Finding NE and SPNE

Question

Given the diagram find SPNE and NE:

In my opinion NE is: (6,4),(8,5) and (4,6) but i have problem with SPNE. How to find SPNE ?

Answer 1

To find the Subgame Perfect Nash equilibrium, we need to solve for the nash equilibria of each subgame. This game has 3 subgames:

1. The game 2 plays if 1 plays A.
2. The game 1 plays if 1 plays B.
3. The whole game.

In the subgame identified in 1, player 2 plays C, because $4>2$.

In the subgame identified in 2, $(E,X)$ is the unique nash equilibrium. This is an equilibrium, because if player 1 deviates, he gets 2 instead of 4, and if player 2 deviates, her payoffs don't change. Therefore neither player has a profitable deviation. There are no equilibria where player 1 plays F, since then it would be optimal for player 2 to play X, but then player 1 would have a profitable devaition to E.

Finally, we think about the whole game. We know that if player 1 plays A, they reach subgame 1, in which case they get 1. If they play B, they reach subgame 2, where they get 4. So player 1 should play B.

So the SPNE is player 1 plays $(B,E)$, player 2 plays $(C,X)$. Note that I have written this in terms of strategies, not payoffs, and specified strategies at information sets that aren't reached in equilibrium. This is important, because it uniquely identifies what happens in the nash equilibrium and in each subgame, unlike writing down payoffs.

To find all the nash equilbrium, it is probably easiest to write down the normal form version of this game. This is a 4x4 game (player 1s strategies are AE, AF, BE, BF).