The Jailhouse Phone-Call Conjecture:

In a head-up poker situation on any given street before the showdown, given a pot size and the size of hero's and villain's stacks, and and ranges of possible holdings for both hero's and villain's hands, there is a strategy for the hero for the rest of the hand that is game-theory optimal, and the details of that strategy do not depend on the past history of the hand. In particular, it does not matter that this particular branch of the game tree would be unreachable if even one of the two players had been playing optimally up to this point; there is an optimal line of play from this point onward.

Handwaving argument as to why this is true: All of the information from the past history of the hand that is pertinent to the subsequent optimal strategy is contained in the pot size and the two players' ranges. Either or both ranges may well be polarized and/or unbalanced as a result of the earlier play, but the details of the history in terms of who has bet and raised or just checked and called at whatever point up to now translate into current ranges, and it is those ranges that should play into our decision-making.

In the limit where both hero's and villain's ranges collapse into single hands, the Jailhouse Phone-Call Conjecture becomes equivalent to Sklansky's Fundamental Theorem of Poker.

I offer this as a motivation for as-played discussion in strategy posts, even for situations where we think we would never find ourselves because we would never botch the play on earlier streets so badly.

tl;dr: Given board, stack sizes, and ranges, it doesn't matter how badly we botched the hand up to now, there is still a correct way to finish the hand.

Calling this a conjecture is...distasteful.

How about you call it a corollary to the minimax theorem instead?

From my point of view it's a conjecture because I haven't proved it. Moreover, I think I'm asserting something that seems true to me but I think requires work to prove: that if we know the ranges we don't need to know the history.

There isn't any work to do because you are defining what "game theory optimal" means in this context of players who have played non-equilibrium strategies up to now.

As far as I can tell, your definition is "Treat the situation as a fresh game where the players have been dealt hands from XYZ distributions, and the pot is N units. Then GTO means the equilibrium strategy for that game." Then trivially, such strategies exist by the minimax theorem.

If you want to make some stronger claim concerning the relationship of such strategies on the partial tree to the equilibrium strategies on the same partial tree that would have occurred had the players played in equilibrium on the full tree up till now, forget it, you can't.

"Yes, I was correct; it is obvious!"

I'm glad Jerrod posted here because +1. I think the entire original post is wrong and in limit holdem I think it is very very important how we arrived at the street we are analyzing.

DD's most recent post is a prime example of this. OTR and myself disagree with the river play precisely because we disagree on pre-flop ranges... 3 streets prior. How DD plays the river in a vacuum is completely irrelevant since it doesn't exist. Right now I think OTR is wrong on his river play in that thread because I think pre-flop is tighter than he does, but I also think I might be too tight 10 handed so I look forward to how the discussion progresses but I think the conjecture in the OP of this thread is not relevant to limit holdem in the slightest. I am happy to see that Mr. GT agrees.

Now wait a second. I didn't say there wasn't any value in examining the river in a vacuum, or saying "well here are the distributions and pot size, what now?" In fact, I think there is a ton of value in doing that in practice, because maybe you exploited them previously and now you just want to lock up that value because maybe you believe that now they aren't exploitable any more. Or maybe you just haven't refined your total strategy down every branch; maybe the other dude did something weird like limping on the button and you don't have a highly developed optimal strategy for dealing with it.

Doing this doesn't come with the ironclad guarantees of playing optimally throughout the hand, but I'd hardly say that it "isn't relevant to limit holdem in the slightest."

If we follow this conjecture, I think it goes to dispel the myth of 'initiative' or 'tempo.'

In general, the bettor (with initiative) will have a stronger than the caller. However, if a turn/river card shifts these ranges in favour of the caller, and both players realize this, the caller should donk in these situations.

The turn is a whole new puzzle: we shouldn't restrict ourselves by worrying about tempo, and such an action is not 'exploitable' as many believe.

I understand that it's obvious that there exists a GTO strategy for the "fresh game", but I have a problem with the concept of Subgame perfect equilibrium.

and...

So I was supposing that the equilibrium should be part of an existing "original" subgame perfect equilibrium. Or may be I am misunderstunding everything...

What are initiative or tempo except a representation of prior action and its implication for ranges? A cbet is a cbet not because of some magical property that gives the bet authority, but because aggression on the previous round strengthens the cbettor's range.

So I think we're agreeing this is subject to change as cards come out. Maybe I'm just late to the party or otherwise not getting the argument that you should continue to bet a weaker perceived range than your opponent because "Initiative!" Are people saying that?

So here you can't actually break the game into subgames, because when you arrive at a situation, you actually have arrived at an information set, not a subgame. You can look up these terms and stuff if you want to know more. You also might want to look at the concept of perfect Bayesian equilibrium which is the analogue of subgame perfect equilibrium for games of imperfect information. Wikipedia has an article about it.

I think I got it, thanks. Then by the subgame definition, the only subgames we can consider in a poker game is when preflop it is folded to each position, and this only if we disregard folded hands card removal effects.

This does not limit the utility of the toy games to model the equilibrium strategies? We can assert that the results obtained has any similarity with the global Nash equilibrium?

I've seen that reasoning a lot in this forum. "We have initiative, we should bet." It's a myth that should be dispelled.

Some players even make [poor] raises just to get the initiative: because initiative is good, "it makes the hand easier to play," or some other junk reason like that.

Having just had a big to-do over in small stakes over this sort of a situation, I think I am in position to observe the following:

1. On the one hand, earlier streets do matter. Obviously it is true that sometimes you are in the situation you are in because of earlier streets, and that's worth exploring.

2. At the same time, I have found that 2+2'ers who repeatedly insist on focusing on earlier streets often vastly understate the possibility that useful information can be gained by examining the later street as played. My thread demonstrated this-- sure I may have screwed up pre-flop, but the idea that someone who never screws up pre-flop could never face the situation I faced on the turn in that hand is just crazy. It is actually a common situation.

Also there are such things as misclicks. They happen more often online, but they can happen live as well. "I raise! Wait, he already raised?", and now you've 3-bet K8 suited on the button, or whatever.

In other words, even a good poker player will have legitimate questions about how to play a hand where mistakes have already been made. (Even if that king-8 suited hand should have been folded, once you 3-bet, your opponent has a range and there should be a correct way to play it once the flop hits.)

Either of those could be valid, but only if, respectively, (a) one's perceived range has increased through "initiative" such that a bet would be profitable (b) there's value to increasing one's perceived range on the next round.

I can't begin to tell you how glad I am that I don't have to worry about my equilibrium strategy on a partial tree after having climbed the full tree and looking down at the pack of wolves that actually understand wtf you guys are talking about.

"Fifteen birds in five fir trees...."

I skimmed Mathematics of Poker but didn't think about it all since I would never play any of the stupid AKQ games and such that they are writing about.

If you cannot master a stupid AKQ game, how can you be brave enough to approach game as complex as full-blown 5 street poker with a 52 card deck and wager real money there ?

One reason is our opponents are weaker in the 52 card 5 street game....

Yeah, not really. I mean, there are mostly no guarantees about the usefulness of abstracting and solving and deabstracting. In fact, there is a paper that proves that having a "finer" abstraction can result in a worse strategy in the real game. So even beyond toy games, whatever you do isn't guaranteed to be right in the full game. Nevertheless, it seems empirically to be useful. Also, empirically, finer abstractions produce better strategies in general. But I don't think there are theorems. I don't always read every paper on this though, so one of the academic poker GT researchers may know more than I do.

sorry for re-awakening this thread, but I think that after some investigation I found something important.

The papers

Waugh, K.; Schnizlein, D.; Bowling, M.; and Szafron, D. 2009. Abstraction pathology in extensive games.

and

Johanson, M.; Waugh, K.; Bowling, M.; and Zinkevich, M. 2011. Accelerating best response calculation in large extensive games.

show that the abstracted game equilibrium is not necessarily the least exploitable strategy of the subgame strategy space.

Anyway, in a game where a player uses abstraction and the other does not the abstracted player's strategy is by definition the least exploitable strategy that can be represented in the abstracted space.

This may look not very usefull when solving toy games like the various [0,1] from MoP, but for the not so complex games like the Jam or Fold the bb is unabstracted!

So, to be sure we find "least exploitable strategies" we shall study games in which only one player is abstracted. For example, ending with a Jam or a Fold after a certain number of bets/streets. This shall allow to use a foe essentially unabstracted in virtual play (or CFR) solvable Toy Games.

All jail phone calls are recorded and never hyphenated.

Is there an accessible introduction to this stuff for those of us who aren't ready to make an academic career of it? Should I just buy Mathematics of Poker and find resources to work my way through it?

Particularly in this post, I think have an intuitive understanding of what you mean by abstraction and deabstraction, but not really, and it's not immediately defined here.

the deucescracked series 'math attacks' by bellatrix is a useful companion to 'mathematics of poker.' it helped me a lot as i worked thru the book. there are also plenty of useful 'intro to game theory' type resources out there - some books at any bookstore, i'd guess, or look for a well-reviewed one on amazon. i know that deucescracked just released two short videos by danzasmack which are basically an intro to gto/game theory. disclaimer: i'm affiliated with deucescracked, so am most familiar with their material. there's likely good material on the other training sites also.