How to compute Nash equilibria for this 3x3 matrix

Question

\begin{matrix} &P2 \\ P1&0,0 & 7,6 & 6,7 \\ &6,7 & 0,0 & 7,6 \\ &7,6 & 6,7 & 0,0 \end{matrix}

Then:

For P1: \begin{matrix} & A &B&C \\ A&0,0 & \underline{7},6 & 6,7 \\ B&6,7 & 0,0 & \underline{7},6 \\ C&\underline{7},6 & 6,7 & 0,0 \end{matrix}

For P2: \begin{matrix} & A &B&C \\ A&0,0 & 7,6 & 6,\underline{7} \\ B&6,\underline{7} & 0,0 & 7,6 \\ C&7,6 & 6,\underline{7} & 0,0 \end{matrix}

Now merge: \begin{matrix} & A &B&C \\ A&0,0 & \underline{7},6 & 6,\underline{7} \\ B&6,\underline{7} & 0,0 & \underline{7},6 \\ C&\underline{7},6 & 6,\underline{7} & 0,0 \end{matrix}

And now what? I need to select the cell that both values are selected but there is no cell like that.

It doesn't seem there is a pure equilibrium. You might need to consider mixed strategies. — Thomas Andrews, Sep 01 '23 at 15:19
And the equilibrium mixed strategy seems fairly obvious from the rotational symmetry — Henry, Sep 01 '23 at 15:22
Just to be sure, row B column A is really supposed to be $0,7,$ and not $6,7?$ $6,7$ would make the question slightly easier. — Thomas Andrews, Sep 01 '23 at 15:44
@henry There isn't a rotational symmetry - there is row B, column A, which is at odds. That was why I wondered if there was an error in copying the matrix. — Thomas Andrews, Sep 01 '23 at 16:27
@ThomasAndrews You have eagle eyes. It is $6,7$ in the top table — Henry, Sep 01 '23 at 16:31

Thomas Andrews · Answer 1 · 2023-09-01T16:58:30.553

You can think of this as a Rock/Paper/Scissors sort of game, where a match is a tie, with no payoff for either player, but when not a tie, the winner gets $7$ points and the loser get $6.$

As with Rock Paper Scissors, it helps not in the least for a player to prefer one option over the other. So the best option is for each player to pick each option with equal probability, which is indeed a Nash equilibrium.

Showing there is no other Nash equilibrium might be a little work, but (1) not much work, and (2) you didn't ask for all.

How to compute Nash equilibria for this 3x3 matrix

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