Finding the pure Nash equilibria for a bimatrix game

Question

I have to find the pure Nash equilibria for the bimatrix game

\begin{align}\begin{pmatrix}11,10 & 6,9 &10,9\\ 11,6 & 6,6 & 9,6\\ 12,10 & 6,9 & 9,11\end{pmatrix}\end{align}

Let's denote the strategies for player 1 by $A$, $B$ and $C$ and the strategies of player 2 by $X$, $Y$ and $Z$. I tried different combinations of strategies, but for every strategy player 1 plays, player 2 has a strategy that it prefers. For this strategy, player 1 will play another strategy.

Can anyone help me find the pure Nash equilibria without eliminating weakly dominated strategies?

Answer 1

If $\ A= \pmatrix{11&6&10\\11&6&9\\12&6&9}\ $ and $\ B=\pmatrix{10&9&9\\6&6&6\\10&9&11}\ $ are the players' payoff matrices, then we have $$ a_{22}=6 \ge a_{i2}\ \mbox{ for all } i\in\left\{1,2,3\right\}\ \mbox{, and}\\ b_{22}=6\ge b_{2j}\ \mbox{ for all } j\in\left\{1,2,3\right\}\ ,\ \ \ \ \ \ $$ so the pure strategies $\ i=2\ $ for the first player (i.e. the row selector), and $\ j=2\ $ for the second player (column selector) constitute a Nash equilibrium for the game.