# Mitchell Reid

625 points

Yes, very often. In fact in HU spots, one player will always realize more than 100% equity. It depends on a variety of factors, the main ones being

-What each players range is

-Who has position

-The stack to pot ratio

There's more, but they're the main ones

In terms of your example it highly depends on the board obviously. Or was your question regarding on average? If so, it's around 112%-115% depending how you set up your sim

### March 27, 2018 | 12:11 a.m.

I'm really sorry about this. I think I mentioned this in my email, but I had some unfortunate personal circumstances to attend to during this time, hence the late reply. Normally I strive to reply ASAP (average 8.2 hours reply time going by times on emails), but this was a really bad week for me

### March 22, 2018 | 5:50 p.m.

Putting multiple sizes in the same sim isn't the best way to find optimal size. You should do multiple separate sims each with one size and see which tree returns the highest WR. It depends which pack you buy on whether you get this or not

### March 22, 2018 | 5:46 p.m.

Sorry for the late reply, don't come on here much. At the time of the OP, PioCLOUD wasn't free and costed 2-6 times more than ours did. So we did this to give people a more fair price

As of the present time, PioCLOUD solutions are a good option for people with low budgets, but they are really limiting. PioSOLVER can only solve HU spots so it's only BB vs RFI and relies on accurate assumptions of preflop ranges. That's why our new packs are multi-way solves, so it's not just RFI vs BB and has all spots in the preflop game tree which is far more applicable and more accurate

### March 22, 2018 | 5:45 p.m.

What position is the virtual player? What did he have to do vs non all ins?

### March 4, 2018 | 7:42 a.m.

For someone at your level I'd recommend spending a lot of time on these forums (posting and reading other peoples posts). Imo they're the best forums when it comes to quality advice

### March 4, 2018 | 7:31 a.m.

Hi Ben. I agree with you exploitative conclusions with the KK hand that you analyzed with Pio in the end. In general when you're studying and you conclude the best action is different to what the solver suggests (like you did in this hand). How do you tell the difference between the times your reasoning and assumptions are sound, and the times you're reaching so that you don't have to admit to yourself you made a bad play?

### March 4, 2018 | 6:26 a.m.

Their solutions are actually free now, but they specifically asked for people not to share links in forums as they don't want too many downloads happening at once. They shouldn't be too hard to find via google though

The limit to their preflop solutions for 6-max is PioSOLVER can only solve BBvRFI. If you're after full 6-max solutions then these are the 3 sites i know of: site 1, site 2, site 3

### March 4, 2018 | 6:07 a.m.

I'm not sure I know of the solution you're talking about. Some more details will help with getting an answer

### March 3, 2018 | 8:10 p.m.

Also, the squeeze % is very important. If the fish is passive and doesn't squeeze much then calling is much higher EV, even if they play well postflop. That means we can get away with looser calls preflop, as Kevin said

### March 2, 2018 | 10:16 p.m.

Some great points here. I think one thing that's under-considered when people normally talk about this spot is whether the BB has a CC range vs the 3-bet, and how loose that CC range is. We should play very different if that is or isn't the case.

If they do have a CC range then 3-betting is slightly more attractive as we get to play vs a fish who has a capped, face up range in a large pot. Then postflop they're unlikely to play near optimally given the range they called with preflop, increasing our edge winrate in the game

### March 2, 2018 | 10:13 p.m.

I'm not saying to solve the spot. I'm saying put multiple hands in at once to see the EV of all of them. If you put the fishes entire strategy in, then you can find all hands such that taking action A is best, and all hands taking action B is best

### March 2, 2018 | 7:21 a.m.

There's different approaches depending on what you're trying to learn. In theory we should be seeking to maximize our EV with each individual combo. So if you're just after an answer to whether you played the hand well or not, just inputting the one hand is enough.

If it's a line that happens often, you're probably best to input your whole range. Reason is you'll learn much more than if you only put in the one hand. If you can see the best way to play multiple hands rather than just the one, and try to conclude the reasoning for why such actions are best for each hand, then you'll have a much better understanding of the spot moving forward, and make much better decisions in game

I think a study mistake that people make far too often is just checking to see if they played a hand right or not, rather than using the tools to develop their understanding of the spot, and seeking how to best play the spot in sessions moving forward

If it's an uncommon spot though, then all the extra work is unnecessary and a waste of time

### March 2, 2018 | 5:47 a.m.

As I explained above. Standard deviation over multiple IID random variables doesn't add like variance does

So if your SD over n hands is S. Then your SD over 1 hand is S/sqrt(n)

Plugging in $50 for S and 100 for n we get 50/sqrt(100)=50/10=5

### March 2, 2018 | 12:15 a.m.

What's the difference between a std of 100, 80, 50? Thanks!

Your standard deviation depends on the game you play and the strategy you play. You can see what this is by taking a look at your database

### March 1, 2018 | 11:10 p.m.

In this example, shouldn't the SD in $/hand at 50NL be 4?

So in this example our winrate is 8bb/100 and our SD is 100bb/100. That makes our SD $50 per 100 hands, or $5 per hand.

Our winrate is 8bb/100 or $4 per 100 hands. Which is $0.04 per hand

### March 1, 2018 | 11:08 p.m.

Sure thing. Here's the link to the spready

### Feb. 28, 2018 | 7:51 p.m.

Standard deviation doesn't add for random variables, however variance does. Therefor we need to square-root to get back into SD

var(H1 + H2 + ... + HN) = N x var(H) where all Hn are IID random variables

sd(H1 + H2 + ... + HN) = sqrt(N) x sd(H)

Apologies for the notation. You can't subscript in these forums. Also stars turn things to italic so I can't use the multiplication symbol

### Feb. 28, 2018 | 7:50 p.m.

I should probably do a couple examples to help a bit more

Let's say you are playing 25nl and want to know when you can move up to 50nl. Let's say your winrate is 10bb/100 and 8bb/100 respectively. Standard deviation of 100bb/100 is pretty standard. So at 25nl your $/hand is 0.025 and your SD in $/hand is 2.5. At 50nl your $/hand is 0.04 and your SD in $/hand is 5.

Therefor c=(5^2-2.5^2)/(2(0.04-0.025))=$625. So play 50nl when your BR is above $625 and play 25nl when your BR is below $625

Here's a picture of a spreadsheet to see how that can change throughout stakes

As you can see, optimal BRM is much more aggressive than people think. Also, many people say a certain number (like 50 buy ins) is optimal, but it should change heavily depending on your winrate and the standard deviation of each game

### Feb. 28, 2018 | 7:19 p.m.

One issue you'll find with the 5% ROR is people will quit much sooner than going broke. So if you need 30 buy ins for a 5% ROR then in practice people will fail closer to 30-50% of the time depending on a variety of things. Meaning a much larger initial bankroll is required for them to succeed

### Feb. 28, 2018 | 7:05 p.m.

This can actually be solved for if you know exact winrates and variance's in each stake. We rarely do but that doesn't mean we can't get very good guidelines

The first thing to think about is what are we trying to optimize? Your first thought might be we are trying to maximize EV or minimize variance. However after a bit of thought you'll see this will get you no where. If we maximize EV that means we should play the highest stakes we can afford (assuming we are beating those stakes for a higher $/hand than all lower stakes). This clearly isn't the answer. If we instead try to minimize variance we'll see that to do that we should try to play the lowest stakes possible, which clearly isn't correct either

So instead we need to optimize some other formula that strikes some balance between these two. I spoke about this a bit in another thread, but that formula is what we call the utility function and that has been solved to be *U(x)=ln(x)*, where U is the utility function and ln is the natural logarithm

Then, when making a bet we then desire to maximize this function instead of anything else.

**How does this apply to bankroll management?**

Well using some basic undergrad maths and some approximations (I don't think including the derivation will be helpful), we can arrive at the formula *c=(var(X)-var(Y))/(2(E[X]-E[Y]))*

*c* is the cutoff bankroll (move up when your bankroll is above this, move down when it's below)

*var(X)* is the variance (standard deviation squared) of the bigger game, *var(Y)* is the variance of the smaller game

*E[X]* is your winrate in the bigger game, *E[Y]* is your winrate in the smaller game

There's some important things to keep in mind when using this. This assumes you're willing to regularly move up and down stakes. If you have some ego problems with moving down then a) you're an idiot, and b) use a more conservative bankroll management system. The other thing to consider is this is trying to maximize you bankroll growth in the long run, it doesn't take into consideration living costs or anything that's regularly getting pulled out of the bankroll.

If you're interested in more about this I would recommend checking out The Mathematics of Poker by Bill Chen and Jerrod Ankenman

Here's a picture of the formula which might be easier to read

### Feb. 28, 2018 | 6:24 p.m.

How you'd measure it is another problem

### Feb. 4, 2018 | 6:29 a.m.

Irrelevant from chipEV perspective

### Feb. 1, 2018 | 2:15 a.m.

I apologize then, I honestly didn't mean it that way

### Jan. 31, 2018 | 2:23 p.m.

I think you've misinterpreted my comment. I'm simply suggesting to play more tables

### Jan. 29, 2018 | 4:44 p.m.

No problem, happy to help

I should probably add that this stuff is much more useful for bankroll math and deciding what stake to play, than deciding how much running it twice is worth

### Jan. 29, 2018 | 9:52 a.m.

Yes that's correct. Basic version is enough if it's updated. As stated above it's not required though

### Jan. 29, 2018 | 8:28 a.m.

Hi mate. You in fact don't need PioSOLVER to view the ranges, hence why we don't specify that on the site. We just put them in the PioSOLVER saved tree format for people who prefer that. For people don't have a paid version of PioSOLVER we offer them in two alternate formats (both which use PioVIEWER), which only needs the free version

We recently included an FAQ in the packs that explains how to use the files.

Would you be able to email [email protected], or message me your email address. It's possible you received none of the above if you were given an old link

### Jan. 29, 2018 | 8:26 a.m.

Are you aware you can play more than 1 table? Just play enough reg tables such that your hands/hr is the same as zoom

This process will not work. Not all hands realize the same amount of equity. The amount each hand realizes depends on the range as well, and the equilibrium solver is changing the range constantly at every iteration. You need to buy a full preflop solver to solve this.

Thankfully, HUSNG spots are available for free, but the distributor has specifically asked for people not to post links on forums. So inbox me and I'll link them for you

## March 27, 2018 | 6:18 a.m.