The replicator–mutator dynamics has become a powerful mathematical framework for the modelling and analysis of complex biological, economical and social systems. It has been employed in the study of, among other applications, population genetics [
14], autocatalytic reaction networks [
33], language evolution [
23], the evolution of cooperation [
18] and dynamics of behaviour in social networks [
24]. Suppose that in an infinite population there are
n types/strategies
\(S_1,\ldots , S_n\) whose frequencies are, respectively,
\(x_1,\ldots , x_n\). These types undergo selection; that is, the reproduction rate of each type,
\(S_i\), is determined by its fitness or average pay-off,
\(f_i\), which is obtained from interacting with other individuals in the population. The interaction of the individuals in the population is carried out within randomly selected groups of
d participants (for some integer
d). That is, they play and obtain their pay-offs from a
d-player game, defined by a payoff matrix. We consider here symmetric games where the pay-offs do not depend on the ordering of the players in a group. Mutation is included by adding the possibility that individuals spontaneously change from one strategy to another, which is modelled via a mutation matrix,
\(Q=(q_{ji}), j,i\in \{1,\ldots ,n\}\). The entry
\(q_{ji}\) denotes the probability that a player of type
\(S_j\) changes its type or strategy to
\(S_i\). The mutation matrix
Q is a row-stochastic matrix, i.e.
$$\begin{aligned} \sum _{j=1}^n q_{ji}=1, \quad 1\le i\le n. \end{aligned}$$
The replicator–mutator is then given by, see, e.g. [
19‐
21,
25],
$$\begin{aligned} {\dot{x}}_i=\sum _{j=1}^n x_j f_j({\mathbf {x}})q_{ji}- x_i {\bar{f}}({\mathbf {x}})=:g_i(x),\qquad i=1,\ldots , n, \end{aligned}$$
(1)
where
\({\mathbf {x}}= (x_1, x_2, \dots , x_n)\) and
\({\bar{f}}({\mathbf {x}})=\sum _{i=1}^n x_i f_i({\mathbf {x}})\) denotes the average fitness of the whole population. The replicator dynamics is a special instance of (
1) when the mutation matrix is the identity matrix.
In this paper, we are interested in properties of the equilibrium points of the replicator–mutator dynamics (
1). Note that we are concerned with dynamic equilibria almost exclusively. There might be a dynamic equilibrium which is not a Nash equilibrium of the game. These dynamic equilibrium points are solutions of the following system of polynomial equations:
$$\begin{aligned} {\left\{ \begin{array}{ll} g_i(x)=0, \quad i=1,\ldots , n-1,\\ \sum _{i=1}^n x_i=1. \end{array}\right. } \end{aligned}$$
(2)
The second condition in (
2), that is the preservation of the sum of the frequencies, is due to the term
\(x_i{\bar{f}}({\mathbf {x}})\) in (
1). The first condition imposes relations on the fitnesses. We consider both deterministic and random games where the entries of the payoff matrix are, respectively, deterministic and random variables. Typical examples of deterministic games include pairwise social dilemmas and public goods games that have been studied intensively in the literature, see, e.g. [
15,
16,
27,
32,
35]. On the other hand, random evolutionary games are suitable for modelling social and biological systems in which very limited information is available, or where the environment changes so rapidly and frequently that one cannot describe the pay-offs of their inhabitants’ interactions [
9‐
11]. Simulations and analysis of random games are also helpful for the prediction of the bifurcation of the replicator–mutator dynamics [
20,
21,
25]. Here, we are mainly interested in the number of equilibria in deterministic games and the expected number of equilibria in random games, which allow predicting the levels of social and biological diversity as well as the overall complexity in a dynamical system. As in [
20,
21,
25], we consider an independent mutation model that corresponds to a uniform random probability of mutating to alternative strategies as follows:
$$\begin{aligned} q_{ij}=\frac{q}{n-1},~~i\ne j,~~q_{ii}=1-q,~~1\le i,j\le n. \end{aligned}$$
(3)
In particular, for two-strategy games (i.e. when
\(n=2\)), the above relations read
$$\begin{aligned} q_{12}=q_{21}=q,~~ q_{11}=q_{22}=1-q. \end{aligned}$$
The parameter
q represents the strength of mutation and ranges from 0 to
\(1-\frac{1}{n}\). The two boundaries have interesting interpretation in the context of dynamics of learning [
21]: for
\(q=0\) (which corresponds to the replicator dynamics), learning is perfect and learners always end up speaking the grammar of their teachers. In this case, vertices of the unit hypercube in
\({\mathbb {R}}^n\) are always equilibria. On the other hand, for
\(q=\frac{n-1}{n}\), the chance for the learner to pick any grammar is the same for all grammars and is independent of the teacher’s grammar. In this case, there always exists a uniform equilibrium
\({\mathbf {x}}=(1/n,\ldots , 1/n)\) (cf. Remark
1). Equilibrium properties of the replicator dynamics, particularly the probability of observing the maximal number of equilibrium points, the attainability and stability of the patterns of evolutionarily stable strategies have been studied intensively in the literature [
2,
3,
12,
13,
17]. More recently, we have provided explicit formulas for the computation of the expected number and the distribution of internal equilibria for the replicator dynamics with multi-player games by employing techniques from both classical and random polynomial theory [
4‐
7]. For the replicator dynamics, that is when there is no mutation, the first condition in (
2) means that all the strategies have the same fitness which is also the average fitness of the whole population. This benign property is no longer valid in the presence of mutation making the mathematical analysis harder. In a general
d-player
n-strategy game, each
\(g_i\) is a multivariate polynomial of degree
\(d+1\); thus, (
2) is a system of multivariate polynomial equations. In particular, for a two-player two-strategy game, which is the simplest case, (
2) reduces to a cubic equation whose coefficients depend on the payoff entries and the mutation strength. For larger
d and
n, solving (
2) analytically is generally impossible according to Abel’s impossibility theorem. Nevertheless, there has been a considerable effort to study equilibrium properties of the replicator–mutator dynamics in deterministic two-player games, see for instance [
19‐
21,
25]. In particular, with the mutation strength
q as the bifurcation parameter, bifurcations and limit cycles have been shown for various classes of fitness matrices [
19,
25]. However, equilibrium properties for multi-player games and for random games are much less understood although in the previously mentioned papers, random games were employed to detect and predict certain behaviour of (
1).
In this paper, we explore further connections between classical/random polynomial theory and evolutionary game theory developed in [
4‐
7] to study equilibrium properties of the replicator–mutator dynamics. For deterministic games, by using Descartes’ rule of signs and its recent developments, we are able to fully characterize the equilibrium properties for social dilemmas. In addition, we provide a method to compute the number of equilibria in multi-player games via the sign changes of the coefficients of a polynomial. For two-player social dilemma games, we calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player two-strategy random games whose pay-offs are independently distributed according to a normal distribution, we obtain explicit formulas to compute the expected number of equilibria by relating it to the expected number of positive roots of a random polynomial. Interestingly, due to mutation, the coefficients of the random polynomial become correlated as opposed to the replicator dynamics where they are independent. The case
\(q=0.5\) turns out to be special and needs different treatment. We also perform extensive simulations by sampling and averaging over a large number of possible payoff matrices, to compare with and illustrate analytical results. Moreover, numerical simulations also show interesting behaviour of the expected number of equilibria when the number of players tends to infinity or when the mutation goes to zero. It would be challenging to analyse these asymptotic behaviours rigorously, and we leave it for future work.