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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    |
             ----------------------------
share|improve this question
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

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