# Finding Nash Equilibrium

Where would the Nash equilibrium lie in the pictured scenario? Whats confusing me is that both persons best response changes depending on the choice of the other person (there is no dominant strategy). Am I correct in thinking the Nash equilibrium is at (-1,-1) as both A and B want to betray as it would be most beneficial, but as they both betray they end up at (-1,-1)

The two (pure) Nash equilibrium in this game is (Betray, Silent) and (Silent, Betray).

Let us see why (Betray, Silent) is an equilibrium.

Let us look at person A.

Person B is playing Silent.

1. If she plays Betray, she gets $$2$$.
2. If she deviates to Silent, she gets $$1$$. So she would not play Silent.

Now consider person B.

Person A is playing Betray.

1. If she continues playing Silent, she gets $$0$$.
2. If she deviates to Betray, she gets $$-1$$. So she would not play Betray.

No player has an incentive to switch strategies. So this is an equilibrium.

The same argument holds for the (Silent, Betray) equilibrium.