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

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    |
----------------------------

-
 It's not obvious to me what the payoff of a $3$-player version of the game would be. Wikipedia has a long article on the game but doesn't mention a version with more than $2$ players. Two possible payoff definitions might be that a) a point is awarded for each of the three pairs according to the regular rules or b) a player wins a point if and only if her move beats both of the other moves. – joriki Jan 13 at 20:03 I've added the payoff matrices that I'm thinking about. – Wesley Tansey Jan 13 at 20:27 How on earth did you expect us to guess that that was what you had in mind? – joriki Jan 13 at 21:16 I suppose it was the most intuitive payoff matrix to me. It's similar to poker, where you have a single pot and draws result in split pots. – Wesley Tansey Jan 13 at 21:20