Hey Miranda,
This is a pretty standard problem asking for the normal form of an extensive form game. In general, to solve questions like this you can follow these steps:
1. Determine how may information sets each player has in the extensive form game.
In this example, player 1 has two info sets, one is the starting nodes, the other one is the info set in the right-bottom corner with two decision nodes included in it. Player 2 has 2 info sets, and both of them are singleton (i.e. a info set that contain only one decision node.)
2. Define the action set of the corresponding normal form game as the Cartesian product of action sets of a player on each info set.
In this example, player 1 has two info sets. One the first info set, player 1's action set is A1 = {A, B}. On the second info set, player 1's action set is A2 = {a, b}. So in the corresponding normal form game, player 1's action set is A1 x A2 . (x means Cartesian product.) That is, player 1's action set in the normal form game is A1' = A1 x A2 = {A, B} x {a, b} = {(A, a), (A, b), (B, a), (B, b)}.
Similarly, player 2's action set in the normal forma game is A2' = {X, Y} x {Z, W} = {(X, Z), (X, W), (Y, Z), (Y, W)}.
3. Construct the corresponding normal form game using players' action sets.
In this example, we see player 1 has 4 actions, and player 2 also has 4 actions. So we can draw a 4 by 4 table to show the payoffs as following:
4. Then we can solve for pure or mixed strategy NE using the corresponding normal form. (I guess I'll leave this step to you.)
I hope this helps.
-Chuck
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