Does a $3 \times 3$ zero-sum game with a pure equilibrium point always have a dominated strategy?

Question

In a $2 \times 2$ zero-sum game, if pure equilibrium strategies exist then one of the players has a (weakly) dominated strategy.

Is this also true in a $3 \times 3$ game?

Answer 1

Actually this is not true for 2x2 games: For a constant matrix, say all zeroes, all strategy profiles are Nash equilibria.

OK, so let's assume that there is no payoff equivalent strategies in the game. I presume by "equilibrium point" you mean a pure strategy equilibrium, otherwise the statement is still false for 2x2 games (e.g., take matching pennies, which has a unique completely mixed equilibrium).

I do agree that for 2x2 games, if there is a pure equilibrium then there exists a (weakly) dominated strategy. For 3x3 this is not true. Here's a counter example:

3 x 3 Payoff matrix A:

  2  3  3
  1  4  0
  1  0  4

3 x 3 Payoff matrix B:

  -2  -3  -3
  -1  -4   0
  -1   0  -4

EE = Extreme Equilibrium, EP = Expected Payoff

Decimal Output

  EE  1  P1:  (1)  1.000000  0.000000  0.000000  EP=  2.0  P2:  (1)  1.000000  0.000000  0.000000  EP=  -2.0

Rational Output

  EE  1  P1:  (1)  1  0  0  EP=  2  P2:  (1)  1  0  0  EP=  -2

Connected component 1:
{1}  x  {1}