Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

Question

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question:

Is $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ the only Nash equilibrium in a 3-player game of Rock Paper Scissors? How can we discover this analytically?

Edit: Payoff matrices below, in terms of P1 payoff.

P1=Rock
                         P3

                Rock    Paper   Scissors
             ----------------------------
    Rock     |   0   |   -1   |    0.5  |
             |--------------------------|
P2  Paper    |  -1   |   -1   |    0    |
             |--------------------------|
    Scissors |  0.5  |    0   |    2    |
             ----------------------------

P1=Paper
                         P3

                Rock    Paper   Scissors
             ----------------------------
    Rock     |   2   |   0.5  |    0    |
             |--------------------------|
P2  Paper    |  0.5  |    0   |   -1    |
             |--------------------------|
    Scissors |   0   |   -1   |   -1    |
             ----------------------------

P1=Scissors
                         P3

                Rock    Paper   Scissors
             ----------------------------
    Rock     |  -1   |    0   |   -1    |
             |--------------------------|
P2  Paper    |   0   |    2   |   0.5   |
             |--------------------------|
    Scissors |  -1   |   0.5  |    0    |
             ----------------------------