I've read around that the GTO action is the one that makes your opponent indifferent to something. Could anyone be as precise as possible stating this? Indifferent to any action? Indifferent to changing the frequencies of his actions given the situation and range he has? Indifferent to small changes in his frequencies?
Long rant: thinking out loud
I'm trying to understand the principles of GTO, and I'll use this post to clarify my own understanding and doubts, and ask for confirmation and guidance. In particular, the "indifference" principle is confusing me.
By "strategy", I understand some set of rules that dictate the action to take given any situation where the player doesn't know what the other players cards, but does know his own. It might be deterministic if the action is always the same given the situation, or mixed if you inject some randomness in it.
- GTO is the strategy that maximizes your EV against the best an opponent that knows your strategy can do
- GTO is also (one of) the best counter-strategy to GTO. In other words, there won't be some deterministic strategy that can do better against GTO (that doesn't adapt) than GTO. I can kiiinda see why this could the case for strategies close to GTO but I don't see it in general yet. Is it? Let's say I buy it.
- For any fixed strategy (mixed or not), the best counter-strategy is deterministic. If there are more than one, then surely any combination of both, random or not, is also optimal. This I totally buy.
So if 2. and 3. are true, then GTO, being a mixed strategy, is a random combination of best determinist strategies against itself. This means that if you knew you were playing against perfect GTO, then you could play any of these deterministic strategies and not lose (though you would lose against an exploitative style). This also means that, given that I've heard GTO is mixed, if the opponent will definitely play GTO, then there are always/usually/often(?) more than one deterministic best strategies. Is this just for when he plays GTO or its the case for any strategy he picks? If its just for GTO, maybe this is the justification for the indifference principle:
- If you are playing GTO, your opponent is indifferent to folding/betting (or is it any action? or is it "changing the frequencies of his actions assuming these are already GTO?)
At least some possible interpretations of 4. ring some alarms, so I don't want to get this wrong, and I'd appreciate feedback. For example, for any fixed (but possibly mixed) strategy of mine, there are best counter-strategies, and for any of these, changing slightly the frequencies of each action shoudn't change the outcome (as in, when you are at a function's maximum you don't mind moving a tiny bit). If this is what 4. means, then the suggestion of "play what you think would put your opponent indifferent to changing his current strategy a bit" would be a bad tip. It means: play the strategy that makes your current opponent's strategy the best one. So this can't be it.
Now, if it means, "put your opponent in a spot where NO action he takes makes any difference (as opposed to 'slight changes in the frequencies of his actions')", then
is this always possible? My intuition tells me no, but in rocks-papers-scissors it is.
if this is possible and one plays this way, how could we EVER get any edge even if he plays the worst poker ever? Wouldn't we be dooming ourselves to lose on rake? So again, this can't be it.
Or maybe it means "put your opponent in a spot where he's in doubt between 2 actions". But that would be weird. An action could be 'raise 3 pots' and another one 'raise 3.1 pots'. What would I gain my making him indifferent between two things that both leave me in a bad spot?
Also, the fact that it's a "partial information" game and the opponent strategy cannot possibly depend on my exact cards as long as my strategy doesn't depend on them either, makes this more complex.
In sum, what exactly does that indifference principle I've read loosely stated mean? Is there anything I'm clearly getting confused about here?
I'm hoping that once this is clear, the "range balancing" idea and its importance will fit into place with a stronger mathematical intuition that my current "it's good to not give much information to the opponent so he eats your bluffs and your value bets sometimes".
Simplest example that illustrates all the relevant principles
In a rock-paper-scissors game where the bet is doubled if the winner has rocks, one would be tempted to choose rock more often. However, this is the opposite of GTO, since a guy that always chooses paper would get positive EV. If I'm not mistaken, one should choose paper with double the frequency of the other two. This gives an EV of 0, no matter the strategy of the opponent. He can't exploit us, but he can't make mistakes either. Does this also extend to GTO poker, where if played perfectly you just can't win until you deviate from it a bit, or is this an oversimplified example that misses some crucial property? If the former, then GTO suddenly sounds less interesting, given that it boycotts every chance you get of getting an edge. But if the latter, what could be the simplest example of a game that catches all the basic properties here?