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Old 01-24-2019, 04:14 AM   #76
plexiq
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Re: Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by robert_utk View Post
I decline. The forum is for learning, not excommunication. Hang around more often so that you can learn.
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.
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Old 01-24-2019, 04:43 AM   #77
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Re: Can Someone Do This Calculation I Put In My Book

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:

Quote:
Originally Posted by robert_utk View Post
The deciding factor in determining a call, is where does the call break even.

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

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

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

Now, I admit, that did send me for a loop. However, after studying range combinatorics, I see the answer.

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


Quote:
Originally Posted by robert_utk View Post
This is as close as I can get, within a few combos (non-mixed), and within 4 significant digits digits, over 500 million trials.

Rake% = 26.21 percent
Rake$ = 55.04 dollars

Pot = 10
Dark Shove = 100
Call = 100
Pot after rake = 154.959
Rake per player if tied = 22.5205



Caller calls with a range of: 66+, A5s+, K9s+, QTs+, JTs, A9o+, KTo+, QTo+



When the game runs with these parameters this happens (using Equilab):

Caller calls with frequency .17948718
Caller folds with frequency .82051282
The Caller wins with frequency .6354
The caller loses with frequency .3446
The players tie with frequency .0200


The expectation value of the game to the caller is:

(.17948718)(54.959)(.6354) - (.17948718)(100)(.3446) - (.17948718)(.0200)(22.5205) = .00189 -->.189 cents

The expectation value to the dark shove is:

10(.82051282) + (54.959)(.3446)(.17948718) - (100)(.6354)(.17948718) - (.17948718)(.0200)(22.5205) = .11895 --> 11.895 cents


If the rake goes any lower, the caller will start to add combos and the shover goes negative.

Last edited by robert_utk; 01-24-2019 at 04:50 AM. Reason: removed a word 'matching'
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Old 01-24-2019, 05:04 AM   #78
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Re: Can Someone Do This Calculation I Put In My Book

https://en.wikipedia.org/wiki/Nash_e...mal_definition
Quote:
Informal definition Nash Equilibrium

Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"

If any player could answer "Yes", then that set of strategies is not a Nash equilibrium.
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.

Last edited by plexiq; 01-24-2019 at 05:23 AM.
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Old 01-24-2019, 05:07 AM   #79
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Re: Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by robert_utk View Post
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.
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.

Quote:
Originally Posted by robert_utk View Post
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
Rake is the reason why the pusher has higher EV here than the caller.
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Old 01-24-2019, 06:52 AM   #80
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Re: Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by ZKesic View Post
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.
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Old 01-24-2019, 09:55 AM   #81
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Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by plexiq View Post
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.)
Yes, I have pointed out several times that our equations are different. Algebra matters.



Quote:
Originally Posted by plexiq View Post
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.

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)_/¯
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Old 01-24-2019, 10:10 AM   #82
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Re: Can Someone Do This Calculation I Put In My Book

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Originally Posted by robert_utk View Post
Yes, I have pointed out several times that our equations are different. Algebra matters.
Equations are equivalent, only the equity numbers we plug in for the ranges are different.

Quote:
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.
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?
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Old 01-24-2019, 10:18 AM   #83
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Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by plexiq View Post
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?
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Old 01-24-2019, 10:25 AM   #84
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Re: Can Someone Do This Calculation I Put In My Book

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Originally Posted by ZKesic View Post
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.

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

Quote:
Originally Posted by ZKesic View Post
Rake is the reason why the pusher has higher EV here than the caller.


Only at rake levels above equilibrium. That is the point.
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Old 01-24-2019, 10:28 AM   #85
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Re: Can Someone Do This Calculation I Put In My Book

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
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Old 01-24-2019, 11:10 AM   #86
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Re: Can Someone Do This Calculation I Put In My Book

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Originally Posted by robert_utk View Post
Maybe read some more wikipedia?

¯\_(ツ)_/¯
Quote:
Originally Posted by robert_utk View Post
Really, man. How much longer are you going to keep going?
Please don't resort to addressing other posters like this. Say your piece and let it stand for itself.

Last edited by RustyBrooks; 01-24-2019 at 11:41 AM.
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Old 01-24-2019, 11:21 AM   #87
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Can Someone Do This Calculation I Put In My Book

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Originally Posted by RustyBrooks View Post
Please don't resort to addressing other posters like this. Say your piece and let it stand for itself.


Understood. I apologize to plexiq.

Last edited by robert_utk; 01-24-2019 at 11:22 AM. Reason: Corrected spelling.
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Old 01-24-2019, 12:04 PM   #88
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Re: Can Someone Do This Calculation I Put In My Book

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.
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Old 01-25-2019, 01:46 AM   #89
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Re: Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by plexiq View Post
[This just slightly expands on MonkeyT1lt's calcs to find the exact cut-off.]

The cut-off is when K9s becomes a profitable call around 20.6% rake, increasing the call range from (14.2%, 55+ A7s+ A9o+ KTs+ KJo+ QJs) to (14.5%, 55+ A7s+ A9o+ K9s+ KJo+ QJs), at which point pushing becomes unprofitable for 72o, 32o, 42o(?).

K9s is approx. 59.9885% equity against ATC.

So the lowest rake where pushing ATC is still optimal is at slightly above the break-even point for K9s:
x = 1 - 100$ / (210$ * 0.599885) = 20.6197%

(Of course, the ATC pusher is +EV for much longer, but at lower rake the pusher's EV can be increased by slightly tightening the range.)
Yep yep. 20.6197% is the answer then.

Quote:
Originally Posted by ZKesic View Post
Yes, that's how we were calculating it. 27o becomes break even when its EV is 0 not 10.

If we average the BE equilibriums we get ~20.555%, which we could say is the final solution to this problem, right?

We can end the thread with that imo.
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%.
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Old 01-26-2019, 11:46 AM   #90
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Re: Can Someone Do This Calculation I Put In My Book

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Originally Posted by David Sklansky View Post
It shouldn't be hard if you program commuters. I will acknowledge your help in the book. (Probably titled The Theory of Poker Applied To No Limit).

There is ten dollars in the pot. Me and you get a two card starting holdem hand.

I bet 100 all in or fold. You call or fold.

Th house rakes a giant amount from the 210 dollar pot if you call. Nothing if you don't. If it the rake is high enough you can't call with two aces. If it is somewhat lower, say fifty bucks, and I bet in the dark every hand, there are a few hands you would call with, namely any hand that is more than a 100-60 favorite over a random hand. But even though you would call with these hands I should still bet every time because I will steal the ten so often and sometimes win when I don't.

At what rake should I stop pushing everything in the dark? (I say "in the dark" because your I want your strategy to be based on my hand being random.)

(I want to know because it analogizes to certain tournament bubble situations)



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?
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Old 01-26-2019, 08:50 PM   #91
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Re: Can Someone Do This Calculation I Put In My Book

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.
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Old 01-26-2019, 11:30 PM   #92
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Re: Can Someone Do This Calculation I Put In My Book

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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).
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Old 01-27-2019, 04:01 AM   #93
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Re: Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by robert_utk View Post
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
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.
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Old 01-27-2019, 04:37 AM   #94
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Re: Can Someone Do This Calculation I Put In My Book

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Originally Posted by browni3141 View Post
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.
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:
Quote:
Originally Posted by robert_utk View Post
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.
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."
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Old 01-27-2019, 09:32 AM   #95
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Re: Can Someone Do This Calculation I Put In My Book

Hi ZKesic,

Thank you for your comment.

Quote:
Originally Posted by ZKesic View Post
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.

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.

Quote:
Originally Posted by ZKesic View Post
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.
See above.
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Old 01-27-2019, 09:52 AM   #96
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Re: Can Someone Do This Calculation I Put In My Book

Quote:
Originally Posted by plexiq View Post
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.
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.
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Old 01-27-2019, 10:03 AM   #97
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Re: Can Someone Do This Calculation I Put In My Book

When the caller uses a 5.7% calling range, is the pusher's EV with ATC higher or lower than in your proposed equilibrium?
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Old 01-27-2019, 10:20 AM   #98
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Re: Can Someone Do This Calculation I Put In My Book

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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.
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Old 01-27-2019, 10:24 AM   #99
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Re: Can Someone Do This Calculation I Put In My Book

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.
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Old 01-27-2019, 10:51 AM   #100
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Re: Can Someone Do This Calculation I Put In My Book

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Originally Posted by plexiq View Post
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.
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