum_{i=1}^{n}L_{i}(\mathbf{U}_{i},\mathbf{E}_{i},\bm{\lambda},\bm{\mu}_{i})$
(45)

where

$\displaystyle
L_{i}(\mathbf{U}_{i},\mathbf{E}_{i},\bm{\lambda},\bm{\mu}_{i})=-\sum_{t=0}^{T-1}\tilde{f}_{i,t}(\mathbf{U}_{i})+\Phi_{i}(\mathbf{U}_{i})+\sum_{t=0}^{T-1}\lambda_{t}e_{i}(t)+\sum_{t=0}^{T-1}\mu_{i,t}\big{(}h_{i}(\mathbf{u}_{i}(t))+e_{i}(t)-a_{i}(t)\big{)}.$
(46)

Here $\mu_{i,t}\geq 0$ since they correspond to the inequality constraints. We
have used the conventional notation
$\bm{\mu}_{i}=(\mu_{i,0},\dots,\mu_{i,T-1})^{\top}$ and
$\bm{\mu}=(\bm{\mu}_{1}^{\top},\dots,\bm{\mu}_{n}^{\top})^{\top}$.

Finally, letting an optimal dual solution of (42) be
$(\bm{\lambda}^{\ast},\bm{\mu}^{\ast})$, there holds from strong duality

$\displaystyle(\mathbf{U}^{\star},\mathbf{E}^{\star})\in\arg\min\
L(\mathbf{U},\mathbf{E},\bm{\lambda}^{\ast},\bm{\mu}^{\ast})$ (47)

if $(\mathbf{U}^{\star},\mathbf{E}^{\star})$ is a dynamic social welfare
equilibrium. This implies from (4.2) that

$\displaystyle(\mathbf{U}_{i}^{\star},\mathbf{E}_{i}^{\star})\in\arg\min\
L_{i}(\mathbf{U},\mathbf{E},\bm{\lambda}^{\ast},\bm{\mu}^{\ast}).$ (48)

Now, $\bm{\mu}^{\ast}$ is obtained by solving

$\displaystyle\max_{\bm{\lambda},\bm{\mu}}\ \min_{\mathbf{U},\mathbf{E}}\
L(\mathbf{U},\mathbf{E},\bm{\lambda},\bm{\mu})=\max_{\bm{\lambda},\bm{\mu}}\
\min_{\mathbf{U},\mathbf{E}}\
\sum_{i=1}^{n}L_{i}(\mathbf{U}_{i},\mathbf{E}_{i},\bm{\lambda},\bm{\mu}_{i})$
(49)

where the maximization and minimization are taken in their respective domains
for $\bm{\lambda},\bm{\mu},\mathbf{U},\mathbf{E}$. As a result, there must
hold

$\displaystyle\bm{\mu}_{i}^{\ast}\in\arg\max_{\bm{\mu}_{i}}\min_{\mathbf{U}_{i},\mathbf{E}_{i}}L_{i}(\mathbf{U}_{i},\mathbf{E}_{i},\bm{\lambda}^{\ast},\bm{\mu}_{i}).$
(50)

It is worth emphasizing that
$L_{i}(\mathbf{U}_{i},\mathbf{E}_{i},\bm{\lambda}^{\ast},\bm{\mu}_{i})$ is,
precisely, the Lagrangian of (43). Therefore, (50) ensures that
$\bm{\mu}_{i}^{\ast}$ is an optimal dual solution of (43), and then from
strong duality (48) ensures that $(\mathbf{U}^{\star},\mathbf{E}^{\star})$ is
an optimal primal solution of (43). In other words, we have proven
$(\bm{\lambda}^{\ast},\mathbf{U}^{\star},\mathbf{E}^{\star})$ is a competitive
equilibrium.

The proof of the desired theorem is now complete. $\square$

##  5 Numerical Examples

###  5.1 MAS with Static Decisions

In this section, several numerical examples are provided for validation of
Theorem 1, 2, 3 and Proposition 2. These experiments are implemented using the
Python software package CVXPY.

Example 1. Consider a multi-agent system with four agents who have local
resource $(a_{1},a_{2},a_{3},a_{4})=(13,14,4,7)$. Each agent $i$ is associated
with a utility function $f_{i}$ which is represented by
$f_{i}(x_{i})=\min(k_{i}x_{i},\beta_{i})$ with
$(k_{1},k_{2},k_{3},k_{4})=(21,20,23,32)$ and
$(\beta_{1},\beta_{2},\beta_{3},\beta_{4})=(135,600,130,150)$.

(i) Let the multi-agent system be MAS-SALD. The social welfare equilibrium can
be computed by numerically solving the optimization problem (3) as

$\mathbf{x}^{\star}=(6.429,21.232,5.652,4.688)^{\top},$

and the corresponding optimal dual variable is also obtained as
$\lambda^{\ast}=20$. Letting $\lambda^{\ast}=20$, we then compute a
competitive equilibrium that satisfies (1)-(2) as

$\mathbf{x}^{\ast}=(6.429,21.232,5.652,4.688)^{\top}.$

In particular, we obtain $x_{1}^{\ast}=6.429$, $x_{3}^{\ast}=5.652$, and
$x_{4}^{\ast}=4.688$ by solving (1), and further establish
$x_{2}^{\ast}=21.232$ from (2). Clearly there holds
$\mathbf{x}^{\star}=\mathbf{x}^{\ast}$, which is consistent with Theorem 1 .

(ii) Let the multi-agent system be MAS-SALTD. We compute the social welfare
equilibrium $(\mathbf{x}^{\star},\mathbf{e}^{\star})$ by solving the
optimization problem (14)-(17) as

$\mathbf{x}^{\star}=(6.429,21.232,5.652,4.688)^{\top}\quad\mathbf{e}^{\star}=(6.571,-7.232,-1.652,2.313)^{\top}.$

The optimal dual variable $\tau^{\ast}$ corresponding to the equity constraint
(15) can be obtained as $\tau^{\ast}=-20$. We take
$\lambda^{\ast}=-\tau^{\ast}=20$ and establish a competitive equilibrium that
satisfies (12)-(13) as

$\mathbf{x}^{\ast}=(6.429,21.232,5.652,4.688)^{\top}\quad\mathbf{e}^{\ast}=(6.571,-7.232,-1.652,2.313)^{\top}.$

In particular, we compute $(x_{1}^{\ast},e_{1}^{\ast})=(6.429,6.571)$,
$(x_{3}^{\ast},e_{3}^{\ast})=(5.652,-1.652)$ and
$(x_{4}^{\ast},e_{4}^{\ast})=(4.688,2.313)$ by solving (12), and further
obtain $(x_{2}^{\ast},e_{2}^{\ast})=(21.232,-7.232)$ from (13). Again there
holds
$(\mathbf{x}^{\star},\mathbf{e}^{\star})=(\mathbf{x}^{\ast},\mathbf{e}^{\ast})$,
which validates Theorem 2. $\square$

Example 2. Consider a multi-agent system with four agents. The utility
function for agent $i$ is in the quadratic form
$f_{i}=-\frac{1}{2}b_{i}x_{i}^{2}+k_{i}x_{i}$ for $i=1,2,3,4$. We consider two
pairs of system parameters

$\mathbf{b}=(2,5,3,4)^{\top}\quad\mathbf{k}=(21,17,23,13)^{\top};$ (PM.1)
$\mathbf{b}^{{}^{\prime}}=(2,5,3,4)^{\top}\quad\mathbf{k}^{{}^{\prime}}=(25,22,24,14)^{\top}.$
(PM.2)

Let the network resource capacity $C=\sum_{i=1}^{4}a_{i}$ take values in an
interval $(0,40).$ We sample the interval $(0,40)$ uniformly with a step-size
$0.8$ to obtain $50$ different values for $C$. For each $C$, we compute the
optimal prices of the system under MAS-SALD and MAS-SALTD.

For MAS-SALD, the optimal dual variables $\lambda^{\ast\rm(PM.1)}_{\rm SALD}$
and $\lambda^{\ast\rm(PM.2)}_{\rm SALD}$ are computed for $10^{4}$ times
corresponding to each value of $C$ by solving (3), respectively, under the
parameter setting (PM.1) and (PM.2). For MAS-SALTD, the optimal dual variables
$\tau^{\ast\rm(PM.1)}_{\rm SALTD}$ and $\tau^{\ast\rm(PM.2)}_{\rm SALTD}$
related to the equity constraint (15) are also computed for $10^{4}$ times
corresponding to each value of $C$ by solving (14)-(17), respectively, under
the parameter setting (PM.1) and (PM.2), and then we take
$\lambda^{\ast\rm(PM.1)}_{\rm SALTD}=-\tau^{\ast\rm(PM.1)}_{\rm SALTD}$ and
$\lambda^{\ast\rm(PM.2)}_{\rm SALTD}=-\tau^{\ast\rm(PM.2)}_{\rm SALTD}$. In
Figure 1, we plot the $50$ points of optimal prices versus $C$, to obtain an
approximate trajectory of the optimal price as a function of $C$.

From the plot, we easily observe that the optimal price $\lambda^{\ast}_{\rm
SALD}$ in MAS-SALD can indeed take negative values; while the optimal price
$\lambda^{\ast}_{\rm SALTD}$ in MAS-SALTD is always non-negative. These
observations are consistent with Proposition 1 and Proposition 2. Moreover,
for both MAS-SALD and MAS-SALTD, the plot shows that the optimal prices
$\lambda^{\ast}_{\rm SALD},\lambda^{\ast}_{\rm SALTD}$ are decreasing as the
network resource capacity $C$ increases. $\square$

Figure 1: The curves of the optimal prices as functions of the network
resource capacity in Example 2. The optimal prices of the MAS-SALD with (PM.1)
and (PM.2) are represented by $\lambda^{\rm(PM.1)\ast}_{\rm
SALD},\lambda^{\rm(PM.2)\ast}_{\rm SALD}$ respectively, while
$\lambda^{\ast\rm(PM.1)}_{\rm SALTD},\lambda^{\ast\rm(PM.2)}_{\rm SALTD}$ are
the optimal prices of MAS-SALTD with (PM.1) and (PM.2) respectively.

Example 3. Consider a MAS-SALD with three agents and network capacity $C=18$.
Each agent’s utility function is set as the quadratic form
$f_{i}=-\frac{1}{2}b_{i}x_{i}^{2}+k_{i}x_{i}$ for $i=1,2,3$. The system’s
highest pricing for $\lambda^{\ast}$ that agents can accept socially is
assumed to be $\lambda^{{\dagger}}=42.$ Take $b_{\rm\min}=4$, $b_{\rm\max}=6$,
$k_{\rm\min}=40$, and $k_{\rm\max}=50$. We can verify such a configuration of
$(b_{\rm\min},b_{\rm\max},k_{\rm\min},k_{\rm\max})$ is a point in
$\mathscr{S}_{\ast}$ defined in (28).

(i) Let $\mathbf{b}$ be fixed to be $\mathbf{b}=(4,5,6)^{\top}$. Take
$k_{3}=42,44,46,48$, respectively. We sample the space for
$(k_{1},k_{2})\in[40,50]^{2}$ and compute the optimal pricing $\lambda^{\ast}$
by solving the optimal dual variable of (3). Then we plot the contour maps for
the optimal price as a function of $k_{1}$ and $k_{2}$ in Figure 2(a).

(ii) Let $\mathbf{k}$ be fixed to be $\mathbf{k}=(44,46,48)^{\top}$. Take
$b_{3}=4.4,4.8,5.2,5.6$, respectively. We sample the space for
$(b_{1},b_{2})\in[4,6]^{2}$ and compute the optimal pricing $\lambda^{\ast}$
by solving the optimal dual variable of (3). Then we plot the contour maps for
the optimal price as a function of $b_{1}$ and $b_{2}$ in Figure 2(b).

From these plots, we observe that the maximum value for the price
$\lambda^{\ast}$ is $21$, which is lower than $\lambda^{{\dagger}}=42$. This
illustrates all socially admissible utility functions for parameters in the
set $\mathscr{S}_{\ast}$ lead to socially acceptable prices, providing a
validation for Theorem 3(ii). From Figure 2(a), the optimal price is monotone
under the partial order $\preceq$ with respect to $\mathbf{k}$, which is
consistent with Theorem 3(iii). $\square$

(a)

(b)

Figure 2: Contour maps for the optimal prices in Example 3.

###  5.2 MAS with Dynamic Decisions

Example 4. Consider a MAS-DALTD with three agents who have initial states

$\mathbf{y}_{1}(0)=\begin{bmatrix}1\\\ 4\\\
\end{bmatrix},\mathbf{y}_{2}(0)=\begin{bmatrix}2\\\ 5\\\
\end{bmatrix},\mathbf{y}_{3}(0)=\begin{bmatrix}3\\\ 3\\\ \end{bmatrix},$

and local resource

$\mathbf{a}_{1}=[50,\cdots,50]_{1\times
T},\quad\mathbf{a}_{2}=[50,\cdots,50]_{1\times
T},\quad\mathbf{a}_{3}=[30,\cdots,30]_{1\times T}.$

The dynamical state $\mathbf{y}_{i}(t)\in\mathbb{R}^{2}$ of agent $i$ is
described by

$\mathbf{y}_{i}(t+1)=\mathbf{A}_{i}\mathbf{y}_{i}(t)+\mathbf{B}_{i}\mathbf{u}_{i}(t),\quad
t=0,\dots,T-1,$

where

$\mathbf{A}_{1}=\begin{bmatrix}-0.6&0\\\ 0&-0.7\\\
\end{bmatrix},\mathbf{A}_{2}=\begin{bmatrix}-0.5&0\\\ 0&-0.2\\\
\end{bmatrix},\mathbf{A}_{3}=\begin{bmatrix}-0.4&0\\\ 0&-0.8\\\
\end{bmatrix},\mathbf{B}_{1}=\begin{bmatrix}2&0\\\ 0&7\\\
\end{bmatrix},\mathbf{B}_{2}=\begin{bmatrix}4&0\\\ 0&6\\\
\end{bmatrix},\mathbf{B}_{3}=\begin{bmatrix}9&0\\\ 0&3\\\ \end{bmatrix}.$

The utility function of agent $i$ is in the quadratic form

$f_{i}(\mathbf{y}_{i}(t),\mathbf{u}_{i}(t))=\mathbf{y}_{i}^{\top}(t)\mathbf{R}_{i}\mathbf{y}_{i}(t)+\mathbf{W}_{i}\mathbf{y}_{i}(t)+\mathbf{u}_{i}^{\top}(t)\mathbf{Q}_{i}\mathbf{u}_{i}(t)+\mathbf{K}_{i}\mathbf{u}_{i}(t),$

where

$\mathbf{R}_{1}=\begin{bmatrix}-5&0\\\ 0&-8\\\
\end{bmatrix},\mathbf{R}_{2}=\begin{bmatrix}-3&0\\\ 0&-7\\\
\end{bmatrix},\mathbf{R}_{3}=\begin{bmatrix}-2&0\\\ 0&-1\\\
\end{bmatrix},\mathbf{Q}_{1}=\begin{bmatrix}-5&0\\\ 0&-4\\\
\end{bmatrix},\mathbf{Q}_{2}=\begin{bmatrix}-1&0\\\ 0&-6\\\
\end{bmatrix},\mathbf{Q}_{3}=\begin{bmatrix}-3&0\\\ 0&-2\\\ \end{bmatrix},$
$\mathbf{W}_{1}=\begin{bmatrix}200&300\\\
\end{bmatrix},\mathbf{W}_{2}=\begin{bmatrix}200&400\\\
\end{bmatrix},\mathbf{W}_{3}=\begin{bmatrix}450&300\\\
\end{bmatrix},\mathbf{K}_{1}=\begin{bmatrix}50&60\\\
\end{bmatrix},\mathbf{K}_{2}=\begin{bmatrix}50&20\\\
\end{bmatrix},\mathbf{K}_{3}=\begin{bmatrix}80&20\\\ \end{bmatrix}.$

The terminal utility of agent $i$ is also set as the quadratic form

$\Phi_{i}(\mathbf{y}_{i}(T))=\mathbf{y}_{i}^{\top}(T)\mathbf{R}_{i}\mathbf{y}_{i}(T)+\mathbf{W}_{i}\mathbf{y}_{i}(T).$

Upon taking $\mathbf{u}_{i}(t),$ the required resource is determined by

$h_{i}(\mathbf{u}_{i}(t))=\mathbf{u}_{i}^{\top}(t)\mathbf{H}_{i}\mathbf{u}_{i}(t),$

where

$\mathbf{H}_{1}=\begin{bmatrix}5&0\\\ 0&8\\\
\end{bmatrix},\mathbf{H}_{2}=\begin{bmatrix}3&0\\\ 0&7\\\
\end{bmatrix},\mathbf{H}_{3}=\begin{bmatrix}2&0\\\ 0&1\\\ \end{bmatrix}.$

Let the time horizon take the value of $T=30.$ We compute the dynamic social
welfare equilibrium $(\mathbf{U}^{\star},\mathbf{E}^{\star})$ by solving the
optimization problem (36)-(39) and the optimal dual variables
$-\bm{\lambda}^{\ast}$ corresponding to (39). Given $\bm{\lambda}^{\ast}$, we
further compute the dynamic competitive equilibrium
$(\mathbf{U}^{\ast},\mathbf{E}^{\ast})$ by solving (35). In Figure 3, we plot
the dynamic social welfare equilibrium and the dynamic competitive
equilibrium. From the plot, we can see that the dynamic social welfare
equilibrium and the dynamic competitive equilibrium agree, which is consistent
with Theorem 4. The dynamic optimal price for traded resource versus $t$ is
also shown in Figure 4, where the price experiences oscillations both at the
beginning and in the end over the time horizon, and holds a steady value in
between.

Figure 3: The dynamic social welfare equilibrium and competitive equilibrium
in Example 4. Figure 4: The dynamic optimal price for traded resource versus
$t$ in Example 4.

##  6 Conclusions

We studied multi-agent systems with decentralized resource allocation without
external resource supply. For multi-agent systems with static local
allocations, we showed that under general convexity assumptions, the
competitive equilibrium and the social welfare equilibrium exist and agree
using a duality analysis. We also studied the problem of social shaping for
competitive equilibriums, where the pricing under a competitive equilibrium is
associated with an upper bound. We presented an explicit family of socially
admissible utility functions under which the pricing is always socially
acceptable. Finally, a dynamical multi-agent system was studied and
generalized in an optimal control context. In future work, numerical
algorithms for the computation of the socially admissible utility function are
possible.

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