1
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thank you! This is very helpful. Thank you so much. $\endgroup$ – Maths Apr 4 at 20:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.