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
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
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'