# Is there a 2*2 game with a unique NE but not dominance solvable?

Is there a 2*2 game with a unique NE but not dominance solvable? If so, could you give an example of the payoff matrix?

Here is the definition of dominance solvable, "if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a dominance-solvable game. (Reference: https://en.wikipedia.org/wiki/Strategic_dominance#:~:text=In%20any%20case%2C%20if%20by,called%20a%20dominance%2Dsolvable%20game.)"

• Look up matching pennies. May 8, 2023 at 7:33
• @MichaelGreinecker Thank you :)! I am wondering, would there be any such 2*2 game with a unique pure NE, but not dominance solvable? For matching pennies, it has a mixed NE, if I understand correctly.
– sera
May 8, 2023 at 9:33
• There is not. Let's label both players' two strategies $0$ and $1$. Ignoring indifference, the basic idea is this: If neither has a dominant strategy, then their best response depends on the choice of the other player. Without loss of generality, we can assume that the first player wants to match the strategy of the second player. If the second player also wants to match the strategy of the first player, we get a coordination game with two pure equilibria. If player two wants to mismatch, we have matching pennies with no pure equilibrium. May 8, 2023 at 13:39
• @MichaelGreinecker I see, thanks a lot for your explanations!!!
– sera
May 8, 2023 at 18:58