arXiv:1911.06458v2 [math.OC] 6 Dec 2021

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Exponentially convergent distributed Nash
equilibrium seeking for constrained
aggregative games
Shu Liang1,2 , Peng Yi1,2 , Yiguang Hong1,2 and Kaixiang Peng3
1 Department of Control Science & Engineering, Tongji

University, Shanghai, 200092, China.
2 Shanghai Research Institute for Intelligent Autonomous

Systems, Shanghai, 201210, China.
3 School of Automation and Electrical Engineering, University of

Science and Technology Beijing Beijing, 100083, China.
Contributing authors: sliang@tongji.edu.cn;
yipeng@tongji.edu.cn; yghong@iss.ac.cn; kaixiang@ustb.edu.cn;
Abstract
Distributed Nash equilibrium seeking of aggregative games is investigated and a continuous-time algorithm is proposed. The algorithm is
designed by virtue of projected gradient play dynamics and distributed
average tracking dynamics, and is applicable to games with constrained
strategy sets and weight-balanced communication graphs. We obtain an
exponential convergence of the proposed algorithm to the Nash equilibrium. Numerical examples illustrate the effectiveness of our methods.
Keywords: Distributed algorithms, aggregative games, projected gradient
play, weight-balanced graph, exponential convergence

1 Introduction
Distributed Nash equilibrium seeking with game-theoretic formulation and
multi-agent system consideration has received research attention from the control and optimization communities, partially due to its applications in smart
grids, communication networks and artificial intelligence. Various distributed
1

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Distributed Nash equilibrium seeking for constrained aggregative games

algorithms for Nash equilibrium or generalized Nash equilibrium seeking have
been developed, which guide a group of discrete-time or continuous-time agents
to achieve the equilibrium based on local data and information exchange over
a network graph [1–6].
Aggregative games have become an important type of games since the
well-known Cournot duopoly model was proposed [7], where the strategic interaction is clearly characterized via an aggregation term. Recently, aggregative
games have been considered in congestion control of communication networks
[8], public environmental models [9], demand response management of power
systems [10], and multiproduct-firm oligopoly [11]. Because of the large-scale
systems involved in these problems, seeking or computing the Nash equilibrium
in a distributed manner is of practical significance.
We consider distributed Nash equilibrium seeking of aggregative games,
where the aggregation information is unavailable to each local player and the
communication graph can be directed with balanced weights. Similar problems have also been investigated in [10, 12–16]. In this work, an exponentially
convergent algorithm design is proposed for the considered problem. First, a
distributed projected gradient play dynamics is designed, where we replace
the global aggregation by its local estimation to calculate the gradient. Then
an average tracking dynamics is augmented, where the distributed tracking
signals are local parts of the aggregation. We analyze these interconnected
dynamics and prove that our distributed algorithm achieves an exponential
convergence to the Nash equilibrium. The contributions are as follows:
• A distributed Nash equilibrium seeking algorithm for aggregative game is
developed. The algorithm is designed with two interconnected dynamics: a
projected gradient play dynamics for equilibrium seeking and a distributed
average tracking dynamics for estimation of the aggregation. The projected
part can deal with local constrained strategy sets, which generalizes those in
[10, 15]. Also, the distributed average tracking dynamics applies to weightbalanced directed graphs, which improves the algorithm in [13].
• Exponential convergence of the proposed distributed algorithm is obtained,
which is consistent with the convergence results in [10, 14, 17] for unconstrained problems and is stronger than those in [14, 17] for constrained ones.
In other words, this is a first work, to our knowledge, to propose an exponentially convergent distributed algorithm for aggregative games with local
feasible constraints.
The rest of paper is organized as follows. Section 2 shows some basic concepts and preliminary results, while Section 3 formulates the distributed Nash
equilibrium seeking problem of aggregative games. Then Section 4 presents our
main results including algorithm design and analysis. Section 5 gives a numerical example to illustrate the effectiveness of the proposed algorithm. Finally,
Section 6 gives concluding remarks.

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2 Preliminaries
In this section, we give basic notations and related preliminary knowledge.
Denote Rn as the n-dimensional real vector space; denote 1n = (1, ..., 1)T ∈
n
R , and 0n = (0, ..., 0)T ∈ Rn . Denote col(x1 , ..., xn ) = (xT1 , ..., xTn )T as the
column vector stacked with column vectors x1 , ..., xn , k · k as the Euclidean
norm, and In ∈ Rn×n as the identity matrix. Denote ∇f as the gradient of f .
A set C ⊆ Rn is convex if λz1 + (1 − λ)z2 ∈ C for any z1 , z2 ∈ C and
0 ≤ λ ≤ 1. For a closed convex set C, the projection map PC : Rn → C is
defined as
PC (x) , argmin kx − yk.
y∈C

The projection map is 1-Lipschitz continuous, i.e.,
kPC (x) − PC (y)k ≤ kx − yk,

∀ x, y ∈ Rn .

A map F : Rn → Rn is said to be µ-strongly monotone on a set Ω if
(x − y)T (F (x) − F (y)) ≥ µkx − yk2 ,

∀ x, y ∈ Ω.

Given a subset Ω ⊆ Rn and a map F : Ω → Rn , the variational inequality
problem, denoted by VI(Ω, F ), is to find a vector x∗ ∈ Ω such that
(y − x∗ )T F (x∗ ) ≥ 0,

∀ y ∈ Ω,

and the set of solutions to this problem is denoted by SOL(Ω, F ) [18]. When Ω
is closed and convex, the solution of VI(Ω, F ) can be equivalently reformulated
via projection as follows:
x ∈ SOL(Ω, F ) ⇔ x = PΩ (x − αF (x)), ∀ α > 0.
It is known that the information exchange among agents can be described
by a graph. A graph with node set V = {1, 2, ..., N } and edge set E is written
as G = (V, E) [19]. The adjacency matrix of G can be written as A = [aij ]N ×N ,
where aij > 0 if (j, i) ∈ E (meaning that agent j can send its information to
agent i, or equivalently, agent i can receive some information from agent j),
and aij = 0, otherwise. A graph is said to be strongly connected if, for any
pair of vertices, there exists a sequence of intermediate vertices P
connected by
N
edges. For i ∈ V, the weighted in-degree and out-degree are diin = j=1 aij and
P
N
diout = j=1 aji , respectively. A graph is weight-balanced if diin = diout , ∀ i ∈ V.
N ×N
The Laplacian matrix is L = Din −A, where Din = diag{d1in , . . . , dN
.
in } ∈ R
The following result is well known.
Lemma 1 Graph G is weight-balanced if and only if L + LT is positive semidefinite;
it is strongly connected only if zero is a simple eigenvalue of L.

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3 Problem Formulation
Consider an N -player aggregative game as follows. For i ∈ V , {1, ..., N },
the ith player aims to minimize its cost function Ji (xi , x−i ) : Ω → R by
i
, where
choosing the local decision variable xi from a local strategy set Ωi ⊂ RnP
n
x−i , col(x1 , ..., xi−1 , xi+1 , ..., xN ), Ω , Ω1 × · · ·× ΩN ⊂ R and n = i∈V ni .
The strategy profile of this game is x , col(x1 , ..., xN ) ∈ Ω. The aggregation
map σ : Rn → Rm , to specify the cost function as Ji (xi , x−i ) = ϑi (xi , σ(x))
with a function ϑi : Rni +m → R, is defined as
N

σ(x) ,

1 X
ϕi (xi ),
N i=1

(1)

where ϕi : Rni → Rm is a map for the local contribution to the aggregation.
The concept of Nash equilibrium is introduced as follows.
Definition 1 A strategy profile x∗ is said to be an Nash equilibrium of the game if
Ji (x∗i , x∗−i ) ≤ Ji (yi , x∗−i ), ∀ yi ∈ Ωi , ∀ i ∈ V.

(2)

Condition (2) means that all players simultaneously take their own best
(feasible) responses at x∗ , where no player can further decrease its cost function
by changing its decision variable unilaterally.
We assume that the strategy sets and the cost functions are wellconditioned in the following sense.
A1: For any i ∈ V, Ωi is nonempty, convex and closed.
A2: For any i ∈ V, the cost function Ji (xi , x−i ) and the map ϕ(xi ) are
differentiable with respect to xi .
In order to explicitly show the aggregation of the game, let us define map
Gi : Rni × Rm → Rni , i ∈ V as
Gi (xi , ηi ) , ∇xi Ji (·, x−i ) |σ(x)=ηi
1
= (∇xi ϑi (·, σ) + ∇σ ϑi (xi , ·)T ∇ϕi ) |σ=ηi .
N

(3)

Also, let G(x, η) , col(G1 (x1 , η1 ), ..., GN (xN , ηN )). Clearly, G(x, 1N ⊗σ(x)) =
F (x), where the pseudo-gradient map F : Rn → Rn is defined as
F (x) , col{∇x1 J1 (·, x−1 ), ..., ∇xN JN (·, x−N )}.
Under A1 and A2, the Nash equilibrium of the game is a solution of the
variational inequality problem VI(Ω, F ), referring to [18]. Moreover, we need
the following assumptions to ensure the existence and uniqueness of the Nash
equilibrium and also to facilitate algorithm design.

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A3: The map F (x) is µ-strongly monotone on Ω for some constant µ > 0.
A4: The map G(x, η) is κ1 -Lipschitz continuous with respect to x ∈ Ω and
κ2 -Lipschitz continuous with respect to η for some constants κ1 , κ2 > 0. Also,
for any i ∈ V, ϕi is κ3 -Lipschitz continuous on Ωi for some constant κ3 > 0.
Note that the strong monotonicity of the pseudo-gradient map F has been
widely adopted in the literature such as [2–4, 10, 14–16].
The following fundamental result is from [18].
Lemma 2 Under A1-A4, the considered game admits a unique Nash equilibrium
x∗ .

In the distributed design for our aggregative game, the communication
topology for each player to exchange information is assumed as follows.
A5: The network graph G is strongly connected and weight-balanced.
The goal of this paper is to design a distributed algorithm to seek the Nash
equilibrium for the considered aggregative game over weight-balanced directed
graph.

4 Main Results
In this section, we first propose our distributed algorithm and then analyze its
convergence.

4.1 Algorithm
Our distributed continuous-time algorithm for Nash equilibrium seeking of the
considered aggregative game is designed as the following differential equations:


ẋ = PΩi (xi − αGi (xi , ηi )) − xi , xi (0) ∈ Ωi

 i


N

X
aij (ηj − ηi ),
θi (0) = 0m
θ̇i = β


j=1



ηi = θi + ϕi (xi )

(4)

Algorithm parameters α and β satisfy

2µβλ2 − 4κ2 κ3
,
κ2 βλ2 + 2µκ2 κ3
2κ2 κ3
β>
,
µλ2
0<α<

(5)

where
κ , κ1 + κ2 · κ3 ,
(6)
and λ2 is the smallest positive eigenvalue of 12 (L + LT ) (L is the Laplacian
matrix).

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The compact form of (4) can be written as



ẋ = PΩ (x − αG(x, η)) − x, x(0) ∈ Ω
θ̇ = −βL ⊗ Im η,
θ(0) = 0mN


η = θ + ϕ(x)

(7)

where ϕ(x) = col(ϕ1 (x1 ), ..., ϕN (xN )). Furthermore, we can rewrite (7) as


ẋ = PΩ (x − αG(x, η)) − x, x(0) ∈ Ω
η̇ = −βL ⊗ Im η + d ϕ(x), η(0) = ϕ(x(0))
dt

(8)

The dynamics with respect to x can be regarded as distributed projected
gradient play dynamics with the global aggregation σ(x) replaced by local
variables η1 , ..., ηN . The dynamics with respect to η is distributed average
tracking dynamics that estimates the value of σ(x). The design idea is similar
to [10, 13]. Here, we use projection operation to deal with local feasible constraints, and replace the nonsmooth tracking dynamics in [13] by this simple
one to cope with weight-balanced graphs.

4.2 Analysis
First, we verify that the equilibrium of dynamics (8) coincides with the Nash
equilibrium x∗ .
Theorem 1 Under A1 - A5, the equilibrium of dynamics (8) is
   ∗ 

x
x
x∗
=
=
.
η
η∗
1N ⊗ σ(x∗ )

(9)

Proof The equilibrium of (8) should satisfy
0n = PΩ (x − αG(x, η)) − x
0mN = −L ⊗ Im η
d
ϕ(x) as zeros. Since G is strongly connected,
which are obtained by setting ẋ, η̇ and dt
L ⊗ Im η = 0 implies η1 = η2 = · · · = ηN = η ⋄ for some η ⋄ to be further determined.
T
Since G is weight-balanced, 1T
N L = 0N . Combining this property with dynamics
(8) yields
N
N
1 X
d
1 X
η̇i = σ(x),
ηi (0) = σ(x(0)).
N
dt
N
i=1

i=1

As a result,
N

1 X
ηi = σ(x),
N

(10)

i=1

which implies that any equilibrium pair (x⋄ , 1N ⊗η ⋄ ) should also satisfy η ⋄ = σ(x⋄ ).

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Substituting x⋄ , 1N ⊗ η ⋄ into the projected equation for the equilibrium yields
0n = PΩ (x⋄ − αG(x⋄ , 1N ⊗ η ⋄ )) − x⋄
= PΩ (x⋄ − αF (x⋄ )) − x⋄ ,
which indicates x⋄ = x∗ . Therefore, the point given in (9) is the equilibrium of (8).
This completes the proof.


In view of the identity (10) derived from (8), let
y , η − 1N ⊗ σ(x).
Then it follows from L1N = 0N and (8) that


ẋ = PΩ x − αG(x, 1N ⊗ σ(x) + y) − x

d
ẏ = −βL ⊗ Im y +
ϕ(x) − 1N ⊗ σ(x)
dt
T
= −βL ⊗ Im y + ∇ϕ(x) − 1N ⊗ ∇σ(x) ·


PΩ x − αG(x, 1N ⊗ σ(x) + y) − x

(11)
(12)

The whole dynamics with respect to x and y consists of two interconnected
subsystems as shown in Fig. 1. Each dynamical subsystem has its own state

Fig. 1 The interconnection of two subsystems (11) and (12).

variable, equilibrium point and external input.
Our convergence results are given in the following theorem.
Theorem 2 Under A1-A5, the distributed continuous-time algorithm (4) with
parameters satisfying (5) converges to the Nash equilibrium with an exponential
convergence rate.

Proof Let
ω1 ,

2α · µ − α2 · κ2
,
2+α·κ

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ω2 , β · λ2 − α · κ2 · κ3 ,
ξ1 , α · κ2 ,
ξ2 , κ3 (2 + α · κ),

and
γ ∗ , ω1 + ω2 −

q

(ω1 − ω2 )2 + 4ξ1 ξ2 .

We will show that the rate of exponential convergence of our algorithm is γ ∗ . It
follows from (5) that γ ∗ > 0, and
(ω1 −

γ∗
γ∗
) · (ω2 −
) = ξ1 ξ2
2
2

Let
H(x) , x − PΩ (x − αF (x)),
e
H(x,
y) , x − PΩ (x − αG(x, 1N ⊗ σ(x) + y)),
e
ξ(x, y) , H(x,
y) − H(x)

We verify the following three properties.
1) kξ(x, y)k ≤ α · κ2 kyk.
2) The map F is κ-Lipschitz continuous.
3) The map H is ω1 -strongly monotone.
Property 1) holds because

e
kξ(x, y)k = kH(x,
y) − H(x)k
= kPΩ (x − αF (x))

− PΩ (x − αG(x, 1N ⊗ σ(x) + y))k
≤ αkF (x) − G(x, 1N ⊗ σ(x) + y))k
≤ α · κ2 kyk.
Property 2) follows from the fact that
kF (y) − F (x)k
= kG(y, 1N ⊗ σ(y)) − G(x, 1N ⊗ σ(x))k
≤ kG(y, 1N ⊗ σ(x) − G(x, 1N ⊗ σ(x))k
+ kG(y, 1N ⊗ σ(y)) − G(y, 1N ⊗ σ(x))k
≤ κ1 ky − xk + κ2 · κ3 ky − xk.
Property 3) holds because
(x − y)T (H(x) − H(y))
= kx − yk2 − (x − y)T ·
(PΩ (x − αF (x)) − PΩ (y − αF (y)))
≥ kx − yk(kx − yk
− kPΩ (x − αF (x)) − PΩ (y − αF (y))k)
≥ kx − yk(kx − yk − kx − αF (x) − (y − αF (y))k,

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and

kx − yk − kx − αF (x) − (y − αF (y))k
=

kx − yk2 − kx − αF (x) − (y − αF (y))k2
kx − yk + kx − αF (x) − (y − αF (y))k

≥

2α(x − y)T (F (x) − F (y)) − α2 kF (x) − F (y)k2
(2 + α · κ)kx − yk

2α · µ − α2 · κ2
kx − yk.
2+α·κ
In addition, there holds the identity H(x∗ ) = 0, since x∗ is the Nash equilibrium.
Consider the following Lyapunov candidate function
1
V1 (x) = kx − x∗ k2 .
2
Its time derivative along the trajectory of (11) is
≥

e
V̇1 = −(x − x∗ )T H(x,
y)

= −(x − x∗ )T (H(x) + ξ(x, y))
= −(x − x∗ )T (H(x) − H(x∗ )) − (x − x∗ )T ξ(x, y)
≤ −ω1 kx − x∗ k2 + kx − x∗ kkξ(x, y)k
≤ −ω1 kx − x∗ k2 + ξ1 kx − x∗ kkyk.

Next, we focus on dynamics (12). Let
ζ(x, y) ,

d
(ϕ(x) − 1N ⊗ σ(x))
dt

= ∇ϕ(x) − 1N ⊗ ∇σ(x)

T

·


PΩ x − αG(x, 1N ⊗ σ(x) + y) − x ,

where the time derivative ẋ is along the dynamics (11).
Clearly, 1T
N ⊗ Im ζ(x, y) = 0. Also, since

kPΩ x − αG(x, 1N ⊗ σ(x) + y) − x∗ k

≤ kPΩ x − αG(x, 1N ⊗ σ(x) + y)


− PΩ x − αG(x, 1N ⊗ σ(x)) k


+ kPΩ x − αF (x) − PΩ x∗ − αF (x∗ ) k

≤ α · κ2 kyk + kx − x∗ k + α · κkx − x∗ k,
there holds
kζ(x, y)k ≤ k∇ϕ(x) − 1N ⊗ ∇σ(x)k·


kPΩ x − αG(x, 1N ⊗ σ(x) + y) − xk

≤ κ3 kPΩ x − αG(x, 1N ⊗ σ(x) + y) − x∗ k
+ κ3 kx − x∗ k

≤ κ3 · (2 + α · κ)kx − x∗ k + α · κ2 · κ3 kyk.
Let
1
1 1T ⊗ Im y,
N N N
1
b⊥ , (IN − 1N 1T
y
N ) ⊗ Im y.
N
b,
y

9

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T
b+y
b⊥ . Since 1T
Then y = y
N L = 0N , it follows from (12) that

ḃ = 0,
y

As a result,

b = 0,
y(t)

b
y(0)
= 0.

b⊥ (t),
y(t) = y

∀ t ≥ 0.

Consider the following Lyapunov candidate function
1
kyk2 .
2
The time derivative of V2 along the trajectory of (12) is
V2 (y) =

V̇2 = −βy T (L ⊗ Im )y + y T ζ(x, y)


1
(L + LT ) ⊗ Im y + y T ζ(x, y)
= −βy T
2


⊥ T 1
T
b )
b⊥ + y T ζ(x, y)
= −β(y
(L + L ) ⊗ Im y
2
b⊥ k2 + y T ζ(x, y),
≤ −β · λ2 ky

where the last inequality follows from Rayleigh quotient theorem [20, Page 234]. Also,
b⊥ (t), ∀ t ≥ 0,
since y(t) = y
b⊥ k2 + y T ζ(x, y)
V̇2 ≤ −β · λ2 ky
= −β · λ2 kyk2 + y T ζ(x, y)

≤ −(β · λ2 − α · κ2 · κ3 )kyk2
+ κ3 · (2 + α · κ)kykkx − x∗ k
= −ω2 kyk2 + ξ2 kykkx − x∗ k.
Combining V1 and V2 , let
V , ξ 2 V1 + ξ 1 V2
The time derivative of V along the trajectory of (11) and (12) is
V̇ ≤ −ξ2 ω1 kx − x∗ k2 + 2ξ2 ξ1 kx − x∗ kkyk − ξ1 ω2 kyk2
= −γ ∗ V − (ω1 −

γ∗
γ∗
)ξ2 kx − x∗ k2 − (ω2 −
)ξ1 kyk2 + 2ξ1 ξ2 kx − x∗ kkyk
2
2

≤ −γ ∗ V.
Therefore, the algorithm converges to the Nash equilibrium with the exponential
convergence rate γ ∗ .

Remark 1 Exponential convergence of distributed algorithms has become a research
topic in recent years. [21] has designed a distributed discrete-time optimization algorithm and proves its exponential convergence via a small-gain approach, while [22]
has introduced a criterion for the exponential convergence of distributed primal-dual
gradient algorithms in either continuous or discrete time. Theorem 2 provides an
exponential convergence result by analyzing the interconnected subsystems.

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5 Numerical Example
Consider a Cournot game played by N = 20 competitive players. For i ∈ V =
{1, ..., N }, the cost function ϑi (xi , σ) and strategy set Ωi are
ϑi (xi , σ) = ai x2i + bi xi + ci xi σ(x),


1
1 i
+√ ,
Ωi = − 1 − ,
2i 10
i
where
ai = 0.1 + 0.01 ∗ sin(i),

bi =

i − ln(i)
,
1 + i + i3

ci = 0.003 ∗ cos(i),
and

N

1 X
σ(x) =
xj .
N
j=1

It can be verified that the game mode satisfies A1-A4 with constants µ =
0.1770, κ1 = 0.2199, κ2 = 0.0030, κ3 = 1. We adopt a network graph as shown
in Fig. 2, which satisfies A5.

Fig. 2 The communication graph of the agents.

To render condition (5), we assign α = 3 and β = 1. The trajectory of
strategy profile generated by our algorithm is shown in Fig. 3.
In order to make some comparisons, we also use directed cycle graph and
undirected Erdos-Renyi (ER) graph for the algorithm. The performance of the
algorithm with these graphs is shown in Fig. 4.
These results indicate that our distributed algorithm exponentially converges to the Nash equilibrium.

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Distributed Nash equilibrium seeking for constrained aggregative games
1
0.5
0
-0.5
-1
-1.5
0

10

20

30

40

50

Fig. 3 The trajectory of strategy profile generated by our distributed algorithm.

2
original graph
cycle graph
Erdos-Renyi graph

1
0
0

10

20

30

40

50

0

10

20

30

40

50

0
-10
-20

Fig. 4 Performance of the algorithm with different graphs.

Finally, we use the undirected ER graph to compare our algorithm with
the one given in [13]. The numerical results are shown in Fig. 5. It indicates
that our algorithm converges faster than that algorithm. In addition, only our
algorithm applies to directed graphs such as the original graph and the directed
cycle graph.

6 Conclusions
A distributed algorithm has been proposed for Nash equilibrium seeking of
aggregative games, where the strategy set can be constrained and the network
is described by a weight-balanced graph. The exponential convergence has

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2
our algorithm
the algorithm in (Liang et al. 2017)

1
0
0

10

20

30

40

50

0

10

20

30

40

50

0
-5
-10

Fig. 5 Performance comparison of the two distributed algorithms.

been established. The effectiveness of our method has also been illustrated by
a numerical example. Further work may consider generalized Nash equilibrium
seeking problem for aggregative games with coupled constraints.

Declarations
The authors confirm that there are no known conflicts of interest associated
with this publication and there has been no significant financial support for
this work that could have influenced its outcome.
We confirm that the manuscript has been read and approved by all named
authors and that there are no other persons who satisfied the criteria for
authorship but are not listed. We further confirm that the order of authors
listed in the manuscript has been approved by all of us.
This paper was supported in part by National Natural Science Foundation of China under Grant 61903027,72171171,62003239, and in part by
Shanghai Municipal Science and Technology Major Project under grant
2021SHZDZX0100, and in part by Shanghai Sailing Program under Grant Nos.
20YF1453000.

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[3] Salehisadaghiani, F., Shi, W., Pavel, L.: Distributed Nash equilibrium
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