Stability and Stabilization of Nash Equilibrium for Uncertain Noncooperative Dynamical Systems With Zero-Sum Tax/Subsidy Approach

A zero-sum tax/subsidy approach for stabilizing unstable Nash equilibria in pseudo-gradient-based noncooperative dynamical systems is proposed without the information of agents’ personal sensitivity parameters. Specifically, we first present several sufficient conditions for guaranteeing stability of an unstable Nash equilibrium in the face of uncertainty. Furthermore, we develop a framework where a system manager constructs a zero-sum tax/subsidy incentive structure by collecting taxes from some agents and giving the same amount of subsidy in total to other agents so that the agents’ payoff structure is properly modified. Finally, we present several numerical examples to illustrate the utility of the zero-sum tax/subsidy approach.


I. INTRODUCTION
G AME THEORY has widely contributed in last decades for investigating noncooperative multiagent systems where many applications are found in both engineering and economics, for example, wireless sensor networks [1], communication channel allocation [2], signal interference avoidance [3], data security in intelligent transportation systems [4], and electricity market [5], to name but a few. Agents in the noncooperative systems mutually affect the selfish decision making of the other agents through the interconnected relations of their utilities or payoffs.
It is common knowledge that in noncooperative systems, the agents' selfish decision making may degrade the social welfare [6], [7]. For example, the tragedy of the commons describes a social trap involving the conflict between the individual interests and the public interest in the allocation of resources [8]. In such a situation, without a person who is entitled to control the entire noncooperative system, every agent expands its demand independently according to his own self-interest, and the limited resources are destined to be overexploited by the unrestricted demands, which eventually harms the common good of all agents in the common resource systems. For the aggregation of such self-interested agents, it has turned out that the imposition of external policies or explicit incentive mechanisms changes agents' decision-making tendencies and hence, results in the endogenously cooperative behaviors in the noncooperative systems [9]- [11]. As a coercion policy, which agents cannot escape once in place, a tax/subsidy approach (TSA) was proposed by [12] to reward or penalize the deviations from the average contribution of the other competitors to the public goods. In contrast to the coercion policy, Varian [13] investigated a compensation mechanism where agents are allowed to voluntarily subsidize the other agents in the prestage when the other agents' decisions are not made yet. The compensation mechanisms are understood as a liberal solution as agents have freedom to escape the mechanism. In usual, the liberal solution works as a weak external rule to the noncooperative system and is expected to be less efficient than the coercion solution.
In order to describe the state change of noncooperative systems, several models are proposed in the literature. Specifically, agents' dynamic decision behaviors are typically characterized by the best-response dynamics (or called dynamic fictitious play) [14], [15] and myopic pseudogradient dynamics (or called better response dynamics, or dynamic gradient play) [16]- [18] for discrete-time and continuoustime systems, respectively. In the pseudogradient dynamics setup, the agents continuously change their state according to the pseudogradient projection onto their own local state space without having foresight. For example, Singh et al. [19] analyzed agents' behaviors in a noncooperative system with two agents and quadratic payoff functions. Bowling and Veloso [20] investigated the agents' behaviors with a variable learning rate for the case where an agent wins (possesses higher utility than the opponent) in the two-agent noncooperative system. The article [21] proposed a congestion control framework for data traffic with the pseudogradient dynamics for the users on the Internet. The article [22] explored the stability change of a noncooperative system with loss-averse agents while [23] discussed the relationship between the positively invariant set and the set of positive externalities for a This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ pseudogradient-based noncooperative system with two agents and quadratic payoffs.
To improve the social utility level, it is preferable to develop a compensation mechanism that collects taxes from some agents and gives subsidies to some other agents. Specifically, Alpcan et al. [24] modified agents' original payoff functions in order to reach the highest social welfare by adding a pricing term among the agents. For stabilizing minimum latency flows in the Braess graphs, [25] considered the capitation tax and subsidy. Morimoto et al. [26] imposed a subsidy mechanism to achieve stabilization for heterogeneous replicator dynamics. It is necessary to emphasize that in the above works the existence of a system manager is assumed and he/she is characterized as a resource owner or distributor who is able to give additional subsidies. However, the system manger in many economic applications serves merely as a mediator and does not have productivity to pay the additional profits to the agents. In such a case, every subsidy has to be financed by taxes taken from the others [27] and hence, the tax/subsidy mechanism ought to be designed in a zero-sum fashion, for example, [28]. Ideally, the system manager has all the knowledge about the noncooperative system, including the payoff functions and the decision dynamics of the agents. In reality, it is often difficult to observe perfect information about the activities of the noncooperative agents. This hidden information is called private information in economics [29] and this uncertainty can be obstructive for designing the incentive mechanisms. Even though in the existing gradient-based Nash equilibrium seeking problems [24], [30]- [32], the seeking speed is predetermined, the rational agents in a noncooperative dynamical system, in general, change their states according to their own inherent sensitivities which may not be observed by the system manager. The work in [33] provided an explicit mechanism by side payments with the idea of transferring the utility in a two agent system, which induces cooperation and drives the noncooperative system to the socially maximum welfare state, but unfortunately, the case with more agents and the sensitivity parameters is not considered. Indeed, even though for a twoagent noncooperative system, the sensitivity parameters do not change the stability property of Nash equilibria [34], they may change the stability property in the system with more than two agents and bring agents' state to a worse utility state.
In this article, we develop a utility-transfer framework for noncooperative systems to remodel agents' dynamical decision making in the face of agents' private information. Specifically, we assume that that the sensitivity parameters in the pseudogradient dynamics are uncertain to the system manager. Under this uncertainty, the system manager is expected to construct a zero-sum tax/subsidy mechanism to (globally) stabilize a Nash equilibrium. To deal with the uncertainty, we first characterize the stability of the Nash equilibrium for arbitrary values of sensitivity and then investigate the zero-sum tax/subsidy framework without knowing the sensitivity parameters. In the proposed TSA, the system manager defines the utility-transfer structure dividing the agents into subgroups so that the utility transfers are completed within the subgroups in a zero-sum and distributed manner. The amounts of tax (negative incentive) and subsidy (positive incentive) for each agent are determined by quadratic incentive functions with well-chosen control parameters. It turns out from the numerical examples that the proposed framework can guarantee global asymptotic stabilizability for some noncooperative systems with nonquadratic payoff functions.
The article is organized as follows. In Section II, we characterize the pseudogradient-based noncooperative dynamical systems. In Section III, we discuss the stability of a Nash equilibrium for multiagent noncooperative systems without knowing agents' sensitivity parameter. In Section IV, we first introduce our zero-sum tax/subsidy mechanism for twoagent noncooperative systems, and then extend it to more general multiagent systems. Furthermore, in Section V, we present a couple of illustrative numerical examples. Finally, in Section VI, the conclusion is given.
Notations: We use the following notations in this article. We write Z for the set of positive integers, R for the set of real numbers, R + for the set of positive real numbers, R m×n for the set of m×n real matrices, and R n for the set of n×1 real column vectors. Furthermore, we write det(·) for determinant, (·) T for transpose, and diag[ · ] for diagonal matrices. Finally, we write the identity matrix and the ones vector of dimension n by I n and 1 n , respectively.

II. PRELIMINARY, MOTIVATIONS, AND PROBLEM STATEMENT A. System Description
Consider the noncooperative system with payoff functions J i : R N → R for agent i ∈ N , where N {1, . . . , N} denotes the set of agents. Each agent i ∈ N controls its state (strategy) x i ∈ R, i ∈ N . Let x = (x i , x −i ) ∈ R N denote all agents' state (strategy) profile, where x −i ∈ R N−1 denotes the agents' state profile except agent i. In this article, we suppose that each agent i aims to increase its own payoff where J i may depend on all the agents' state. We denote the noncooperative system by G(J) with J {J i } i∈N .
Definition 1 [35]: For the noncooperative system G(J), the state profile x * ∈ R N is called a Nash equilibrium of G(J) if At a Nash equilibrium, no agent has any incentive to deviate unilaterally from the equilibrium state if the other agents' state does not change.
Note that the noncooperative system G(J) may not possess any Nash equilibrium. Some sufficient conditions for existence of a Nash equilibrium with the closed convex domain can be found in [16] and [36,Ch. 2]. However, in general, guaranteeing the existence of a Nash equilibrium for an unbounded state space is a complicated problem. In this article, we suppose that there exists at least one Nash equilibrium. In this case, under Assumption 1, since the Nash equilibrium x * satisfies Moreover, it is important to note that the Nash equilibrium is characterized independent of the underlying dynamics.

B. Myopic Pseudogradient Dynamics
In this article, we suppose that each agent continuously changes its state (strategy) of the noncooperative system G(J) in the unbounded state space R N in order to increase its own payoff. Specifically, we assume that the state profile x(·) is available for all the agents and each agent follows the pseudogradient dynamics given bẏ where α i , i ∈ N , are agent-dependent positive constant parameters representing sensitivity to the increasing/decreasing payoff per unit state change [16]. In this case, agents selfishly concern their own payoffs and myopically change their states (strategies) according to the current information without any foresight on the future state of the other agents. The pseudogradient dynamics are widely used as the dynamics for rational and selfish agents [19]- [22], [33], [34], [37]. The agents' moving rates given by (3) are characterized to be proportional to the projection of the gradient of J i (x) onto x i -axis, which is called the pseudo-gradient, but the sensitivity parameters α i , i ∈ N , which decide how fast the agents move, are in many cases private so that they are not observed. It is important to note that at the Nash equilibrium x * ,ẋ(t) = 0 since (2) holds.

C. Motivations and Problem Statement
1) Motivation: Some of the Nash equilibria may be unstable in the noncooperative system G(J), since agents' payoff functions are generally different from each other. For instance, Fig. 1 shows the payoff functions of each agent in a two-agent noncooperative system with an unstable Nash equilibrium. Assume there is a system manager, for example, the governor of the markets, who controls the amount of tax and subsidy (negative and positive incentives, respectively) and demands to stabilize around a Nash equilibrium for encouraging agents to converge to it. Assuming all the information of the payoff functions J i (x), i ∈ N , is known, we suppose that the system manager chooses the Nash equilibrium possessing the largest social utility from the set of Nash equilibria of G(J) as the target Nash equilibrium. A fundamental question is how the system manager designs an incentive mechanism to stabilize the possibly unstable target Nash equilibrium with uncertain sensitivity parameters α i , i ∈ N .
Assumption 2: There exists a known Nash equilibrium which is the target equilibrium such that the system manager wishes to guarantee stability around x * .
Note that the computation of the Nash equilibrium for the noncooperative system G(J) is beyond the scope of this article. The relevant methods for calculating Nash equilibria can be found in [37]- [41] and the references therein.
2) Problem: Consider the the target Nash equilibrium x * with uncertain sensitivity parameters α i , i ∈ N , for the system manager. Our main objectives are two folds: 1) find the condition for determining the stability property of the Nash equilibrium x * with arbitrary α i , i ∈ N and 2) design an explicit incentive mechanism to stabilize the possibly unstable Nash equilibrium x * with the unknown sensitivity parameters

III. STABILITY ANALYSIS OF NASH EQUILIBRIUM WITH UNKNOWN SENSITIVITY PARAMETERS
In this section, we characterize stability properties of the Nash equilibrium of the noncooperative system G(J). Specifically, we first present the results for the general N-agent case, and then specialize the results to 3-agent and 2-agent cases. For the statement of the following results, let α (α 1 , . . . , α N ) and define: Note that under Assumption 1, since the functions J i (x), i ∈ N , are twice continuously differentiable, the matrix (4) is a continuous function with respect to x. Moreover, under Assumption 2, the diagonal terms This fact is used in the analysis of the following results.

A. Stability Analysis for N-Agent Noncooperative Systems
The sensitivity parameters α i , i ∈ N , are inherent to each of the agents and are not exactly observed. Without knowing the value of α for the N-agent noncooperative system, the following results provide several ways to determine stability of the Nash equilibrium.
then the Nash equilibrium x * is unstable for any positive The result is a direct consequence of Lyapunov's indirect method. Specifically, consider the characteristic equa- it follows from (5) that a 0 < 0. Hence, it follows from the Routh or Hurwitz criterion that the Nash equilibrium x * is unstable. The fictitious sensitivity 1 N in (5) can be replaced by anŷ α ∈ R N + to determine instability because it does not change the sign of the determinant of A(J, ·, x * ).
Relation of payoff dependency between the agents can be characterized by defining a graph. For specific graph structures, we can specialize condition (5) as shown in the following examples.
Example 1: Consider the noncooperative system with the payoff dependency given by the center-sponsored star network illustrated in Fig. 2(a), where agent 1 is the center of the network. In this case, note that since the left-hand side of (5) is given by Noting that Assumption 2 implies that then the Nash equilibrium x * is unstable for any positive constants α i , i ∈ N . Example 2: Consider the noncooperative system with the payoff dependency given by the directed ring network illustrated in Fig. 2(b). In this case, note that since then the Nash equilibrium x * is unstable for any positive constants α i , i ∈ N . Now, a sufficient condition is provided to guarantee stability without knowing α i , i ∈ N , in the following theorem.
Theorem 1: Consider the Nash equilibrium x * ∈ R N for the N- then the Nash equilibrium x * is locally asymptotically stable for any positive constants is satisfied, it follows using the linearized dynamics (6) that: around x * and hence, the Nash equilibrium x * is asymptotically stable for all positive sensitivity parameters α i , i ∈ N . Remark 1: The result in Theorem 1 appears to be similar to [16, Ths. 8 and 9] but it is certainly different in that Theorem 1 guarantees asymptotic stability for arbitrary α by evaluating the sign definiteness of A T (J,α, x * ) + A(J,α, x * ) for a particularα. To determine whether suchα exists, we can address the linear matrix inequality (LMI) feasibility problem given by (12) assuming that all the information of J are known.
Remark 2: Because of the continuity of A(J,α, x) with respect to x, (10) implies that there exists a connected set T denote the vector field of the pseudogradient dynamics and let V(x) f T (x)Pf (x). It is important to note that a subset of the region of attraction can be characterized by with the maximum attainable δ ∈ R + such that D δ 2 ⊆ Dα 1 and Dδ 2 is connected in the neighborhood of x * for allδ < δ. This is because V(x) is understood as a Lyapunov function and it satisfiesV( It is important to note that the estimated region of attraction D δ 2 depends on the choice ofα in A(J,α, x * ) and can be substantially smaller than the actual region of attraction. But for the special case where A T (J,α, x) + A(J,α, x) < 0 holds for all x ∈ R N , since it can be shown that f (x) = 0 only when x = x * in R N , it follows that the Nash equilibrium x * is globally asymptotically stable for arbitrary α. For instance, if the payoff functions are quadratic, then (10) guarantees global asymptotic stability as (4) is a constant matrix.
Remark 3: For the noncooperative system with quadratic payoff functions where (∂ 2 J i (x * )/∂x i ∂x j ) ≥ 0, i, j ∈ N , i = j, it follows from the properties of the Metzler matrix that (10) in Theorem 1 is also a necessary condition for the Nash equilibrium x * to be asymptotically stable for arbitrary α.
Example 3: Consider the N-agent noncooperative system with the payoff dependency given by the center-sponsored star network illustrated in Fig. 2(a). To investigate the conditions for the payoff functions J i (x), i ∈ N , such thatα ∈ R N + exists to satisfy (10), note that the kth-order leading principal minor of A T (J,α, x * ) + A(J,α, x * ) withα 1 = 1 is given by (10) is equivalent to for k = 2, . . . , N. Therefore, since all the terms in the righthand side are negative, the existence problem ofα in satisfying (10) is equivalent to finding a solutionα = (1,α 2 , . . . ,α N ) for (15) with k = N. Now, suchα exists if and only if the simple condition Remark 4: Note that the local stability of the Nash equilibrium x * under the dynamics (3) can also be directly derived if the matrix A(J, α, x * ) [or, equivalently, [31]. The proof is based on Gershgorin's circle theorem [42].

B. Stability Analysis for 3-Agent Noncooperative Systems
Recall that based on Lyapunov's stability method, Theorem 1 requires us to look forα to make the symmetric part of A(J,α, x * ) negative definite to guarantee stability. For the case of N = 3, it is possible to characterize a different set of stability conditions on the payoff functions based on the Hurwitz criterion.

Remark 5:
The conditions in Proposition 1 provide different sufficient conditions from the one in Theorem 1.
⎦ , but in this case, the condition in (19) is false. For a special case of the payoff dependency, it is interesting to observe that the conditions in Proposition 1 are equivalent to (10) in Theorem 1. In such a case, (17)-(19) guarantee the existence ofα for A T (J,α, x * ) + A(J,α, x * ) < 0 as shown in the following remark.
Remark 6: Consider the 3-agent noncooperative system with the payoff dependency given by the undirected serial graph, which is a special case of the centersponsored star network discussed in Example 3. Note that (∂ 2 J 2 (x * )/∂x 2 ∂x 3 ) = (∂ 2 J 3 (x * )/∂x 3 ∂x 2 ) = 0 because J 2 (x) and J 3 (x) are not the functions of x 3 and x 2 , respectively. In this case, inequality (19) is automatically satisfied. Furthermore, note that Hence, the conditions (17)- (19) are satisfied if and only if where the right-hand side is same as (16). Therefore, for this special case of the payoff dependency, Proposition 1 provides exactly the same sufficient conditions as the one given in Theorem 1.

C. Stability Analysis for 2-Agent Noncooperative Systems
Now, we assume N = 2 for the noncooperative system G({J 1 , J 2 }). The following results are investigated in [34] and fundamental in constructing the incentive function that we develop in Section IV. First, we note that stability can be determined by the sign of the determinant of A.
Proposition 2 [34]: Consider the Nash equilibrium x * ∈ R 2 for the 2-agent noncooperative system G({J 1 , J 2 }) with pseudogradient dynamics (3). If the payoff functions J 1 (x) and then the Nash equilibrium x * is asymptotically stable for any positive constants α 1 , α 2 > 0. Remark 7: The undirected graph topology of the payoff dependency for the 2-agent system is a special case of the center-sponsored star network discussed in Example 3. Note that (24) is equivalent to (16) by letting N = 2, and hence, (24) represents the necessary and sufficient condition for the existence ofα in Theorem 1.
It follows from Corollary 1 (for N = 2) and Proposition 2 that if det A({J 1 , J 2 }, {1, 1}, x * ) > 0 (resp., < 0), then the Nash equilibrium x * is asymptotically stable (resp., unstable). This fact implies that the existence ofα for (10) is in fact the necessary and sufficient condition for stability of x * assuming that there is no eigenvalue of A({J 1 , J 2 }, {α 1 , α 2 }, x * ) on the imaginary axis. In the case where det A({J 1 , J 2 }, {1, 1}, x * ) = 0 implying that at least one of the eigenvalues of A({J 1 , J 2 }, {α 1 , α 2 }, x * ) is zero, the Nash equilibrium x * of (3) may be stable or unstable depending on the payoff functions that the agents are associated with. For an example of addressing the center manifold to determine stability, see [34]. The next result shows the fact that the eigenvalues of the 2 × 2 Jacobian matrix of an unstable Nash equilibrium does not possess complex conjugate eigenvalues.
Here, we define a noncooperative system G({J 1 , J 2 }) with the quadratic payoff functions given by where A i a i 11 a i 12 a i 12 a i 22 ∈ R 2×2 is symmetric with T ∈ R 2 , and c i ∈ R, i = 1, 2. Note that different from the noncooperative system with nonquadratic payoff functions, if the Jacobian matrix is singular, then there may exist infinitely many Nash equilibria.
Example 4: Consider the 2-agent noncooperative system G({J 1 , J 2 }) with the quadratic payoff functions (25). Since det A({J 1 , J 2 }, {1, 1}, x * ) = 4(a 1 11 a 2 22 −a 1 12 a 2 12 ), it follows from Proposition 2 that if the payoff functions J 1 (x) and J 2 (x) satisfy a 1 11 a 2 22 < a 1 12 a 2 12 (resp., a 1 11 a 2 22 > a 1 12 a 2 12 ), then the Nash equilibrium x * is unstable (resp., asymptotically stable). Three typical examples showing the vector fields with different combinations of eigenvalues are given in Fig. 3, and the payoff functions of each agent for the unstable case [ Fig. 3(a)] are shown in Fig. 1 above. Notice that when a 1 11 a 2 22 = a 1 12 a 2 12 , the red and the blue lines in Fig. 3, which represent the best response state of agents 1 and 2, respectively, coincide with each other, and the Nash equilibrium x * is Lyapunov stable (all the trajectories converge to the line in this case).

IV. STABILIZATION OF EXISTING NASH EQUILIBRIUM WITH ZERO-SUM TAX/SUBSIDY APPROACH
In this section, we characterize the stabilization method, which is called a TSA around the target Nash equilibrium x * for the noncooperative system without the knowledge of the sensitivity parameters α i , i ∈ N . In this framework, the system manager imposes an incentive mechanism so that the possibly unstable Nash equilibrium state x * is stabilized by transferring the utility between the agents in a zero-sum fashion, that is, the payoff functions of agents are changed toJ In this case, the pseudogradient dynamics (3) are consequently changed tȯ and the corresponding Jacobian matrix (4) at the Nash equilibrium is given by A(J, α, x * ). Here, we suppose that the amount of tax/subsidy affects the agents' utility in the additive way. We begin by characterizing the TSA for the simple 2-agent noncooperative systems, and then extend the approach to more general N-agent systems.

A. Tax/Subsidy Approach for 2-Agent Case
In this section, we discuss the TSA for the 2-agent noncooperative system G({J 1 , J 2 }). Specifically, consider the noncooperative system G({J 1 ,J 2 }) with the adjusted payoff functions J 1 (x),J 2 (x) given bỹ where p k : R 2 → R denotes an incentive function, which is twice continuously differentiable, k is a scalar parameter, and J 1 (x) and J 2 (x) are the original payoff functions satisfying Assumption 2. The incentive function p k (x) can be considered to be a feedback that is designed by the system manager. Note that p k (x) should be determined in such a way that x * remains the Nash equilibrium of G({J 1 ,J 2 }) andJ 1 (x),J 2 (x) should be still partially strictly concave at the desired Nash equilibrium x * , that is Furthermore, p k (x) should satisfy for all k ∈ R, which guaranteeJ i (x * ) = J i (x * ) and ∂J i (x * )/∂x i = 0, i = 1, 2. This framework indicates that the system manager collects tax p k (x) from one agent and gives the same amount to the other agent as subsidy, so that the respective payoff functions are accordingly changed to stabilize the possibly desirable Nash equilibrium. Note that (30) implies that there is no compensation once the agents reach the target Nash equilibrium. Corollary 2: Consider the 2-agent noncooperative system G({J 1 , J 2 }) with TSA (28) and the pseudogradient dynamics (27). If p k (x) in (28) satisfies then the Nash equilibrium x * is stabilized for any positive constants α 1 and α 2 .
Proof: The result is a direct consequence of Proposition 2.
As a typical form of the TSA, we consider the case with a simple quadratic incentive function given by which satisfies (29) and (30) for all k ∈ R. In this case, (31) for k to stabilize the Nash equilibrium is given by where Similarly, consider the case with a simple quadratic incentive function which satisfies (29), (30) for all k ≤ 0. In this case, , and hence, the state profile x * remains the Nash equilibrium of G(J). Moreover, the condition (31) for k to stabilize the Nash equilibrium is given by For the case where the original payoff functions are quadratic as given in (25), the stabilizing condition of k for the incentive function (32) [resp., (36)] is given by (33) with γ 1 = a 1 12 − a 2 12 − (a 1 12 + a 2 12 ) 2 − 4a 1 11 a 2 22 , γ 2 = a 1 12 − a 2 12 + (a 1 12 + a 2 12 ) 2 − 4a 1 11 a 2 22 (resp., k < a 1 11 + a 2 22 − (a 1 11 − a 2 22 ) 2 + 4a 1 12 a 2 12 ).

B. Distributed Tax/Subsidy Approach for N-Agent Case
In the following, we extend the TSA characterized in the previous section to a higher dimensional system G(J) with N = {1, . . . , N}. In particular, we suppose that the system manager decomposes the agents into several subgroups and installs distributed controllers (computers) for each of the subgroups. Each of the distributed controllers defines a utility transfer structure represented by a graph within the subgroup, which we call the tax/subsidy adjustment graph, such that the graph is weakly connected. Even though the controllers work in a distributed manner, the system manager needs to know, a priori, the information of the payoff functions of all the agents before the operation.
We suppose that the number of subgroups is c and the tax/subsidy adjustment graphs G 1 , . . . , G c are chosen as undirected graphs in such a way that there is no isolated agent that is free from the compensation mechanism. It is important to note that each distributed controller η ∈ {1, . . . , c} transfers the utilities between the agents consisting of G η with the information from the same set of the agents, that is, where V η denotes the set of nodes constituting the tax/subsidy adjustment graph G η . Henceforth, let N i be the set of neighbor agents for agent i. Now, consider the adjusted payoff functions given bỹ with the quadratic incentive functions where and N i is the number of the agents in N i . Note that p K i (x) depends only on part of the agents' state x i , i ∈ V η , in the subgraph G η . Furthermore, if there are multiple subgroups, then K can be transformed to a block-diagonal matrix by reordering the labels of the agents. Notice that the incentive functions given by (39) are a generalization of the combined functions of (32) and (36). Furthermore, (39) implies for all i ∈ N . In this case, since (39) implies and hence, the state profile x * remains the Nash equilibrium of G(J). Consequently, the Jacobian matrix of the adjusted pseudogradient dynamics is written as A(J, α, The following result provides a way to determine K ∈ K in the incentive functions given by (39) for the N-agent noncooperative system. Corollary 3: Consider the N-agent noncooperative system G(J) and the pseudogradient dynamics (27). If the matrix K ∈ K in (39) satisfies one of the following two sets of conditions.
for someα ∈ R N + , then the Nash equilibrium x * is stabilized by the TSA (38), (39) for any positive constants α i , i ∈ N . Proof: Note that since k ij = 0, j ∈ V η , N number of inequalities characterized by 1) make A(J, 1 N , x * ) strictly diagonally dominant (i.e., i (x * )/∂x i ∂x j )| for all i ∈ N ) and the inequality in 2) makes A T (J,α, Hence, the two results are direct consequences of Gershgorin's circle theorem and Theorem 1, respectively. Remark 8: Corollary 3 indicates that with the information of agents' original payoff functions J 1 , . . . , J N , the system manager can command the distributed controllers to process the tax/subsidy framework (38), (39) by transmitting the information of corresponding elements of a well-chosen matrix K to the distributed controllers. As such, the system manager can stabilize the target Nash equilibrium x * for arbitrary α i , i ∈ N , even though the sensitivity parameters α i , i ∈ N , are unknown to him/her. It can be easily found that N number of inequalities characterized by 1) are always solvable for K ∈ K such that A(J, 1 N , x * ) is strictly diagonally dominant, because k ii , i ∈ N , can be taken to be sufficiently small so that each agent's own utility is dominant compared to the effect by the other agents. Moreover, even though the inequality characterized in 2) is a special linear matrix inequality with the constraint K ∈ K, it is possible to make 2) [i.e., A(J,α, x * ) + A T (J,α, x * )] strictly diagonally dominant to make sure that it is negative definite, that is for x = x * . It is interesting to see that (42) can determine {k ij } i∈N ,j∈{i+1,...,N} with a givenα satisfyingα i −α j = 0, i ∈ N j , j ∈ N i , and k ii ≤ 0, i ∈ N . Furthermore, when (42) is satisfied for all x ∈ R N , it can be shown that the possibly unstable Nash equilibrium x * is globally asymptotically stabilized. Remark 9: The condition 1) in Corollary 3 also indicates that each distributed controller η ∈ {1, . . . , c} can independently choose parameters is given. In other words, each distributed controller η can work in a decentralized way even for the case where the number of the agents is large.
Remark 10: In the case where the number N of the agents is so large that the calculation of the target Nash equilibrium x * is infeasible, our proposed framework can be similarly implemented without calculating the Nash equilibria for G(J). Specifically, by settingx * as the target state, the incentive functions for the subgroup η, η ∈ {1, . . . , c}, are given by with β i ∈ R, i ∈ V η , satisfying and {k ij } i,j∈V η satisfying the condition 2) in Corollary 3 with x * replaced byx * . Note that when the target statex * is not the original Nash equilibrium x * in G(J), the linear terms β i (x i −x * i ) − j∈N i β j (x j −x * j )/N j of the incentive functions (43) with β i ∈ R, i ∈ V η , satisfying (44), contribute to make the target statex * a Nash equilibrium in G(J). In such a case, it is understood that the original Nash equilibrium x * in G(J) is moved to the target statex * in G(J) under the proposed TSA. Alternatively, when the target statex * happens to be the same as the original Nash equilibrium x * in G(J), the condition (44), which is met by the distributed controller, requires β i = 0 in order for (43) to reduce to (39). It is worth noting that the establishment of (43) does not force the system manager to collect global information of the payoff functions J i (x), i ∈ N , since the target statex * is not necessary to be the original Nash equilibrium x * in G(J).

V. ILLUSTRATIVE NUMERICAL EXAMPLES
In this section, a couple of numerical examples are presented for illustrating the results and the conditions concerning the Fig. 4. Trajectories of the states with and without the zero-sum TSA under ten different sets of sensitivity parameters α 1 ∈ [20, 50] and α 2 ∈ [30, 85]. The trajectories of agents' states diverge without the TSA but converge to the target Nash equilibrium x * with the proposed TSA for the same set of sensitivity parameters.

VI. CONCLUSION
We investigated the Nash equilibrium stabilization problem for noncooperative dynamical systems through a TSA. In the proposed TSA, a system manager collects some taxes from some of the agents and gives the same amount in total as subsidies to the neighbor agents in the tax/subsidy adjustment graphs. To deal with the uncertainty in terms of the private information, we explored the stability conditions of Nash equilibria without knowing the private information, and also obtained the conditions under which the state trajectory converges to the originally unstable Nash equilibrium using incentive functions. Finally, we provided the numerical examples for demonstrating stabilization of unstable Nash equilibrium for two-agent and five-agent noncooperative systems.
There still remain several open problems on the analysis and stabilization of agent's selfish behaviors in the noncooperative dynamical systems; for example, the state jumps in the combined dynamics with myopic pseudo-gradient dynamics and best-response dynamics, cognitive hierarchy and risk-averse behavior in pseudogradient dynamics, stabilization under timevarying communication links with unknown number of agents, hierarchical structure of the payoff dependencies, and security problems with malicious attackers, to name but a few.

ACKNOWLEDGMENT
The first author acknowledges the financial support by Chinese Scholarship Council.