Exactly my reason for the offer, I don't believe you are here to learn at all. I already pointed out your mistake, twice. You doubled down without
giving it a second thought and demand that I spoon feed it to you.

Idk if you are just very very stubborn or deliberately poisoning this thread. You declining the bet gives us some info in that regard though.

Anyway, here you go:
https://en.wikipedia.org/wiki/Nash_e...mal_definition

Your calling range is in blatant conflict with that definition, you already acknowledged that it contains -EV hands. How does the caller's EV change if he stops including these -EV hands? If the answer is that the caller's EV increases then your range can't be part of a Nash Equilibrium.

Up until writing this post, I was under the assumption that the OP knows everything I am about to write.


However, I must respectfully set aside that assumption because I do not know where or how the method by which you all have come to the incorrect answer came from. Your calling strategy is picking a single combo that is minimally +EV and calling everything better as well. Anyone who constructs ranges that way is seriously overfolding, by a mile. This would cost a LOT of money at a poker table, and lead to the world’s worst red line. This bettor has a LOT of junk in his shove. You should have a lot of combos that win against that junk.

You took the original incorrect answer of 20.61 percent and lowered it to 17.48 percent just to make it fit your incorrect strategy. That is not how Nash Equilibriums are calculated.

So, here and now, I will explain what a preflop Nash Equilibrium in holdem poker is, how to calculate it, and what RANGE VS RANGE combinatorics means in No-Limit Texas Holdem.

I have showed you the correct method already. Any further clarification was covered when I said:

The deciding factor in determining a call, is where does the call break even.

When someone attacks you with a betting range, you defend yourself with a calling range. When next it is your turn to attack, you do it and then the other guy has to defend himself. THAT IS POKER. This needs no further explanation.

In this particular puzzle, this is a preflop call decision of range vs. range.

Can’t emphasize this point enough. It’s ranges. Not individual hands. You call when your hand is located within your entire call range. You calculate your entire call range to defend your equity against attacking opponents.

You guys are calling too narrowly, leaving your call way too +EV.

The OP has pointed out that at very high rake, the pusher is guaranteed a profit. Any rake above 44 percent and the shover gets the whole 10 bucks in antes. At 44 it starts to pay him less than ten, all the way down to zero at equilibrium. Yet, your strategy for the caller at any rake level below 44 is to also turn a profit. At your supposed equilibrium, the caller has a net profit of 2.22!

(.6277)(73.292)(276/1326) - (100)(.3507)(276/1326) - (13.354)(276/1326)(.0216) = 2.21611

Do you think it is reasonable for the pusher, at equilibrium, to risk his tournament life for a few cents, only to guarantee his opponent +2.22? There is no way. NO WAY that the shover is going to be shoving at your incorrectly suggested equilibrium. Why? Just to hand over 2 bucks to you? Do you think that is winning poker? In GTO the shover knows your call range. If he knows you are a nit with an incorrectly constructed nitty range, he will not shove with ATC. He will pick hands to fold.

The shover will always be all in and always with any two cards. His 100 is always added to the pot. THAT is the decision point for the caller. There is no other decision point in this puzzle. OP asked at what rake does the shover start to lose money with ATC. Only below that point would the shover be choosing a hand to fold, and only below that point would a shove no longer be a viable strategy. So, when does the shover lose money?

Well, the actual correct answer depends on a caller that actually knows how to call. Hence this post.

The high rake in this puzzle is what represents the ICM considerations of the call. There are no reasons to fold other than not to lose money. If this game is repeated over and over, you call as many times as possible with any minimal +EV. THAT IS THE POINT. When the pot is 110 and the high rake makes it less than 110, then you defend your equity in that smaller pot. If you overfold you give away money. Actually, you dispense it like an ATM. At your suggested call range, the shover would choose a range that maximizes his portion of the antes, not hand over 2 bucks to you. If he knows you muck individual combos by EV and not an ENTIRE RANGE according to EV, then he will win that 10 bucks way more often than he should.

Defend your equity.

How? By calling as often as possible, without losing money. If you call less often, the opponent will bet less often.

Well, there is a point at which that 10 bucks gets equally divided between the shover and the caller. And when both players are playing a GTO strategy the EV to both players will be the same. That is what a preflop Nash Equilibrium looks like. EQUAL.

The way you defend your equity with a call in holdem poker is to call with an ENTIRE CALLING RANGE that makes THE ENTIRE BETTING RANGE of the bettor indifferent. You call, as often as you can, without losing money. If your EV of calling is .00000000001, you call. Period. That is what GTO calling is all about. You know your calling range, you locate your actual dealt hand either within that range or without. In range? CALL. Does anyone really think that a perfectly executed GTO call makes money? The bettor made you indifferent or he would not have bet. You keep him indifferent by calling with a net EV of zero. You call less often than that, he wins more often. He is not playing poker to hand you money.


Your complaint about my range is that it contains combos that are, individually, -EV vs ATC.

I know why it *appears* to be so. Apparently, this insight is not common knowledge. The exact explanation of this answer is the only thing that I am not putting in this post. It is sufficient for me to simply say that you do not call with a single hand. You call with a range. Figure out the rest on your own, it will be eye opening.

It really should not need to even be said here in a high quality poker theory forum, but I will say it anyway.

In the game of No-Limit Texas Holdem, when there are only two players dealt, the EV of a GTO preflop shoving range is equal to the EV of a GTO preflop calling range, adjusted for the blinds. Here there are no blinds, the players are exactly equal. When they are in equilibrium, they both in fact have EV ~equal to zero, with perfectly executed GTO betting and calling. How could it be otherwise, in a zero sum 2-person non-cooperative game? Where would any extra EV for either player come from?

Since these are uniform ranges, this puzzle has a particularly pleasant likely final answer in mixed strategy. This will probably be 25%(105/100) = 26.25 percent rake. DUCY? <<<That time I could not resist>>>


I stand by my range, with pure strategy, within 4 combos.

https://en.wikipedia.org/wiki/Nash_e...mal_definition
Caller's EV with your range:
[66+ A5s+ A9o+ K9s+ KTo+ QTs+ QTo+ JTs]
Equity: 64.5698%
Combos: 238 / 1326
EV_Caller = 238 / 1326 * (210$ * (1.0 - 0.2621) * 0.645698 - 100$) = 0.010$

(This roughly matches your own calc, you have slightly different equities. Does not matter.)

Caller's EV with alternative range:
[77+ ATs+ AKo]
Equity: 72.1264%
Combos: 76 / 1326
EV_Caller= 76 / 1326 * (210$ * (1.0 - 0.2621) * 0.721264 - 100$) = 0.674$

So the caller can switch to the 5.7% range and increase his own EV, while the pusher's strategy is fixed. This is a direct violation of the Nash requirement, so your range is not part of a NE.

PS: I'm not correcting your wall of text above. A Nash Equilibrium is not what you think it is. It's absolutely amazing that after all the time you spent in threads arguing about GTO/Nash, you've never actually bothered to read/understand the definition.

You understand that ranges don't actually exist, right?
When we play poker we always only hold one hand at the same time and have to choose what the highest EV play is for that exact hand.

Rake is the reason why the pusher has higher EV here than the caller.

Rake plays a part in this specific example, but more generally his claim that pusher's and caller's EV need to be the same in a Nash Equilibrium is fundamentally wrong.

Yes, I have pointed out several times that our equations are different. Algebra matters.




If you try to inflict your non-equilibrium calling strategy on the pusher in your suggested equilibrium, yet force him to push full range, then he will simply push at a mixed frequency that gives you no money.

Only at my equilibrium is the solution pure strategy.

Maybe read some more wikipedia?

¯\_(ツ)_/¯

Equations are equivalent, only the equity numbers we plug in for the ranges are different.

The pusher is at 100% because pushing is profitable with every single hand. You have him push 100% against a much wider range, but once the caller tightens to 5.7% your adjustment is that he pushes less even though his pushes just became more profitable. How does this make any sense to you?

Helping you would mean harming himself. He will not claim a prize of 6 cents, if it means awarding you $2.22.

He does not play a poker tournament just to pass out tournament equity to other players.

Really, man. How much longer are you going to keep going?

Range vs. Range. Can’t explain it any easier than that.



Only at rake levels above equilibrium. That is the point.

One of us is going to take a hiatus from this subforum, it's not up to me who it's going to be. Have a nice day

Please don't resort to addressing other posters like this. Say your piece and let it stand for itself.

Understood. I apologize to plexiq.

Rusty, I've no problem with a snarky remark or smiley here and there.

But I strongly believe that mod action should be taken against posters who deliberately poison threads by posting wrong info. Beginners used to look into this subforum as reference and having this stuff in here uncorrected is really bad imo.

Robert's posts are filled with basic, fundamental mistakes and misunderstandings. Stuff that he has been corrected on dozens of times. He's been in the Nash/GTO threads for a long time, it's completely implausible that he is posting like this in good faith.

There needs to be stronger modding. If threads ultimately end up like this one then it's a complete waste of time to contribute here, imo.

Yep yep. 20.6197% is the answer then.

To answer the question literally ("At what rake should I stop pushing everything in the dark?") we would go with 20.6197%. To play the GTO strategy below that you would need to check your hand, see whether you have 72o, and shove with the correct frequency.

Though I've heard that in some alternate reality, the answer is 26.25%, in this reality the final answer: 20.6197%.

David, don't you think it would be better to participate in your threads with meaningful discussion? Or, are your threads some sort of poker theory thunderdome where you stand outside and watch the participants battle to the death?

Robert, are you trying to minimize villain’s EV rather than maximize your own? This would be incorrect because this is not a zero sum game.

Hi browni3141,

Thank you for your comment, and your participation.

The question of whether or not the free 10 bucks in antes makes this game non-zero sum, is best answered by considering that the shover is not really a player, but just a predetermined sequence of events. Since all previous actions in this game are explicitly known to all players, this becomes more simply a game of perfect information.

Since the shover is always shoving in all cases, the shover's 100 stack is always added to the pot, and the rake is always deducted from the pot. There is just a pot, and a call decision to make. The game will simply stop running if the EV of the shover goes negative.

This is obviously a model of tournament poker, and the caller is trying to prevent the shover from shoving any more frequently than is possible by optimal calling strategy.

It is the high rake that is variable.

This allows the caller, when the rake drops to equilibrium, to enforce an equilibrium that is in fact zero sum. At equilibrium, If there is a dominant strategy, S*, that a player can implement that causes the game to be both zero-sum and fair, then the game is in fact zero-sum and fair.

They each start with 100, and they end with ~100.

So, the question of zero-sum is actually a question of EV. Remember, if the shover shoves and gives his opponent an EV greater than his own, then he would suffer a net loss of utility (here the utility is tournament equity).

Not every hand in the shovers range has the same EV, but for some reason you're treating it like that's the case.

If the EV of the shover goes negative, the game won't "stop". Even though the EV of him shoving any 2 cards would be negative, the EV of him shoving AA would still be positive. Therefore he should still be shoving AA.

The same way, even if the EV of shoving any 2 cards is positive, it's possible that the EV of the bottom of the range isn't, and the pusher therefore shouldn't be pushing those bottom hands.

If the EV of pushing any two is zero, it means that the pusher may as well just be open folding any two cards and the EV wouldn't change. That's terrible.
It would mean that the pusher is making +EV shoves with the top ~half of his range and -EV shoves with the bottom half. It would be better in this case for him to just fold those bottom hands instead of shoving.

You are right that this game is not zero sum (or constant sum for that matter), but that's not his mistake.

If you think this is a mistake made in good faith then I'll just leave you with this gem:
Just to spell this out explicitly: If the pusher's EV is too high, because the caller is calling too tight, Robert suggests that the pusher's response should be to lower his own EV by pushing less. I guess this account is amusing to follow if you pretend he is taking part in a forced game of counterfactuals: "Describe a universe where a pusher's and caller's EV need to be equal in a Nash Equilibrium."

Hi ZKesic,

Thank you for your comment.


When I said that "the game stops running", what I meant was that the pusher stops pushing with ATC. Sorry for the confusion. The pusher would pick a more optimal pushing range.

See above.

It would seem that you are attributing this statement to me, as a summary of my argument.

The bolded is incorrect and has never been said by me. At the equilibrium in your answer, it is the caller's EV that is too high. This is because you are forcing the shover to shove with pure strategy ATC when he would not in GTO.

If you inflict a non-GTO calling strategy on a GTO bettor, yet force the bettor to use all the same combos in his bet, then the bettor will use those combos, but at mixed frequency. So, he would bet AA always, and 72o less than always.

This would reduce the caller's EV to zero, and maintain the bettors's EV the same. At only slightly less total vpip by the shover, he is still getting those 10 dollars free very often. And with a slightly stronger range, he loses much less money by showdown.

When the caller uses a 5.7% calling range, is the pusher's EV with ATC higher or lower than in your proposed equilibrium?

At rake levels at or above 26.21 percent, where the shover should by all acounts be shoving, the shover's EV is always positive and even higher when the caller overfolds.

At rake levels above equilibrium, but below ~44 percent, the caller can pick any nitty incorrect call range he wants, but he gives away extra tournament equity whenever he overfolds and awards the pot to the shover too often.

So you agree that the pusher's EV is higher against 5.7% calling. Can you explain why he is supposed to change to a less profitable strategy then? Pushing less will decrease his EV.

The pusher is always pushing at any rake above 26.21 percent (pure caller) or 26.25 percent (mixed caller).

That is all he can do, he cant push any less often, with a pure strategy push.