Assuming we know how much information each player has in an IIEFG, then, while constructing the information sets, do we need each information set to contain all the nodes that it can contain?
For instance, for a 2-player IIEFG where neither player can see which actions the other player has played till now, but where each player remembers which action they played at each juncture:
Tree representation of the extensive form game
Is this representation (w.r.t. the information sets) correct?
My query now is: will $I_{2}^{2}$ and $I_{2}^{4}$ actually be in the same information set (since, if player 2 doesn't know what player 1 played in their first turn and their second turn, then they cannot distinguish whether they're in $I_{2}^{2}$ or $I_{2}^{4}$). Same question for $I_{2}^{3}$ and $I_{2}^{5}$.
Or is it up to us, the question of which node to put in which information set? (even after having a description of how much information each player has)
The issue is that in the current representation, the information sets $I_{2}^{2}$ and $I_{2}^{4}$ aren't maximal (and they show player 2 to have more information than player 2 actually has.