Finding Nash Equilibria directly from the extensive form/game tree

Question

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

Answer 1

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.