I´d go with no, as the bb gets a freeroll postflop.

Isn´t GTO the best response against the best response though?,
which likely wont be 100% 3bet and because of that you dont need to defend whatever number you would need in your example, but less ?

Or do I need to reread the thread?

If we assume a finite stack size and a finite smallest bet size (i.e. can't bet half a penny) then a HU game of NLHE will have an enormous but finite amount of possible situations. A strategy set for player X will have individual strategies for each possible hand of player X for each possible situation player X could be put in. Many of these individual strategies will be mixed rather than pure (raise 30%, fold 70%, etc). You are correct in stating that this will be a staggering result, but it will still be finite and known.

Im confident the answer to this is yes....

^ Yes, absolutely.
I was a winning player at 100NL on Cake and I never heard of GTO in my life until this thread.

who won the prop bet?

I disagree on: 1. a practical level 2. a theoretical level.


On a practical level there are players who beat very high 5 Card Draw and 5 Card Low Ball using game theory type systems in California. Their uninformed strategies were powerful enough to beat the real-life games that they played, without adjustment. Of course they adjusted their play based on player information, and so on Understanding possible cheating was a concern, however.

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I do not think what you seem to be saying is true on a theoretical level for the entire 2-round game (excluding possible weird multi-player situations). Although It may be argued that a given players math-based strategy when dissected is not GTO, and exploitable

Of course there are second round situations where a GTO player seeks the play that loses him the least against his opponents range; however, hopefully he was compensated for being in this spot by the first round betting..

In Pot Limit Low Ball it could be correct to throw away a pat hand on the first round that is a small favorite if an expert opponent flipped up a joker wheel draw; reason: the pat hand loses too much in the betting on the second round.

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Idea to keep in mind: In the game Rock-Paper-Succors, I can tell you I will pick my option using a 33% randomizing devise after you secretly pick your option. If I have to pay more than my opponent to end up in this spot; well,,

"GTO, that's a car... right?"

I have a friend who won at those stakes picking his games and he thought GTO was a car.

OK let me phrase my thought in detail.. my understanding i that GTO strategy is one thing, Nash equilibrium strategy is another.

Nash tells us that there exists a strategy "a" of player A (bot) and a strategy "b" of player B (human), such that noone is better of changing their strategy if the other doesn't change his. So suppose that our bot knows strat "a" but the human doesn't know strat "b" and plays "c". So by definition of the Nash equilibrium the human now loses to the bot. So YES theoreticaly we could program a bot with strat "a" and the bot would NEVER lose (at best is would break even), no matter what strategy "c" the human plays. The bot wouldn't need to change its strat based on opponents strat.

However, this doesn't guarantee that strat "a" will be maximally profitable against any strat the human plays. In order to program a bot to play the most profitable strat at any given point in time it would need to make adjustments based on the way the opponent plays (which as I described in my previous posts seems insanely hard). The only case where the maximally profitable strat is the Nash equilibrium strat is the case where our opponent plays his Nash equilibrium strat.

So creating a bot that never loses is far but possible IMO. Creating a bot that maximally exploits any opponent (although theoretically possible too) seems inconceivable, something thats light years away. So yes your bot would win money from Sauce, Ike, Kanu etc. but it would be likely to have lower win rates than me against a massive fish (it would still not lose obv.)

This is where the problem is. When I speak of GTO - a term I don't particularly like, as stated before - I actually mean a (Nash) equilibrium strategy. I was under the impression that this holds for most other poker theorists as well, and this impression is supported by many posts in this thread.

I totally agree with the rest of your post.

Adding to bware's response:

I look at this from a mathematician's point of view. A programmer would have problems if the strategy doesn't fit into memory, or if the time to compute an answer is measured in eons. A mathematician (and some computer scientists) are more likely to say: The problem is finite, so in principle we can solve it.

Of course poker strategies are complex - otherwise where would skill come in? If the strategy is influenced by exterior data like the cocktail waitress, you can either incorporate them into the game description, or just say that they influence the choice of strategy.

If you come up with any situation in HUNL you could ask Ike: "What would you do here?" This defines a strategy.

I used the random generator just for dealing cards.

You can define randomized (mixed) strategies if for every situation you assign probabilities to all strategic options (=legal moves). In theory this is properly defined.

i havent read every post. but have been following this epic thread on and off. so apologies if this has been touched upon earlier.

how is gto play effected if we are playing pot limit omaha?

due to the extra 2 holecards, how may that effect our thinking?

its all about considering the possible parameters imo.
the more exact you know about them, the more exact you can extrapolate the correct move. no matter u have 2 holecards or 4 or 25

On a theoretical level nothing changes. Of course the games are totally different, so you can't even compare the strategies. However, if we knew what equilibria for HUNL and for HUPLO looked like, and if we somehow had to describe them to humans (as Jerrod suggests), then my wild guess would be that there are some similarities. However since we don't know either solution this is pure speculation.

I'm not an expert on the nitty gritty of game theory, but I think in ring games there are multiple nash equilibrium strategies because of the possibility of implicit collusion by coalitions of players.

I'm not sure whether they'd be nash equilibriums (is it possible for a nash equilibrium to not be the same ev for all players given the rules are the same for all of them?) but I think it's more likely that most of the spots arise from a set of strategies not being a NE. Say it's a 3-handed game I'm player A now both B and C are playing sub-optimally such that if only player B changed their strategy or only player C changed their strategy they could win, but based on the strategies they are using now my perfect strategy might still be -ev. This could happen intentionally or unintentionally.

all that math stuff is certainly sexy. Im sure you guys will apreciate this.

no i realise that.

was just hoping you/someone could save me a lot of headache/effort, and tell me how to play 'fundamentally solid' HUPLO.

that gives me a starting point to then imagine how one could develop exploitative strategies in that game.

but you havent. so i hate you well in the poker sense

if we had an infinitely powerful computer, would we already know what to have it do to find a GTO solution for deep stack hunlhe, or would it just have to execute every possible strategy vs. every other possible strategy?

i mean like i said i only skimmed the thread, and dont totally understand the whole gto debate (but i'm getting there), the fact that NL utilises two cards makes the parameters of the game much smaller than PLO.

and just from an imaginative p.o.v., i kind of can see what gto play MIGHT look like for a HUNL match. i was hoping somebody well-versed in PLO might be able to aid my plo thinking by attempting to describe how gto in plo would work.

firstly, you have the extra 2 cards. then you have the pot limit betting structure being different NL. but the flop turn river structure is similar. but then the number of possible holecard combinations is much larger. but theoretically the number of possible flops you'll see and board run-outs is smaller so there's that too.

Because many strategies are known, are fundamentally known or educated guesses, etc

You say this too casually. Yes, like a waitress. Or a waitress who looks to the left - or an Asian waitress, or a waitress who happens to be wearing green... The point is these influences are probably uncountable (I've only modified your one external example) because we haven't worked out how brain-states function, or even if it's just random neurons firing. It's currently undefined. I've never heard a game theorist say that all poker are forms of games; only really simple examples of poker like Kuhn Poker.

So Nash's theorem need not necessarily apply. It might - I just believe that everyone being so cocky about it is being a little premature.

The human mind is not a perfect RNG, yes. But since the external influences are also random to the point of being incalculable, these theories would still apply.

Edit: I'm talking about notion that we are theoretically mixing strategies perfectly randomly. The fact that humans are poor RNGs and thus make seemingly random decisions based on external factors like the cocktail waitress does not disprove the existence of a Nash eq strategy or disprove the assumption that poker is a game.

Sometimes I state things casually since many people get turned off by formal mathematical reasoning.

Let us assume that the waitress has no influence on the cards. Then we can ignore her, solve the waitress-less game, and play our equilibrium. If our opponent is somehow influenced by her, she causes him to deviate from his GTO, and therefore probably costs him money. In a heads-up situation, this means that we win.

So we can restrict ourselves to the actual game of poker, which is finite. From our perspective the opponent's deviations are purely random. Of course if we want to maximally exploit him we need to know his turn check-raising range in the presence of an Asian waitress with green eyes and cup size C.

BTW, "uncountable" has a precise mathematical meaning, which certainly does not apply here.

If there were poker games that are not games then something would be seriously wrong with the mathematical definition of a game. If you came up with such a game I doubt we would recognize it as Poker.

This might be interesting to some.

I think you're just describing nash equilibriums for multiplayer games. I think unlike HU, there is no "perfect strategy" except in the context of your opponent's strategies.

I suspect that if you play an equilibrium strategy in a multiplayer game, then "most" deviations by your opponents will be good for you. However you're not guaranteed to break even anymore.

I liked the 5 Card Draw example too where standing pat could be a way to exploit a bot. But if the bot plays GTO, the GTO decision would probably be to ignore if the opponent stands pat and use the mathematically programmed decision based on hand, pot size and bet sizing.