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.

The rake has to be under 17,48%. At that point the caller starts calling A5o (if he plays perfectly), which makes the blind pushing strategy -EV (it would be more profitable for the pusher to just fold any two cards).


EV for blind pushing when the rake is 17,48%:


EV for blind pushing when the rake is 17,47%:


EV for calling A5o when the rake is 17,48%:


Calling range when the rake is 17,48%:

The formula for calculating the pusher's EV at 17,48% rake (to double check):

1050/1326*10+276/1326*(.3611*(.8252*210-100))-276/1326*(.6389*100)= 0.13

It checks out.

That's not correct if the pusher plays fixed 100% and the caller plays best response. Neither the pusher nor the caller are playing a mixed strategy in that scenario, so we don't get that 0EV range you describe.

In our scenario the bottom of the pusher's range becomes -EV at 20.619% as calculated above. Any slight deviation of the rake from that exact point will make K9s either +EV or -EV (and therefore the bottom of the pushers range -EV / +EV).

It doesn't make sense for the limit rake% here to be bigger than the one 27o had (~ 20-21%).

Also, your calling range doesn't make much sense again, since some hands with 57% equity are calling and some with 60%+ are folding.

You're right, the 27o pusher is shoving a fixed 100% range.
MonkeyT1lt made that comment about the 20.61%-20.50% range and I didn't really question it too much. This range of the same EV for the caller can't exist in this case though, so 20.619% rake is the only point at which K9s is a 0 EV call and 27o is a 0 EV shove.

Some kickers block more straights against a range of any two cards, than vs a range of 72o.

The exact combinatorics of a few combos is debatable, that was the beginning of my answer.

More importantly, is why does your answer vary so far from mine, since mine has the EV calculation included, which is different than yours.

I just got confused about the way rake affects each player's EV. My first post was wrong but I can't change it now.

David Sklansky - I sent you a PM

You have a 17.9% calling range at ~26% rake and your range includes K9s. I posted the math for K9s above and it needs rake below 20.619% to be a profitable call.

It looks like you mistakenly constraint the range of the caller to be >0EV overall, but that means there will be plenty -EV calls in that range. Every single hand needs to be individually >=0EV to call.

There is a reason that Q8o is nicknamed 'The Computer Hand'.

If my answer is wrong, then simply post a correct EV for both players at equilibrium, that uses pure strategy and is more than 4 combos different than mine.

I posted the calculation for K9s and it directly follows that all hands need >=59.9885% equity at 20.619% rake to be profitable calls.

Your posted range includes plenty of hands with less equity at higher rake, which is clearly wrong. Additionally, by your own calculation, the caller's EV is .189 cent for the game. This obviously can't be right if every single hand in the calling range were +EV on its own.

The caller is defending his tourney equity. At what +EV would you suggest he do so?

Why would a caller have hands in his range that lose him money on the call? Every individual hand needs to be at least zero EV to be a call, and that's not the case for the range you posted.

I showed how your K9s call is losing money in your range. Show me a single non-optimal hand in the calling ranges I posted and I'm happy to continue from there. That's either a +EV hand that's not included in the range, or a -EV hand that is included. If you can't then they are optimal as posted.

I think I will stand pat.

Not getting paid to give homework support here. Have a nice evening

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.

Now, normally this is where one might say 'DUCY?'.

However, given who the OP is and the importance the he see it also, that would be inappropriate.


-Rob

This is not a Nash Equilibrium, not even remotely close.

I'm happy to do an exile bet with you:
If I'm mistaken and this is indeed a Nash Equilibrium for the game at 26.21% rake then I'll refrain from posting in this subforum for 1 year. If it is not a NE then you will refrain from posting in this subforum for 1 year. Quote within 24h to book.

Why are you all still arguing about the Nash Equilibrium of this spot? I posted it already, it's 17,48%.

If you think I'm wrong, find one mistake in my calcs.

I already pointed out the mistake in his calculations. Hint: I don't think he is posting in good faith. Hence the exile bet proposal.

Too risky to loose you for a year. Quite glad to see you posting in this sub forum again.

Within what percentage? My answer with pure strategy is within 4 combos of NE at 26.21 percent rake.

The first question is asking: At what point does it become -EV for the pusher to shove the bottom of his range.

The second question is asking: At what point does it become -EV for the pusher to blindly shove any two cards.

I am currently writing a refutation of your answer. Which one again is supposed correct? Ah, no matter. The refutation will cover them both.

Please confirm that we are booked for the exile bet under your proposed conditions: At rake 26.21% your claimed calling range of 66+, A5s+, K9s+, QTs+, JTs, A9o+, KTo+, QTo+ is not within 4 combos of a NE. Pure strategies only.

I decline. The forum is for learning, not excommunication. Hang around more often so that you can learn.