risk, bankroll and optimal betting (math warning ON)

[BillC]

We've just finished an article that addresses bankroll, risk of ruin, and other risk parameters. It addresses and interrelates the topics in the subject. I believe that it properly addresses bankroll requirements, which are improperly addressed on this forum and by Malmuth in GTAOT (plus he didn't send me the book). The relevant result is:

Let k be your Kelly fraction in blackjack (which is equivalent to a utility function, which in turn specifies your risk tolerance). The value of k is equivalent to having a risk of ruin tolerance of exp(-2/k). This in turn is equivalent to always having a bankroll B of v/kr, where v is the variance s^2.

Assume r/s is fixed constant, where r is your hourly win rate and s is the standard deviation. Theoretically, if you always choose a game with r, s so that B=v/kr, your bankroll is the same (as a random process) as a k times Kelly bettor in blackjack with win rate r/s and unit standard deviation.

Example: Let bb=Big Bet. Assume a benchmark range of games with pure ratio s/r=10 (e.g. s=10 bb and r=1 bb). Thus v/r=100 bbs. It is optimal (w.r.t. geometric rate of bankroll growth) if you always have 100 bbs (this is k=1 or full Kelly betting). Having k=1 is optimal but entails wild swings (e.g. you will eventually get halved with probabilty .5) Most blackjack teams are more risk averse and set k between .25 and .4. For poker, k should probably be smaller, e.g. k=1/6, because of lack of certainty about r,s. If k=.25, your bankroll will have to be 400 BBs;
for k=1/6, your bankroll will have to be 600 bbs. Some experts counsel even smaller values of k, e.g. k=1/10. Some futures traders suggest k=1/6 (they call it optimal f ). For holdem, 300 bbs is not really enough for most folks.

Your bankroll is all your assets minus expenses. It is not your separate "gambling bank". For the latter artificial sort of bankroll, using a value of k closer to 1 is recommended (e.g. k=.5), B=200 bbs.

These are facts from the paper. If you want to learn more more about Kelly betting, utility, etc. a good place to start is bjmath.com. Also see R. Epstein's book. The paper will be posted soon. I will email you a copy if you were at least a math or related major. It will be over your head otherwise. The material is based on a diffusion model from stochastic calculus. nigelc21@hotmail.com

[DcifrThs]

does the utility function include the possibilty of moving around in limits??

if so then what are the implications of its inclusion?
-Barron

[BillC]

The point is that you do resize if your BR changes. You resize to game with variance/(win rate times k) equal to your BR. If you sit down and play and get a good win, then you might move up. If you lose, you must consider a move down. (Of course this depends on game availability) You ideally always keep the same BR as a function of s,r and your risk tolerance as parameterized by k. For simplicity, I assume s/r is a fixed ratio, typically around 10 (a pure unitless ratio)

If you follow this set-up, you are k times Kelly betting, where a lot of stuff is known. There are formulas for various risks, like the risk of getting halved, etc. Importantly, one should never bet more than the optimal k=1. I.e., ALWAYS have more than v/r (=100 big bets), even if you are a risk tolerant as hell. It seems that picking a value of k, 0<k<1 is more sensible than trying to pick a risk of ruin (that nearly infinitesimal)

This analysis seems to be new in the poker world.

[Ralle]

What is your opinion about Sileo's article, where the bankroll estimate is ln(1/r)*(s^2)/(2m)? [r=risk of ruin, s=standard deviation, m=expectation]

If you don't think that expression holds, please explain why.

[BillC]

The formula that Sileo rediscovered is fine, and the derivation is reasonable. We show how it follows from a
more general risk formula, that has long been in the graduate textbooks. Here r is the risk of ruin. It is just one risk parameter; there are others as we indicate.

The problem is identifying your r. Is it 2%, .1% or 13%??

How do you feel about losing half your bankroll? a third?
The formula is this: given a kelly fraction k, the prob. that you will eventually be cut down to a fraction a, 0<a<1 of your starting bankroll is a^(2/k-1). E.g. at k=1, it is just a, at k=2 it is a^3. The corresponding risk of ruin (assuming no subsequent resizing) is exp(-2/k). It is gotten from the ruin formula (Sileo), assuming an inital k times Kelly bet.

[Cyrus]

"Your bankroll is all your assets minus expenses."

Are you sure about that definition? (Is that the NPV of future expenses outflows?)

Perhaps you mean Current Assets minus Current Liabilities. But then what about future inflows of income? The paycheck coming in at the end of this month is something that increases my (liquid) assets. It is also something that I can borrow against right now and then I can use the borrowed money to play poker. That payroll money, discounted at the cost of borrowing, should be counted as being part of my bankroll.

[BillC]

Yeah, my defn is a little short. Most people use an artificial (or differently defined) BR. BJ teams use a "team BR", which is not really the collective BRs of the investors. The BR is the NPV of all assets including future inflows and expenses and, say, your car and house minus taxes and your kids college tuition and grocery bills. If you are going to inherit a million in 20 years, discount, and go out and play cards.

[Louie Landale]

I'm not quite clever enough to do these calculations myself, but it doesn't seem right that if your bankroll requirements BB are inversely linerarly related to your risk-tolerance K. No; your bankroll requirements will jump drastically as your risk tolerance diminishes.

I would like to point out that these calculations presume you continue to gamble at a level even after taking big losses. No, most people can and will drop to a lower level. If so, your BB requirements need to be smaller.

I would also like to point out that unlike blackjack where the rules are the same and therefore your R and S are the same at the various levels, at poker R generally goes down and S generally goes up as you advance in Levels.

I would also like to point out that "assets minus expenses (and minus liabilities)" is a pretty bad measure of bankroll: I'm sure the Spouse of a gambler would NOT appreciate the House being counted: the gambler must quite and get a job long before he loses the house. No, your "bankroll" is the amount of money you can lose before quitting; and that's not "artificial" even if it is subjective and variable.

- Louie

[BillC]

Louie, thanks for your remarks. I'll add mine.

"I'm not quite clever enough to do these calculations myself, but it doesn't seem right that if your bankroll requirements BB are inversely linerarly related to your risk-tolerance K. No; your bankroll requirements will jump drastically as your risk tolerance diminishes."

the formula is B=v/kr. as k diminishes, risk tolerance decreases, and B goes up.

"I would like to point out that these calculations presume you continue to gamble at a level even after taking big losses. No, most people can and will drop to a lower level. If so, your BB requirements need to be smaller."

The point is that if one adjusts ones level according to the formula above, then you are doing (fractional) Kelly betting and the well-developed theory applies. We ARE assuming limit resizing if the bank changes. You comment may apply to those using the risk of ruin parameter; but no: if you have a fixed risk of ruin tolerance, then you must drop down to keep that risk of ruin tolerance constant. The optimal k=1 corresponds to having a constant risk of ruin of about 13.5%

We refer to risk of ruin as "instantaneou risk of ruin" in analogy with velocity. It is the risk of ruin assuming (contrary to fact) no future resizing.
"I would also like to point out that unlike blackjack where the rules are the same and therefore your R and S are the same at the various levels, at poker R generally goes down and S generally goes up as you advance in Levels."

No, BJ games vary a lot b/c of rules, penetration and no. of decks. It is critical of course to be aware of how s and r change. Sometimes as r goes up so too does s (e.g. as a game loosens up.

Our model tells you how your results depend on r and s precisely (as a stochastic process), The assumption that r/s is fixed is a theoretical one to optain the geometric Brownian motion (as in blackjack and the stock market)

"I would also like to point out that "assets minus expenses (and minus liabilities)" is a pretty bad measure of bankroll: I'm sure the Spouse of a gambler would NOT appreciate the House being counted: the gambler must quite and get a job long before he loses the house. No, your "bankroll" is the amount of money you can lose before quitting; and that's not "artificial" even if it is subjective and variable."

These are good points should be discussed further. I would say that the house in this case should be subtracted as an "expense" in this case.

I take back what I said about the article being too hard. If you skip the hard math stuff, it still contains some useful info.

[Louie Landale]

Nit-Pick: "expenses" means to me "ongoing expenditures" like food or rent and imply an "over time" variable. I'm sure your equations don't take how long it takes to get ruined multiplied by your ongoing expenses over that time. Responsible gamblers can and will put aside house, car, and expense money (say for 6 months), leaving them with a realistic "bankroll" with which to gamble and is very useful in your equations. Consider changing "expenses", perhaps to "liabilities".

Yes, there is very useful information in this stuff for those of us who can follow the concepts but cannot do the math (just like I can understand a bridge column but cannot play bridge).

Consider adding some clarification that your equations can help gamblers decide what to play and at what level, and for poker can adjust strategy (e.g. play very tight on a short bankroll). But once you are ACTUALLY gambing you need to forget about all that and focus on sound tactics and strategy.

- Louie