In the game shown below what strategies can player 2 adopt in a subgame-perfect equilibrium?

Question

What difference would it make to possible predictions of how rational states play this game if player 2 does not know player 1’s move when it chooses (i.e. node2 and node3 belong to the same information sets; equivalently players move simultaneously).

Answer 1

If we write the game in normal form, we will see that in the simultaneous-moves version, Player 1 has a strictly dominant strategy, the "Up" strategy. So, under rationality, Player 2 "knows" what Player 1 will play, even though they play simultaneously. So the solution here will be $\{Up, In\}$ with payoffs $\{1,2\}$.

In the sequential version, backwards induction tells is that Player 2 will select "In" if he is in node 2, while he will choose "Out" if he is in node 3. In both these cases, Player 1 has the same payoff ($1$), so she is indifferent between choosing "Up" or "Down" - the strategies here are equivalent in terms of payoffs for Player 1.

Note that we make the standard assumption that each player cares about his/her own payoffs only.