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# BRM - Strange result!

• Bronze
Joined: 16.08.2007
Hi all! I'm not very active on these forums, but here's something I'd like to share with you. It might as well be interesting, or lead me to better understand BRM.

I just wrote a program in python which simulates BRM for SnGHU, DoN or such simple stuff. My aim was to simulate the effects of BRM on my winning rate, and take into account the risk of ruin. My idea was that someone might be losing a lot of money in the longrun compared to optimal BRM if he really is to far from it (either to large or too tight) so went for a few calculations.

So basically, I took the SnG HU buy-ins of pokerstars (up to 200\$), added in the rake in my country (6.6% ), and decided that our bot would have 54% itm on every level. So, simple thing so far.

As for the brm, I simulated simple BRM at first:
when chosing to play a game, the script chooses the higher BI for which BI*BRM is inferior to our actual bankroll. Everytime I broke, I respawn to starting BR and remember that I lost the money (removing (prob to broke)*(restarting bankroll) from the end bankroll.
I also make sure that the results are accurate. I chose to run something like 10 000 games.
In order to get a very good accurate value, the results I get are the average results from 5000 runs (each run being 10000 games). The result is that I have very little dispersion.

My results are somewhat... unexpected.
With 80\$ starting BR, and a 5BI BRM, I won 2446\$ (over 10000 games, in average)
With 80\$ starting BR, and a 50BI BRM, I won 2452\$.

With 5\$ starting BR and 5BI BRM, I won 241\$
With 5\$ starting BR and 50BI BRM I won 242\$

I run the same number of games for every simulation. And apparently, it seems that the BRM does not matter on how much I win in the longrun. I do'nt understand this result.
What I expected was a stupid risk of ruin with 5BI BRM but (since the bot has an edge) much more earnings.

Just in case it comes from a script mistake:
 code: ```#!/usr/bin/python #python 2.6x import random br = float(raw_input("Starting bankroll?")) respawn = br brm = int(raw_input("Number of BI for BRM?")) bi = [[200., 45],[100., 54],[50., 54],[30., 54],[20., 54],[10., 54],[5., 54],[3., 54],[1., 54], [0.5, 54]] rake = 0.066 #a function simulating a poker SnG HU game def game(br, bi, rake): if random.randint(1, 100) >= bi[1]: br = br - bi[0] else: br = br + (1+rake)*bi[0] return br #a function to choose which BI to play #simple BRM for the moment def brmsim(brm, startbr, bi, rake): for elem in bi: if br >= brm*elem[0]: return game(startbr, elem, rake) return game(startbr, bi[9], rake) #SIMULATION FUNCTION def simu(brm, br, bi, rake, games, respawn): maxbr = 0 broke = 0 for j in range(games): br = brmsim(15, br, bi, rake) if br >= maxbr: maxbr = br if br < 0.5: broke = broke + 1 br = respawn return broke, br #MAIN print "Starting bankroll:", br print "Chosen BRM:", brm pbroke = 0 avgbr = 0 for i in range(5000): loss, brsim = simu(brm, br, bi, rake, 5000, respawn) pbroke = pbroke + loss avgbr = avgbr + brsim avgbr = avgbr / 5000 pbroke = pbroke / 5000. print "You broke", pbroke, "on average, and won \$", avgbr - respawn*pbroke print "Ratio: Won/RoR =", (avgbr-br)/pbroke```

What this might suggest is really counterintuitive: when we have an edge, the BRM we apply doesn't matter in the long run.
Is the conclusion correct? Incorrect? Would anyone have anything that would help me about this?
• 19 replies
• Super Moderator
Super Moderator
Joined: 02.09.2010
I think that what you are seeing is a good illustration of the fact that BRM is not a money consideration, but rather it is a psychological consideration.

If you have a larger bankroll, loosing a BI is no big deal.
If you have a small bankroll, it can be.

If you are playing with "scared money" it will inevitably affect your play. Your bankroll needs to be big enough at any level so that you simply don't care about each individual loss. It needs to be big enough that you can always focus on the the big picture.

Can you put a factor into your program that adjusts the winrate based on the size of the bankroll?

This will still be somewhat artificial, but let's say that when your hypothetical player has a 50 BI roll, then his winrate is 55% ITM, and drop the winrate by 1% for every 10BI, so at only 10 BI, he is ITM 51% only.

At some point, the downswing becomes sharper.
• Bronze
Joined: 08.09.2009
This is why jumping to bigger stakes (if you can handle it) is always better
• Bronze
Joined: 10.10.2010
I don't like a couple of assumptions you made to get to that conclusion.

1. Your BR is essentially infinite, since you can reload it every time you go broke. Thus making BRM rules of 5 and 50BI irrelevant.
2. As VorpalF2F mentioned, your winrate should adjust as you move up.

So in the end you're not factoring in "scared money" and the fact that if you move up really, really quickly - you will soon play at a level where you don't have an edge and you'll go broke.

I think that good BRM saves you from reloading and also gives you enough time to learn and solidify your skills as you move up. It's comforting to know that when you have a bad day you can keep playing the game, because you have enough BI's to do that. That's good for the mental aspects of the game.
• Bronze
Joined: 14.02.2008
Originally posted by conquistadorrr
I don't like a couple of assumptions you made to get to that conclusion.

1. Your BR is essentially infinite, since you can reload it every time you go broke. Thus making BRM rules of 5 and 50BI irrelevant.
2. As VorpalF2F mentioned, your winrate should adjust as you move up.

So in the end you're not factoring in "scared money" and the fact that if you move up really, really quickly - you will soon play at a level where you don't have an edge and you'll go broke.

I think that good BRM saves you from reloading and also gives you enough time to learn and solidify your skills as you move up. It's comforting to know that when you have a bad day you can keep playing the game, because you have enough BI's to do that. That's good for the mental aspects of the game.
+1

also o.p. doesnt seem to get the huge difference between doing math with a paper and a pencil ( or pc for that matter) and actually playing that.
• Bronze
Joined: 10.06.2012
I think it is a valuable study and tool you are working on.
There are a few errors or oversights in the thinking, the code and others comments. So yeah, it is incorrect, but with a little work it could be a very valuable study and tool.

The main error here must be an error in the code on choosing which BI to play. Not increasing leverage (moving up in stakes) as your bankroll progresses. If this were the case a more aggressive BRM would perform better.
So you must code a way to increase stakes as bankroll increases (and decrease as it decreases).

But...
Not considering risk of ruin is a large oversight.

If you go broke a lot more of the time with the more aggressive buyin method, it will behind. 'You have to bet to win, you can't bet if you have no chips'.

A separate, but complimentary and extremely useful tool would be to calculate your risk of ruin with different BRM. Then add a fluctuating edge (could test for various standard deviations away).

Then one could plug in their edge and what confidence interval (with their risk of ruin) they would be happy with and come up with an 'ideal' BRM to achieve their goals.

Also, there are many other pitfalls that only experience and commonsense can bring that could be added into the coding. For example, as you move up in stakes, your edge will decrease or go negative.
• Bronze
Joined: 16.08.2007
Vorpal:
Yep, it's easy to adjust the itm% in relation to the number of BI we have at a level (or the number of games played at that level as well). However I was not interested in the psychological side, because it would be just too hard to simulate (your play is also affected when you lost 10 in a row, when you won 10 in a row, when you have to go peeing but can't...etc...). I just wanted to correlate the "%growth of bankroll per game" with the BRM (which "should" be correlated, at least that's what I though).

Originally posted by conquistadorrr
I don't like a couple of assumptions you made to get to that conclusion.

1. Your BR is essentially infinite, since you can reload it every time you go broke. Thus making BRM rules of 5 and 50BI irrelevant.
2. As VorpalF2F mentioned, your winrate should adjust as you move up.

So in the end you're not factoring in "scared money" and the fact that if you move up really, really quickly - you will soon play at a level where you don't have an edge and you'll go broke.

I think that good BRM saves you from reloading and also gives you enough time to learn and solidify your skills as you move up. It's comforting to know that when you have a bad day you can keep playing the game, because you have enough BI's to do that. That's good for the mental aspects of the game.
1- In fact I deducted the reloading from the final earnings. Because I don't care going broke more times with an aggressive BRM if it is better in terms or middle-term earnings (taking into accounts the payback of the reload).
2- I was not interested in simulating my (nor any human being) bankroll growth over the years, I think it would be ridiculous to write a program to see "what my earnings would be if I had an edge on everyone". I'm interested in understanding how a BRM can be effective, and how a "bad" or "average" BRM would slow down the results to a significant extent.

I think that good BRM saves you from reloading
I'm not really sure about this. Even if that's what I had in mind for a few years, that's the reason of my "calculations".
Think about playing an MTT. At some point, you are 10bb deep. You have to go and take some 55/45. Either you lose, and reload in another MTT were you are more likely to win money. Or you win, and are more likely to win money. We all know that in MTT this is basically the way to go: trying to get as much longterm value even if the shorterm is goind bust.
Why not having the bankroll being the MTT, and the stake at which we play being the final table? Maybe there are configurations were taking the risk of ruin has a higher EV (because basically if you broke with an aggressive BRM it might cost you 15\$ to get back on track in a few months, but at the same time increases a lot your probability to win more money in the same few months). I'm not saying this idea is good, neither a more aggressive BRM has a higher EV on the stakes we beat (the reason of the 54% ITM in all the stakes simulated), but I'm investigating the possibility of this being true.

In fact if you look at the code
(lines 36-38): when the bankroll gets smaller than the smallest buy-in, I keep track that I reloaded, then reloads.
(lines 47-49): I keep track of how much times I got broke
(line51): I translate this number into a probability to get broke, then display it to the screen a few lines after. I expect it to be the "risk of ruin" we all talk about (maybe this is incorrect though?)

I did not put everything in here, but I also tried to display the ratio "(money earned)/(risk of ruin)", and even though I had a lot of dispersion with this ratio, it seemed to be constant (or at least the numbers seemed to be in the same range with each BRM I tried, from 100BI to 10BI when I had this displayed)
However there might be a problem in the code as well.

Thank you for your comments anyway, I think I'll keep working on this. I'm pretty sure that BRM could be drastically improved than the good old "30BI rules" I'm applying right now for SnG HU.

Also, there are many other pitfalls that only experience and commonsense can bring that could be added into the coding. For example, as you move up in stakes, your edge will decrease or go negative.
This is true, but it does not correspond to what I want to simulate, as explained earlier in this post. My first tries did consider a decreasing edge over the stakes. When I saw that all BRM had the same earnings in that configurations, I considered that if there was a stake at which I don't have an edge, it was kind of normal that the progression would stop there. This is also interesting, but not what I was trying to investigate, hence my decision to choose a marginal edge over each stake.
It might also be interesting to get from a slightly losing configuration on a stake and make the edge rise as we play games at this stake or higher. Today, I think that with such an assumption, a conservative BRM might be more effective.
• Bronze
Joined: 21.07.2009
This is a pretty obvious conclusion imo, just because our BR is lower in terms of BI that doesn't make our long term winrate any lower, however if does mean that we'll go broke many, many times.

The fact that we can reload if using 5BI BRM means that it doesn't matter, and since our winrate is the same in both trials the long term winnings are quite obviously going to be the same.

One interesting spin-off you could do of this is for HU SnG BRM. I have no knowledge of coding or anything, so it would be very interesting for me if you could do this:

Say we use a BRM of 100BI for one trial, with a winrate of around 57% or something.

For the other trial we use a dynamic BRM system where we start with 50BI and shoot at the higher limit when we reach 40BI for that limit, moving down again when we have 35BI.

So say we play the \$7s with a BR of \$350, we shoot at the \$15s with \$600 but move down again at \$525. We move up to the \$30s at \$1200 and move down again at \$1050.

But with this one you can't reload if you go broke, it would be very interesting to see how much faster (if any) your BR grows with this BRM method, and also how many times you go broke.
• Bronze
Joined: 16.08.2007
As to a more dynamic BRM, that's exactly what I was aiming to simulate in the first place. I had a few ideas of strange but maybe accurate BRM I wanted to simulate just to see how it is possible to "uncorrelate" risk of ruin and expected money and exploit that.

Originally posted by Wriggers
The fact that we can reload if using 5BI BRM means that it doesn't matter, and since our winrate is the same in both trials the long term winnings are quite obviously going to be the same.
As to your conclusions, they seem a bit fast to me:
=> I only simulate 5000 games. If you go broke at 2k games you only have 3k games from the reload to make your earnings.
=> The winrate is the same BI-wise, except that you're playing higher buy-ins with a more aggressive BRM. Hence I was expecting more earnings with a more aggressive BRM (with the same winrate, logical since I kept the same itm% and rake for each level).

Actually, the fact that the winrate is the same and therefore the same earnings does to be a biased logic to me. I think that when you go for a more aggressive BRM, the effect of the increased risk of ruin just compensates the extra \$ won because we are playing on a higher limit with an marginal edge. The conclusion seems to remain the same however.
I'll work a little bit more on this in a few days I think, today and tomorrow I'll work on session reviews and my mental game.

Thank you for your interests.
• Bronze
Joined: 21.07.2009
Ahhh ok, yeah I didn't take into account the fact you are using 5BI BRM all the way through (stupid me )

Do you think you could run the program using the BRM method I suggested? the reason i'm so interested is because i'm currently using that BRM and would like to see if it's actually worth it

Because it would seem, if you are disciplined with moving down when needed, that the risk of ruin for the BRM strategy i'm using has a similar risk of ruin to the 100BI flat BRM system, but the potential for much higher earnings. Just wanting to prove that
• Bronze
Joined: 16.08.2007
As I said, I won't work on this before two or three days. But yeah, as I said this is the kind of BRM I would put to test, so as soon as I'll take the time to code more sophisticated BRM be sure I'll try your one as well. I'll post the results in here.

(I'm no programmer nor I have a computing background, and learned to code basic things as a hobby. That's why I linked the script in the first post: just so that if the strange results come from a mistake people have the tools in hands to fustigate me .
It also means that it takes some time to code even simple BRM as the one you suggest (something like one hour maybe, then the simulation time).

I will first have to review the code in depths to be sure that there are no mistakes in it also.
And yeah, I also think that discipline about moving down should be enough in any sound (even in an agressive) BRM to avoid ruin.
I had a few ideas of dynamic BRM as well a few years ago, and never tried them out because my roll was healthy and I had other things to work on (I was already playing at my "highest" winning stake, and therefore using BRM to move up was not a good idea).
• Bronze
Joined: 21.07.2009
Ahh yes I missed those parts of your post, i'm busy and was skim-reading sorry Will be sure to keep an eye on this thread.
• Bronze
Joined: 16.08.2007
Thinking a bit more about it, I really think there is a problem in my code, there is no reason to win the same amount in the longrun when we have the same edge on each stake but play different stakes with the same bankroll.

Ex:
\$1000 bankroll, playing 1000 \$1 HU games with a ROI of 10% => get to \$1100
\$1000 bankroll, playing 1000 \$2 HU games with a ROI of 10% => get to \$1200
In both cases the risk of ruin should be negligible (2nd case we have 500BI BRM with a 10% ROI). I can't see why the negligible RoR increase would kill our profits.

I'll have a trip in train in a few hours, I'll work on this, there must be a problem in my script.
• Super Moderator
Super Moderator
Joined: 02.09.2010
I suggest:
ROI should decrease at higher limits. Reason: Reflects real world.
If possible, calculate winrates in BB/100 using typical winrates from the various levels. Reason: Results more easily understood
When BR < X BI, start decreasing winrate, and decrease it geometrically as BR decreases. Reason: Adjusts for "scared money" and reflects how people tend to behave when on a downswing.

I think that this is a great project.
I wish I had thought of it!
• Bronze
Joined: 16.08.2007
Okay, so I rewrote the script and it now behaves normally.
I made a first bunch of calculations in the train.

The BRM chosen is simple:

br = starting bankroll
brm = we play a game at the highest stake possible for which we have "bankroll > brm*stake"
\$ = average bankroll at the end of 5k games following the BRM
brokes = percentage of brokes.

average of 2k runs of 5k games, better edge=60% itm:
br: 50 brm: 1000 \$: 323 SD: 59 \$, brokes: 0.0 %
br: 50 brm: 500 \$: 323 SD: 18 \$, brokes: 0.0 %
br: 50 brm: 200 \$: 447 SD: 96 \$, brokes: 0.0 %
br: 50 brm: 150 \$: 602 SD: 168 \$, brokes: 0.0 %
br: 50 brm: 100 \$: 1815 SD: 1120 \$, brokes: 0.0 %
br: 50 brm: 75 \$: 6594 SD: 85 \$, brokes: 0.0 %
br: 50 brm: 62 \$: 16793 SD: 19783 \$, brokes: 0.0 %
br: 50 brm: 50 \$: 35178 SD: 17608 \$, brokes: 0.0 %
br: 50 brm: 30 \$: 66723 SD: 12418 \$, brokes: 0.0 %
br: 50 brm: 10 \$: 95583 SD: 11467 \$, brokes: 0. %
br: 50 brm: 5 \$: 92607 SD: 20175 \$, brokes: 0.73 %
br: 50 brm: 3 \$: 42032 SD: 71514 \$, brokes: 8.57 %
br: 50 brm: 2 \$: 16039 SD: 96027 \$, brokes: 14.92 %

So far, a few things, quite logical can be taken out of this:
-> a too nit BRM will slow down our progression.
-> a too aggressive BRM will also slow down our progression. The average takes into account the "0\$" out of the brokes, and therefore shows that too aggressive doesn't mean better.
-> for a player with 60% itm and 6.6% rake on his network, a BRM of 10BI seems around optimal. It seems quite logic to have an aggressive BRM as optimal for a stake we already beat.

Next will come calculations in "one stake" only, just to try to correlate our expected ROI with an optimal BRM to start a new stake.
After this, it will be quite easy to guess something close to optimal BRM for moving up stakes as fast and secure as possible, if able to estimate our ROI at the next stake.
And finally, I'll compare these "optimal BRM" found to BRM commonly used.
That's gonna be a bit of work though, so of course I don't know how long it'll take

My first calculations were done with 54% ITM. Actually with the standard deviation displayed I understood why the first results were strange: the standard deviation was about the same as the "end" bankroll", meaning we had a value totally unexploitable. That's why I decided to get the ITM up.

ROI should decrease at higher limits. Reason: Reflects real world.
If possible, calculate winrates in BB/100 using typical winrates from the various levels. Reason: Results more easily understood
When BR < X BI, start decreasing winrate, and decrease it geometrically as BR decreases. Reason: Adjusts for "scared money" and reflects how people tend to behave when on a downswing.
I agree with the first point. However I was aiming at first looking if BRM had a significant impact or not, so a constant ROI was necessary imo.
Calculating the bb/100 has no use imo to a SnG HU player. I'll later on maybe translate this tool for cash game, but that's not my idea since I don't play CG anymore.
As for the third point, it would be quite easy to do so, but I'm not interested in. I'd argue that it this should already be comprised in the itm% chosen. Also, our ROI is a estimation, and therefore it doesn't make sense to add such thin details on something that is already not precise at all.
I mean, if we know for sure our winrate over more than 1k games at a stake, I'm not really sure that an estimating tool for BRM is needed to chose a BRM.

However of course these things can be implemented in
• Coach
Coach
Joined: 19.11.2010
Great discussion!
• Bronze
Joined: 16.08.2007
Allright, back in this thread for the results of some more calculations.
My PC is still running on the scripts to extract more values, but here are a few results:

I decided to run my script on a single stake, in order to get the risk of ruin on this particular stake depending from our itm% and the rake (it does not depends directly on the ROI, since I believe that the same ROI with different itm% (which basically would mean different rakes) would slightly disturb the results.
The calculations were done with a 6.6% rake (on my networks I have to play these ...)

In these calculations, we start playing 1\$ SnG with a certain amount of buy-ins ("br"), and get out of this the percentage of going broke at that stake. The results are averages of 1k runs of 200 games (for each itm and starting bankroll). Diminished the numbers of simulated games to gain some computing time. however since we are now only playing on one stake we need much less games to evaluate accurately the BRM.

So here are some selected results for comparison.
-> ITM=54% i.e. ROI = 0.8%
br:7, brokes:64.4%
br:9, brokes:51%
br:23, brokes:10.2%

-> ITM=55% i.e. ROI = 2.7%
br:5, brokes:66.8%
br7:, brokes:57%
br8:, brokes:46%
br:20, brokes:9.7%

-> ITM=56% i.e. ROI = 4.6%
br:4, brokes:66.8%
br:7, brokes:48.6%
br:17, brokes:10.4%

-> ITM=57% i.e. ROI = 6.4%
br:3, brokes:72.7%
br:5, brokes:52.2%
br:14, brokes:10.1%

-> ITM=58% i.e. ROI = 8.3%
br:3, brokes:62.8%
br:4, brokes:52.9%
br:13, brokes:10.2%

-> ITM=59% i.e. ROI = 10.2%
br:2, brokes:73.8%
br:4, brokes:46.9%
br:10, brokes:10.2%

-> ITM=60% i.e. ROI = 12%
br:2, brokes:67.3%
br:3, brokes:53.2%
br:9, brokes:11.7%

-> ITM=61% i.e. ROI = 13.9%
br:2, brokes:65.9%
br:3, brokes:45.7%
br:8, brokes:12.5%

Bolded is the closest point to 50% brokes.
I will later try to add in this the average won money at each itm/BRM point, so that we'll be able to define an optimal longterm BRM at each ROI%
• Super Moderator
Super Moderator
Joined: 02.09.2010
Hi,
Just coincidentally the video:
http://www.pokerstrategy.com/video/25526/
Came up this week.

bogdan uses a simulation tool to simulate runs of various lengths, with various winrates and standard deviations.

Your tool looks at SnGs, and the video discusses cash games, but you're both looking at more-or-less the same question:

How does BRM affect long term success?
• Bronze
Joined: 16.02.2011
Hey I am a math geek myself and I can explain the 'surprising result' to you. I haven't gone through all the comments so I don't know if this has been covered yet.

Lets consider 2 things:
EV
Variance(Standard Deviation is a more correct term but poker players use variance)

Think of it like this, suppose I have a coin which gives a head 51% of time and tails 49% of time.I win \$1 for a head and lose \$1 for every tails. So my EV in this situation is +1*.51+(-1)*.49=.02

So on average I expect to make 2 cents per coin flip.
Having said that my standard deviation in this spot is approximately \$1.

Now if I start with a BR of \$1 there is a 49% chance that I go broke on my first flip and a 99.9% chance that I go broke in my first 10 flips.

Whereas if I have \$10 there is less than 1% chance of going broke and if I have a \$100 there is almost no chance of going broke.

So in theory if you have an infinite bankroll then in a +EV game there is no chance of going broke but no one have infinite bankroll hance good bankroll management prevents you from going broke.

Whats wrong with your simulation is that you have only taken the net outcome of 10000 trails but you have not considered the fact that the player with a 5 BI BR might have gone broke in the process of 10000 trails and hence would never complete his 10000 trials.Instead do a simulation for the no. of trials required for both players to go broke and you will see the roll BRM plays.

I hope this makes it clear.
• Bronze
Joined: 16.08.2007
Thank you for your interest.
The subject of your "objection" is basically th reason why I fixed gears, and decided to decide that when we get broke, our bankroll stays at that level for the run. The core of the exploitable calculations are in my last result post.
I did not edit the frst posts, even though the code is obsolete, for the sake of clarity.