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If I have a game that goes as follow:

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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)

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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.

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    $\begingroup$ Small precision: "invest" is a weak dominant strategy. $\endgroup$ – Yann Nov 21 '16 at 16:28
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Do you have a formal notion of what 'stable' means? Nash equilibria are often thought of informally as the strategies that support stable outcomes. If that's all that the term 'stable' means, then of course the outcomes $(5,10)$ and $(10,5)$ are also stable.

As Lee Sin notes above, the outcome $(5,5)$ is also a Nash equilibrium outcome, and so is stable in that sense.

However, if you are defining stable as an outcome supported by an equilibrium in weakly dominant strategies (which it seems like you are), then $(5,5)$ is the unique stable outcome.

Note also that invest is only a weakly dominant strategy -- it is sometimes strictly better but never strictly worse than another strategy. I suspect the sense of stability you mean to invoke involves strictly dominant strategies, of which there are none in the given game.

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  • $\begingroup$ Invest/Invest may not be strictly dominant, but it has more stability than the other two Nash equilibria in the sense that Invest is the optimal strategy if there is any chance that the other player is irrational. $\endgroup$ – Henry Nov 22 '16 at 13:10
  • $\begingroup$ Sure, but then the notion of stability you're now invoking is in the sense of trembling hand-perfection, which is why I asked the OP what notion of stability they were using exactly. $\endgroup$ – Theoretical Economist Nov 22 '16 at 15:26

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