1 Introduction

Evolutionary game theory is the study of frequency-dependent natural selection in which the fitness of individuals is not constant, but depends on frequencies of the different phenotypes in population. This theory was introduced by Maynard Smith and Price [28] and further developed by Maynard Smith [26, 27]. Since then, there has been a veritable explosion of interest by economists and social scientists in evolutionary game theory [4, 6, 7, 9,10,11, 17, 20, 22,23,24, 36, 43, 45, 53]. Classical non-cooperative game theory, which was invented by Von Neumann and Morgenstern [52], typically analyzes an interaction between two players and studies their behavior in strategic and economic decisions. The main problem is how the players can maximize their payoffs in a game given that the players are aware of the structure of the game and consciously try to predict the moves of their opponents. This depends on the cognitive abilities of the players in which the concept of rationality plays an important role. However, evolutionary game theory differs from classical non-cooperative game theory in focusing more on the dynamics of strategy change. It does not rely on rationality. It is presumed that a population of players are randomly interacted and the strategy of a player is fixed which is biologically encoded and heritable that controls its action. However, they have no control over their strategy and need not to be aware of the game. In other words, the players do not choose their strategy and cannot change it: they are born with a strategy and their offspring inherit that same strategy. Evolutionary game theory interprets payoff as biological fitness and success as reproductive success. Hence, the payoff represents reproductive success and the success of a strategy is determined by how good the strategy is in the presence of competing strategies and of the frequency with which those strategies are used. Strategies that do well reproduce faster and strategies that do poorly are outcompeted.

The Nash equilibrium, which was invented by Nash [34, 35], is the solution concept in classical non-cooperative game theory in which each strategy in the Nash equilibrium is the best response to all other strategies in that equilibrium. In other words, it is a strategy profile in which no player can do better by unilaterally changing their strategy to another strategy. An evolutionarily stable strategy (ESS) is akin to the Nash equilibrium in classical non-cooperative game theory. An ESS is a strategy of game dynamics for which another mutant strategy cannot successfully enter the population to disturb the existing dynamics. If it is adopted by a population in a given environment then it is unbeatable which means that it cannot be invaded by any alternative (mutant) strategy that are initially rare. The ESS is an equilibrium refinement of the Nash equilibrium that is “evolutionarily” stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The ESS must be resistant to these alternatives. Therefore, the ESS must be effective against any alternative (mutant) strategy when it is initially rare and successful when it is eventually abundant.

The primary way to study the evolutionary dynamics in games is through replicator equations. They are used to describe the evolutionary dynamics of an entity called replicator which has means of making more or less accurate copies of itself. The replicator can be a gene in population genetics, a molecule in prebiotic evolution, an organism in population ecology, a strategy in evolutionary game. The replicator equation, which was first introduced into models of animal behavior of a single species by Taylor and Jonker [48, 49], is the cornerstone of evolutionary game dynamics. The replicator equation shows the growth rate of the proportion of players using a certain strategy for which that rate is equal to the difference between the average payoff of that strategy and the average payoff of the population as a whole. The general idea is that replicators whose fitness is larger (smaller) than the average fitness of population will increase (decrease) in numbers. The static approach of evolutionary game theory has been complemented by a dynamic stability analysis of rest (stationary) solutions of the replicator equations. The replicator equation satisfies the Folk Theorem of Evolutionary Game Theory (see [7, 23, 24]) which asserts the following four statements for all “reasonable” dynamics of an evolutionary game:

  1. (i)

    A Nash equilibrium is a rest (fixed) point;

  2. (ii)

    A stable rest (fixed) point is a Nash equilibrium;

  3. (iii)

    A strictly Nash equilibrium is asymptotically stable;

  4. (iv)

    Any interior convergent orbit evolves to a Nash equilibrium;

In this paper, we are aiming to show that the Folk Theorem of Evolutionary Game Theory is also true for a model of evolutionary game in which the nonlinear payoff functions are defined by discrete population models for a single species such as Beverton–Holt’s model [5], Hassell’s model [18], Maynard Smith–Slatkin’s model [29], Ricker’s model [39], Skellam’s model [46]. Indeed, we consider a general nonlinear model of evolutionary game in which for any two different pure strategies, a biological fitness of a pure strategy which is frequent in number is better/worse than other one. Particularly, the nonlinear payoff functions defined by discrete population models mentioned above satisfy this hypothesis. In order to observe some evolutionary bifurcation diagram, the nonlinear payoff functions are controlled in two different regimes: positive and negative. We also study the dynamics and stability analysis of the discrete-time replicator equation governed by the proposed nonlinear payoff functions. Namely, we describe all rest (fixed) points, Nash equilibria, and ESSs of the proposed discrete-time replicator equation in both regimes. One of the interesting feature of the model is that if we switch the controlling parameter from positive to negative regime then the set of ESSs changes from one set to another one. In the long-run time, the following scenario can be observed: (i) in the positive regime, the active dominating pure strategies will outcompete other strategies and only they will survive forever; (ii) in the negative regime, all active pure strategies will coexist together and they will survive forever.

The paper is organized as follows: in the next section, we provide all necessary notions and notation from evolutionary game theory which will be used throughout this paper. In Sect. 3, we state the main results (see Theorems A, B, and C) of the paper. In Sect. 4, we describe the sets of the rest points, Nash equilibria, and local ESSs of the proposed discrete-time replicator equation. In Sects. 5 and 6, we study the dynamics and stability analysis of the discrete-time replicator equation in positive and negative regimes, respectively. The applications are presented in Sect. 7. Finally, we finish this paper with some concluding remarks in Sect. 8.

2 Preliminaries

In this paper, we consider a symmetric two-player normal form game that is a game of a single population (no difference of being player-1 or player-2) in which all players share the same strategy set and payoff function. Namely, we consider a unit mass of players each of whom chooses a pure strategy from the set \(\mathbf {I}_m=\{1,2,\ldots ,m\}\). A mixed strategy is a probability vector \(\mathbf {x}=(x_1,\ldots ,x_m)\), i.e., \(x_1+\cdots +x_m=1\) and \(x_i\ge 0\) for all \(i\in \mathbf {I}_m\) where \(x_i\) is the probability that the player will choose the pure strategy i. A pure strategy i can be also seen as a mixed strategy \(\mathbf {e}_i=(0,\ldots ,0,1,0,\ldots ,0)\) with 1 at the ith place which means that the player uses the strategy i with probability 1. The set of mixed strategy of player is the simplex \(\mathbb {S}^{m-1}=\{\mathbf {x}\in \mathbb {R}_{+}^{m}: \sum _{i=1}^{m}x_i=1\}\). Sometimes, the simplex \(\mathbb {S}^{m-1}\) can be also considered as the set of population states which describe densities of the population playing pure strategies. Namely, a probability vector \(\mathbf {x}=(x_1,x_2,\ldots ,x_m)\) is a population state where \(x_i\) is the frequency of the strategy i.

A continuous (possibly nonlinear) function \(f_i(\mathbf {x})\) is a payoff to a strategy \(i\in \mathbf {I}_m\) in a population state \(\mathbf {x}\in \mathbb {S}^{m-1}\). A continuous mapping \(F(\mathbf {x}):=(f_1(\mathbf {x}),\ldots ,f_m(\mathbf {x}))\) is a payoff of the game when the population state is \(\mathbf {x}\in \mathbb {S}^{m-1}\). A payoff to the strategy \(\mathbf {y}\in \mathbb {S}^{m-1}\) when the population state is \(\mathbf {x}\in \mathbb {S}^{m-1}\) is a bivariate continuous function \(\mathcal {E}_F(\mathbf {y},\mathbf {x}):=\sum _{i=1}^{m}y_if_i(\mathbf {x})\) which is linear in the first argument. It is easy to see that \(\mathcal {E}_F(\mathbf {e}_i,\mathbf {x})=f_i(\mathbf {x})\) for any \(i\in \mathbf {I}_m\). A average fitness of the population when it is in a state \(\mathbf {x}\in \mathbb {S}^{m-1}\) is \(\mathcal {E}_F(\mathbf {x},\mathbf {x})=\sum _{i=1}^{m}x_if_i(\mathbf {x})\).

Sometimes, we call \(\mathbf {y}\) a reply strategy to a strategy \(\mathbf {x}\) in an expression like \(\mathcal {E}_F(\mathbf {y},\mathbf {x})\). With this presumption, we say that the strategy \(\mathbf {y}\) is better reply to the strategy \(\mathbf {x}\) than the strategy \(\mathbf {z}\) if one has \(\mathcal {E}_F(\mathbf {y},\mathbf {x})>\mathcal {E}_F(\mathbf {z},\mathbf {x})\). The strategy \(\mathbf {y}\) is called a best reply to the strategy \(\mathbf {x}\) if one has \(\mathcal {E}_F(\mathbf {y},\mathbf {x})\ge \mathcal {E}_F(\mathbf {z},\mathbf {x})\) for any \(\mathbf {z}\in \mathbb {S}^{m-1}\). The set of all best replies to \(\mathbf {x}\) is denoted by

$$\begin{aligned} \mathsf {BR}(\mathbf {x}):=\left\{ \mathbf {y}\in \mathbb {S}^{m-1}: \mathcal {E}_F(\mathbf {y},\mathbf {x})=\max \limits _{\mathbf {z}\in \mathbb {S}^{m-1}}\mathcal {E}_F(\mathbf {z},\mathbf {x})\right\} . \end{aligned}$$

Definition 2.1

(Nash equilibrium) A strategy \(\mathbf {x}\) is called a Nash equilibrium if it is a best reply to itself, i.e., one has \(\mathcal {E}_F(\mathbf {x},\mathbf {x})\ge \mathcal {E}_F(\mathbf {y},\mathbf {x})\) for any \(\mathbf {y}\in \mathbb {S}^{m-1}\). A strategy \(\mathbf {x}\) is called a strictly Nash equilibrium if it is the unique best reply to itself, i.e., one has \(\mathcal {E}_F(\mathbf {x},\mathbf {x})> \mathcal {E}_F(\mathbf {y},\mathbf {x})\) for any \(\mathbf {y}\in \mathbb {S}^{m-1}\) with \(\mathbf {y}\ne \mathbf {x}\).

It was pointed out by Pohley and Thomas [37] that the original definition of an evolutionarily stable strategy (ESS) is inappropriate for the games with nonlinear payoff functions. The main reason for this was the global character of the original conditions that define an ESS. These conditions always cover complete spaces of strategies which do not allow to have more than one ESS in the interior of the whole space. In the series of papers [37, 50, 51] (for further developments also see [2, 47]), in order to avoid such constraints, the concept of a local ESS was introduced for the games with nonlinear payoff functions for which the conditions that define an ESS must hold only locally, i.e., within a small neighborhood of an ESS. It turns out that for linear models the local ESS is equivalent to the original global ESS (see [37]).

Definition 2.2

(Local ESS) A strategy \(\mathbf {x}\) is called a local evolutionarily stable strategy (a local ESS) if the following conditions hold true:

  1. (i)

    \(\mathbf {x}\) is a Nash equilibrium, i.e., \(\mathcal {E}_F(\mathbf {x},\mathbf {x})\ge \mathcal {E}_F(\mathbf {y},\mathbf {x})\) for any \(\mathbf {y}\in \mathbb {S}^{m-1}\);

  2. (ii)

    There is a small neighborhood \(U(\mathbf {x})\subset \mathbb {S}^{m-1}\) of \(\mathbf {x}\) such that \(\mathcal {E}_F(\mathbf {x},\mathbf {y})>\mathcal {E}_F(\mathbf {y},\mathbf {y})\) for all \(\mathbf {y}\in U(\mathbf {x}){\setminus }\{\mathbf {x}\}\).

The replicator equation which was originally developed for symmetric games with finitely many strategies (see [48, 49]) is the most important evolutionary game dynamics (see also [1, 8, 15, 19, 38]). In the continuous case, the replicator and other types of equations are thoroughly studied in the literature [6, 7, 22, 23, 36, 43,44,45, 53].

In this paper, we consider a discrete-time replicator equation \(\mathcal {R}_F:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\)

$$\begin{aligned} \left( \mathcal {R}_F(\mathbf {x})\right) _k=x_k\left( 1+f_k(\mathbf {x})-\mathcal {E}_F(\mathbf {x},\mathbf {x})\right) , \quad \forall \ \mathbf {x}\in \mathbb {S}^{m-1} \ and \ \ k\in \mathbf {I}_m. \end{aligned}$$
(2.1)

Remark 2.3

In replicator equation (2.1), the relative growth rate \(\frac{\left( \mathcal {R}_F(\mathbf {x})\right) _k-x_k}{x_k}\) of the player using a strategy k is equal to the difference between the payoff \(f_k(\mathbf {x})\) of that strategy k and the average payoff \(\mathcal {E}_F(\mathbf {x},\mathbf {x})=x_1f_1(\mathbf {x})+\cdots +x_mf_m(\mathbf {x})\) of the population as a whole. The key idea is that a replicator whose fitness is larger (smaller) than the average fitness of population will increase (decrease) in numbers. Obviously, in order the simplex to be an invariant set for replicator equation (2.1), we must have some constraints on payoff functions. For example, the following condition

$$\begin{aligned} 0\le \max _{k\in \mathbf {I}_m}\max _{\mathbf {x}\in \mathbb {S}^{m-1}}f_k(\mathbf {x})-\min _{k\in \mathbf {I}_m}\min _{\mathbf {x}\in \mathbb {S}^{m-1}}f_k(\mathbf {x})\le 1 \end{aligned}$$

is sufficient for the simplex to be invariant. Indeed, since

$$\begin{aligned} \min _{k\in \mathbf {I}_m}\min _{\mathbf {x}\in \mathbb {S}^{m-1}}f_k(\mathbf {x})\le \mathcal {E}_F(\mathbf {x},\mathbf {x})\le \max _{k\in \mathbf {I}_m}\max _{\mathbf {x}\in \mathbb {S}^{m-1}}f_k(\mathbf {x}), \end{aligned}$$

we obtain for all \(\mathbf {x}\in \mathbb {S}^{m-1}\) and \(k\in \mathbf {I}_m\) that

$$\begin{aligned} 1+f_k(\mathbf {x})-\mathcal {E}_F(\mathbf {x},\mathbf {x})\ge 1+f_k(\mathbf {x})-\max _{k\in \mathbf {I}_m}\max _{\mathbf {x}\in \mathbb {S}^{m-1}}f_k(\mathbf {x})\ge f_k(\mathbf {x})-\min _{k\in \mathbf {I}_m}\min _{\mathbf {x}\in \mathbb {S}^{m-1}}f_k(\mathbf {x})\ge 0. \end{aligned}$$

Hence, we have \(\sum _{k=1}^m\left( \mathcal {R}_F(\mathbf {x})\right) _k=\sum _{k=1}^mx_k=1\) and \(\left( \mathcal {R}_F(\mathbf {x})\right) _k\ge 0\) for all \(\mathbf {x}\in \mathbb {S}^{m-1}\) and \(k\in \mathbf {I}_m\). It is worth mentioning that the model which we are going to propose satisfies the condition given above.

In order to study the stability of rest solutions (fixed points) of the replicator equation, we employ a Lyapunov function.

Definition 2.4

(Lyapunov function) A continuous function \(\varphi :\mathbb {S}^{m-1}\rightarrow \mathbb {R}\) is called a Lyapunov function if the number sequence \(\{\varphi (\mathbf {x}), \varphi (\mathcal {R}(\mathbf {x})), \ldots , \varphi (\mathcal {R}^{(n)}(\mathbf {x})), \cdots \}\) is a bounded monotone sequence for any initial point \(\mathbf {x}\in \mathbb {S}^{m-1}\).

Definition 2.5

(Stable and attracting points) A rest (fixed) point \(\mathbf {y}\in \mathbb {S}^{m-1}\) is called stable if for every neighborhood \(U(\mathbf {y})\subset \mathbb {S}^{m-1}\) of \(\mathbf {y}\) there exists a neighborhood \(V(\mathbf {y})\subset \mathbb {S}^{m-1}\) of \(\mathbf {y}\) such that an orbit \(\{\mathbf {x}, \mathcal {R}(\mathbf {x}), \ldots , \mathcal {R}^{(n)}(\mathbf {x}), \cdots \}\) of any initial point \(\mathbf {x}\in V(\mathbf {y})\) remains inside of the neighborhood \(U(\mathbf {y})\). A rest (fixed) point \(\mathbf {y}\in \mathbb {S}^{m-1}\) is called attracting if there exists a neighborhood \(V(\mathbf {y})\subset \mathbb {S}^{m-1}\) of \(\mathbf {y}\) such that an orbit \(\{\mathbf {x}, \mathcal {R}(\mathbf {x}), \ldots , \mathcal {R}^{(n)}(\mathbf {x}), \cdots \}\) of any initial point \(\mathbf {x}\in V(\mathbf {y})\) converges to \(\mathbf {y}\). A rest (fixed) point \(\mathbf {y}\in \mathbb {S}^{m-1}\) is called asymptotically stable if it is both stable and attracting.

We use the following notions and notations throughout this paper.

Some Notations: Let \(\mathbf {x}=(x_1,\ldots ,x_m)\in \mathbb {R}^m\) and \(\Vert \mathbf {x}\Vert _1:=\sum _{k=1}^{m}|x_k|\). We say that \(\mathbf {x}\ge 0\) (resp. \(\mathbf {x}> 0\)) if \(x_{k}\ge 0\) (resp. \(x_{k}>0\)) for all \(k\in \mathbf {I}_m\). We set \((\mathbf {x},\mathbf {y}):=\sum _{i=1}^{m}x_iy_i\) for any \(\mathbf {x},\mathbf {y}\in \mathbb {R}^m\). Let \(\mathbb {S}^{m-1}=\{\mathbf {x}\in \mathbb {R}^m: \Vert \mathbf {x}\Vert _1=1, \ \mathbf {x}\ge 0\}\) be the standard simplex. We let \(supp (\mathbf {x}):=\{i\in \mathbf {I}_m: x_i\ne 0\}\) and \(null (\mathbf {x}):=\{i\in \mathbf {I}_m: x_i=0\}\) for \(\mathbf {x}\in \mathbb {S}^{m-1}\). The vertex \(\mathbf {e}_i:=(0,\ldots ,0,1,0,\ldots ,0)\) with 1 at the ith place is the pure strategy \(i\in \mathbf {I}_m\). Let \({\mathbb {S}}^{|\alpha |-1}:=conv \{\mathbf {e}_i\}_{i\in \alpha }\) for \(\alpha \subset \mathbf {I}_m\) where \(conv (\mathbf {A})\) is the convex hull of \(\mathbf {A}\). Let \(int {\mathbb {S}}^{|\alpha |-1} :=\{\mathbf {x}\in {\mathbb {S}}^{|\alpha |-1}: supp (\mathbf {x})=\alpha \}\) and \(\partial \mathbb {S}^{|\alpha |-1}:= \mathbb {S}^{|\alpha |-1}{\setminus }int \mathbb {S}^{|\alpha |-1}\) be, respectively, an interior and boundary of the face \({\mathbb {S}}^{|\alpha |-1}\). The center \(\mathbf {c}_\alpha :=\frac{1}{|\alpha |}\sum _{i\in \alpha }\mathbf {e}_i\) of the face \({\mathbb {S}}^{|\alpha |-1}\) is the equally distributed population state of the active pure strategies \(i\in \alpha \). We define a function \(\mathcal {M}_{\alpha , k}(\mathbf {x}):=\max \limits _{i\in \alpha }\{x_{i}\}-x_k\) for \(k\in \alpha \subset \mathbf {I}_m\). Particularly, when \(\alpha =\mathbf {I}_m\), we write \(\mathcal {M}_k(\mathbf {x}):=\max \limits _{i\in \mathbf {I}_m}\{x_{i}\}-x_k\) for \(k\in \mathbf {I}_m\). We define the sets \(\mathsf {MaxInd}_\alpha (\mathbf {x}):=\{k\in \alpha : x_k=\max \limits _{i\in \alpha }\{x_{i}\}\}\) and \(\mathsf {{MinInd}}_\alpha (\mathbf {x}):=\{k\in \alpha : x_k=\min \limits _{i\in \alpha }\{x_{i}\}\}\) for \(\alpha \subset \mathbf {I}_m\). Particularly, we write \(\mathsf {MaxInd}(\mathbf {x})\) and \(\mathsf {MinInd}(\mathbf {x})\) for the set \(\alpha =\mathbf {I}_m\). An orbit (trajectory) of an initial point \(\mathbf {x}\in \mathbb {S}^{m-1}\) is defined as \(\{\mathbf {x}, \mathcal {R}(\mathbf {x}), \ldots , \mathcal {R}^{(n)}(\mathbf {x}), \cdots \}\). An omega limiting set \(\omega (\mathbf {x})\) of the orbit is defined as \(\omega (\mathbf {x}):=\bigcap \limits _{n\in \mathbb {N}}\overline{\bigcup \limits _{k\ge n}\{\mathcal {R}^{(k)}(\mathbf {x})\}}\). A rest (fixed) point set of the replicator operator is \(\mathbf{Fix} (\mathcal {R})=\{\mathbf {x}\in \mathbb {S}^{m-1}: \mathcal {R}(\mathbf {x})=\mathbf {x}\}\). We denote by \(\mathbf{NE} (F)\) and \(\mathbf{ESS} (F)\) the sets of all Nash equilibria and all ESSs, respectively, of the payoff mapping F.

3 The Main Results

In this paper, we would like to consider the following evolutionary game. Let an evolutionary game be identified by a nonlinear payoff mapping \(F(\mathbf {x}):=\left( F_1(\mathbf {x}),\ldots , F_m(\mathbf {x})\right) \) for which \(F_k(\mathbf {x})\) is a payoff (biological fitness) of a pure strategy \(k\in \mathbf {I}_m\) in a population state \(\mathbf {x}\in \mathbb {S}^{m-1}\). Then for any two different pure strategies, say i and j, a payoff (biological fitness) \(F_i(\mathbf {x})\) of a pure strategy which is frequent in number, say i, is better/worse than a payoff (biological fitness) \(F_j(\mathbf {x})\) of a pure strategy j. Mathematically, it means that for any \(\mathbf {x}\in \mathbb {S}^{m-1}\) and \(i,j\in \mathbf {I}_m\) we always have either one of two conditions:

  • \(F_i(\mathbf {x})>F_j(\mathbf {x})\) (resp. \(F_i(\mathbf {x})<F_j(\mathbf {x})\)) whenever \(x_i>x_j\) (resp. \(x_i<x_j\));

  • \(F_i(\mathbf {x})<F_j(\mathbf {x})\) (resp. \(F_i(\mathbf {x})>F_j(\mathbf {x})\)) whenever \(x_i>x_j\) (resp. \(x_i<x_j\)).

We study the proposed evolutionary game when the payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) is defined as follows

$$\begin{aligned} F(\mathbf {x})=\left( \varepsilon f(x_1)g(\mathbf {x}),\ \varepsilon f(x_2)g(\mathbf {x}), \ \cdots \ ,\ \varepsilon f(x_m)g(\mathbf {x})\right) \end{aligned}$$
(3.1)

where \(\varepsilon \in (-1,1)\) is a parameter, \(f:[0,1]\rightarrow [0,1]\) is a strictly increasing continuous function such that \(f(0)=0\) and the function xf(x) is continuously differentiable on [0, 1], and \(g:\mathbb {S}^{m-1}\rightarrow (0,1)\) is a smooth non-vanishing function. In this case, we have \(C_1\le g(\mathbf {x})\le C_2\) for any \(\mathbf {x}\in \mathbb {S}^{m-1}\) for some constants \(0<C_1<C_2<1\).

We now consider the discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) defined by the nonlinear payoff mapping (3.1)

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1+\varepsilon \left( f(x_k)-\sum \limits _{i=1}^mx_if(x_i)\right) g(\mathbf {x})\right] , \quad \forall \ k\in \mathbf {I}_m. \end{aligned}$$
(3.2)

Here, \(\varepsilon \in (-1,1)\) is a controlling regime of the replicator equation. We always assume that \(\varepsilon \ne 0\). We will see that the dynamics and the stability analysis of replicator equation (3.2) will depend on the controlling regime \(\varepsilon \). Namely, if we switch \(\varepsilon \) from positive to negative regime then the set of ESSs changes from one set to another one. This is a quite interesting feature of this replicator equation (3.2). This feature can be also seen in the following case.

Remark 3.1

If g is constant, i.e., \(g(\mathbf {x})=C>0\) for all \(\mathbf {x}\in \mathbb {S}^{m-1}\) then we get

$$\begin{aligned} F_0(\mathbf {x})=\left( \varepsilon Cf(x_1),\ \varepsilon Cf(x_2), \ \cdots \ ,\ \varepsilon Cf(x_m)\right) \end{aligned}$$
(3.3)

It is easy to check that

$$\begin{aligned} \left( F_0(\mathbf {x})-F_0(\mathbf {y}),\mathbf {x}-\mathbf {y}\right) =\varepsilon C\sum \limits _{k=1}^m\left( f(x_k)-f(y_k)\right) (x_k-y_k), \quad \forall \ \mathbf {x},\mathbf {y}\in \mathbb {S}^{m-1}. \end{aligned}$$

Since f is an increasing function and \(\mathbf {x},\mathbf {y}\in \mathbb {S}^{m-1}\), it is clear \(\left( f(x_k)-f(y_k)\right) (x_k-y_k)\ge 0\) for any \(k\in \mathbf {I}_m\). Consequently, for any \(\mathbf {x},\mathbf {y}\in \mathbb {S}^{m-1}\) we have that if \(\varepsilon >0\) then \(\left( F_0(\mathbf {x})-F_0(\mathbf {y}),\mathbf {x}-\mathbf {y}\right) \ge 0\) and if \(\varepsilon <0\) then \(\left( F_0(\mathbf {x})-F_0(\mathbf {y}),\mathbf {x}-\mathbf {y}\right) \le 0\).

An evolutionary game is called a stable game if the nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) satisfies the following condition \(\left( F(\mathbf {x})-F(\mathbf {y}),\mathbf {x}-\mathbf {y}\right) \le 0\) for any \(\mathbf {x},\mathbf {y}\in \mathbb {S}^{m-1}\). The dynamics of the continuous-time replicator equation

$$\begin{aligned} \dot{x}_k=x_k\left( f_k(\mathbf {x})-\mathcal {E}_F(\mathbf {x},\mathbf {x})\right) , \quad \forall \ \ \mathbf {x}\in \mathbb {S}^{m-1} \ and \ k\in \mathbf {I}_m \end{aligned}$$

governed with the stable game has been studied in the paper [21] (also see [43]).

A nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) is called monotone if one has that \(\left( F(\mathbf {x})-F(\mathbf {y}),\mathbf {x}-\mathbf {y}\right) \ge 0\) for any \(\mathbf {x},\mathbf {y}\in \mathbb {S}^{m-1}\). The dynamics of the discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\)

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left( 1+f_k(\mathbf {x})-\mathcal {E}_F(\mathbf {x},\mathbf {x})\right) , \quad \forall \ \mathbf {x}\in \mathbb {S}^{m-1} \ and \ \ k\in \mathbf {I}_m \end{aligned}$$

governed with nonlinear monotone mappings \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) subject to the constraint \(\mathcal {E}_F(\mathbf {x},\mathbf {x})=0\) for all \(\mathbf {x}\in \mathbb {S}^{m-1}\) has been studied in the papers [12,13,14, 31,32,33, 42].

Hence, if g is constant then the nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) given by (3.1) is either monotone or stable depending on the controlling regime \(\varepsilon \in (-1,1)\). In general, if g is not constant then the nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) given by (3.1) is neither monotone nor stable. In this paper, we are aiming to study the dynamics of the discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) given by (3.2) for any non-constant function g (for some examples see Application section).

We now describe the sets \(\mathbf{NE} (F)\) and \(\mathbf{ESS} (F)\) of all Nash equilibria and all local ESSs, respectively, of the nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) given by (3.1) and the set \(\mathbf{Fix} (\mathcal {R})\) of fixed (rest) points of the discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) given by (3.2).

Theorem A

(Rest points, Nash equilibria, and ESS) Let \(\mathbf {e}_i\) be the vertex of the simplex \({\mathbb {S}}^{m-1}\) for \(i\in \mathbf {I}_m\), let \(\mathbf {c}_\alpha :=\frac{1}{|\alpha |}\sum _{i\in \alpha }\mathbf {e}_i\) be the center of the face \({\mathbb {S}}^{|\alpha |-1}\) for all \(\alpha \subset \mathbf {I}_m\), and let \(\mathbf {c}:=\mathbf {c}_{\mathbf {I}_m}=(\frac{1}{m},\ldots ,\frac{1}{m})\) be the center of the simplex \({\mathbb {S}}^{m-1}\). Then the following statements hold true:

  1. (i)

    One has \(\mathbf{Fix}\mathbf (\mathcal {R})=\bigcup \limits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\) for any nonzero \(\varepsilon \in (-1,1)\);

  2. (ii)

    If \(\varepsilon \in (0,1)\) then \(\mathbf{NE}\mathbf (F)=\mathbf{Fix}\mathbf (\mathcal {R})\) and if \(\varepsilon \in (-1,0)\) then \(\mathbf{NE}\mathbf (F)=\{\mathbf {c}\}\subset \mathbf{Fix}\mathbf (\mathcal {R})\);

  3. (iii)

    If \(\varepsilon \in (0,1)\) then \(\mathbf{ESS}\mathbf (F)=\{\mathbf {e}_1,\mathbf {e}_2, \ldots , \mathbf {e}_m\}\subset \mathbf{NE}\mathbf (F)\) and if \(\varepsilon \in (-1,0)\) then \(\mathbf{ESS}\mathbf (F)=\mathbf{NE}\mathbf (F)\).

We define the following constant

$$\begin{aligned} \mu :=\max \limits _{\mathbf {x}\in \mathbb {S}^{m-1}}g(\mathbf {x})\cdot \max \limits _{x\in [0,1]}\frac{d}{dx}\left( xf(x)\right) . \end{aligned}$$

We are now ready to state the main results of this paper.

Theorem B

(Positive regime) Let \(\varepsilon \in (0,1)\) and let \(\mathbf {e}_i\) be the vertex of the simplex \({\mathbb {S}}^{m-1}\) for \(i\in \mathbf {I}_m\). Then the following statements hold true:

  1. (i)

    A stable rest point \(\mathbf {e}_i\) for \(i\in \mathbf {I}_m\) is a Nash equilibrium;

  2. (ii)

    A strictly Nash equilibrium \(\mathbf {e}_i\) for \(i\in \mathbf {I}_m\) is asymptotically stable;

  3. (iii)

    Any interior convergent orbit evolves to a Nash equilibrium.

Theorem C

(Negative regime) Let \(\varepsilon \in (-\frac{1}{\mu },0)\cap (-1,0)\) and let \(\mathbf {c}=(\frac{1}{m},\ldots ,\frac{1}{m})\) be the center of the simplex \({\mathbb {S}}^{m-1}\). Then the following statements hold true:

  1. (i)

    A stable rest point \(\mathbf {c}\) is a Nash equilibrium;

  2. (ii)

    There is no any strictly Nash equilibrium;

  3. (iii)

    Any interior convergent orbit evolves to a Nash equilibrium.

4 The Rest Points, Nash Equilibria, and ESS

4.1 The Proof of Theorem A

(i) We first show \(\mathbf{Fix} (\mathcal {R})=\bigcup \nolimits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\). It is obvious that \(\bigcup \nolimits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\subset \mathbf{Fix} (\mathcal {R})\). Let \(\mathbf {x}\in \mathbf{Fix} (\mathcal {R})\) be a fixed point. We set \(\alpha :=supp (\mathbf {x})\). Since \(\varepsilon \ne 0\) and \(g(\mathbf {x})>0\) for any \(\mathbf {x}\in \mathbb {S}^{m-1}\), it follows from (3.2)

$$\begin{aligned} f(x_k)=\sum \limits _{i\in \alpha }x_if(x_i), \quad \forall \ k\in \alpha . \end{aligned}$$

This means that \(f(x_{k_1})=f(x_{k_2})\) for any distinct \(k_1,k_2\in \alpha \). Since f is strictly increasing, we obtain \(x_{k_1}=x_{k_2}\) for any \(k_1,k_2\in \alpha \). Hence, we get \(\mathbf {x}=\mathbf {c}_\alpha \). This shows \(\mathbf{Fix} (\mathcal {R})=\bigcup \nolimits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\).

(ii) We now describe the set \(\mathbf{NE} (F)\) of all Nash equilibria of the nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\).

Let \(\varepsilon >0\). We show \(\mathbf{NE} (F)=\bigcup \nolimits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\). Since for any \(\mathbf {x}\in \mathbb {S}^{m-1}\) and \(\alpha \subset \mathbf {I}_m\)

$$\begin{aligned} \mathcal {E}_F(\mathbf {c}_\alpha ,\mathbf {c}_\alpha )=\varepsilon f\left( \frac{1}{|\alpha |}\right) g(\mathbf {c}_\alpha )\ge \varepsilon \left( \sum _{i\in \alpha }x_i\right) f\left( \frac{1}{|\alpha |}\right) g(\mathbf {c}_\alpha )=\mathcal {E}_F(\mathbf {x},\mathbf {c}_\alpha ), \end{aligned}$$

we get \(\bigcup \nolimits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\subset \mathbf{NE} (F)\). Let \(\mathbf {x}\in \mathbf{NE} (F)\) and \(\alpha :=supp (\mathbf {x})\). We have for \(\mathbf {z}\in \mathbb {S}^{m-1}\)

$$\begin{aligned} \varepsilon \left( \sum _{i\in \alpha }z_i\right) \min \limits _{i\in \alpha }\{f(x_i)\}g(\mathbf {x})\le \mathcal {E}_F(\mathbf {z},\mathbf {x})\le \varepsilon \left( \sum _{i\in \alpha }z_i\right) \max \limits _{i\in \alpha }\{f(x_i)\}g(\mathbf {x}). \end{aligned}$$

Since \(\mathbf {x}\in \mathbf{NE} (F)\), we must have

$$\begin{aligned} \mathcal {E}_F(\mathbf {x},\mathbf {x})=\max \limits _{\mathbf {z}\in \mathbb {S}^{m-1}}\mathcal {E}_F(\mathbf {z},\mathbf {x})=\varepsilon \max \limits _{i\in \alpha }\{f(x_i)\}g(\mathbf {x}). \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \mathcal {E}_F(\mathbf {x},\mathbf {x})=\varepsilon \left( \sum _{i\in \alpha } x_if(x_i)\right) g(\mathbf {x})\le \varepsilon \max \limits _{i\in \alpha }\{f(x_i)\}g(\mathbf {x}). \end{aligned}$$

Consequently, we obtain \(f(x_k)=\max \nolimits _{i\in \alpha }\{f(x_i)\}\) for any \(k\in \alpha \). Since f is strictly increasing, we obtain \(x_{k_1}=x_{k_2}\) for any \(k_1,k_2\in \alpha \) or \(\mathbf {x}=\mathbf {c}_\alpha \). This shows \(\mathbf{NE} (F)=\bigcup \nolimits _{\alpha \subset \mathbf {I}_m}\{\mathbf {c}_\alpha \}\) for \(\varepsilon >0\).

Let \(\varepsilon <0\). We show \(\mathbf{NE} (F)=\{\mathbf {c}\}\). It is obvious that \(\mathbf {c}\in \mathbf{NE} (F)\).

Let \(\mathbf {x}\in \mathbf{NE} (F)\) and \(\alpha :=supp (\mathbf {x})\). We then want to prove that \(\alpha =\mathbf {I}_m\). We assume the contrary, i.e., \(\alpha \ne \mathbf {I}_m\). Let \(k\in \mathbf {I}_m{\setminus }\alpha \) (since \(\mathbf {I}_m{\setminus }\alpha \ne \emptyset \)). Since \(\varepsilon <0\), \(g(\mathbf {x})>0\), and \(\mathbf {x}\in \mathbf{NE} (F)\), we face the contradiction

$$\begin{aligned} 0=\mathcal {E}_F(\mathbf {e}_k,\mathbf {x})\le \mathcal {E}_F(\mathbf {x},\mathbf {x})=\varepsilon \left( \sum _{i\in \alpha } x_if(x_i)\right) g(\mathbf {x})<0. \end{aligned}$$

This shows \(\alpha =\mathbf {I}_m\). In this case, we have for any \(\mathbf {z}\in \mathbb {S}^{m-1}\)

$$\begin{aligned} \varepsilon \max \limits _{i\in \mathbf {I}_m}\{f(x_i)\}g(\mathbf {x})\le \mathcal {E}_F(\mathbf {z},\mathbf {x})\le \varepsilon \min \limits _{i\in \mathbf {I}_m}\{f(x_i)\}g(\mathbf {x}). \end{aligned}$$

Since \(\mathbf {x}\in \mathbf{NE} (F)\), we must have

$$\begin{aligned} \mathcal {E}_F(\mathbf {x},\mathbf {x})=\max \limits _{\mathbf {z}\in \mathbb {S}^{m-1}}\mathcal {E}_F(\mathbf {z},\mathbf {x})=\varepsilon \min \limits _{i\in \mathbf {I}_m}\{f(x_i)\}g(\mathbf {x}). \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \mathcal {E}_F(\mathbf {x},\mathbf {x})=\varepsilon \left( \sum _{i=1} ^mx_if(x_i)\right) g(\mathbf {x})\le \varepsilon \min \limits _{i\in \mathbf {I}_m}\{f(x_i)\}g(\mathbf {x}). \end{aligned}$$

Consequently, we obtain \(f(x_k)=\min \nolimits _{i\in \mathbf {I}_m}\{f(x_i)\}\) for any \(k\in \mathbf {I}_m\). Since f is strictly increasing, we obtain that \(x_{k_1}=x_{k_2}\) for any \(k_1,k_2\in \mathbf {I}_m\) or \(\mathbf {x}=\mathbf {c}\). This shows \(\mathbf{NE} (F)=\{\mathbf {c}\}\) for \(\varepsilon <0\).

(iii) Finally, we describe the set \(\mathbf{ESS} (F)\) of all local ESSs of the nonlinear payoff mapping \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) given by (3.1). According to Definition 2.2 (see item (i)), we have \(\mathbf{ESS} (F)\subset \mathbf{NE} (F)\).

Let \(\varepsilon >0\). We first show \(\{\mathbf {e}_1,\mathbf {e}_2, \ldots , \mathbf {e}_m\}\subset \mathbf{ESS} (F)\). It order to accomplish it, we want to show that there is a sufficiently small neighborhood \(U(\mathbf {e}_k)\subset \mathbb {S}^{m-1}\) of the vertex \(\mathbf {e}_k\) for \(k\in \mathbf {I}_m\) such that \(\mathcal {E}_F(\mathbf {e}_k,\mathbf {x})>\mathcal {E}_F(\mathbf {x},\mathbf {x})\) for all \(\mathbf {x}\in U(\mathbf {e}_k){\setminus }\{\mathbf {e}_k\}\). Since \(U(\mathbf {e}_k)\subset \mathbb {S}^{m-1}\) is sufficiently small, we have \(\mathsf {MaxInd}(\mathbf {x})=\{k\}\) and \(supp (\mathbf {x}){\setminus }\{k\}\ne \emptyset \) for all \(\mathbf {x}\in U(\mathbf {e}_k){\setminus }\{\mathbf {e}_k\}\). Consequently, we get for all \(\mathbf {x}\in U(\mathbf {e}_k){\setminus }\{\mathbf {e}_k\}\)

$$\begin{aligned} \mathcal {E}_F(\mathbf {x},\mathbf {x})= & {} \varepsilon \left( \sum _{i\in supp (\mathbf {x})} x_if(x_i)\right) g(\mathbf {x})\\< & {} \varepsilon \max \limits _{i\in supp (\mathbf {x})}\{f(x_i)\}g(\mathbf {x}) =\varepsilon f(x_k)g(\mathbf {x})=\mathcal {E}_F(\mathbf {e}_k,\mathbf {x}). \end{aligned}$$

This shows \(\{\mathbf {e}_1,\mathbf {e}_2, \ldots , \mathbf {e}_m\}\subset \mathbf{ESS} (F)\). We now show \(\mathbf{ESS} (F)\cap \mathbf{NE} (F)=\{\mathbf {e}_1,\mathbf {e}_2, \ldots , \mathbf {e}_m\}\).

Let \(\mathbf {c}_\alpha \in \mathbf{NE} (F)\) with \(|\alpha |\ge 2\) be any Nash equilibrium. Let \(\alpha :=\{i_1,i_2,\ldots ,i_{|\alpha |}\}\) such that \(i_1<i_2<\cdots <i_{|\alpha |}\). We want to show that for any sufficiently small neighborhood \(U(\mathbf {c}_\alpha )\subset \mathbb {S}^{m-1}\) of the point \(\mathbf {c}_\alpha \) there always exists \(\bar{\mathbf {x}}\in U(\mathbf {c}_\alpha ){\setminus }\{\mathbf {c}_\alpha \}\) such that \(\mathcal {E}_F(\mathbf {c}_\alpha ,\bar{\mathbf {x}})<\mathcal {E}_F(\bar{\mathbf {x}},\bar{\mathbf {x}})\). Let \(\delta >0\) be any sufficiently small number and \(\Vert \bar{\mathbf {x}}-\mathbf {c}_\alpha \Vert _1<\delta \) such that \(supp (\bar{\mathbf {x}})=\alpha \) and \(\bar{x}_{i_1}<\bar{x}_{i_2}<\cdots <\bar{x}_{i_{|\alpha |}}\) (it is always possible to choose such \(\bar{\mathbf {x}}\in U(\mathbf {c}_\alpha )\) for any \(\delta >0\)). Since f is strictly increasing, it follows from \(f(\bar{x}_{i_1})<f(\bar{x}_{i_2})<\cdots <f(\bar{x}_{i_{|\alpha |}})\) and Chebyshev’s sum inequality (see [30])

$$\begin{aligned} \mathcal {E}_F(\mathbf {c}_\alpha ,\bar{\mathbf {x}})=\varepsilon \left( \frac{1}{|\alpha |}\sum _{j=1}^{|\alpha |}f(\bar{x}_{i_j})\right) g(\bar{\mathbf {x}})<\varepsilon \left( \sum _{j=1}^{|\alpha |}\bar{x}_{i_j}f(\bar{x}_{i_j})\right) g(\bar{\mathbf {x}})=\mathcal {E}_F(\bar{\mathbf {x}},\bar{\mathbf {x}}). \end{aligned}$$

It means that \(\mathbf {c}_\alpha \in \mathbf{NE} (F)\) (\(|\alpha |\ge 2\)) is not a local ESS. Consequently, we obtain \(\mathbf{ESS} (F)=\{\mathbf {e}_1,\mathbf {e}_2, \ldots , \mathbf {e}_m\}\) for \(\varepsilon >0\).

Let \(\varepsilon <0\). According to Definition 2.2, we have \(\mathbf{ESS} (F)\subset \mathbf{NE} (F)=\{\mathbf {c}\}\). We show \(\mathbf {c}\in \mathbf{ESS} (F)\).

Let \(U(\mathbf {c})\subset \mathbb {S}^{m-1}\) be a sufficiently small neighborhood of the center \(\mathbf {c}\) of the simplex \({\mathbb {S}}^{m-1}\). Since f is strictly increasing, two vectors \((x_1,\ldots ,x_m)\) and \((f(x_1),\ldots ,f(x_m))\) are the same ordered, i.e., if \(x_i>x_j\) (resp. \(x_i<x_j\)) then \(f(x_i)>f(x_j)\) (resp. \(f(x_i)<f(x_j)\)), Since \(\varepsilon <0\), it then follows from Chebyshev’s sum inequality (see [30]) that

$$\begin{aligned} \mathcal {E}_F(\mathbf {c},{\mathbf {x}})=\varepsilon \left( \frac{1}{m}\sum _{i=1}^{m}f(x_i)\right) g({\mathbf {x}})>\varepsilon \left( \sum _{i=1}^{m}{x}_{i}f(x_i)\right) g({\mathbf {x}})=\mathcal {E}_F({\mathbf {x}},{\mathbf {x}}) \end{aligned}$$

for any \(\mathbf {x}\in U(\mathbf {c}){\setminus }\{\mathbf {c}\}\) (for our case, the equality in Chebyshev’s sum inequality holds only for \(\mathbf {x}=\mathbf {c}\)). It means \(\mathbf{ESS} (F)=\mathbf {c}\) for \(\varepsilon <0\). This completes the proof of Theorem A.

5 The Regime \(\varepsilon \in (0,1)\)

5.1 The Lyapunov Function

We study the dynamics of the discrete-time replicator equation (3.2) by means of a Lyapunov function.

Proposition 5.1

Let \(\varepsilon \in (0,1)\). Then the following statements hold true:

  1. (i)

    \(\mathcal {M}_k(\mathbf {x}):=\max \limits _{i\in \mathbf {I}_m}\{x_{i}\}-x_k\) for \(k\in \mathbf {I}_m\) is an increasing Lyapunov function for \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1};\)

  2. (ii)

    \(\mathcal {M}_{\alpha , k}(\mathbf {x}):=\max \limits _{i\in \alpha }\{x_{i}\}-x_k\) for \(k\in \alpha \subset \mathbf {I}_m\) is an increasing Lyapunov function for \(\mathcal {R}:int \mathbb {S}^{|\alpha |-1}\rightarrow int \mathbb {S}^{|\alpha |-1}\).

Proof

Here, we only present the proof of the part (i). The part (ii) can be similarly proved. Its proof is omitted here.

We show that \(\mathcal {M}_k(\mathbf {x})=\max \limits _{i\in \mathbf {I}_m}\{x_{i}\}-x_k\) for \(k\in \mathbf {I}_m\) is an increasing Lyapunov function along an orbit (trajectory) of the operator \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1}\).

Let \(\mathbf {x}\in int \mathbb {S}^{m-1}\). It follows from (3.2) for any \(k,t\in \mathbf {I}_m\) that if \(x_t=x_k\) then \((\mathcal {R}(\mathbf {x}))_t=(\mathcal {R}(\mathbf {x}))_k\) and if \(x_t\ne x_k\) then

$$\begin{aligned} (\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k =(x_t-x_k)\left[ 1+\varepsilon \left( \frac{x_tf(x_t)-x_kf(x_k)}{x_t-x_k}-\sum \limits _{i=1}^mx_if(x_i)\right) g(\mathbf {x})\right] \end{aligned}$$

which yields the following equality

$$\begin{aligned} (\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k\\ =(x_t-x_k)\left[ 1-\varepsilon g(\mathbf {x})+ \varepsilon \left( \frac{x_tf(x_t)-x_kf(x_k)}{x_t-x_k}+\sum \limits _{i=1}^mx_i(1-f(x_i))\right) g(\mathbf {x})\right] \end{aligned}$$

Since \(0<\varepsilon , g(\mathbf {x})<1\), \(0\le f(x_i)\le 1\) for all \(i\in \mathbf {I}_m\) and the function xf(x) is increasing on [0, 1], i.e.,

$$\begin{aligned} \left( x_tf(x_t)-x_kf(x_k)\right) (x_t-x_k)\ge 0, \quad \forall \ \mathbf {x}\in int \mathbb {S}^{m-1}, \end{aligned}$$

we obtain \(\mathsf {Sign}\left( (\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k\right) =\mathsf {Sign}(x_t-x_k)\) for any \(k,t\in \mathbf {I}_m\) and

$$\begin{aligned} \mathsf {MaxInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x}), \quad \forall \ \mathbf {x}\in int \mathbb {S}^{m-1}. \end{aligned}$$

Moreover, if \(t\in \mathsf {MaxInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x})\) then for all \(k\in \mathbf {I}_m\)

$$\begin{aligned}&(\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k\\&\quad =(x_t-x_k) +\varepsilon (x_t-x_k)\left( \frac{x_k\left( f(x_t)-f(x_k)\right) }{x_t-x_k}+\sum \limits _{i=1}^mx_i\left( f(x_t)-f(x_i)\right) \right) g(\mathbf {x}). \end{aligned}$$

This means \(\mathcal {M}_k\left( \mathcal {R}(\mathbf {x})\right) \ge \mathcal {M}_k(\mathbf {x})\) for all \(k\in \mathbf {I}_m\).

By repeating this process, we get for all \(k\in \mathbf {I}_m\), \(n\in \mathbb {N}\), and \(\mathbf {x}\in int \mathbb {S}^{m-1}\) that

$$\begin{aligned} \mathsf {MaxInd}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x}), \quad \mathcal {M}_k\left( \mathcal {R}^{(n+1)}(\mathbf {x})\right) \ge \mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) . \end{aligned}$$

This shows that \(\mathcal {M}_k(\mathbf {x})\) for \(k\in \mathbf {I}_m\) is a Lyapunov function over the set \(int \mathbb {S}^{m-1}\). This completes the proof. \(\square \)

5.2 The Stability Analysis

Theorem 5.2

If \(\varepsilon \in (0,1)\) then an orbit of the replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) starting from any initial point \(\mathbf {x}\in \mathbb {S}^{m-1}\) converges to the center of the face \(\mathbb {S}^{|\mathsf {MaxInd}(\mathbf {x})|-1}\).

Proof

Without loss of generality, we may assume that \(\mathbf {x}\in int \mathbb {S}^{m-1}\). Otherwise, we choose a suitable Lyapunov function given in the part (ii) of Proposition 5.1 depending on an initial point \(\mathbf {x}\in \mathbb {S}^{m-1}\).

We fix \(t_0\in \mathsf {MaxInd}(\mathbf {x})\). As shown in Proposition 5.1, we obtain for any \(n\in \mathbb {N}\) that \(\mathsf {MaxInd}(\mathbf {x})=\mathsf {MaxInd}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \). Since the sequence \( \left\{ \mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \right\} _{n=0}^\infty \) where

$$\begin{aligned} \mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) =\left( \mathcal {R}^{(n)}(\mathbf {x})\right) _{t_0}-\left( \mathcal {R}^{(n)}(\mathbf {x})\right) _k \end{aligned}$$

is convergent for each \(k\in \mathbf {I}_m\), the sequence \(\left\{ \left( \mathcal {R}^{(n)}(\mathbf {x})\right) _{t_0}\right\} _{n=0}^\infty \) where

$$\begin{aligned} \left( \mathcal {R}^{(n)}(\mathbf {x})\right) _{t_0}=\frac{1}{m}\left( 1+\sum _{k=1}^m\mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \right) \end{aligned}$$

is also convergent (m is the number of pure strategies \(\mathbf {I}_m=\{1,2,\cdots m\}\)). Hence, the sequence \(\left\{ \left( \mathcal {R}^{(n)}(\mathbf {x})\right) _{k}\right\} _{n=0}^\infty \) for each \(k\in \mathbf {I}_m\) where

$$\begin{aligned} \left( \mathcal {R}^{(n)}(\mathbf {x})\right) _k=\left( \mathcal {R}^{(n)}(\mathbf {x})\right) _{t_0}-\mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \end{aligned}$$

is also convergent. This means that the trajectory \(\left\{ \mathcal {R}^{(n)}(\mathbf {x})\right\} _{n=0}^\infty \) is convergent and its omega limiting point is some fixed (rest) point \(\mathbf {x}^{*}\). We now want to show that

$$\begin{aligned} \mathsf {MaxInd}(\mathbf {x}^{*})=\mathsf {MaxInd}(\mathbf {x}). \end{aligned}$$

Since \(\mathsf {MaxInd}(\mathbf {x})=\mathsf {MaxInd}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \) and \(\mathcal {M}_k\left( \mathcal {R}^{(n+1)}(\mathbf {x})\right) \ge \mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \) for any \(n\in \mathbb {N}\) and \(k\in \mathbf {I}_m\), we have that \(\mathsf {MaxInd}(\mathbf {x}^{*})\supset \mathsf {MaxInd}(\mathbf {x})\). We now show \(\mathsf {MaxInd}(\mathbf {x}^{*})\subset \mathsf {MaxInd}(\mathbf {x})\). Indeed, if \(k_0\in \mathsf {MaxInd}(\mathbf {x}^{*})\) then we get that

$$\begin{aligned} 0\le \mathcal {M}_{k_0}\left( \mathbf {x}\right) \le \mathcal {M}_{k_0}\left( \mathcal {R}(\mathbf {x})\right) \le \cdots \mathcal {M}_{k_0}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) \le \cdots \le \mathcal {M}_{k_0}\left( \mathbf {x}^{*}\right) =0 \end{aligned}$$

This means \(\mathcal {M}_{k_0}\left( \mathbf {x}\right) =0\) or \(k_0\in \mathsf {MaxInd}(\mathbf {x})\). Hence, we prove \(\mathsf {MaxInd}(\mathbf {x}^{*})=\mathsf {MaxInd}(\mathbf {x})\). Consequently, due to Theorem A, among the centers of all faces of the simplex the only fixed point \(\mathbf {x}^{*}\) which satisfies the property \(\mathsf {MaxInd}(\mathbf {x}^{*})=\mathsf {MaxInd}(\mathbf {x})\) is the center of the face \(\mathbb {S}^{|\mathsf {MaxInd}(\mathbf {x})|-1}\). This completes the proof. \(\square \)

5.3 The Proof of Theorem B

(i) We show that the stable rest points are only the vertices of the simplex.

On the one hand, according to Theorem 5.2, if \(|\alpha |>1\) for \(\alpha \subset \mathbf {I}_m\) then the center \(\mathbf {c}_\alpha \) of the face \(\mathbb {S}^{|\alpha |-1}\) (which is the rest (fixed) point) is not stable. Indeed, for any small neighborhood \(U(\mathbf {c}_\alpha )\) of the center \(\mathbf {c}_\alpha \) there are points \(\mathbf {x}\in U(\mathbf {c}_\alpha )\) and \(\mathbf {y}\in U(\mathbf {c}_\alpha )\) such that \(|\mathsf {MaxInd}(\mathbf {x})|=1\), \(|\mathsf {MaxInd}(\mathbf {y})|=1\), and \(\mathsf {MaxInd}(\mathbf {x})\ne \mathsf {MaxInd}(\mathbf {y})\) for which the orbits of the points \(\mathbf {x}\) and \(\mathbf {y}\) converge to two different vertices of the simplex \({\mathbb {S}}^{m-1}\). Therefore, the center \(\mathbf {c}_\alpha \) of the face \(\mathbb {S}^{|\alpha |-1}\) is not stable whenever \(|\alpha |>1\) for \(\alpha \subset \mathbf {I}_m\).

On the other hand, according to Theorem 5.2, the vertex \(\mathbf {e}_k\) for \(k\in \mathbf {I}_m\) of the simplex \({\mathbb {S}}^{m-1}\) is stable. Indeed, for any small neighborhood \(U(\mathbf {e}_k)\) of the vertex \(\mathbf {e}_k\) one has \(\mathsf {MaxInd}(\mathbf {x})=\{k\}\) for any \(\mathbf {x}\in U(\mathbf {e}_k)\). As shown in Proposition 5.1 that \(\mathsf {MaxInd}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x})=\{k\}\) for any \(n\in \mathbb {N}\). Since \(\varepsilon >0\), it follows from (3.2) and \(k\in \mathsf {MaxInd}(\mathbf {x})\) that

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1+\varepsilon \sum \limits _{i=1}^mx_i\left( f(x_k)-f(x_i)\right) g(\mathbf {x})\right] \ge x_k. \end{aligned}$$

Hence, we obtain \(\Vert \mathcal {R}(\mathbf {x})-\mathbf {e}_k\Vert _1=2(1-\left( \mathcal {R}(\mathbf {x})\right) _k)\le 2(1-x_k)=\Vert \mathbf {x}-\mathbf {e}_k\Vert _1\) which implies \(\mathcal {R}(U(\mathbf {e}_k))\subset U(\mathbf {e}_k)\) and consequently \(\mathcal {R}^{(n)}(U(\mathbf {e}_k))\subset U(\mathbf {e}_k)\). This means that the vertex \(\mathbf {e}_k\) for \(k\in \mathbf {I}_m\) of the simplex \({\mathbb {S}}^{m-1}\) is stable.

(ii) We first show that the vertex \(\mathbf {e}_k\) for \(k\in \mathbf {I}_m\) of the simplex \({\mathbb {S}}^{m-1}\) is a strictly Nash equilibrium, i.e., the strategy \(\mathbf {e}_k\) is the unique best reply to itself, \(\mathsf {BR}(\mathbf {e}_k)=\{\mathbf {e}_k\}\) for all \(k\in \mathbf {I}_m\). Indeed, for all \(\mathbf {x}\in {\mathbb {S}}^{m-1}{\setminus }\{\mathbf {e}_k\}\) we have that

$$\begin{aligned} \mathcal {E}_F(\mathbf {e}_k,\mathbf {e}_k)=\varepsilon f\left( 1\right) g(\mathbf {e}_k)>\varepsilon x_kf\left( 1\right) g(\mathbf {e}_k)=\mathcal {E}_F(\mathbf {x},\mathbf {e}_k). \end{aligned}$$

As we already showed that the vertex \(\mathbf {e}_k\) for \(k\in \mathbf {I}_m\) of the simplex \({\mathbb {S}}^{m-1}\) is both stable (due to part (i)) and attracting (due to Theorem 5.2). Hence, a strictly Nash equilibrium \(\mathbf {e}_i\) for \(i\in \mathbf {I}_m\) is asymptotically stable.

(iii) Due to Theorem 5.2, an interior orbit starting from any initial point \(\mathbf {x}\in int \mathbb {S}^{m-1}\) converges to the center of the face \(\mathbb {S}^{|\mathsf {MaxInd}(\mathbf {x})|-1}\) which is a Nash equilibrium. This completes the proof of Theorem B.

6 The Regime \(\varepsilon \in (-1,0)\)

We first define the following constant

$$\begin{aligned} \mu :=\max \limits _{\mathbf {x}\in \mathbb {S}^{m-1}}g(\mathbf {x})\cdot \max \limits _{x\in [0,1]}\frac{d}{dx}\left( xf(x)\right) . \end{aligned}$$

Since the function xf(x) is continuously differentiable on [0, 1], \(\mu \) is well-defined.

6.1 The Lyapunov Function

We study the dynamics of the discrete-time replicator equation given by (3.2) by means of a Lyapunov function.

Proposition 6.1

Let \(\varepsilon \in (-\frac{1}{\mu },0)\cap (-1,0)\). Then the following statements hold true:

  1. (i)

    \(\mathcal {M}_k(\mathbf {x}):=\max \limits _{i\in \mathbf {I}_m}\{x_{i}\}-x_k\) for \(k\in \mathbf {I}_m\) is a decreasing Lyapunov function for \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1}\);

  2. (ii)

    \(\mathcal {M}_{\alpha , k}(\mathbf {x}):=\max \limits _{i\in \alpha }\{x_{i}\}-x_k\) for \(k\in \alpha \subset \mathbf {I}_m\) is a decreasing Lyapunov function for \(\mathcal {R}:int \mathbb {S}^{|\alpha |-1}\rightarrow int \mathbb {S}^{|\alpha |-1}\).

Proof

Here, we only present the proof of the part (i). The part (ii) can be similarly proved. Its proof is omitted here.

We show that \(\mathcal {M}_k(\mathbf {x})=\max \limits _{i\in \mathbf {I}_m}\{x_{i}\}-x_k\) for \(k\in \mathbf {I}_m\) is a decreasing Lyapunov function along an orbit (trajectory) of the operator \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1}\).

Let \(\mathbf {x}\in int \mathbb {S}^{m-1}\). It follows from (3.2) for any \(k,t\in \mathbf {I}_m\) that if \(x_t=x_k\) then \((\mathcal {R}(\mathbf {x}))_t=(\mathcal {R}(\mathbf {x}))_k\) and if \(x_t\ne x_k\) then

$$\begin{aligned} (\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k =(x_t-x_k)\left[ 1+\varepsilon \left( \frac{x_tf(x_t)-x_kf(x_k)}{x_t-x_k}-\sum \limits _{i=1}^mx_if(x_i)\right) g(\mathbf {x})\right] . \end{aligned}$$

Since \(-\frac{1}{\mu }<\varepsilon <0\) and for all \(t,k\in \mathbf {I}_m\)

$$\begin{aligned} \left| \frac{x_tf(x_t)-x_kf(x_k)}{x_t-x_k}\right| \le \max \limits _{x\in [0,1]}\frac{d}{dx}\left( xf(x)\right) , \end{aligned}$$

we obtain \(\mathsf {Sign}\left( (\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k\right) =\mathsf {Sign}(x_t-x_k)\) for any \(k,t\in \mathbf {I}_m\) and

$$\begin{aligned} \mathsf {MaxInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x}), \quad \forall \ \mathbf {x}\in int \mathbb {S}^{m-1}. \end{aligned}$$

Moreover, if \(t\in \mathsf {MaxInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x})\) then for all \(k\in \mathbf {I}_m\)

$$\begin{aligned}&(\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k\\&\quad =(x_t-x_k) +\varepsilon (x_t-x_k)\left( \frac{x_k\left( f(x_t)-f(x_k)\right) }{x_t-x_k}+\sum \limits _{i=1}^mx_i\left( f(x_t)-f(x_i)\right) \right) g(\mathbf {x}). \end{aligned}$$

Since \(\varepsilon <0\), we obtain that \(\mathcal {M}_k\left( \mathcal {R}(\mathbf {x})\right) \le \mathcal {M}_k(\mathbf {x})\) for all \(k\in \mathbf {I}_m\).

By repeating this process, we get for all \(k\in \mathbf {I}_m\), \(n\in \mathbb {N}\), and \(\mathbf {x}\in int \mathbb {S}^{m-1}\) that

$$\begin{aligned} \mathsf {MaxInd}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x}), \quad \mathcal {M}_k\left( \mathcal {R}^{(n+1)}(\mathbf {x})\right) \le \mathcal {M}_k\left( \mathcal {R}^{(n)}(\mathbf {x})\right) . \end{aligned}$$

This shows that \(\mathcal {M}_k(\mathbf {x})\) for \(k\in \mathbf {I}_m\) is a Lyapunov function over the set \(int \mathbb {S}^{m-1}\). This completes the proof. \(\square \)

6.2 The Stability Analysis

Theorem 6.2

If \(\varepsilon \in (-\frac{1}{\mu },0)\cap (-1,0)\) then an orbit of the replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) starting from any initial point \(\mathbf {x}\in \mathbb {S}^{m-1}\) converges to the center of the face \(\mathbb {S}^{|supp (\mathbf {x})|-1}\).

Proof

Without loss of generality, we may assume that \(\mathbf {x}\in int \mathbb {S}^{m-1}\). Otherwise, we choose a suitable Lyapunov function given in the part (ii) of Proposition 6.1 depending on an initial point \(\mathbf {x}\in \mathbb {S}^{m-1}\).

Since \(\mathcal {M}_k(\mathbf {x})\) for \(k\in \mathbf {I}_m\) is a Lyapunov function over the set \(int \mathbb {S}^{m-1}\), one can show by using the same technique implemented in Theorem 5.2 that the orbit (trajectory) \(\left\{ \mathcal {R}^{(n)}(\mathbf {x})\right\} _{n=0}^\infty \) of the operator \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1}\) starting from any initial point \(\mathbf {x}\in int \mathbb {S}^{m-1}\) is convergent and its omega limiting point is some fixed (rest) point \(\mathbf {x}^{**}\). Moreover, we now show that the orbit (trajectory) \(\left\{ \mathcal {R}^{(n)}(\mathbf {x})\right\} _{n=0}^\infty \) of the operator \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1}\) is separated from the boundary \(\partial \mathbb {S}^{m-1}\) of the simplex \(\mathbb {S}^{m-1}\), i.e., for some \(\delta _0>0\) we have

$$\begin{aligned} \min \limits _{k\in \mathbf {I}_m}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) _k\ge \delta _0, \quad \forall \ n\in \mathbb {N}. \end{aligned}$$

Indeed, as we already showed in the part (i) of Proposition 6.1 that

$$\begin{aligned} \mathsf {Sign}\left( (\mathcal {R}(\mathbf {x}))_t-(\mathcal {R}(\mathbf {x}))_k\right) =\mathsf {Sign}(x_t-x_k) \end{aligned}$$

for any \(k,t\in \mathbf {I}_m\), we obtain \(\mathsf {MinInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MinInd}(\mathbf {x})\) for all \(\mathbf {x}\in int \mathbb {S}^{m-1}\).

Since \(\varepsilon <0\), it follows from (3.2) and \(k\in \mathsf {MinInd}(\mathbf {x})\) that

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1-\varepsilon \sum \limits _{i=1}^mx_i\left( f(x_i)-f(x_k)\right) g(\mathbf {x})\right] \ge x_k. \end{aligned}$$

This means that \( \min \limits _{k\in \mathbf {I}_m}\left( \mathcal {R}(\mathbf {x})\right) _k\ge \delta _0\) for \(\delta _0:=\min \limits _{k\in \mathbf {I}_m}x_k>0\) (since \(\mathbf {x}\in int \mathbb {S}^{m-1}\)).

By repeating this process, we obtain that

$$\begin{aligned} \mathsf {MinInd}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) =\mathsf {MinInd}(\mathbf {x}), \quad \min \limits _{k\in \mathbf {I}_m}\left( \mathcal {R}^{(n)}(\mathbf {x})\right) _k\ge \delta _0, \quad \forall \ n\in \mathbb {N}. \end{aligned}$$

Since the trajectory \(\left\{ \mathcal {R}^{(n)}(\mathbf {x})\right\} _{n=0}^\infty \) of the operator \(\mathcal {R}:int \mathbb {S}^{m-1}\rightarrow int \mathbb {S}^{m-1}\) is separated from the boundary \(\partial \mathbb {S}^{m-1}\) of the simplex \(\mathbb {S}^{m-1}\), its limiting point \(\mathbf {x}^{**}\) belongs to the interior \(int \mathbb {S}^{m-1}\) of the simplex \(\mathbb {S}^{m-1}\). Then the only fixed point in the interior \(int \mathbb {S}^{m-1}\) of the simplex \(\mathbb {S}^{m-1}\) is its center. Hence, the omega limiting point of the trajectory starting from any initial point \(\mathbf {x}\in int \mathbb {S}^{m-1}\) is the center of the simplex \(\mathbb {S}^{m-1}\). This completes the proof. \(\square \)

6.3 The Proof of Theorem C

(i) We show that the only stable rest point is the center \(\mathbf {c}=(\frac{1}{m},\ldots ,\frac{1}{m})\) of the simplex \(\mathbb {S}^{m-1}\) which is, due to Theorem A, a Nash equilibrium.

On the one hand, according to Theorem 6.2, if \(|\alpha |<m\) for \(\alpha \subset \mathbf {I}_m\) (which means \(\alpha \ne \mathbf {I}_m\)) then the center \(\mathbf {c}_\alpha \) of the face \(\mathbb {S}^{|\alpha |-1}\) (which is the rest point) is not stable. Indeed, for any small neighborhood \(U(\mathbf {c}_\alpha )\subset \mathbb {S}^{m-1}\) of the center \(\mathbf {c}_\alpha \) there is a point \(\mathbf {x}\in U(\mathbf {c}_\alpha )\cap int \mathbb {S}^{m-1}\) such that an orbit of the point \(\mathbf {x}\) converges to the center \(\mathbf {c}\) of the simplex \({\mathbb {S}}^{m-1}\). Therefore, the center \(\mathbf {c}_\alpha \) of the face \(\mathbb {S}^{|\alpha |-1}\) is not stable whenever \(|\alpha |<m\) for \(\alpha \subset \mathbf {I}_m\) (which means \(\alpha \ne \mathbf {I}_m\)).

On the other hand, according to Theorem 6.2, the center \(\mathbf {c}\) of the simplex \(\mathbb {S}^{m-1}\) is stable. Namely, for any ball \(U_R(\mathbf {c})\subset int \mathbb {S}^{m-1}\) of radius R there exists a (small) ball \(U_r(\mathbf {c})\subset int \mathbb {S}^{m-1}\) of radius r (we may choose \(r=\frac{R}{m}\)) such that \(\Vert \mathcal {R}^{(n)}(\mathbf {x})-\mathbf {c}\Vert _1\le R\) for any \(n\in \mathbb {R}\) whenever \(\Vert \mathbf {x}-\mathbf {c}\Vert _1\le r\). Indeed, it is clear that \(\max _{i\in \mathbf {I}_m}\{x_i\}-\min _{i\in \mathbf {I}_m}\{x_i\}\le \Vert \mathbf {x}-\mathbf {c}\Vert _1\le r\). Since \(\varepsilon <0\), for \(k_1\in \mathsf {MinInd}(\mathbf {x})\) and \(k_2\in \mathsf {MaxInd}(\mathbf {x})\) it follows from (3.2) that

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _{k_1}=x_{k_1}\left[ 1-\varepsilon \sum \limits _{i=1}^mx_i\left( f(x_i)-f(x_{k_1})\right) g(\mathbf {x})\right] \ge x_{k_1},\\ \left( \mathcal {R}(\mathbf {x})\right) _{k_2}=x_{k_2}\left[ 1+\varepsilon \sum \limits _{i=1}^mx_i\left( f(x_{k_2})-f(x_{i})\right) g(\mathbf {x})\right] \le x_{k_2}. \end{aligned}$$

Since \(\mathsf {MinInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MinInd}(\mathbf {x})\) and \( \mathsf {MaxInd}\left( \mathcal {R}(\mathbf {x})\right) =\mathsf {MaxInd}(\mathbf {x})\), we obtain

$$\begin{aligned} \min _{i\in \mathbf {I}_m}\{x_i\}\le \min _{i\in \mathbf {I}_m}\{\left( \mathcal {R}(\mathbf {x})\right) _i\}\le \max _{i\in \mathbf {I}_m}\{\left( \mathcal {R}(\mathbf {x})\right) _i\}\le \max _{i\in \mathbf {I}_m}\{x_i\},\\ \max _{i\in \mathbf {I}_m}\{\left( \mathcal {R}(\mathbf {x})\right) _i\}-\min _{i\in \mathbf {I}_m}\{\left( \mathcal {R}(\mathbf {x})\right) _i\}\le \max _{i\in \mathbf {I}_m}\{x_i\}-\min _{i\in \mathbf {I}_m}\{x_i\}\le r. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} \Vert \mathcal {R}(\mathbf {x})-\mathbf {c}\Vert _1\le m \left( \max _{i\in \mathbf {I}_m}\{\left( \mathcal {R}(\mathbf {x})\right) _i\}-\min _{i\in \mathbf {I}_m}\{\left( \mathcal {R}(\mathbf {x})\right) _i\}\right) \le m\cdot r=R. \end{aligned}$$

By repeating this process, we obtain that \(\Vert \mathcal {R}^{(n)}(\mathbf {x})-\mathbf {c}\Vert _1\le R\) for any \(n\in \mathbb {R}\) whenever \(\Vert \mathbf {x}-\mathbf {c}\Vert _1\le r\). This means that the center \(\mathbf {c}\) of the simplex \(\mathbb {S}^{m-1}\) is the stable rest point.

(ii) We show that the center \(\mathbf {c}\) of the simplex \(\mathbb {S}^{m-1}\) which is the unique Nash equilibrium is not a strictly Nash equilibrium. Namely, we show \(\mathsf {BR}(\mathbf {c})=\mathbb {S}^{m-1}\). Indeed, for all \(\mathbf {x}\in \mathbb {S}^{m-1}\) we have that

$$\begin{aligned} \mathcal {E}_F(\mathbf {c},\mathbf {c})=\varepsilon f\left( \frac{1}{m}\right) g(\mathbf {c})=\varepsilon \left( \sum _{i=1}^{m}x_i\right) f\left( \frac{1}{m}\right) g(\mathbf {c})=\mathcal {E}_F(\mathbf {x},\mathbf {c}). \end{aligned}$$

(iii) Due to Theorem 6.2, an interior orbit starting from any initial point \(\mathbf {x}\in int \mathbb {S}^{m-1}\) converges to the center \(\mathbf {c}\) of the simplex \(\mathbb {S}^{m-1}\) which is a Nash equilibrium. This completes the proof of Theorem C.

7 Applications

In replicator equation (3.2), the fitness function \(F:\mathbb {S}^{m-1}\rightarrow \mathbb {R}^m\) is defined by a strictly increasing continuous function \(f:[0,1]\rightarrow [0,1]\) and a smooth non-vanishing function \(g:\mathbb {S}^{m-1}\rightarrow (0,1)\).

In this section, we choose population models for a single species such as Beverton–Holt’s model [5], Hassell’s model [18], Maynard Smith–Slatkin’s model [29], Ricker’s model [39], Skellam’s model [46] as a strictly increasing continuous function \(f:[0,1]\rightarrow [0,1]\). In general, there are two types of population models for a single species: density-independent and density-dependent population models. In its own turn, density-dependent population models can be split into two classes:

  • under-compensating in which the models are given by monotone functions;

  • over-compensating in which the models are given by unimodal functions.

Over-compensating population models which are given by unimodal (non-monotone) functions may produce cycles and chaos (see [16]). However, in this paper, we only consider under-compensating population models which are given by monotone functions.

Meanwhile, as a smooth non-vanishing function \(g:\mathbb {S}^{m-1}\rightarrow (0,1)\), for example, we can choose the following functions

  • \(g(\mathbf {x})=a_1x^{1+r_1}_1+\cdots +a_mx^{1+r_m}_m\) for any \(\mathbf {x}\in \mathbb {S}^{m-1}, \ a_i\in (0,1), \ r_i\in \mathbb {R}_{+}, \ i\in \mathbf {I}_m\);

  • \(g(\mathbf {x})=a_1x_1e^{-b_1x_1}+\cdots +a_mx_me^{-b_mx_m}\) for any \(\mathbf {x}\in \mathbb {S}^{m-1}, \ a_i,b_i\in (0,1), \ i\in \mathbf {I}_m\);

  • \(g(\mathbf {x})=a_1x_1h_1(x_1)+\cdots +a_mx_mh_m(x_m)\) for any \(\mathbf {x}\in \mathbb {S}^{m-1}, \ a_i\in (0,1)\) and \(h_i:[0,1]\rightarrow [0,1]\) is any smooth function such that \(h_i(t)>0\) for all \(t>0, \ i\in \mathbf {I}_m\).

However, we do not specify a function \(g:\mathbb {S}^{m-1}\rightarrow (0,1)\) in any examples given below.

Example 7.1

Let

$$\begin{aligned} f(x)=\frac{ax^r}{(1+bx^q)^s} \end{aligned}$$

where \(0<a\le (1+b)^s\), \(b>0\), \(q>0\), \(r>0\), \(s>0\), and \(0<bsq\le (1+b)r\).

  1. (i)

    If \(q=r=s=1\) then we obtain Beverton–Holt’s model [5];

  2. (ii)

    If \(r=q=1\) then we obtain Hassel’s model [18];

  3. (iii)

    If \(r=s=1\) then we obtain Maynard Smith–Slatkin’s model [29];

The discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) takes the following form

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1+\varepsilon \left( \frac{ax_k^r}{\left( 1+bx_k^q\right) ^s}-\sum \limits _{i=1}^m\frac{ax_i^{r+1}}{\left( 1+bx_i^q\right) ^s}\right) g(\mathbf {x})\right] , \quad \forall \ k\in \mathbf {I}_m. \end{aligned}$$

Example 7.2

Let

$$\begin{aligned} f(x)=ax^re^{b(1-x)} \end{aligned}$$

where \(0<a\le 1\) and \(0<b\le r\). If \(r=1\) then we obtain Ricker’s model [39]. The discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) takes the following form

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1+\varepsilon \left( ax_k^re^{b(1-x_k)}-\sum \limits _{i=1}^max_i^{r+1}e^{b(1-x_i)}\right) g(\mathbf {x})\right] , \quad \forall \ k\in \mathbf {I}_m. \end{aligned}$$

Example 7.3

Let

$$\begin{aligned} f(x)=ax^r(1-be^{c(1-x)}) \end{aligned}$$

where \(a,r>0\), \(\max \{1-\frac{1}{a},0\}< b\le 1\) and \(0<c \le \ln b^{-1}\). If \(r=0\) then we obtain Skellam’s model [46]. The discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) takes the following form

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1+\varepsilon \left( ax_k^r\left( 1-be^{c(1-x_k)}\right) -\sum \limits _{i=1}^max_i^{r+1}\left( 1-be^{c(1-x_i)}\right) \right) g(\mathbf {x})\right] , \ \forall \ k\in \mathbf {I}_m. \end{aligned}$$

Example 7.4

Let

$$\begin{aligned} f(x)=a_nx^{r_n}+a_{n-1}x^{r_{n-1}}+\cdots a_3x^{r_3}+a_2x^{r_2}+a_1x^{r_1} \end{aligned}$$

where \(a_1, \ \ldots , \ a_n > 0\) such that \(0<\sum \nolimits _{k=1}^{n}a_k\le 1\) and \(r_n>r_{n-1}>\cdots>r_2>r_1>0\). Some special cases have been already studied: \(n=1\) with \(r_1=1\) in [25, 40, 41] and \(n=1\) with any \(r_1\in \mathbb {N}\) in [32, 33]. The discrete-time replicator equation \(\mathcal {R}:\mathbb {S}^{m-1}\rightarrow \mathbb {S}^{m-1}\) takes the following form

$$\begin{aligned} \left( \mathcal {R}(\mathbf {x})\right) _k=x_k\left[ 1+\varepsilon \left( \sum \limits _{j=1}^{n}a_jx_k^{r_j}-\sum \limits _{i=1}^m\sum \limits _{j=1}^{n}a_jx_i^{r_j+1}\right) g(\mathbf {x})\right] , \quad \forall \ k\in \mathbf {I}_m. \end{aligned}$$

8 Conclusions

The basis for evolutionary game theory is the Folk Theorem of Evolutionary Game Theory (see [7, 23, 24]) which asserts the following three statements for all “reasonable” dynamics (particularly, the replicator equation) of an evolutionary game:

  1. (i)

    A stable rest point is a Nash equilibrium;

  2. (ii)

    A strictly Nash equilibrium is asymptotically stable;

  3. (iii)

    Any interior convergent orbit evolves to a Nash equilibrium;

In the continuous case, the replicator and other types of equations are thoroughly studied for various kinds of evolutionary games such as potential games, stable games, supermodular games, imitation dynamics, best response dynamics, and monotone dynamics in the literature [6, 7, 14, 22,23,24, 32, 33, 36, 43,44,45, 53].

In this paper, we have considered a discrete-time replicator equation. Namely, we have proposed a nonlinear model of evolutionary game in which for any two different pure strategies, a biological fitness of a pure strategy which is frequent in number is better/worse than other one. Particularly, the nonlinear payoff functions defined by discrete population models such as Beverton–Holt’s model, Hassell’s model, Maynard Smith–Slatkin’s model, Ricker’s model, Skellam’s model satisfy this hypothesis. In order to observe some evolutionary bifurcation diagram, we also control the nonlinear payoff function in two different regimes: positive and negative. It has been shown that the Folk Theorem of Evolutionary Game Theory is true for a discrete-time replicator equation governed by the proposed nonlinear payoff function. In the long-run time, the following scenario can be observed: (i) in the positive regime, the active dominating pure strategies will outcompete other strategies and only they will survive forever; (ii) in the negative regime, all active pure strategies will coexist together and they will survive forever.

As an application, we also showed that nonlinear payoff functions defined by under-compensating population models for a single species satisfy the hypothesis of the proposed model. It is of independent interest to study the dynamics of replicator equations governed by nonlinear payoff functions of over-compensating population models for a single species. Since over-compensating population models which are given by unimodal functions may produce cycles and chaos (see [16]), probably, one could expect complicated and non-convergent dynamics of the associated discrete-time replicator equations.

Finally, we would like to mention that it is also plausible to consider a payoff function which takes on negative values. For example, let us consider the simple highway congestion game in the economics of transportation (see [3, 43]). A pair of towns, Home and Work, are connected by m different roads \(\mathbf {I}_m=\{1,2,\ldots , m\}\). To commute from Home to Work, an agent must choose a road connecting two towns, Home and Work. Obviously, driving on a road increases the delays experienced by other drivers on that road. Therefore, a highway congestion involves negative externalities and the payoff of choosing a road is the negation of a delay on that road. Meanwhile, a delay on a road is an increasing function of the number of agents driving on that road. To formalize it mathematically, let \(\mathbf {x}=(x_1,\ldots ,x_m)\) be a probability vector for which \(x_k\) is a frequency of agents driving on a road k. Let \(f(x_k)\) be a delay on a road k which is an increasing function of a variable \(x_k\) and \(g(\mathbf {x})\) be an external cost function which has an equal impact on a usage of each and every road. The payoff \(F_k(\mathbf {x})\) of choosing a road k is the negation (which is \(\varepsilon <0\)) of the product of a delay function \(f(x_k)\) on a road k and a common external cost function \(g(\mathbf {x})\), i.e., \(F_k(\mathbf {x})=\varepsilon f(x_k)g(\mathbf {x})\). Hence, it is plausible to consider the negative payoff functions \(F(\mathbf {x})=\left( \varepsilon f(x_1)g(\mathbf {x}),\ \varepsilon f(x_2)g(\mathbf {x}), \ \cdots \ ,\ \varepsilon f(x_m)g(\mathbf {x})\right) \) for \(\varepsilon <0\) in the economics of transportation (for a detailed discussion, the reader may refer to [3, 43]).