This quote is a gem:
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And by the way, on page 27 of No-Limit Hold 'em for Advanced Players by Matt Janda under the heading "A Note About the Bet Sizes" it says:
As a partial answer, when you look at statistical distributions their minimums and maximums tend to be broad. For example, instead of the maximum of a statistical distribution coming to a sharp peak where the ascent to the maximum is steep and the descent from the maximum is also steep, the (graphed) distribution will usually look more like a bell curve with a rounded top and moderate slopes on each side near the maximum point.
The set of functions that qualify as probability distributions/densities is really, really, really big and it's just silly to say most of them (and we're talking about continuous distributions right? Pretty sloppy to leave off that detail) when graphed look the way he's describing. He's ignoring skew and kurtosis.
I know he says "tend" and he's not intending to make a strong mathematical statement, but just for a really simple counterexample, what about the Uniform[0,1] distribution? How about Poisson(1)?
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Using solvers to test pre-flop bet sizing, I found that for similar bet-sizes, the differences in EV are quite small and even indifferent in many situations. For example, going from 2.25bb to a 2.3bb open size won't have a significant EV impact in your bottom line.
This is such BS because for one, yeah of course a difference of a mere 2% is not gonna have an appreciable difference on EV. What about 2x vs 3x? That's a much more valuable test.
And how the hell did he "solve" this problem in multiway spots? I wish he'd share this multiway GTO solver dream machine with me because I'm stuck with a HU only one (of course he doesn't actually have that software).
No mention of ranges? Antes or not? Stack sizes? Cash or MTT?
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I only have equilibrium ranges for 6max but that should be enough to illustrate the point. Can someone run this math? I am unsure of how to calculate independent events.
LJ opens--->HJ 3bets 8.1% of the time
LJ opens--->CO 3bets 8.5% of the time
LJ opens--->BTN 3bets 7.5% of the time (it is less because BTN calls more)
LJ opens--->SB 3bets 7.6% of the time
LJ opens--->BB 3bets 6.0% of the time
LJ doesn't get 3bet 91.9% of the time
LJ doesn't get 3bet 91.5% of the time
LJ doesn't get 3bet 92.5% of the time
LJ doesn't get 3bet 92.4% of the time
LJ doesn't get 3bet 94.0% of the time
BTN opens--->SB 3bets 14.8% of the time
BTN opens--->BB 3bets 13.9% of the time
14.35% on average we get 3bet
Odds we do not get 3bet by SB-->85.2%
Odds we do not get 3bet by BB-->86.1%
85.65% on average we do not get 3bet
Not equilibrium, if you purchased these ranges under the impression they are true equilibrium ranges then you were misled.
The multiway solvers take all sorts of liberties to simplify the game tree (because even a 3-player preflop tree, with a reasonable postflop strategy profile, and a reasonable number of flop subsets is gonna be MASSIVE), and I do believe the creators of the multiway solvers even go as far as explicity saying they make no guarantee that their solutions are converging properly.
Not at all saying these ranges you have are bad, but caveat emptor.
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After playing 100,000 hands, players start to get an idea of what their actual win rate looks like, but a sample of at least 1,000,000 hands is required to be statistically significant”
This is definitely just pulled out of thin air.
Seems pretty silly to think anyone can ever know their true winrate.
In fact I'm not sure it's even statistically valid to think of winrate in any other terms except "greater than some hypothesis" or "less than some hypothesis" (i.e. you'd need a hypothesis test to determine that) but that's pretty tricky too because there is no way the underlying distribution of wins/losses is normal, it's probably not even close to normal (I believe the Kolmogorov-Smirnov Test does that), and I'm pretty sure you'd need some exotic hypothesis test. Can someone correct me if I'm wrong?
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The required samples size depends on the player’s standard deviation, the confidence level and the acceptable deviation. Relying on the Central Limit Theorem for normality of a sample mean you can show that a sample size of about 1.2 million is required to have 90% confidence of the mean being within 1bb of the true value if the SD is 85bb/100. However, it is less than half of that at 525,000 if the player accepts a deviation of 1.5bb from the true value.
At 80% confidence, the required sample sizes are about half of those stated above. The confidence level to use along with the deviation specified will naturally depend on how the result will be used. I agree with the concern about the likely change in the playing environment.
Do you know how to derive required sample size and if so would you be willing to quickly do that for me? I totally forgot how to do that!