bankroll concepts - risk of ruin
 

if i assume a given win rate and standard deviation, then the larger my bankroll, the lower my risk of ruin, correct?

by the same token, if i require a y% risk of ruin, then I need a $x bankroll.

now, let's say i run the calculation and fund a BR with the required $x for a y% risk of ruin, and head to the tables. suppose i lose money in the first month: is my risk of ruin now higher than it was when i started?

if so, then isn't a BR formula basically saying "you need a $x bankroll to have a y% RoR (given the WR and SD), unless you ever (i.e., initially) run bad enough that your BR drops below $x"? that is, if you lose in the first month, you should be prepared to replenish your bankroll to the starting amount to maintain a y% RoR going forward?

i'm having a hard time matching that line of thought to the idea that a BR calculation gives a definitive, "one time" amount that should be enough to withstand swings and avoid going broke all but y% of the time.

Re: bankroll concepts - risk of ruin
 

Your reasoning is correct, but you're putting too much faith in the calculation.

If you always replenish your bankroll to $x, then your risk of ruin is zero (unless you lose all $x before replenishment). The risk of ruin calculation says when you have $x, if you keep playing regardless of bankroll, there is y% chance that you will go broke. Every time you win or lose any money, your risk of ruin changes.

Most people use this as a theoretical result only. If your bankroll is large enough that your risk of ruin is, say, 1%, most people can play comfortably without worrying about bankroll. For most people, that means they can play their best poker and enjoy life. If the risk of ruin gets up to, say, 25%, you take some remedial action: move to lower stakes, play more conservatively or get some more money.

In that case, your risk of ruin never was 1%, because you never planned to play without adjustment all the way to ruin. But the number gave you the confidence to ignore bankroll for a while. Similarly, the 25% wasn’t real either, it was just an alarm signal to say, “start worrying about your bankroll.” You then adjusted your play to reduce the risk of ruin to an acceptable level again.

The reason people do this kind of thing is to avoid thinking about bankroll every day and in every poker decision. You establish a comfort zone for risk of ruin, then live and play as if the risk of ruin is zero. When you get outside your comfort zone, you fix things to get back in. Worrying about bankroll every day isn’t fun, and worrying about bankroll in a poker decision hurts your play

Re: bankroll concepts - risk of ruin
 

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let's say i run the calculation and fund a BR with the required $x for a y% risk of ruin, and head to the tables. suppose i lose money in the first month: is my risk of ruin now higher than it was when i started?

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Team X was on the books as a 2:1 favorite over team Y. At halftime team Y is ahead by a lot. Is team X a 2:1 favorite now?

Re: bankroll concepts - risk of ruin
 

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In that case, your risk of ruin never was 1%, because you never planned to play without adjustment all the way to ruin. But the number gave you the confidence to ignore bankroll for a while. Similarly, the 25% wasn’t real either, it was just an alarm signal to say, “start worrying about your bankroll.” You then adjusted your play to reduce the risk of ruin to an acceptable level again.


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thanks for the reply. that really helped to clarify things for me.

one question: is 25% a reasonable point to "start worrying about your bankroll"? i wasn't sure if you were using that number as a sort of established standard, or just threw it out there.

the basic dilemma for me is this: once i decide on a reasonable size bankroll for the game i intend to play, at what point should i *really* be concerned that i'm no longer properly bankrolled for that game if i start off in the new game with a downswing.

just as an example: when i run my BR calculation for 10/20, given my expected WR and SD I find that i should have a $5300 BR for 10/20 @ 1% RoR. i wouldn't hit the 25% RoR mark until i was down to $1600. if i'm down $2500-3000 in the first few weeks, should i still feel comfortable playing as if my RoR was zero?

Re: bankroll concepts - risk of ruin
 

Your risk of ruin is approximated by a function of your bankroll, expectation and variance. Therefore as you bank changes, so does your risk of ruin. There is no "once and for all" number.

If your risk of ruin exceeds 13%, then if you can move down in stakes, you certainly should. Probably you want to keep your ROR under 1%. These numbers all come from fractional Kelly betting theory.
For more on this topic see bjmath.com, especially the risk formula paper by BillC and Marc Ingenoso.

Re: bankroll concepts - risk of ruin
 

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If your risk of ruin exceeds 13%, then if you can move down in stakes, you certainly should.

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I think this is too strong of a statement.

I know where it's coming from: (1) If you are currently playing at full Kelly, and will never resize bets (i.e. change stakes), then your ROR is around 13% (or 13.5%, whatever). (2) If you are resizing your stakes/bets as your bankroll changes, then you don't want to bet more than full Kelly (to max your growth).

I think you are using (2) and (1) together to make your statement that "ROR above 13% is always bad". But note that the two statements apply to different scenarios: whether or not you resize.

A person who wants high EV and is less concerned about ruin may very well have a ROR above 13%. I don't think it's common, nor would I advise anyone to do this in most situations. Nevertheless, I don't think we should proclaim 13% ROR as the highest permissible one, absolutely.

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Probably you want to keep your ROR under 1%.

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This is very conservative for people who can replenish their bankrolls. It would often be better to set a higher ROR, but then resize (change stakes) if your bankroll drops significantly. In this case, your true ROR is lower than the the ROR you originally calculated, but your EV is going to be much higher.

If one is a retiree who plays to supplement his income, then a conservative ROR is certainly appropriate.

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These numbers all come from fractional Kelly betting theory.
For more on this topic see bjmath.com, especially the risk formula paper by BillC and Marc Ingenoso.

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Am I correct that you are a different BillC?

Re: bankroll concepts - risk of ruin
 

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just as an example: when i run my BR calculation for 10/20, given my expected WR and SD I find that i should have a $5300 BR for 10/20 @ 1% RoR. i wouldn't hit the 25% RoR mark until i was down to $1600. if i'm down $2500-3000 in the first few weeks, should i still feel comfortable playing as if my RoR was zero?

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If you're down 150BBs in the first few weeks of playing that level I would suggest that you reevealuate your expected win rate, as well as BR, when calculating RoR.

Re: bankroll concepts - risk of ruin
 

I am making some assumptions to invoke Kelly of course. Some fool might want to bet his whole BR on a coin flip, but Kelly is optimal in well-known senses, which one can read about.

Bankrolls can't be replenished by definition. The replenishment would be included in the initial BR. If I am going to inherit a million dollars in 30 years, I would include the present value in my BR.
The 1% rule is in line with the usual 300-500 big bet bankroll often recommended for limit holdem for a "skilled player".

You are incorrect--I am the same BillC

Re: bankroll concepts - risk of ruin
 

I agree with KurtSF that if you lose you should worry more about your expected win rate than your risk of ruin. After all, if your original ROR was 1%, only 4% of the time should you ever see it balloon to 25%. Therefore, even a confident player should take a hard look at her game.

I don't buy the Kelly logic in this situation for several reasons. One is that, as BillC says, it defines bankroll as everything you could get over your life, not just your cash in hand today. Second, as alThor says, it assumes you don't change your variance as a percentage of bankroll. Finally, it assumes everything is scalable, that you can bet as little or much as you like without cost.

In poker, you can't easily adjust your stake. Moving down in limit may not improve your ratio of win rate to standard deviation, and it may have costs to your reputation, or giving up accumulated knowledge about players. Playing more hands may mean playing more poker than you want, and your level of play might suffer. Turning down close bets hurts your win rate.

I personally think 25% ROR is a reasonable level to reassess, but reasonable people can differ. It will depend on your circumstances and personality. The point is to escape constant adjustment, to concentrate on poker and not on bankroll.

Re: bankroll concepts - risk of ruin
 

thank you for the replies, this has been a very helpful thread for me.

let's assume that we have enough experience to feel confident that our estimate of winrate is accurate for the game we're going to be playing. i understand that true winrate is not something that is easy to be confident in. suppose we've played, i dunno, 1 million hands in this game and we're using our results over those hands to estimate our winrate.

messing around with the RoR calculation i find that i would need to lose 200BB for my RoR to go from 1% to 25%. is a 200BB downswing really enough to start questioning a winrate that has been established over a very large number of hands?

i can definitely see this being a good trigger point to start considering my course of action given my smallish bankroll. but it seems like my main concern at this point would be my high RoR, moreso than doubt about my estimate of winrate... this seems even more true for a 150BB downswing.

now, when moving to a new game, it seems much more reasonable to start questioning your estimated winrate if you have a bad start. in this circumstance, my question is this: is 150-200BB too much to lose before revising your estimated winrate in a new game? if so, what size of loss (in BB) would be a reasonable "trigger point" for readjusting your winrate assumptions/estimations when making RoR calculations? also, suppose that 100BB is a reasonable trigger point. isn't this much moreso the case if the 100BB is lost in 2000 hands than if it takes 10,000+ hands to accumulate that loss?

now that i think about it i think i've seen some forumlas for this type of thing (i.e., "your 100BB loss means that your winrate is between x and y with z margin of error"). i'll start searching for that now, but if you have any links and/or additional thoughts for me i'd much appreciate it. =)

Re: bankroll concepts - risk of ruin
 

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messing around with the RoR calculation i find that i would need to lose 200BB for my RoR to go from 1% to 25%. is a 200BB downswing really enough to start questioning a winrate that has been established over a very large number of hands?

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Probably not. If you've played a million hands at this level, then in the next 10k hands lose 200BB, that is a blip on the radar. I wouldn't think twice about losing that much with as much confidence as 1000k hands sample will give you.

OTOH, very few people have that kind of sample to base their expectations on. Especially because when they "get serious" they will often rack up a number of hands that dwarfs whatever sample they were basing their expectations on. The key is the "one long session" idea. As long as you didn't change games, or improve, or otherwise change the "assumptions", everything should be considered together... don't split it up. If you play 20k hands at 1BB/100, then the next 1K hands you lose 200BB, you are not a winner with a losing streak. You are aparently a break even player, but don't really know. If you play 100K hands at 2BB/100 and then 200k at -1BB/100, again, you are apparently a break even player but don't really know.

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i can definitely see this being a good trigger point to start considering my course of action given my smallish bankroll. but it seems like my main concern at this point would be my high RoR, moreso than doubt about my estimate of winrate... this seems even more true for a 150BB downswing.

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That risk of ruin calculator is nifty, but you you really should know that the number you are plugging in for winrate is an estimate. How confident are you? How many hands?

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now, when moving to a new game, it seems much more reasonable to start questioning your estimated winrate if you have a bad start. in this circumstance, my question is this: is 150-200BB too much to lose before revising your estimated winrate in a new game? if so, what size of loss (in BB) would be a reasonable "trigger point" for readjusting your winrate assumptions/estimations when making RoR calculations? also, suppose that 100BB is a reasonable trigger point. isn't this much moreso the case if the 100BB is lost in 2000 hands than if it takes 10,000+ hands to accumulate that loss?

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I would consider both amount and time. If you lose 100BB in 1k hands, who cares. Lose 20BB over 100k hands and the lossrate is tiny, but its much more likly to be real. If you do something that can potentially have a big impact on your rate like move to a new game, you should keep close tabs on your bankroll, etc. Big swings can hurt.

This is very prescient example for me. I beat a game to the tune of 5BB/100 over 100k hands. Moved up in levels and over the next 15k hands lost 500BB. Whoops! Guess I wasn't as good as I thought I was!

As a "rule of thumb", I would say you should start being aware around 200BB. Any less is pretty standard, but around there is where it starts becoming pretty unlikly for a winning player to have a sustained losing streak.

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now that i think about it i think i've seen some forumlas for this type of thing (i.e., "your 100BB loss means that your winrate is between x and y with z margin of error"). i'll start searching for that now, but if you have any links and/or additional thoughts for me i'd much appreciate it. =)

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You should do that for sure, to give yourself an idea of where you stand. Anyone can have a 200BB losing streak. Easily. But its definitly worth reevaluating your expected win rate if you do. No links, sorry.

Good luck.

Re: bankroll concepts - risk of ruin
 

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This is very prescient example for me. I beat a game to the tune of 5BB/100 over 100k hands. Moved up in levels and over the next 15k hands lost 500BB. Whoops! Guess I wasn't as good as I thought I was!

As a "rule of thumb", I would say you should start being aware around 200BB. Any less is pretty standard, but around there is where it starts becoming pretty unlikly for a winning player to have a sustained losing streak.


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ok, thanks!

yeah i mean, when you're unsure of your winrate estimate in a new game, it really sucks to get to -200BB, start questioning your assumptions at that point, and keep going until you get to -3,4,500BB before you say "yep, guess i'm not a 2BB/100 winner in this game" lol.

seems like maybe somewhere between -100-200BB is the time to just stop, move back to the game you're a winner in, replenish the roll and work on improving anything you think might have been your weakness before giving the new game another shot....

Re: bankroll concepts - risk of ruin
 

1. You must take into account future bankroll supplements when discussing risk of ruin. If you do not, any discussion seems absurd. If you have an alternate bankroll model you will just have to end up converting it to the traditional notion - I think. If you have another way to do it, I'd love to hear it.

2. Kelly model is a good continuous approximation to discrete reality. You readjust your betting levels in stages, approximating Kelly or fraction thereof. The approximation is very good in practical terms.

I think it is best to start with a Kelly model for bankroll and factor other considerations in. I mean, what are you going to do otherwise? Just use seat-of-the-pants numbers like 25%?

If you cannot drop down in stakes, at some point you might be well-advised to stop playing. The cardinal sin of the Kelly criterion is to overbet you bankroll. Betting twice kelly (ROR=1/e) or more leads to negative bankroll growth.

Of course having confidence you are winning is of paramount importance. Obviously most are losers.
People complain about all this splitting of hairs over bankroll, when they should be talking about poker. But some of us split hairs for a living.

Re: bankroll concepts - risk of ruin
 

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If you cannot drop down in stakes, at some point you might be well-advised to stop playing. The cardinal sin of the Kelly criterion is to overbet you bankroll. Betting twice kelly (ROR=1/e) or more leads to negative bankroll growth.


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i need to do some reading on the kelly stuff. but there's one thing that's hard for me to grasp about RoR as it applies to poker:

suppose i run the calculation given my WR and SD and find that a 250BB BR = 1% RoR. well, when i only have 25BB on the table and i'm betting in increments of 1BB, my RoR is 0%. if i lose all those chips, then my BR is now smaller and my RoR is higher when i start my next session.

so when you talk about "betting twice kelly or more", how does that really apply to poker, when in poker we're not really "betting" much at all (relative to BR)?

Re: bankroll concepts - risk of ruin
 

If you are engaged in an activity that say gives you a certain win rate r and variance v per unit of time (or e.g. 100 hands), betting k times Kelly translates into having a bankroll of v/(kr). The attendant risk of ruin given k is exp(-2/k).

So if your v/r is 100 big bets, betting Kelly means having 100 big bets as a BR. Twice Kelly would be a BR of 50 units. You certainly want to readjust before getting below the 50 unit mark, and probably before going below the 100 unit mark.

You see, Kelly betting applies to a macro game, which averages over whatever goes on from hand to hand at the table. It is a continuous approximation which only depends on win rate and SD. There is a way to apply Kelly theory to micro decisions such as drawing and pot odds, and blackjack playing indices, but that is for another thread.