Ah, looks like I misunderstood the definition of the term "subgame". I thought that this would be one subgame: "I have KK preflop on the button, and my opponent's range is X". And another subgame might be: "I have 75o preflop on the button, and my opponent's range is X". From my opponent's perspective, these 2 situations would occur on the same level of the game tree and my opponent has no way of distinguishing between these (at this stage of the game tree) due to the hidden information, thus I thought they would be called subgames. But you're telling me that the concept of subgame applies in both directions (not only imperfect information by my opponent's perspective, but also imperfect information in my perspective). So when I say "my opponent's presumed range is X", that sentence doesn't describe 1 subgame, that sentence actually describes multiple subgames.

My earlier point was that if an opponent's strategy is fixed, then I don't have to balance how I play that KK compared to how I play that 75o. That I can play each hand "in a vacuum", in a strategy where the EV of each hand is maximized. Note that this idea is only relevant if my opponent is not playing perfect GTO (contrary to your example). For example, my opponent may have a weakness where their preflop fold frequency is the same regardless of how whether I raise 3BB or 2BB. Against this weakness the optimal play would be to raise 3BB with KK and raise 2BB with 75o. Obviously, this would be a bad strategy if my opponent was allowed to adapt their strategy to me, but it's the optimal play if my opponent's strategy is locked.

To clarify further:

- "exploitative style" poker strategy is made at the level "My hand is 96s, and my opponent's range is presumed to be Y"

- "GTO style" poker strategy is made at the level "My range is X, and my opponent's range is presumed to be Y"

If the opponent's strategy is fixed, then our optimal strategy is to play 100% exploitative. If the opponent's strategy is not fixed, then the optimal strategy will be a mix with elements from both exploitative and GTO strategies. (Again, note that this only makes sense if the opponent is not playing perfect GTO. If the opponent is playing perfect GTO, then obviously the optimal strategy is to also play GTO.)

I thought that Noam Brown was making a point related to this concept. It seems that I was mistaken and his point was related to something else entirely.

> But you're telling me that the concept of subgame applies in both directions (not only imperfect information by my opponent's perspective, but also imperfect information in my perspective). So when I say "my opponent's presumed range is X", that sentence doesn't describe 1 subgame, that sentence actually describes multiple subgames.

Yes; consider a chess move, every move is a subgame - a subtree of that game. He just can't safely say tree, because there are games that are better described as a graph than a tree. Actually, Two Envelope is such a game. The action switch, in Two Envelope, leads to the subgame in which you are at I[null] and need to decide whether to keep or switch - the same situation as you started in. Switch again and you arrive at the subgame I[null]. It contains itself.

> If the opponent's strategy is fixed, then our optimal strategy is to play 100% exploitative. If the opponent's strategy is not fixed, then the optimal strategy will be a mix with elements from both exploitative and GTO strategies. (Again, note that this only makes sense if the opponent is not playing perfect GTO. If the opponent is playing perfect GTO, then obviously the optimal strategy is to also play GTO.)

I agree.