# Is it possible to find a nash equilibrium that is not an equilibrium in weakly dominant strategy?

I know that it is possible to have a Nash equilibrium which is not an equilibrium in dominant strategy, but is it also applicable for equilibrium in weakly dominant strategy (i.e. a Nash equilibrium which is NOT an equilibrium in weakly dominant strategy)? If yes, what would be an example?

$$\begin{array}{|c|c|c|} \hline &L&R\\\hline T&1,1&0,0\\\hline B&0,0&0,0\\\hline \end{array}$$

In the game above, there are two pure strategy Nash equilibria:

• $$(T,L)$$ is an equilibrium in weakly dominant strategies;
• $$(B,R)$$ is an equilibrium in weakly dominated strategies.

Noting that "dominant" and "dominated" are two different words, I believe the above example answers both the question in your post as well as your question in the comment below.

• This is not the question I asked. I asked about Nash equilibrium which is NOT an equilibrium in weakly dominated strategies. – Aqqqq Nov 21 '19 at 22:08
• @Aqqqq Actually, this is the answer to the question you asked! You edited the question later. This is clear from the edit history. – Giskard Nov 21 '19 at 22:09
• @Giskard I edited the question to make it clearer for understanding. No meaning was changed. The only thing I added was the content in the bracket. – Aqqqq Nov 21 '19 at 22:11
• @Aqqqq: Your question asks for NE that's not in weakly dominant strategies. $(B,R)$ is NOT a Nash equilibrium in weakly dominant strategies, it's in weakly dominated strategies. – Herr K. Nov 21 '19 at 22:12
• @Aqqqq I believe you. Herr K's answer does answer both your questions though. – Giskard Nov 21 '19 at 22:15