  In this paper, we study the distribution and behaviour of internal equilibria
in a $d$-player $n$-strategy random evolutionary game where the game payoff
matrix is generated from normal distributions. The study of this paper reveals
and exploits interesting connections between evolutionary game theory and
random polynomial theory. The main novelties of the paper are some qualitative
and quantitative results on the expected density, $f_{n,d}$, and the expected
number, $E(n,d)$, of (stable) internal equilibria. Firstly, we show that in
multi-player two-strategy games, they behave asymptotically as $\sqrt{d-1}$ as
$d$ is sufficiently large. Secondly, we prove that they are monotone functions
of $d$. We also make a conjecture for games with more than two strategies.
Thirdly, we provide numerical simulations for our analytical results and to
support the conjecture. As consequences of our analysis, some qualitative and
quantitative results on the distribution of zeros of a random Bernstein
polynomial are also obtained.
