Robust Consensus of Networked Evolutionary Games with Attackers and Forbidden Profiles †

Using the algebraic state space representation, this paper studies the robust consensus of networked evolutionary games (NEGs) with attackers and forbidden profiles. Firstly, an algebraic form is established for NEGs with attackers and forbidden profiles. Secondly, based on the algebraic form, a necessary and sufficient condition is presented for the robust constrained reachability of NEGs. Thirdly, a series of robust reachable sets is constructed by using the robust constrained reachability, based on which a constructive procedure is proposed to design state feedback controls for the robust consensus of NEGs with attackers and forbidden profiles. Finally, an illustrative example is given to show that the main results are effective.

board in Figure 1 of [21], where the authors showed that "the profile C2 → B3 is a forbidden move for the black king by the rules of chess." Thus, it is meaningful to investigate the strategy consensus problem of NEGs with attackers and forbidden profiles.
In this paper, we study the robust consensus of NEGs with attackers and forbidden profiles by using the ASSR approach. It should be pointed out that our NEG model only considers synchronized strategy updates and average incomes, which is different from NEG models with random sequential strategy updates and total incomes. The main innovation point of this paper are twofold. For one thing, we propose the concept of robust constrained reachability for NEGs, which is effective in dealing with attackers and forbidden profiles. For another, we establish a constructive procedure to design state feedback controls for the robust consensus of NEGs with attackers and forbidden profiles, which is easily operated with the tool of MATLAB (R2016a, The MathWorks, Natick, US State).
In the sequel, the matrix product is the semi-tensor product of matrices, which is defined as: Given two matrices M ∈ R m×n and N ∈ R p×q . Set α = lcm(n, p) be the least common multiple of n and p. Then, where ⊗ is the Kronecker product. When n = p, STP is equivalent to the conventional matrix product. Therefore, we omit the symbol " " if no confusion arises in the following. For the detailed properties of STP, please see [22,23]. It is noted that the conventional matrix product requires n = p, while STP is applicable to any two real matrices. Thus, STP is a new matrix product. When considering a finite-valued system such as an NEG, if we identify each strategy as a canonical vector, then one can multiply these canonical vectors by STP (in this case, the conventional matrix product is not valid).
In this way, one can convert the dynamics of an NEG into a linear form, which establishes a bridge between NEGs and classic control theory [14]. This is also the motivation for why we use STP to study the robust consensus of NEGs with attackers and forbidden profiles. The rest of this paper is organized as follows. Section 2 formulates the problem investigated in this paper. In Section 3, the main results of this note are given. In Section 4, an illustrative example is given to show the obtained new results, which is followed by a brief conclusion in Section 5.
The notations of this paper are standard. N, Z + and R denote the sets of natural numbers, positive integers and real numbers, respectively. D k := {1, 2, · · · , k}.
, which is briefly expressed as A = δ n [j 1 j 2 · · · j t ]. Denote the set of n × t logical matrices by L n×t . Blk l (M) denotes the l-th n × n block of an n × mn matrix M. For M, N ∈ R n×r , the Khatri-Rao product of M and N is defined as where Col s (M) denotes the s-th column of the matrix M.

Problem Formulation
A networked evolutionary game, denoted by ((N, E), G, Π), consists of: where N denotes the set of vertices (players), and E denotes the set of edges.
where N 1 is the set of ordinary players, N 2 is the set of pseudo players who can control the evolutionary game, and N 3 is the set of attackers (In an NEG, attackers are selfish nodes in the network graph who use the network but do not cooperate. Attacker is different from stochastic player in NEGs with "Fermi rule", where stochastic player is also an ordinary player who may cooperate.) who may destroy the evolutionary game. Set |N 1 | = n, |N 2 | = m and |N 3 | = q. • Fundamental networked game, G, such that if (i, j) ∈ E, then i and j play the FNG repetitively with the strategy set S. Without loss of generality, for |S| = k, we let S = D k . Denote the strategies of each player in N 1 , N 2 and N 3 at time t by z i (t) ∈ D k , w j (t) ∈ D k and ξ l (t) ∈ D k , respectively, where i = 1, · · · , n, j = 1, · · · , m and l = 1, · · · , q. • Strategy updating rule, Π. Denote the λ-th step neighborhood of each player P i ∈ N by U λ (i).
When λ = 1, we briefly denote by U(i) the one step neighborhood of P i . At each time instance, each player P i plays the FNG with its neighbors in U(i), and its average payoff, denoted by c i , has the following form: where c ij : S × S → R denotes the payoff of P i playing with its neighbor P j , j ∈ U(i). Throughout this paper, the strategy updating rule is described by the following fundamental evolutionary equation: where f i is determined by the following SUR (Unconditional Imitation with Fixed Priority): P i (t + 1) is selected as the best strategy from strategies of its neighbors in U(i) at time t. Precisely, if j * = arg max j∈U(i) c j (P j , P k |k ∈ U(j)), then P i (t + 1) = P j * (t). When the neighbors with maximum payoff are not unique, say, arg max j∈U(i) c j (P j , P k |k ∈ U(j)) := {j * 1 , · · · , j * r }, we choose j * = min{j * 1 , · · · , j * r }.
We give an example to demonstrate how to use the SUR to determine the fundamental evolutionary equation.

Example 1.
Consider an NEG consisting of five players, in which the set of players is denoted by N = {P 1 , P 2 , P 3 , P 4 , P 5 }, and the network graph of the game is shown in Figure 1. The basic game of this NEG is the snowdrift game [14], whose payoff matrix is given in Table 1, where "cooperate" and "defect" are denoted by "1" and "2", respectively. Hence, all the players have the same strategy set S = {1, 2}. In this NEG, P 4 is assumed to be a pseudo player who can freely choose its own strategy at each step, while P 5 an attacker who may destroy the evolutionary game. Denote the strategies of P 1 , P 2 , P 3 , P 4 and P 5 at time t by P 1 (t), P 2 (t), P 3 (t), P 4 (t) and P 5 (t), respectively.
Noting that each column of M κ corresponds to a prescribed value of f κ in Table 2.
In the following, motivated by Example 1, we establish the algebraic form of NEGs with attackers. Identify each strategy λ ∈ S = D k as the canonical vector form δ λ For each evolutionary dynamic equation (3), one can draw a table like Table 2. From the table, one can find a matrix M i ∈ L k×k m+n+q such that where M i is called the structural matrix of f i . Multiplying all the n Equations in (5) together, we obtain the algebraic form of NEGs with attackers as follows: where L = M 1 * M 2 * · · · * M n ∈ L k n ×k n+m+q . In this paper, we assume that the set of strategy profiles in N 1 takes values from the following forbidden profiles set (In an NEG, forbidden profiles set is a set strategy profiles which are illegal according to rules, laws and regulations of the game.): where 1 ≤ i 1 < i 2 < · · · < i r ≤ k n and |C z | = r. Now, we introduce the robust consensus problem studied in this paper.

Definition 1.
Consider the NEG (6) with attackers and forbidden profiles set C z . Let η ∈ ∆ k and η n ∈ C z be given. The NEG is said to achieve robust consensus at η ∈ ∆ k , if there exist a positive integer τ and a control sequence {w(t) : t ∈ N} such that We aim to design a state feedback control in the form of where b i : D n k → D k are k-valued logical functions, which needs to be determined, under which the NEG (6) with attackers and forbidden profiles set C z achieves robust consensus at η ∈ ∆ k .
Assume that the structural matrix of b i is B i , i = 1, · · · , m. Then, by using the Khatri-Rao product of matrices, the state feedback control (8) can be described in the following form: where B = B 1 * B 2 * · · · * B m ∈ L k m ×k n is called the state feedback gain matrix. Thus, our objective becomes how to design the state feedback gain matrix B ∈ L k m ×k n such that the robust consensus achieves.

Main Results
In this section, we firstly present a necessary and sufficient condition for the robust constrained reachability of NEGs with attackers and forbidden profiles, based on which we propose a constructive procedure to design the state feedback gain matrix B for the robust consensus of NEGs with attackers and forbidden profiles.
Firstly, we give the definition for the robust constrained reachability of NEGs with attackers and forbidden profiles, which is crucial to the robust consensus of NEGs.

Definition 2.
Consider the NEG (6) with attackers and forbidden profiles.
(i) z d ∈ C z is said to be one step robustly reachable from z 0 ∈ C z , if there exists a control w ∈ ∆ k m such that z d = L ξ w z 0 holds for any ξ ∈ ∆ k q . (ii) A nonempty set Ω ⊆ C z is said to be one step robustly reachable from z 0 ∈ C z , if there exist a control w ∈ ∆ k m and z ξ ∈ Ω (depending on ξ) such that z ξ = L ξ w z 0 holds for any ξ ∈ ∆ k q .
In the following, we present a criterion for the robust constrained reachability of NEGs with attackers and forbidden profiles.
Consider the NEG (6). Split L ∈ L k n ×k m+n+q into k q blocks as where L s ∈ L k n ×k m+n , s = 1, 2, · · · , k q . Split each L s into k m blocks as where L j s ∈ L k n ×k n , j = 1, 2, · · · , k m . Define where L s = [ L 1 s L 2 s · · · L k m s ] ∈ R r×rk m , and Obviously, L j s is obtained from L j s by deleting all the elements in the rows and columns with indexes {1, 2, · · · , k n } \ {i 1 , i 2 , · · · , i r }. Proof. On one hand, it is easy to see from δ i α k n = L j s δ i β k n that (δ k n [i 1 i 2 · · · i r ]) T δ i α k n = (δ k n [i 1 i 2 · · · i r ]) T L j s δ i β k n . On the other hand, a simple calculation shows that Therefore, δ α r = L j s δ β r . This completes the proof.
Based on Definition 2 and Lemma 1, we have the following result on the robust constrained reachability of NEGs with attackers and forbidden profiles. Theorem 1. Consider the NEG (6) with attackers and forbidden profiles set C z .
(i) z d = δ i α k n ∈ C z is one step robustly reachable from z 0 = δ i β k n ∈ C z , if and only if there exists a positive integer 1 ≤ j ≤ k m such that (ii) A nonempty set Ω ⊆ C z is one step robustly reachable from z 0 = δ i β k n ∈ C z , if and only if there exists a positive integer 1 ≤ j ≤ k m such that Proof. We firstly prove conclusion (i).
(Necessity) Suppose that z d = δ i α k n ∈ C z is one step robustly reachable from z 0 = δ i β k n ∈ C z . Then, there exists a control w = δ j k m such that δ i α k n = L δ s k q δ j k m δ i β k n holds for any s = 1, 2, · · · , k q . By Lemma 1, one can see that δ α r = L j s δ β r holds for any s = 1, 2, · · · , k q . Thus, ( L j s ) α,β = 1 holds for any s = 1, 2, · · · , k q , which implies that (14) holds.
(Necessity) Assuming that Ω is one step robustly reachable from z 0 = δ i β k n ∈ C z , then there exist a control w = δ j k m and z ξ = δ k n holds for any ξ = δ s k q ∈ ∆ k q . By Lemma 1, we know that δ α(ξ) r = L j s δ β r holds for any s = 1, 2, · · · , k q . Since Col β ( L j s ) is a logical vector, one can see that ∑ δ iα k n ∈Ω L j s α,β = 1 holds for any s = 1, 2, · · · , k q , which implies that (15) holds.
(Sufficiency) Suppose that (15) holds for some integer 1 ≤ j ≤ k m . Since Col β L j s ∈ L r×1 , we know that ∑ δ iα k n ∈Ω L j s α,β = 1 holds for ∀ s = 1, 2, · · · , k q . Therefore, for each ξ = δ s k q ∈ ∆ k q , there r holds for any ξ = δ s k q ∈ ∆ k q . By the construction of L j s , we can obtain that where Γ is given in (16). It is easy to see from Γ δ holds for any ξ = δ s k q ∈ ∆ k q . By Definition 2, Ω is one step robustly reachable from z 0 = δ i β k n ∈ C z . This completes the proof.
Based on the robust constrained reachability of NEGs with attackers and forbidden profiles, we inductively construct a series of robust reachable sets as follows. Let η n = δ c k n ∈ C z , where η ∈ ∆ k and c is uniquely determined by η. For example, if η = δ 1 k , then c = 1; if η = δ k k , then c = k n . Define Ω γ (η) = δ i α k n ∈ C z : there exists an integer 1 ≤ j ≤ k m such that where Ω 1 (η) represents the set of states that can robustly reach η n = δ c k n in one step, and Ω γ (η) is the set of states that can robustly reach Ω γ−1 (η) in one step. Then, based on a simple calculation, we have the following results.
Finally, we prove that the condition (19) is also necessary for the robust consensus of NEGs with attackers and forbidden profiles. Theorem 3. If the NEG (6) with attackers and forbidden profiles set C z achieves robust consensus at η ∈ ∆ k , then there exists an integer 1 ≤ τ ≤ r such that (19) holds.
Proof. Assume that the NEG (6) with attackers and forbidden profiles set C z achieves robust consensus at η ∈ ∆ k . Then, one can obtain that (i) η n is one step robustly reachable from itself in one step. (ii) There exists a positive integer τ such that η n is robustly reachable from any z 0 ∈ C z at the τ-th step.

An Illustrative Example
Consider an NEG consisting of five players, in which the set of players is denoted by N = {P 1 , P 2 , P 3 , P 4 , P 5 } and the network graph of the game is shown in Figure 2. The basic game of this NEG is the Boxed Pigs Game [14], whose payoff matrix is given in Table 3, where "Press" and "Wait" are denoted by "1" and "2", respectively. Hence, all the players have the same strategy set S = {1, 2}. In this NEG, P 4 is assumed to be a control, while P 1 is assumed to be an attacker. We suppose that P 1 , P 3 and P 5 denote small pigs, while P 2 and P 4 big pigs. Denote the strategies of P 1 , P 2 , P 3 ,P 4 and P 5 at time t by x 1 (t), x 2 (t), x 3 (t), x 4 (t) and x 5 (t), respectively. Network graph of the NEG, where P 1 , P 3 and P 5 denote small pigs, while P 2 and P 4 big pigs. P 1 and P 4 are assumed to be attacker and control, respectively. According to the SUR of this paper, we have the following evolutionary dynamic equations: where f 1 , f 2 , f 3 are Boolean functions, which can be uniquely determined by the SUR. Let "δ 1 2 " be the vector form of "1" and "δ 2 2 " be the vector form of "2". Using the vector form of x j (t), j = 1, 2, 3, 4, 5 and letting z(t) = x 2 (t) x 3 (t) x 5 (t), w(t) = x 4 (t), ξ(t) = x 1 (t), by drawing a table like Table 2, we can obtain the algebraic form of the NEG as follows: where L = δ 8 [ 1 1 1 1 8 8 8 5 4 1 1 1 8 8 8 8   1 1 1 5 8 5 5 5 4 4 4 5 8 8 8 8].

Conclusions
In this paper, we have considered the robust consensus of NEGs with attackers and forbidden profiles, and presented some new results. Based on the algebraic representation of NEGs with attackers and forbidden profiles, we have proposed a necessary and sufficient condition for the robust constrained reachability of NEGs, which is an effective tool for the robust consensus control design. In addition, by constructing a series of robust reachable sets, we have presented a constructive procedure to design state feedback controls for the robust consensus of NEGs with attackers and forbidden profiles.
It should be pointed out that one can check the robust consensus of NEGs based on the simulation from a table like Table 2. However, the simulation method may be somewhat blind. Compared with this classic method used in game theory, the STP based theoretical framework avoids the blindness of finding a suitable control strategy.