Distributed Finite-Time and Fixed-Time Nash Equilibrium Seeking for Non-Cooperative Game with Input Saturation

: In this paper, a modiﬁed distributed Nash equilibrium seeking problem with input saturation has been investigated. The payoff function of each player is related to the communication topology, which is closer to the actual scenes. The sigmoid function is utilized to limit the range of the input. Through the network communication between players and the gradient of the players’ payoff functions, the ﬁnite-time and ﬁxed-time distributed Nash equilibrium seeking protocols with input saturation are given. Using the Lyapunov stability analysis, it is determined that the action of each player converges to the Nash equilibrium if all the players update their action according to the proposed algorithms. The numerical simulations are also provided to testify the algorithms.

An asynchronous gossip-based strategy for Nash equilibrium seeking was presented in [10].Ref. [11] exploited distributed Nash equilibrium seeking protocols for both quadratic and nonquadratic payoff functions by utilizing the estimates of all players' actions for each player.Ref. [12] proposed a distributed averaging integral algorithm, which does not require the algebraic connectivity of the graph.Distributed event-triggered algorithms with weaker step-size condition for searching for Nash equilibrium were considered in [13].The authors of [14] exploited algorithms with time-varying payoff function subject to a convex constraint.In [15], the authors solved the distributed Nash equilibrium seeking problem with shared coupling constraints through a doubly augmented operator splitting method.Ref. [16] designed algorithms on a time-varying communication network based on partial-decision information.[17] proposed the protocol for high-order systems with convex separable coupling constraints.Distributed Nash equilibrium seeking algorithms under heterogeneous Euler-Lagrange systems were exploited in [18].Ref. [19] utilized a hybrid system approach to handle the distributed Nash equilibrium seeking problem with persistent attacks.The authors of [20] adopted the event-triggered method to reduce the communication frequency between players for distributed Nash equilibrium seeking protocols under strongly connected graphs.The authors of [21] presented distributed algorithms which utilize optimal responses instead of the payoff functions' gradients to update the actions of players.
Finite-time and fixed-time control approaches [22,23] have attracted much interest from researchers, as they are expected to provide a stable control system in a finite settling time for many engineering systems.Finite-time or fixed-time control systems for multiagent systems and distributed optimization have been fully studied [24][25][26][27][28][29]; however, there are few results concerning finite-time or fixed-time distributed Nash equilibrium seeking algorithms [30].At the same time, input saturation is also a common and practical phenomenon due to the physical limitations.The authors of [31] designed the distributed

Preliminaries and Problem Formulation
In this section, the distributed Nash equilibrium problem under undirected and connected graphs with input saturation is formulated.Some related conceptions, assumptions and lemmas are also provided.

Problem Formulation
Consider a game with n players.The set of players is denoted by υ = {1, 2, ..., n}, f i (g i (x)) represents the payoff function of player i, where g i (x) = [l i1 * x 1 , l i2 * x 2 , ..., l in * x n ] T , l ij denotes the element of the Laplacian matrix L, x = [x 1 , x 2 , ..., x n ] stands for the vector of the players' actions, x i ∈ R m is the action of player i and m is the dimension of x i .In the game, each player aims to min The distributed algorithm is proposed to make all the players find Nash equilibrium through the action exchange between players under undirected and connected graphs with input saturation when Nash equilibrium of the game exists.

Remark 1.
Existing results about distributed Nash equilibrium algorithms extend generalized Nash equilibrium to distributed systems directly and assume that the payoff function of each agent is determined by all the players and are designed to estimate the decisions of all the player.However, each player can not perceive any other players except itself and its neighbors in the distributed algorithms normally.So it is more reasonable and realistic that the payoff function of each player is determined by itself and its neighbors.In this paper, the payoff function of each player is related to the communication topology, which means that the payoff function of each player is determined by itself and its neighbors.

Graph Theory
Let G = (υ, ε) be a graph, where υ = {1, ..., N} is the node set and ε ⊆ υ × υ is the edge set.An edge from i to j is denoted by (i, j), which means that player i can transmit information to player i.The graph is undirected if (i, j) ∈ ε for anytime (j, i) ∈ ε.The graph is unilaterally connected if there exists a sequence of directed path to go from at least one of the nodes to the other for any two nodes.The graph is strongly connected if there exists at least one path between any two nodes.A = [a ij ] ∈ R N×N is a weighted adjacency matrix related to G , such that a ij = 1 if (j, i) ∈ ε; otherwise, a ij = 0. D = diag{d 1 , ..., d n } ∈ R n×n denotes the degree matrix of the graph G , where d i = ∑ N j=1 a ij .The Laplacian matrix of G is described as L = D − A. sgn(•) represents the signum function.sig(x) a ∆qsgn(x)|x| a .

Nash Equilibrium
, where
(i) If there exists a positive-definite function V(x, t), such that where c > 0 and 0 < α < 1.Then, V(x, t) approaches zero in finite time, and the finite time T is estimated by Then, V(x, t) approaches zero in fixed time, and the fixed time T is estimated by Lemma 3. ∀x ∈ R, e x − 1 >= x and the equal sign holds if and only if x = 0. sgn(e x − 1) = sgn(x).
Thus, e x − 1 >= x and the equal sign holds if and only if x = 0. Notice that e x − 1 > 0 if and only if x > 0, e x − 1 = 0 if and only if x = 0 and e x − 1 < 0 if and only if x < 0, we have sgn(e x − 1) = sgn(x).The proof is complete.
Assumption 1.The graph of the communication topology is undirected and connected.
Assumption 2. The payoff function of each player f i (x) is twice-continuously differentiable.

Assumption 3. The payoff function of each player
for all x, z ∈ R N .

Distributed Finite-Time Nash Equilibrium Seeking Protocol with Input Saturation
In this section, a distributed finite-time Nash equilibrium protocol under undirected and connected graphs with input saturation is given.
Players update their actions by the following equations ẋi = −k i sig(y i ) α , where k i > 0 is the possible maximum input for player i, α < 1 is a positive parameter.
Theorem 1. Suppose Assumptions 1-4 are satisfied; the players update their action according to (3).Then, the actions of all the players converge globally to the Nash equilibrium x * with input limit k i in finite time.
Proof.With Assumption 3, it follows that the Nash equilibrium point for the game is unique.Notice that y i monotonically increases with the increase in z i .Thus, the control system (3) is stable if and only if the equation Finite time: Take the candidate Lyapunov function as With Lemma 3, it follows that sgn( From Assumption 3 and 4, we have We have |z where . With Lemma 3, it follows that κ > 0. Combining ( 4)-( 7), yield Since 0 < α < 1, according to Lemma 1, it follows that From ( 8) and ( 9), we have Vi ≤ −cV . From Lemma 2, it is determined that the actions of all the players converge to the Nash equilibrium x * in finite time.
Input limit: The proof is complete.Remark 2. In this paper, the sigmoid function is utilized to limit the range of the input.Compared to the method used in [31] to realize bounded control, the advantages of the sigmoid function are threefold.
(1) The sigmoid function is continuous and smooth, which makes the actuator achieve a better control effect.(2) Due to the property of the sigmoid function, the deadband of the bounded control system is avoided.
(3) The definition domain and the value domain of the sigmoid function correspond one to one.If the sensor is accurate enough, we can get the input before saturation handling from the input processed by sigmoid function so as to observe the control system better.

Distributed Fixed-Time Nash Equilibrium Seeking Protocol sith Input Saturation
In Theorem 1, the finite-time distributed Nash equilibrium seeking protocol under undirected and connected with bounded control is investigated.The fixed-time case is studied in this section.
The actions of players are determined using the following equations ẋi = − k i 2 (sig(y i ) α + sig(y i ) β ), where 0 < α < 1, β > 1 and k i > 0 are the possible maximum inputs for player i.
Theorem 2. Suppose that Assumptions 1-4 are satisfied, and the players update their action according to (10).Then, the actions of all the players converge globally to the Nash equilibrium x * with input limit k i in fixed time 1) .
Proof.Similar to the analysis in Theorem 1, the control system (10) is stable if and only if the equation

Fixed time:
The candidate Lyapunov function is selected as

and it follows that
With Lemma 3 and Assumption 3 and 4, it follows that sgn( (x where η > 0 is the upper bound of |x where κ = 1−e −θ i η θ i η(1+e −θ i η ) . Combining with (11)-( 14), we have With Lemma 1, it follows that From ( 15) and ( 16), yield . Using Lemma 2, it follows that the action of each player converges to the Nash equilibrium in fixed time with the setting time 1) .
Input limit: Based on Theorem 1, we have y Remark 3. In this paper, the payoff function of each player is relative to the communication topology.An example is provided to illustrate how the communication topology affects the payoff function and Nash equilibrium.The communication topology of three players is shown in Figure 3. Player 1 is the neighbor of player 2, player 2 is the neighbor of player 3, and player 3 is the neighbor of player 1.The corresponding Laplacian matrix of the communication topology is Communication topology of three players.
It follows that The payoff functions of the players are designed as The payoff function of each player is just determined by the actions of itself and its neighbors, which is more consistent with the actual situation.Player i searches for the action that can minimize their payoff function along the decreasing direction of the partial derivative, i = 1, 2, 3.At the same time, the actions of their neighbors also affect the action of player i.Finally, the actions of all players reach a balance; that is, the distributed Nash equilibrium mentioned in this paper is achieved.
Remark 4.Although this paper studies the modified distributed Nash equilibrium seeking under undirected and connected graphs, the results in this paper can also extend to the case under unilateral connected graphs.Different from the distributed Nash equilibrium seeking problem studied in [11][12][13][14][15], the payoff function of each player in this paper is determined by the actions of the player and their neighbors, and the condition of undirected and connected needed to estimate the actions of all the player can be weakened for unilateral connected graphs.An example is given as follows to illustrate the application of unilateral connected graphs in non-cooperative games and the modified distributed Nash equilibrium problem in realistic situations.
Example 1: The diagram of commodity sales network is shown in Figure 4. p i , w i and r i represent producers, wholesalers and retailers, respectively.We use the distributed Nash seeking equilibrium problem under unilateral connected graphs to build the model of the commodity price changing.The producer p 1 sells the good to the wholesalers at the price l 1 , the wholesalers w i sells the good to the retailers r j (j = 1, ..., 6) at the price m i , i = 1, 2. Notice that the price of the good for the retailers selling to the customers is affected by its neighbors' prices and upstream prices due to the market regulation.The modified distributed Nash equilibrium problem in this paper can explain some practical problems to a certain extent.

Simulation
In this section, numerical examples are given to validate the finite-time and fixed-time distributed Nash equilibrium seeking protocols with input saturation under undirected and connected graphs.
The communication topology between players is shown in Figure 5.The Laplacian matrix of the communication topology is With the communication topology and g i (x), the payoff functions of the players are designed as The initial actions of the players are set as x(0) = [1, 2, 0, −1, 0, 0] T .It is derived that the unique Nash equilibrium is x * = [−4.2038,−0.5853, −1.0110, −0.6918, 0.3725] T , based on direct calculation.For comparison, the normal distributed Nash equilibrium seeking algorithm without finite-time and fixed-time handling is given as follows: ẋi = −k i y i , the distributed fixed-time Nash equilibrium seeking algorithm with input saturation using piecewise function is given as follows: where ∆ i > 0, Φ ∆ i : R m → R m is a saturation function with a saturation level ∆ i , which means, for q = [q 1 , q 2 , . . ., q m ] T ∈ R m , Take the parameters as 05. Plots of players' actions under algorithms (3), (10), ( 17) and ( 18) are shown in Figure 6, Figure 7, Figure 8 and Figure 9, respectively.The convergence times of the three algorithms are 16.24 s , 27.35 s, 32.64 s and 14.52 s , respectively, which means the finite-time algorithm (3) and the fixed-time algorithm (10) have more advantages in terms of convergence speed and time compared with the normal algorithm (17).As for the input limit, trajectories of the input states under (3), (10), ( 17) and ( 18) are given in Figure 10, Figure 11, Figure 12 and Figure 13, respectively.The input ẋi ranges from −k i to k i all the time for all i = 1, 2, ..., 6.Compared with the input plot in Figure 13, although ẋi in Figure 11 spends more time to change to zero due to the transformation of sigmoid function, the plot in Figure 11 is smoother and avoids the deadband.The algorithms realize the distributed Nash equilibrium seeking with input saturation in finite time and fixed time.

Conclusions
This paper investigates a modified distributed Nash equilibrium seeking problem.The payoff function of each player is determined by the actions of the player and their neighbors.A finite-time protocol and a fixed-time protocol with input saturation are proposed.The sigmoid function is utilized to limit the range of the inputs for the player.The proposed algorithms are theoretically verified using Lyapunov stability analysis.Finally, numerical simulations are given to validate the algorithm; the convergence time of finite-time, fixedtime and normal algorithms is given to illustrate the advantages of the proposed algorithms.Future work will include designing distributed Nash equilibrium algorithms with switching topologies and communication delay.

Figure 4 .
Figure 4.The diagram of commodity sales network.

Figure 5 .
Figure 5. Communication topology for six players.

6 Figure 10 .
Figure 10.Plot of the input ẋi for the players under (3).

6 Figure 11 .
Figure 11.Plot of the input ẋi for the players under (10).

6 Figure 12 .
Figure 12.Plot of the input ẋi for the players under (17).

6 Figure 13 .
Figure 13.Plot of the input ẋi for the players under (18).