# Finding Nash Equilibria directly from the extensive form/game tree

How can you find the Nash equilibrium of a game directly from the extensive form game/game tree of a game. You can find Nash equilibria from the strategic form (normal form table), but finding it directly from the extensive form seems very interesting as well. A position/strategy profile is a Nash equilibrium if it is a best response to other strategies of the other players. (The photo is in the link). Thank you.

How can we do this for the 3 player game tree below:enter image description here

It's a bit tedious to describe, but with enough practice, you should be able to follow the steps below to quickly identify NEs (not only the subgame perfect ones) from a game tree.

The idea is that you first suppose player 1 plays a certain strategy. Then you find out the best responses of the other players. Lastly you check whether the initially supposed strategy for player 1 is a best response to the other players' best responses (to it). If it is, you have a profile of mutually best responding strategies, hence a NE; if it isn't, then you don't have any NE with the initially supposed strategy of player 1.

1. Suppose player 1 plays $$A$$

• Player 2's best response to $$A$$ is $$w$$ and player 3's best response to $$A$$ is both $$S$$ and $$T$$
1. Suppose player 3 plays $$S$$
• Player 1's best response to $$(w,S)$$ is $$A$$, same as the initial supposition. NE: $$(A,w,S)$$
2. Suppose instead player 3 plays $$T$$
• Player 1's best response to $$(w,T)$$ is $$C$$, not $$A$$. No NE.
2. Suppose player 1 plays $$B$$

• Player 2's best response to $$B$$ is $$\{u,v,w\}$$ and player 3's best response to $$B$$ is $$\{S,T\}$$
• Check whether $$B$$ is a best response to each of $$\{(u,S),(u,T),(v,S),(v,T),(w,S),(w,T)\}$$. If it is, we have a NE; if not, we don't.
• We can verify that only $$(B,v,S)$$ is a NE
3. Suppose player 1 plays $$C$$
• Player 2's best response to $$C$$ is $$\{u,v,w\}$$ and player 3's best response to $$C$$ is $$S$$
• Check whether $$C$$ is a best response to each of $$\{(u,S),(v,S),(w,S)\}$$. If it is, we have a NE; if not, we don't.
• Only NE is $$(C,v,S)$$.

Overall, there are three NEs: $$(A,w,S)$$, $$(B,v,S)$$, $$(C,v,S)$$, with the first being the only SPE.

• Thank you! This is very helpful. Thank you so much. – Maths Apr 4 '19 at 20:40