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Can mixed strategy Nash equilibria include weakly dominated strategies?

Is there a mixed strategy Nash equilibrium in this game? (see attachment)

Since players are rational, they will only play strategies which aren't dominated. BL strictly dominates AL and AR for player 1. Thus, B weakly dominates A for player 2. After which, BL strictly dominates BR for player 1.

Since there is only one rationalizable strategy for each player there can be no mixed strategy Nash equilibrium.

But can mixed strategy Nash equilibria include weakly dominated strategies? If this is case then each player would have two strategies, BL and BR for player 1; A2 and B2 for player 2.

Thanks in advance.
Screen Shot 2017-05-22 at 19.27.17.png16.9KB
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Avatar for Speckle
Speckle
18
Yes, Nash equilibria can include weakly dominant strategies.

I know this isn't exactly what you asked, but the (pure strategy) Nash equilibria here are (BL, B) and (BR, A). Wasn't sure you got to this conclusion from your post.

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