Game Theory Nash equilibrium

Question

How do we do in order to find Nash Equilibrium in a $3$ players game with $3$ possible strategies for each player?

Answer 1

I'm going to assume you're looking for pure-strategy NE, as mixed is actually a bit difficult (well, tedious, not that difficult).

Here's an example of a three-player game. It should be somewhat similar to the one you're working on except that here P1 and P2 have 2 possible actions rather than 3. P3 does have 3 here. P1 chooses between T and B. P2 chooses between L and R. P3 chooses between X,Y and Z.

In any case, to find the pure-strategy NE. Simply underline the best responses to each profile of opponent actions. So, start with P1. We need to find his BR for each profile of P2 and P3 actions. If P2 and P3 play R and X respectively, then P1's best response is either T or B (they both give payoff of 0) - so underline both of those zero payoffs for P1. If P2 and P3 play L and X respectively, then B is the unique best response as 1>0, so we underline just that 1. We do the same for LY, RY, LZ, and RZ, as all are feasible profiles of opponent actions.

Next we do the same for P2. We underline his best responses to each of the possible profiles of P1 and P3 actions: TX, BX, TY, BY, TZ, BZ.

Next, we do the same for P3. We underline his best responses toe ach of the possible profiles of P1 and P2 actions: TL, TR, BL, BR. Note, for this one, you are actually comparing P3's payoff's across the three matrices.

Finally, a pure-strategy NE exists wherever there is a cell in which payoffs for each of the three players is underlined. Give it a go on this game if you want and suggest the NE you find in a comment below and I'll let you know if you're on the right track.