By [Theorem 3.1](http://www.cs.ubc.ca/~condon/papers/condon-lipton89.pdf#page=6) and the proof's IPS being one-way, that problem remains [RE](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:R#re)-hard even when player_to_move is always 1 (i.e., only 1 player is needed) <br>and<br> s<sub>0</sub> ≠ s<sub>a</sub> and s<sub>a</sub> is not in Range(next_state_table) <br> (i.e., it's literally impossible for the player to lose) <br>and<br> p1_info and p2_info and number_of_choices are independent of the state <br> (i.e., the player's _only_ feedback is whether or not it just won) .