Two players, A and B play the following game. First A must choose IN or OUT. If A chooses OUT, then the game ends, and the payoffs are: A gets 2 and B gets 0. If A chooses IN, then B observes this and must then choose IN or OUT. If B chooses OUT, then the game ends, and the payoffs are: A gets 0 and B gets 2. If A chooses IN and B chooses IN, then they play the following simultaneous move game:
Player 2
L R
Player 1: U 3, 1 0, -2
D -1, 2 1, 3
Find all the pure-strategy subgame perfect Nash equilibria of the game.
I'm not sure if my answer is correct: A will choose OUT, B will choose IN when A and B selects D and R respectively, B will choose OUT when A and B select U and L respectively.
Your answer is correct, as can be confirmed by backward induction starting at the simultaneous move. The formulations "will choose" and "when A and B select" are somewhat misleading, since these choices will not actually occur, as the game ends after A chooses OUT. These are merely the players' strategies for these choices.