  In this paper, we study the number of equilibria of the replicator-mutator
dynamics for both deterministic and random multi-player two-strategy
evolutionary games. For deterministic games, using Decartes' rule of signs, we
provide a formula to compute the number of equilibria in multi-player games via
the number of change of signs in the coefficients of a polynomial. For
two-player social dilemmas (namely, the Prisoner's Dilemma, Snowdrift, Stag
Hunt, and Harmony), we characterize (stable) equilibrium points and
analytically calculate the probability of having a certain number of equilibria
when the payoff entries are uniformly distributed. For multi-player random
games whose payoffs are independently distributed according to a normal
distribution, by employing techniques from random polynomial theory, we compute
the expected or average number of internal equilibria. In addition, we perform
extensive simulations by sampling and averaging over a large number of possible
payoff matrices to compare with and illustrate analytical results. Numerical
simulations also suggest several interesting behaviour of the average number of
equilibria when the number of players is sufficiently large or when the
mutation is sufficiently small. In general, we observe that introducing
mutation results in a larger average number of internal equilibria than when
mutation is absent, implying that mutation leads to larger behavioural
diversity in dynamical systems. Interestingly, this number is largest when
mutation is rare rather than when it is frequent.
