Holonomy

The Central Limit Theorem doesn't say that for any number of samples the sample mean is normally distributed. It says as the number of samples tend to infinity the distribution of the sample mean tends to the normal distribution (technically assuming finite variance). How quickly this happens (and therefore how quickly it become a good approximation is data dependent). For 2k hands we only have 20 100 hand samples and these samples have bigger tails than a normal distribution so I am unsure how good an approximation it would be at that stage. I don't think it invalidates the analysis - it's just that the real confidence intervals are even wider.

This becomes more of an issue when people look at 2k hand samples on PokerDope and conclude they have run 4 sigma below EV which should happen extremely rarely with a normal distribution - but is not that rare when dealing with heavier tails. (Not to mention the obvious selection bias!)

Yeah that is how you convert the standard deviation of groups of n hands to a single sample - divide by root(N). Be aware that smaller groupings of hands can have higher kurtosis (bigger tails) which mean you need to be careful when using confidence intervals on small numbers of hands. (Basically it ends up not really being valid as you are assuming a normal distribution). Even 100 hand samples are not normally distributed, so if you have 2k hands you only have 20 samples of something. So this means that the actual confidence intervals are even wider than the use of standard deviation tells you.

Not sure what you mean by this. A Nash equilibrium always means the same thing. No player may unilaterally improve his equity (key being unilaterally). Or do you mean the fact that these algorithms do not converge to a Nash equilibrium (they converge to a coarse correlated equilibrium although these are still strong strategies).

That said what makes you sure that your deviations are better than the solve? Empirical data? I guess that get tied in with an effective population exploit as well.

My expectation is that you won't be able to have a really big impact on the variance, 10% at most, I would be surprised if it was even this high. I don't think variance is something you should be optimising for.

Yeah as pointed out by Trinity QQ-JJ are the same hand in your example. Is it possible there would be no exploit? Maybe, but I suspect unlikely. You certainly can't guarantee that there won't be one. Not sure if it was precluded by your question but if you had one of your value hands betting 50% and one of your bluffs betting 50% and you symmetrically deviated here you would obviously be exploitable as your B:V ratio is off.

in the HU there was a TD hand where Jungle 3bet OOP w/T6543 I think and Yuri broke a 9. I would be interested in hearing that one reviewed. Nice vid btw!

Definitely keen to see the HU as well!

So this printing comment on 1/3 is not quite true. If you look at PIO you will find the BB overfolds to various sizings of cbets even in an optimal strat. You have to compare the EV generated from the overfold to the EV generated from the X back line. So if you had a 0% equity hand and BB overfolds then you should probably bet this hand - but the truth is that hardly any hands truly have 0 equity and they may have even more EV in a delayed cbet line.

I think the rule as stated is worth remembering but like all things the truth is more complicated.

Yeah the algorithms are pretty well known and the papers can be found on the arxiv generally. The trick is in implementing them correctly and making them run fast. PIO uses a proprietary algorithm I believe but it will still converge to the same answers. To be clear neural net machine learningy type solutions generally do not have guarantees about converging to the nash (or coarse correlated in the case of multiway) equilibrium. so the solvers as we know them are brute force decision trees (often with abstractions).

All "reasons" for things are stuff that we (humans) come up with for why the mathematical solution is what it is. A lot of people reason backwards - eg why doesn't the solver care about board coverage here etc. The point is it never cared - it just solved the mathematical problem, your heuristic just doesn't work here. If you want my "reasons" as to why 54 is preferred to 53s or 64s:

1) 54s makes better pairs than 53s
2) it makes more straights than 64s
3) if you started to have 64s and 54s in the same tight range you might have too many 4s in some sense and are starting to dilute the frequency of big cards too much allowing an opponent to exploit you. Also would mean you woudl be les lilely to cash in on 44x boards as the opponent knows you have a stronger range.

I would be folding this hand pre. KQs fine to 3bet.

Actually doesn't preflop rake reduce the incentive to take the flop down pre vs a game that doesn't rake preflop? It tightens you up though so hard to make a direct comparison.

To be clear I haven't run the sims. (I presume they don't rake the raise if not called).

Ok great, well best to him and thank you for the information.

Same question again 7 months later :)

Monker let’s you input various sizes and see which one it chooses. Bigger oop is a thing. Pot sized raises (100bb) are too small. You basically allow the ip to take a cheap flop and then correctly over fold once they have seen the flop.

I meant easiness of the games live!

You could probably get a small safety boost from shortstacking (as it will drop variance a bit) until you hit 4000bb. But I would also say that a chunk of the EV live comes from big mistakes in large pots (inelasticity of calling ranges etc)

I am not totally convinced by the if you have a job don’t worry point of view. I think it is useful to treat that part of your life as a professional as well and is what lets you sustainably grow that part of your wealth. It also ensures you have the right mentality when that becomes (hopefully) a significant piece of your personal worth.

As for the lower buyin being the amount you bought in for - it is more
Complicated than that. It’s really the variance that you care about and that doesn’t drop as quickly as the buyin you are using. I think it is better to think in terms of if you have enough bb in your bankroll. 4000-10000 is solid. Maybe you can go a bit lower accounting for the easiness of the games online and the fact you can rebuild if necessary. 2000 would be an absolute floor for me.

If people are not raising enough it's not obvious that the mix hand become folds - you would need Monker to figure this out. A bit like how on the flop if opponents don't raise enough we can small bet more.

But be a bit careful with frequency locking as it can do some clever stuff with its range allocation. You might need to lock it manually.

If they were calling too wide and raising correctly then I think the mix hand become folds.

+1

Quite the win rate - poker not dead... Even a lower confidence bound looks pretty good.

26:51 Thought it was interesting that KK only calls as OOP and AJ bluff raises! You really need to block those quads...

You can’t use PIO in this situation as it’s multi-way which is a very different situation. KQ probably mostly check.

Also I am Not sure you have included the chance that the opponent is just bluffing and your 7 is already good?

I think your framing is the wrong way around. It is not that you need to extend the concept of pot odds. The EV at the nash equilibrium just is calculated by playing all possible hands according to the frequencies. That is the correct answer for the EV. Odd and implied odds are one tool for approximating what the correct strategy might be.

It’s not really logic in a classic mathematical sense. It is basic high school algebra - mostly solving simultaneous equations (but applied tersely and with the application foremost in the mind).

I would move to journals section you might get more following.

Hi Luke, great I will check those out.

Yeah really like this format. Compressed takeways / highlights /heuristic are very valuable imho. I think the possible way to make it more entertaining is the way imba did it - with the review being of recorded play and then the video paused with deep thought given where necessary. It gives natural breaks in the teaching process but the density of info is probably lower. (I realise that is a huge amount of work though).

If the match could be mix too that would be amazing! But definite +1 for any format of this!

It's an OOP min bet so it's not reopening the action? I see your broader point though. It is possible that by altering the size the opponent can effectively bet from 33 and 75 to 45 only it is having that effect for a very small cost.