Because their no anti strategy that can profit. Your only choices are to play to a draw or lose.

A question i have is If player A has a simple GTO strategy that uses 2 bet sizings per a street and player B has a strat that uses 10 per a street doesn't player B win?

Yes. None of those 2 strategies are GTO, one is just closer

Generally speaking, the strategies of both players are only GTO within their respective abstract games where no other bet sizes are allowed. They are not GTO for the actual full game.

If you confront these players with a sizing that isn't present in their solution then they will have to somehow approximate a reasonable strategy for that unknown sizing. How much of an edge they give up there depends entirely on the type of process that is used to approximate the strategy for the new line "on the fly".

Let's say the 2 sizing sim could resolve the new sizing it's facing along the 2 default ones. Even then, it would lose EV to the 10 sizing one.

Depends. If all of player A's sizings are covered in player B's solution then player B will have an edge, I guess that's the scenario you have in mind.

But if A's sizes are not fully covered in B's solution then it's perfectly possible that player A could have an edge, even when using fewer sizes.

Yeah I was talking about a case where they are covered

This has become a very interesting thread!

The OP's reference to RPS and how it is not really a good game to use as an intuition-sharpener for poker got me thinking about how a few years ago I was in a thread where we were talking about exactly that, and I posted about a different game, one that is maybe better for understanding the types of stuff that people are talking about in this thread. So with minor editing, here it is.

(Note: in what follows, I use the term "admissible" to mean any pure strategy which is played with nonzero frequency in a Nash equilibrium. "Inadmissible" is the opposite.)

In Two-Finger Morra, there are 4 identical strategies available to each player. You can play one or two fingers, and simultaneously, you also guess one or two fingers. Your "guess" means you're guessing how many fingers your opponent will play.

So the strategies are ordered pairs of a play and a guess: (1,1), (1,2), (2,1), and (2,2). The first number is your play and the second number is your guess.

Now here is how "money" changes hands in Two-Finger Morra:

1. If both guesses are incorrect or both guesses are correct, it is automatically a tie and each player gets 0.
2. If your guess is correct and your opponent's guess is incorrect, you win the sum of the fingers that have been played (and vice versa--if your opponent guesses right and you don't, you lose the sum of the fingers that have been played).

So for example, if you play (1,1) and your opponent plays (1,2), you have each "played" 1. But you have guessed correctly and your opponent has guessed incorrectly. That means you win 2 (because you each "played" 1).

Here's the entire payout matrix, with the payouts shown for the row player (and my apologies for the poor formatting):

_________(1,1)_________(1,2)_______(2,1)________(2 ,2)

(1,1) -------(0) ------------- (2) ---------- (-3) -------------- (0)

(1,2) ------(-2) ------------ (0) ------------ (0) ------------- (3)

(2,1) ----- (3) ------------ (0) ------------ (0) ------------- (-4)

(2,2) ----- (0) ------------ (-3) ------------ (4) ------------- (0)

So there are a couple things we can already see from the payout matrix, before talking about Nash equilibria. First, in a similar manner to RPS, no strategy in this game is dominated. The reason is because every strategy has a corresponding strategy that only it can beat. So there's kind of a "circle of exploitation" like there is in RPS. In this game it goes

(1,1) beats (1,2) --> (1,2) beats (2,2) --> (2,2) beats (2,1) --> (2,1) beats (1,1)

But unlike RPS, the payouts are different! This is going to be important.

Now let's talk about the Nash equilibria for this game. I'm using the plural because TFM does not have only one Nash equilibrium. It actually has infinitely many Nash equilibria, and therefore infinitely many GTO strategies. They are all related--sort of a "family" of GTO strategies.

The Nash equilibria for this game all involve both players playing a mix of (1,2) and (2,1) only, and the frequency with which they play (1,2) has to be between 4/7 and 3/5 (or if you prefer percentages, 4/7 is a bit higher than 57%, and 3/5 is 60%). Thus the frequency of (2,1) will be between 2/5 and 3/7. So for example, 58% (1,2) / 42% (2,1) is GTO.

While it may not be obvious at first glance where those numbers came from, it'll be obvious if you try to prove that such a strategy is GTO--which you could do by proving that if you play that way, your opponent cannot gain EV by deviating from his own GTO strategy. In other words, your opponent's EV by playing any strategy is either 0 or negative.

So here are some stray observations about this game based on what the Nash equilibria are:

-Unlike RPS, TFM is an example of a game in which, despite no strategy being dominated, some of the strategies are admissible and some are inadmissible. The admissible strategies are (1,2) and (2,1), and the inadmissible strategies are (1,1) and (2,2).

-In RPS, because every strategy is admissible, playing GTO guarantees 0 EV no matter what your opponent does. You never lose, but you also never gain. That is not true for TFM. If your opponent deviates by only playing (1,2) or (2,1)--just not in a GTO balance--you still break even; so it is possible to deviate from GTO and not lose EV against GTO. But if your opponent plays an inadmissible strategy, now you will show a positive EV by playing GTO.

-Here's an interesting one that is also different from RPS. Obviously by definition of GTO (and since there are only 2 players), anytime your opponent isn't playing GTO, you can find an exploitive counter-strategy that wins EV. But there's a big difference between deviating by playing an unbalanced mix of admissible strategies and playing an inadmissible strategy. If your opponent deviates by playing 100% (2,2), you will win by playing GTO but could win more by playing 100% (1,2). But what happens if your opponent sticks to admissible strategies but is unbalanced? Say he plays 100% (1,2). Since this strategy is admissible, it breaks even against GTO. But also, looking at the payout matrix, (1,2) breaks even against all admissible strategies! In other words, if you want to exploit someone playing an admissible strategy, you have to deviate to an inadmissible strategy--in this case (1,1).

Could you explain why only playing (1,2) and (2,1) would be GTO or admissible?
I don't really see a reason why those are better than the other options.

Even if the sizings are covered by the other player it still isn't that straightforward.
Let's say Player A is playing a strategy consisting of 1 sizing of say 25%. They will construct their betting (and checking) ranges in a different way than player B who uses 25/50/75/100 % for example.

The "gto" response vs bet 25% will be different against both players.


So for practical purposes less sizings are usually better. The goal is to solve and study strategies that we can implement. Arguing that more Sizings is closer to "gto" might be technically correct but in the real world it's not about being closer to GTO but rather about implementing your simplified strategies.

TLDR: Stop mashing infinite sizings into the solver and instead learn to execute your (dumbed down) strategy. That part is hard enough.

OP: GTO wins because unlike in rock,paper scissors every hand has an EV for every action you choose. Whether that's bluffing or value betting. Some hands have higher EV bluffing. Some hands have a higher EV as checks.
Simply put you win by taking the highest ev line more often than your opponents.

At risk of derailing the thread, here's a quick explanation of how you can conclude I'm right about what the GTO strategy is.

(1,2) can only be beaten by (1,1), and you lose 2 every time that happens. On the other hand, (2,1) beats (1,1) and you win 3 every time that happens. So you can't be exploited by playing (1,2) as long as you play (2,1) often enough. "Often enough" here means you have to play (2,1) at least 2/5 of the time. At 2/5 (2,1) and 3/5 (1,2), your EV against (1,1) is

(2/5)(3) + (3/5)(-2) = 0

Any more than that and you will show a profit against (1,1).

(2,1) can only be beaten by (2,2) and you lose 4 every time that happens. But (1,2) beats (2,2) and you win 3 every time that happens. So by similar logic as above, if you play (1,2) 4/7 of the time and (2,1) the rest of the time, you can't be exploited by (2,2), since

(4/7)(3) + (3/7)(-4) = 0

Any more than 4/7 and you will show a profit against (2,2).

Now the key observation here is that 4/7 is less than 3/5. So if you pick any number in the interval (4/7, 3/5) and play (1,2) at that frequency and (2,1) the rest of the time, you will break even against (1,2) and (2,1) and you will profit against both (1,1) and (2,2).

If you try to carry out this exercise for the other two strategies, you hit a contradiction (since 2/5 is less than 3/7, you CAN be exploited if you play (1,1) or (2,2) at any frequency).

What? We are not talking about humans, we are talking about 2 sims playing each other. More sizings = less abstacted GTO. Less abstracted GTO will beat the worse abstraction, no way around it. And the main reason is precisely because the worse abstraction's response to sizings will lead to huge EV losses.

GTO doesn't lose to 1 sizing abstraction when it misreads their range, because that's how GTO works, it doesnt care if villain plays different from GTO, it still can't lose
But 1 sizing abstraction does lose a shitload when it misreads GTO's range.

Okay so just making sure I understand you correctly. The way to use PIO is to mash in as many sizings as humanly possible so you have an accurate counter to any sizing your opponent can throw at you?

Because as you put it "gto will respond well to the 1 sizing abstraction". Well I disagree with that.

The GTO response to a 1 sizing abstraction is very to different to the GTO response to a strategy that splits between 2 sizings (even if the sizing is included in your gto sim)

I never once claimed that the smaller sizing abstraction would win in a HU match. That's not even relevant. All I'm saying is you can't keep adding more sizes and claim to have an "optimal response".

edit: You are correct more sizing is closer to GTO. I'm not arguing that. Just saying that it's only GTO against another player playing the same strategy. If they play a different sizing strategy (with more,less or even just different betsizings / raise sizings) your GTO solution won't be the perfect response.

Figured this was important enough to point out because some beginners might get confused and think "more sizings are better" as a result.

I'm not even talking about what's practical, just about theoretical concepts.

But now that we get into the practicality of it, yes, i believe having a few sizings will be very beneficial over just 1, for the main reason that defenses against a bet in a 1 sizing sim are dogshit.

And to touch on the "GTO vs a different strategy" thing. Yes, GTO will not be maximally efficient against a leaky strategy, but in the case of facing a 1 sizing abstraction, it will be a good chunk better, even though there would be a different strategy (Hard splits based on hand strength), that would exploit the 1 sizing sim even more.

See the point is that a 10 sizing strategy isn't GTO either. Please stop calling it "gto".
It's only GTO as long as your opponent cooperates by using the exact sizings you input. Any deviation on their sim input would lead to a different GTO strategy based on their inputs. Whether they use one or 4 sizings in their sims.

For that reason it simply doesn't matter whether you have "their bet sizes covered" in your sim. If that was how it worked poker would be dead because every idiot could build a perfect bot in a couple weeks. Just run a PIO sim with infinite sizings and suddenly you are unbeatable.

I'm trying to understand where we disagree. do you think it's enough to simply cover the betsize in your sim and that will give you an answer against any strategy involving that size? Because well... it's not.

I know that using the word GTO for a bunch of different things gets confusing, but I think you get my point when I say that what I'm matching up is a 10 sizing sim vs a 1 sizing sim.

To your second point. It's very funny that you think if someone could run a pio sim of every single spot with a **** ton of sizings that bot wouldn't be practically unbeatable by a human. Yes it would. And no, no one in their home has the computing power to solve this, that's the only reason why poker is not dead lol.

My point is that a defense in a 1 sizing sim is very exploitable, while a defense on a 2 sizing sim is way less exploitable, and so on.

It's important to be precise when discussing concepts like GTO. Strictly speaking neither sim is GTO against the other sim. They will both be leaking EV (both vs "true gto" and the other players fake GTO) thus they are not optimal.
Even if you include my betsize in your sim. It will still give you the wrong answer against my strategy. Because you didn't solve for my strategy.
"but it's close enough" no. close doesn't exist in GTO. Close = wrong.

As for your other point.Well.... Software like this is out there. (for training purposes) but no professional uses it because it doesn't accurately represent the game tree. They assume a 4 sizing spit on any street and the result is utter nonsense. No river range resembles what a real river range looks like etc etc.
It really isn't as easy as mashing in more sizings = better answers.

Again I agree the 10 sizing sim would win. That's obvious I just strongly disagree that "covering my size" even matters.

In the bet 25% only node the 10 sizing sim will leak EV by responding incorrectly.
edit:

also correct this software would smash a human. but that can be said about even simplified bots with only a few sizings sadly.

The 10 sizing sim is a toygame within NLHE, just as much as the 1 sizing sim is a toygame within the 10 sizing game. Therefor, in that 10 sizing game, the 10 sizing sim is the absolute GTO (leakless) while the 1 sizing sim has leaks. The only way to exploit the 10 sizing sim in NLHE would be to play more sizings than it accounts for.

You are making the understandable mistake of thinking that every response to a bet that's not assuming the correct betting range is just as bad. It's not.

The responses to bets in the "10 sizing GTO" will have break even or positive expectation versus every possible strategy within that 10 sizing game, even if they are misreading the actual betting range (for example, by villain having only 1 betsizing on that node).

On the other hand, the responses to bets by the "1 sizing GTO" will have break even or NEGATIVE expectation in the "10 sizing game".

I can't really say much more to have you understand this, so I'll leave it at that.

EDIT: The reason professional players don't play while looking at cloud sims is because its against the rules and they would get banned. If they were to follow those outputs they would beat people even if those people played differently, it doesn't matter that the river range in the sim is way different from how human villain plays, sim still wins. Plus, the sims are not that great because we don't have the computing power to make great ones yet.

EDIT2: I just reread and saw you meant that no professional uses Training (not RTA) software with 4+ sizings per node because its useles... What are you talking about? I play professionally and I train like this, and I know plenty others who do aswell.

1) Yes. Yes it is. I don't know why this is difficult to understand for you but in order to calculate a counter you need precise inputs. If the betting range is different the optimal counter strategy will be different too. Thus your response to the size is WRONG. It's only right within your sim. It's wrong against any strategy that isn't your sim.
2) Incorrect. b25% only will beat a strategy that includes 25% as one of the sizing options. I mean beating in the sense of "outperforming the optimal counter strategy of b25%"
Again if you want to optimally counter b25% only YOU NEED TO SOLVE B25% ONLY!!!

regarding your 2nd edit. I build training software for MTTs(dto poker) that teaches simplified GTO strategy. I actually do believe that most people would be better of using less sizings in their training sims.

But that's mostly because of what we as humans can implement.

In general bet sizing perfection is hilarious overrated in poker.

edit: I'd also be happy to offer you a bet if you still don't believe me. We can run my b25% pot only on flop against your counter strat calculated based on as many sizings as you like.

DM me if you wanna work out details

Beating is NOT "Outperforming the optimal counter strategy", it means to literally match both strategies up and see who has a positive winrate and who doesn't. I know that to maximally exploit a 1 sizing sim you need to play a 1 sizing sim yourself, but that's not my point.

My point is that not every mistake is as severe. Defending "10-sizing style" vs a 1 sizing betting range is underperforming compared to the maximally exploitative strategy, yes. But not by a long stretch, by a very short one, we're talking in the 0.5%s.
On the other hand, defending "1-sizing style" vs a 10 sizing betting range will get absolutely demolished.

This is just a fact of poker theory, I'm not saying you should play like this. Of course you should try to play in the most profitable way against each villain, but that doesn't change how theory works.

Thankfully the real answer to the problem you describe is simply in the middle. The best way to study sims for practical purpose is to use limited sizings. So no 1 (well maybe on the flop or in some veryyy specific spots) and certainly not 10.
2 flop, 3-4 turn 3-4 river is usually what i recommend.


i'm not recommending 1 size either. Just using this as an argument to point out that even the sim with the most sizings won't give you the optimal answer

Will quantum computers be able to solve poker ?

It's interesting because infinite (decimal) bet sizings are possible therefore is there a true equilibria?

Sent from my HRY-LX1 using Tapatalk

If there are infinitely many sizings then there's actually not even a guarantee NEs exist at all.

In reality we don't have infinitely many sizings though, you can't bet fractions of a cent or fractions of a tournament chip.