A game is symmetric if all players have the same strategy set and the payoff of a given strategy is determined merely by the strategy itself and has nothing to do with who plays it. Symmetric games have been widely studied since the dawn of game theory, and together with zero-sum games, they form one of the most classical subclasses of games. For example, Nash [1,2] proposed a very important concept of equilibrium, called Nash equilibrium, which is a strategy profile in which each player’s strategy is an optimal response to the strategies of the other players. This concept has a nice property that there is at least one Nash equilibrium for every finite game, and every finite symmetric game leads to a symmetric Nash equilibrium (an equilibrium in which all players use the same strategy) [1,2,3,4]. Some researchers were devoted to symmetric games; for additional statements, one can refer to [5,6,7,8,9,10,11,12,13,14,15].

The symmetric game, in particular, is important in the study of economic and biological models [16,17,18,19]. As we know, many interactions have been modeled in terms of one-shot and some of them have been reported as symmetric two-player cooperative dilemmas, including the Prisoner’s Dilemma [20,21], Hawk Dove Game [21], Snowdrift Game [22], and Stag-Hunt Game [23]. If each player has n actions to choose in a 2-player symmetric game, the game can be represented by two n-by-n matrices [24,25]. It is known that the 2-player symmetric game can be formulated as a linear complementary problem. The Lemke-Howson algorithm [24] is a well known method which was designed to solve the above mentioned linear complementary problem.

However, in many real-life situations, decisions are made by groups of people that include more than two individuals [19]. This type of collective action problem is better to be described as an m-person symmetric game [26,27,28]. We prefer to find symmetric equilibria for an m-person symmetric game because asymmetric behavior appears relatively unintuitive [6] and difficult to explain in a one-shot interaction [29]. Nash proved that there is a symmetric equilibrium in every m-person symmetric game [2], but the Nash equilibrium equations for m-person games are non-linear which are difficult to be solved analytically in general.

To better address the m-person game, Huang and Qi [30] extended the classic form of a game to a tensor form by using a tensor to represent the utility function of each player. They reformulated the m-person game as a tensor complementary problem, and showed that finding a Nash equilibrium of the m-person is equivalent to finding a solution to the resulting tensor complementary problem. Abdou et al. [31] presented an efficient method to solve the pure Nash equilibrium by using tensor operations. However, they did not take the Nash equilibrium of a multiplayer symmetric game into consideration.

In this paper, we consider the tensor-based m-person symmetric games. We demonstrate that the payoff tensors of the k-th player in an m-person symmetric game are k-mode symmetric and that the payoff tensors between different players are the transpose of each other. We also reformulate the m-person symmetric games as a tensor complementary problem. It is worth noting that the order of tensors in the tensor complementary problem is less than that resulting in [30]. In addition, we show that finding a symmetric Nash equilibrium of the m-person symmetric game is equivalent to finding a solution to the resulting tensor complementary problem. Finally, we apply a hyperplane projection algorithm to solve the resulting tensor complementary problem and provide some numerical results for solving the symmetric Nash equilibrium.

The rest of this paper is organized as follows: In Section 2, we present some definitions and notations that will be frequently used in the following parts. In Section 3, we provide a brief description of the m-person symmetric game. Some experimental results that can be used to explain our proposed theory’s reliability are shown in Section 4. Finally, we provide a brief summary in Section 5.

In this section, we introduce some definitions and notations, which will be used in the sequel. Throughout this paper, we use small letters (e.g.,a), small bold letters (e.g.,$\mathbf{a}$), capital letters (e.g.,A), and calligraphic letters (e.g.,$\mathcal{A}$) to denote scalars, vectors, matrices, and tensors, respectively. For a positive integer n, let $\left[n\right]=\{1,2,\cdots ,n\}$.

A real m-th order $n_1\times n_2\times \cdots \times n_m$-dimensional tensor is a multidimensional array, and its elementwise form can be denoted as

When $m=2$, $\mathcal{A}$ is an $n_1$-by-$n_2$ matrix. If $n_1=n_2=\cdots =n_m=n$, $\mathcal{A}$ is called a real m-th order n-dimensional tensor. We denote the set of all real m-th order $n_1\times n_2\times \cdots \times n_m$-dimensional tensors by ${\mathbb{R}}^{n_1\times n_2\times \cdots \times n_m}$ and denote the set of all real m-th order n-dimensional tensors by ${\mathbb{R}}^{[m,n]}$. When $m=1$, ${\mathbb{R}}^{[1,n]}$ is simplified as ${\mathbb{R}}^n$, which is the set of all n-dimensional real vectors. We denote the set of all nonnegative n-dimensional real vectors by ${\mathbb{R}}_{+}^n$. Let ${\mathbf{1}}_n$, ${\mathbf{e}}_i$, and 0 denote the n-dimensional vector of all ones, the i-th column of the n-dimensional identity matrix, and the zero vector, respectively. The order $\mathbf{x}$⩾$\mathbf{0}$ means that each component of $\mathbf{x}$ is nonnegative.

The definition of the k-mode product of a tensor with a vector is recalled as follows.

Let $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)\in {\mathbb{R}}^{n_1\times n_2\times \cdots \times n_m}$ and ${\mathbf{u}}^{\left(k\right)}=\left(u_{i_j}^{\left(k\right)}\right)\in {\mathbb{R}}^{n_k}$ with $k\in \left[m\right]$. According to Definition 1, it follows that where $\mathcal{A}{\overline{\times }}_m{\mathbf{u}}^{\left(m\right)}{\overline{\times }}_{m-1}\cdots {\overline{\times }}_2{\mathbf{u}}^{\left(2\right)}$ is an $n_1$-dimensional vector with entries

For simplicity of notion, we use $\mathcal{A}{\mathbf{u}}^{\left(m\right)}\cdots {\mathbf{u}}^{\left(2\right)}{\mathbf{u}}^{\left(1\right)}$ to denote the scale and use $\mathcal{A}{\mathbf{u}}^{\left(m\right)}\cdots {\mathbf{u}}^{\left(2\right)}$ to denote the real $n_1$-dimensional vector

Particularly, when $n_1=n_2=\cdots =n_m=n$, and $\mathbf{u},{\mathbf{u}}^{*}\in {\mathbb{R}}^n$, we take the symbols $\mathcal{A}{\mathbf{u}}^m$, $\mathcal{A}{\mathbf{u}}^{*m-1}\mathbf{u}$, and $\mathcal{A}{\mathbf{u}}^{*m-k}\mathbf{u}{\mathbf{u}}^{*k-1}$ to denote the scales and respectively. The symbol $\mathcal{A}{\mathbf{u}}^{m-1}$ is short for the n-dimensional vector $\mathcal{A}{\overline{\times }}_m\mathbf{u}{\overline{\times }}_{m-1}\cdots {\overline{\times }}_2\mathbf{u}$.

The definitions of symmetric and semi-symmetric tensors are given as follows.

We provide a more general definition about k-mode symmetric of a tensor.

By Definitions 1, 2, and 4, we have the following lemma.

In [35], the definition of transpose of a tensor was introduced as follows:

We recall the psdudomonotone mapping and projection operator, which will be used in Section 4.

In this section, we first provide the definition of a tensor form of an m-person game, adapted from [30] and [31].

Given an m-person game $G=(\left[m\right];{\left\{\left[n_k\right]\right\}}_{k=1}^m;{\left\{{\mathcal{A}}^{\left(k\right)}\right\}}_{k=1}^m)$. A mixed strategy of player k is the probability distribution on pure strategy set $\left[n_k\right]$. Let denote the set of the all probability distributions on $\left[n_k\right]$. For ${\mathbf{u}}^{\left(k\right)}=\left(u_{i_j}^{\left(k\right)}\right)\in {\mathsf{\Omega }}_k$, the probability assigned to $i_j$-th pure strategy of player k is $u_{i_j}^{\left(k\right)}$.

Using denote the set of strategy profiles. We say that $({\mathbf{u}}^{\left(1\right)},\cdots ,{\mathbf{u}}^{\left(m\right)})\in \mathsf{\Omega }$ is a mixed strategy combination if ${\mathbf{u}}^{\left(k\right)}\in {\mathsf{\Omega }}_k$ is a mixed strategy of player k for any $k\in \left[m\right]$. If a mixed strategy combination $({\mathbf{u}}^{\left(1\right)},\cdots ,{\mathbf{u}}^{\left(m\right)})\in \mathsf{\Omega }$ is played, then the expected payoff of player k is

What we consider here is the m-person symmetric games. In such games, all players play the same role in the game and moreover the payoff of any player depends only upon its strategy and the combination of the strategies of its opponents.

It is well know that the payoff matrices of the 2-person symmetric game are transpose of each other [20]. Furthermore, we come to a similar conclusion for the m-person symmetric game.

Nash proved the existence of equilibrium for m-person games and symmetric equilibrium for symmetric m-person games in 1951 [2], and the statement is summarized in the following Lemma.

Now, we are going to propose two equivalent conditions to ensure that a mixed strategy combination is a symmetric Nash equilibrium, which are shown in Theorems 2 and 3.

Let where $\eta =\underset{i_1,i_2,\cdots ,i_m\in \left[n\right]}{max}{a}_{i_1{i}_2\cdots i_m}^{\left(1\right)}+1$ and $\mathcal{E}\in {\mathbb{R}}^{[m,n]}$ is an m-th order n-dimensional tensor whose all entries are ones. According to ${\mathcal{A}}^{\left(1\right)}$ is 1-mode symmetric we can see that $\mathcal{A}$ is also a 1-mode symmetric tensor with all entries are positive.

We consider the following model: Find a mixed strategy combination $({\mathbf{x}}^{*},{\mathbf{x}}^{*},\cdots ,{\mathbf{x}}^{*})\in \mathsf{\Omega }$ such that ${\mathbf{x}}^{*}$ is an optimal solution to the optimization problem (5). Given a tensor $\mathcal{B}\in {\mathbb{R}}^{[m,n]}$ and a vector $\mathbf{q}\in {\mathbb{R}}^n$, the tensor complementary problem, denoted by TCP$(\mathbf{q},\mathcal{B})$, is to find a vector $\mathbf{z}\in {\mathbb{R}}^n$ such that

The TCP$(\mathbf{q},\mathcal{B})$ was introduced by Song and Qi [38] and was further studied by many researchers [39,40,41,42,43,44,45,46,47,48]. The tensor complementary problem is a generalization of the linear complementary problem (corresponding to $m=2$) [49] and also a special instance of a nonlinear complementary problem, as well as a particular case of a variational inequality problem corresponding to the close convex cone ${\mathbb{R}}_{+}^n$[50]. Denote the solution set of the TCP($\mathbf{q},\mathcal{B}$) by SOL($\mathbf{q},\mathcal{B}$). In this section, we show that the m-person symmetric game can be reformulated as a specific tensor complementary problem.

Let $G=(\left[m\right];\left[n\right];{\mathcal{A}}^{\left(1\right)})$ be an m-person symmetric game and $\mathcal{A}$ defined as (4). We construct a tensor complementary problem as follows,

There is an explicit corresponding relation between the solutions of a symmetric Nash equilibrium of the m-person symmetric game and a solution to the TCP$(-{\mathbf{1}}_n,\mathcal{A})$ (7).

It is well known that the hyperplane projection algorithm [50] is a class of effective methods for solving variational inequalities and complementary problems. In this section, we apply the hyperplane projection algorithm to solve the TCP$(-{\mathbf{1}}_n,\mathcal{A})$ and provide some preliminary numerical results for solving the m-person symmetric game.

Let $G=(\left[m\right];\left[n\right];{\mathcal{A}}^{\left(1\right)})$ be an m-person symmetric game and $\mathcal{A}$ defined as (4). Denote $F\left(y\right)=\mathcal{A}{\mathbf{y}}^{m-1}-{\mathbf{1}}_n$, then we can rewrite the TCP$(-{\mathbf{1}}_n,\mathcal{A})$ (7) as follows,

It is well know that (17) is equivalent to the variational inequality problem: Find a nonnegative vector $\mathbf{y}\in {\mathbb{R}}_{+}^n$ such that and it is equivalent to finding a root of the following equations