Nash Equilibrium and Dominant Strategy If I have a game that goes as follow:



Player 1 is the row player and player 2 is the column player. I think that the Nash Equilibria should be (10, 5) and (5, 10), since neither of the player has incentive to unilaterally deviate given the other's strategy. But then the dominant strategy, I thought, for both player is to always invest. Does this mean that the stable outcome is actually (5, 5) and that it is inconsistent with the NEs, which should be stable? (I only work with pure strategy here)
You're on the right track here. You need to check every outcome for its potential to be a NE. You're correct in stating that outcomes (5,10) and (10,5) are NEs however you didn't identify that (5,5) is also an NE.

(5,5)
If player one deviates he receives a payout of 5. 
If player two deviates he receives a payout of 5.

Therefore no player has an incentive to unilaterally deviate and (5,5) is a Nash equilibrium.

You're also correct that (5,5) is the dominant strategy solution. Just keep in mind that the dominant strategy solution will always be a Nash equilibrium but not all Nash equilibria are dominant strategy solutions.