Risk of Ruin

[Dan Mezick]

I'm curious how many players actually realize their risk of ruin expressed in percentage terms.

Please be honest in your answers to the poll.

Let's discuss the RoR concept.

[jason1990]

I voted for the fifth option, but it probably should have been worded differently. No one can know their RoR exactly because (1) the formulas for RoR are based on models that incorporate simplifying assumptions, and (2) no one can "exactly" know their true winrate and standard deviation (SD).

I've brought this up before, but it's been a while, so I'll bring it up again. We all are well-versed in sample size requirements for winrate. But what about SD? In Mason's essay, Computing Your Standard Deviation (http://www.twoplustwo.com/mmessay8.html), he says

[ QUOTE ]
A good rule of thumb is to have at least 30 observations (playing sessions) for the estimate to be reasonably accurate.

[/ QUOTE ]
It's true that 30 observations are going to give you a pretty accurate confidence interval for your winrate. (I don't mean a small interval, just one whose size accurately corresponds to the given degree of confidence.) The confidence interval for your winrate should ideally be built using your true SD. But in practice, you must use your estimated SD, which is going to have an error. How different is the confidence interval which uses the estimated SD from the confidence interval which uses the true SD? Well, with 30 observations, it's not going to be much different. But it is not because your estimated SD is close to your true SD.

When I started in nano-limits, I did some analysis of my SD. After about 40 sessions, I had an estimated SD of 21 BB/100, but a 95%-confidence interval for my SD was [17,29].

So 30 sessions is plenty if all you want to do is build a confidence interval about your winrate. But it is not enough if you want to get close to your true SD. And accurately estimating your true SD is what you need to do to compute RoR. (Of course, it may be harder to get close to your true winrate, which you also need for RoR, but at least you can try to get one statistic nailed down.)

[jason1990]

[ QUOTE ]
How different is the confidence interval which uses the estimated SD from the confidence interval which uses the true SD? Well, with 30 observations, it's not going to be much different.

[/ QUOTE ]
Okay, this was said in a sloppy, hand-wavy, sweep-it-under-the-rug type way, and, as it stands, it's probably not even true. So I feel compelled to elaborate on where the 30 comes from.

Suppose I play n sessions which, for simplicity, are all the same length, say 100 hands. Let X_i be the number of BBs I won in the i-th session and I will assume each X_i is normal with (unknown) mean w and (unknown) variance s^2. So my true standard deviation is s. Let W and S be my estimated (or observed) winrate and standard deviation, respectively.

If I knew s, then I could build a confidence interval for w by using the fact that (W - w)/(s/sqrt{n}) is normal. But I don't know s, so I must use S. And the problem is that (W - w)/(S/sqrt{n}) is not normal. But when n is at least 30, then this quantity is close to normal, so in this sense we are justified in using S instead of s in our usual construction of the confidence interval. But that does not mean that the numerical values of s and S are necessarily close.

Which means we may not be justified in replacing s with S in other formulas, such as the RoR formula.

[fnord_too]

I selected 4, but none of the options are really correct for me. My risk of ruin is pretty near zero, since I will drop down if my bankroll drops, and I carry a large bankroll for what I play. You cannot know your RoR exactly. It is impossible to know your exact expectation or standard deviation in Poker. You cannot solve explicityly that like you can in craps, say. Maybe 5 was a more apropriate choice, because I am always looking at EV and bankroll and adjusting accordingly. Basically, what I am saying, is that trying to get an exact figure is futile and silly since estimation error is so high. (As the old saying goes: "Measure with a micrometer, mark with chalk, cut with an axe.")

[Dan Mezick]

This is what I am getting at: precisely calculating your (estimated) risk of ruin.

Why?

This is a hugely important calculation, because you must accept the risk to accept responsibility for your results.

I believe we may all reasonably agree that to be able to accept the risk, you must first know the risk. Specifically, you MUST know your risk of ruin. If accurately estimating your RoR is impossible in poker, then accepting responsibility for all your results in poker is also impossible. This risk-acceptance issue is actually a Psych Forum issue, and to address it, RoR must first be determined.

I'm calling on all probability experts on this Forum to help me solve this problem of defining a specific problem: tournament poker RoR. It’s a function of estimating the following player info:

per-session probability of winning,
the average win per session,
the average loss per session, and
the bankroll size.

Let’s say the game is a weekly rebuy tournament where it costs $40 to play and $20 to rebuy. You can rebuy during hour #1 if your chips fall below 50% of the starting chip count.

Let’s say you have a stop-loss rule of $80 per session: if you rebuy twice on top of the $40 ($40+$20+20) and you lose all your chips, you exit the event, EVEN IF it is during the 1st hour when you could rebuy again. Downside risk is thus precisely defined at a max of $80 per session.

Let’s say you have $2000 of bankroll to start with.

Further let’s say you have 30 observations of playing sessions, with wins and losses, as follows:

1 -$40.00
2 -$40.00
3 -$60.00
4 $60.00
5 -$60.00
6 -$80.00
7 -$100.00
8 $960.00
9 -$60.00
10 -$60.00
11 $300.00
12 -$40.00
13 -$60.00
14 -$80.00
15 $150.00
16 -$40.00
17 -$60.00
18 -$40.00
19 $200.00
20 -$40.00
21 $500.00
22 -$40.00
23 -$40.00
24 -$80.00
25 -$60.00
26 -$40.00
27 -$40.00
28 -$40.00
29 -$40.00
30 -$40.00

$890.00 (+)

What’s your estimated RoR? HOw does it change if you make the stop loss $60 or $100 instead of $80?

With these observations you can calculate risk of ruin for various bankroll sizes. And by sizing the bankroll, you can adjust RoR to a RoR-probability acceptable to you, if that probability is not currently acceptable.

Correct? Is 30 enough observations to construct a valid RoR estimate? Is using the last 30 ("rolling") observations the right approach, to incorporate subtle changes in the game and your play in it, over time?

Those interested in solving this problem may be interested in this paper, which discusses achieving these goals in another game of probabilities-- trading.

Position Sizing
http://www.traderscalm.com/psizinginterview.html

Thanks in advance for those who find this an interesting probability problem, and post some answers to this tournament RoR question.

If I have not provided enough info please post that. I believe 30 observations is enough to get it right in terms of a valid estimate of RoR.

[uphigh_downlow]

Based on your strategy of reducing the stakes that you ply at (which most people do)

I would like to ask a question that I hope will get answered sometime. Assuming that you play always play a game where your roR is a certain onstant.

Do you have any idea about at what RoR, you will be a break-even player over the long run(infinite time). Or do we need more variables to answer this more accurately.

idea is that you get stuck in the cycle of changing stakes, and never seem to make a profit. Since you win at lower stakes and lose at higher.

[dabluebery]

I wish there had been another option for me. I HAD calculated all this stuff a while back, but once I realized I have plenty of BB's in my bankroll for my risk levels, I stopped worrying about it.

Now I play 6-max and I'm broke. It's the darndest thing.

Rob

[Dan Mezick]

[ QUOTE ]
idea is that you get stuck in the cycle of changing stakes, and never seem to make a profit. Since you win at lower stakes and lose at higher.

[/ QUOTE ]

This is directly related to RoR and indirectly related to taking responsibility for all results.

If you know by moving up you are increasing your RoR by a factor of 3,4,5 or even 10, you will be much less likely to make the move until and unless all conditions (proper bankroll) are in place for an acceptable RoR at the new level.

Items that get measured tend to get managed, items managed tend to improve. Attention is essential to perception.

This is especially true of risk.

[Dan Mezick]

It's likely you increased your RoR dramatically when you moved to 6-max. Maybe it was a HUGE increase. Knowing your RoR and accepting the risk is essential to taking responsibility.

Taking responsibility is essential to eliminating denial. Eliminating denial is essential to playing well.

Therefore knowing your RoR for a given scenario is essential to playing well.

[pzhon]

Risk of ruin is a simple concept, but in practice you aren't sure of

/images/graemlins/diamond.gif Your win rate.
Of course you may have an estimate, but small differences greatly affect the ROR. If you will change levels, you have to guess what your win rates will be at the other levels. You also have to guess whether games will get tougher or softer in the future, and what bonuses will be offered.

/images/graemlins/diamond.gif Your standard deviation.
This is harder to figure out than you might expect. My SD estimates according to PokerTracker have oscillated wildly.

/images/graemlins/diamond.gif What you will do if you hit a large upswing or downswing.
If you will step down, when will you move back up? What is your plan for taking shots at higher levels?

/images/graemlins/diamond.gif How much is really in your bankroll (as opposed to your balance).
It's hard for me to imagine going on a huge downswing and losing my whole balance. If I did, I think I would put more money into poker. That extra money counts as part of my bankroll now.

With some simple estimates, I find my risk of ruin is under 0.01% (1/10^4), as long as I don't go crazy after I hit a hot streak. I'm not going to worry about calculating it more exactly until I hit a big downswing and it increases greatly.

So, are risk of ruin calculations useless? I don't think so. You can replace the bankroll with a session balance to get a session risk of ruin. You can use this when you are taking shots at a higher level or a different game. Your session balance can start at the amoutn you budget for trying the new level. You can use the session risk of ruin to determine how likely it is that you will have your balance cut in half, which might be the threshold you use for moving down in limits.

[fnord_too]

I don't think 30 sessions is enough in most, if not all, cases. People have some pretty major differences in BB/100 from 100k block to 100k block. The variance here is pretty wild.

Moreover, the game truly is dynamic, both your play and the play of the table you are sitting at. Consider mutual funds for a second. Their mantra is "past performance is no gurantee of future results", and these are institutions who employ a lot of quants to minimize risk and/or maximize expectation. I think there is a lot more predictability in poker than in protfolio management, but the point is that trying to quantify these things to fine levels is, I don't believe, feasible for all the reasons that phzon points out.

(I just realized you linked something to trading, but I don't have time to read that right now. I am reasonably well versed in portfolio management, though.)

(This post is getting into rambling but I have a lot of random thoughts to share.)

Let's look at your stop loss idea though. IF you always play your best game, and exercize good game selection, and can accurately assess the level of competion, and leave when it the game is not sufficiently profitable, then stop losses make no sense. Of course, no one can do all these things, though many may think they can. So stop losses are interesting from a Baysean perspective.

Consider your expectation is 2BB/100, with a 15BB/100 standard deviation. You can figure out the likelyhood that you will be N BB from even in either direction if you assume a normal distribution. (This gets into the central limit theorem and a bunch of statistics that I don't want to think about right now, and that are not really germane to what I am trying to get at.) Further say that you realize your EV varries due to how well you are playing and how the rest of the table is playing against you. Now say you have notionalized some discrete EV's and their likelihoods:
+4 BB/100 25%
+3 BB/100 35%
+2 BB/100 10%
+1 BB/100 5%
0 BB/100 5%
-1 BB/100 5%
-2 BB/100 5%
-3 BB/100 5%
-4 BB/100 5%

(That is actually 1.85BB/100, but I don't feel like playing with the numbers any more).

Now, those prior probabilities are a big assumption, but you can figure the liklihood you are at one of those discrete nodes (another simplification, the discrete part) depending on where you are after so many hands.

For instance, say you are at -20BB after 200 hands, you can work out the liklihood that you are in each mode via conditional probability. Working this stuff out in advance may give you some reasonable stop loss numbers. That is you can set an arbitrary confidence, such as "I will not play if there is a 20% chance I am not in a +2BB/100 or better situation."

So, if you believe you can come up with good prior probabilities, you can do some math to come up with stop losses that suit you. There is going to be a lot of uncertainty in those prior probabilities, but that might be fine.

Of course if you know you start to play like crap after you drop a buy in regardless of the game or your level of fatigue, then you really have no need for math in establishing stop losses, just self awareness and dicipline.

This sort of Baysean inference could result in some interesting RoR math, but its impact may be too small taking all the uncertainties into account.

That is kind of what I was originally trying to say, that the cascading errors in measurement leave you with really broad confidence intervals.

(To paraphrase TJ, I appologize for the long winded rambling nature of this reply, I did not have time to write a short one.)

[dabluebery]

Thanks, I was kidding.

[Dan Mezick]

Thanks for your post. Tell me how you clearly and unambiguously define your risk per session without employing a stop loss. Is there some other way?

[Dan Mezick]

Thanks for your post here.

Session risk is defined as your stop-loss value, say 30BB. I see where you are coming from with all the unknowns you listed. Theoretically if you have enough observations you can calculate RoR based on these plus the defined session risk.

The biggest wild card is the variability of the game and the variability of your own play.

The basic answer from many respondees seems to be "you cannot validly estimate RoR for poker, even in a very simple example such as 30 observations of a weekly NLTH tourney with a min of $40 and a max of $100, using a stop-loss to define the max session risk."

Gee, that makes poker very discretionary. Doesn't it?

It also makes it impossible to accept the risk in a responsible manner, since you do not know what the risk actually is.

Right?

[fnord_too]

[ QUOTE ]
Thanks for your post. Tell me how you clearly and unambiguously define your risk per session without employing a stop loss. Is there some other way?

[/ QUOTE ]

The is no way to (assuming clearly and unambiguously implies accurately, too.) (Well, you can see when you are playing badly or are outclassed if you are able maintain objectivity, but purely by results you cannot know why they are for a session, the standard deviation is too big and the variables myriad.)

Some things I do is watch my play and if I see myself making clearly bad plays for no good reason I quit. I don't play when really tired any more. Also, right now, I am doing a tour of sit and goes, which have a fixed limited down side. When I play no limit, I generally will stop if I have lost a couple of buy ins if I did not lose them on draw outs after all the money went in.

[fnord_too]

Just thought of something. Tvesrky and Khaneman published a lot of studies about how people, even people who know statistics, believe small sample sizes will be representative of the population. A session is a very small sample, to look for it to be representative is falling into a trap.

I am a quant at heart, but I think you need qualitative analysis in session by session situations.

Although, here is another thought: The tendancy to play worse when stuck is well known. (That is, the tendancy in the poker playing population as a whole, not necessarily in a specific individual.) One could do some interesting analysis on say win rate in the second hundred hands of a session when losing greater than or equal to some number of BB in the first 100 hands. Then an individual, after enough data are collected, could have some significant results and draw some correlations that may be helpful. But that would take a lot of work and time, and you would have to assume away any change in the individuals game (e.g. getting better at playing or avoiding tilt).

[uphigh_downlow]

what I meant to suggest was that one always play at a constant RoR. If my bankroll/skill improves/degrades, I move up/down, so my RoR is still the same.

RoR is constant over all games. Now at what RoR does one expect to be a break-even player.

Just trying to compuete a threshold. Or atleast see if there can be a closed form expression for it.

Hoping some can shed some light on it.

[Jerrod Ankenman]

[ QUOTE ]
Risk of ruin is a simple concept, but in practice you aren't sure of

/images/graemlins/diamond.gif Your win rate.
Of course you may have an estimate, but small differences greatly affect the ROR. If you will change levels, you have to guess what your win rates will be at the other levels. You also have to guess whether games will get tougher or softer in the future, and what bonuses will be offered.



[/ QUOTE ]

By the way, in our upcoming book, we have a nifty RoR formula that takes a stab at accounting for the normally very high uncertainty surrounding win rate.

(snip)

[ QUOTE ]

So, are risk of ruin calculations useless? I don't think so. You can replace the bankroll with a session balance to get a session risk of ruin. You can use this when you are taking shots at a higher level or a different game. Your session balance can start at the amoutn you budget for trying the new level. You can use the session risk of ruin to determine how likely it is that you will have your balance cut in half, which might be the threshold you use for moving down in limits.

[/ QUOTE ]

Or you could just calculate your risk of ruin for that bankroll size and then calculate a different one for the game you'll move down to and multiply them together or something.

Risk of ruin is just a model. It doesn't mean "the risk that you will lose all your money playing poker." It means that if you assume:

--You play a fixed, single game forever.
--The distribution of outcomes from the game is fixed and known.
--You have a starting stake, and no other money will be added to that stake from external sources.
--All the money generated from the game will be put back into the bankroll.

Then you can find your chance of losing your stake before you win an arbitrarily large amount. But these assumptions aren't valid for any players that I know. So RoR is just a rough guide. I personally find it much more useful to do RoR calculations on amounts smaller than my bankroll as "what's the chance that I'll be down X at some point?"

If I lose half my bankroll, I'll certainly infer using Bayes' theorem that some assumption I'm making about the games that I'm in is wrong. The likelihood of that probably dwarfs the likelihood that I just happened to have a four sigma negative event for six months.

I don't think it's essential to attempt to know your precise risk of ruin, because most players I know who are interested in risk don't actually just play till they're broke at the same game. The concept of a half-bankroll is also useful here, especially as an off the cuff way of understanding how likely you are to be down N bets starting from X bets, which is the real heart of bankroll requirements. If your RoR is like .000001 or something, you probably have orders of magnitude more RoR from game conditions changing or your win rate estimates being wrong than you do of actually going broke given your assumptions are correct.

Jerrod

[Dan Mezick]

[ QUOTE ]
Now at what RoR does one expect to be a break-even player

[/ QUOTE ]

To determine this you need your actual edge, which will inform your session risk value. It's like sizing a bet in trading: your perceived edge informs your "what percentage of equity to risk" decision. The smaller the edge the smaller the bet (position) size.

Keeping RoR small won't make you break even with a small negative edge. But playing with a RoR that's too high will make you go broke, even if you have a huge edge. RoR is a function of edge and bet size. If you have a good method and a decent edge but your bet size (session risk) is too large you can go broke quite easily. That's a crime when you have a real edge.

Small RoR derived from small session risk will keep you alive long enough to possibly figure out your method is bad, so you can make adjustments before the RoR actually hits you.

[Dan Mezick]

Ok you induced the question. Can you tell more about this book?