Need help with finding pure strategy nash equilbria

Question

In the following game, how can I find the pure strategy Nash equilibria?

The answers are apparently $(b,d)$ and $(b,g)$ but I'm not sure why.

I have realised the following:

• Player one (rows) has no strategy which is weakly dominated.
• If we delete all weakly dominated strategies for player 2 (columns), only d remains.

Answer 1

Deleting weakly dominated strategies in a non $0$-sum game will certaingly leave some equilibria untouched but it can delete some other equilibria as in the game above.

On the other hand you can safely delete strongly dominated strategies. That is the case only for strategy (c) which leaves you with the payoff matrices $$Α=\begin{array}{r|rrrrr|}&d&e&f&g&h\\\hline a&-1&1&-1&0&-1\\b&0&0&1&0&1\end{array}\qquad \text{ and } \qquad B=\begin{array}{r|rrrrr|}&d&e&f&g&h\\\hline a&1&-1&1&0&1\\b&0&0&-1&0&-1\end{array}$$ Now in order to find the pure strategy Nash equilibria (this does not work for the mixed ones) denote in matrix A the best response of player I to each strategy of player II as follows $$Α=\begin{array}{r|rrrrr|}&d&e&f&g&h\\\hline a&-1&1^*&-1&0^*&-1\\b&\color{blue}{0^*}&0&1^*&\color{blue}{0^*}&1^*\end{array}$$ and similarly in B the best response (or responses if more than one) of player II to each of the two strategies of player I as follows $$B=\begin{array}{r|rrrrr|}&d&e&f&g&h\\\hline a&1^*&-1&1^*&0&1^*\\b&\color{blue}{0^*}&0^*&-1&\color{blue}{0^*}&-1\end{array}$$ Now, the entries with a star in both matrices, are the pure strategy Nash equilibria. Indeed these are $$(b,d) \text{ and } (b,g) \qquad \text{ both with payoffs } (0,0)$$

Answer 2

My approach, given Stef's solution:

• Highlight the best responses for $P_{1}$ in red
• Highlight the best responses for $P_{2}$ in blue
• Payoff profiles highlighted in black will represent pure strategy nash equilibria.

$P_{1}$ Best responses

$\:$

$P_{2}$ Best responses

Pure strategy nash equilibria

They are $(b,d)$ and $(b,g)$.