The Kelly Criteria for Bankroll Management

Hitchhiker's Guide to...

Just reading a great book on stock market investment and gambling called "Fortune's Formula" by William Poundstone. It's a bestseller in the US at the moment. Thought i'd summarise a bit about what it says about bankroll management because it has interesting implications that are quite different from a traditional view of bankroll management in poker. This post is a mixture of passing on the information, seeking opinions, and me using the thread to get clear in my head what is being said in the book.

The book is essentially about the "Kelly Criteria". This is a formula for bankroll management that was developed by John Kelly from Bell Labs in the 50's. It was initially developed as a method of managing information flow over communication systems; then it was adapted as a method of managing bankroll for card-counting blackjack players; and it finally has ended up as a stock market investment strategy.

The method is different to traditional bankroll management models such as 'fixed percentage of bankroll' or the 'martingale approach' (doubling up until you get a predetermined profit - obviously not recommended for poker!).

Two quick starting issues:

1. The method is only for +EV investments rather than gambling. It therefore has to be assumed that a poker player following the Kelly Criteria is a sufficiently good player so that they can know beforehand that they will (on average) be +EV in a given poker game. If you are going to be -EV in a poker game then you are just a gambler and there is no point following the Kelly Criteria.

2. The second issue is the question of how to define a poker game. It could either be defined as an individual hand of poker or the whole session that you are seated for. Given the ability to learn throughout a session it is probably best to consider an entire session as a poker game and thus apply bankroll management to the entire session as opposed to individual hands.

The Kelly Criteria

The Kelly Criteria first involves estimating the +EV edge you have (or 'overlay' in bookie terms). Obviously this is not straightforward as you are estimating the probability of something that hasn't yet happened. But, with good notetaking and pokertracker you can be reasonably confident of what your +EV edge is going to be. Let's assume for the purpose of illustration that you have a 20% edge in a particular game. i.e. that you leave an average poker game of this type 20% ahead. This would mean that you should commit 20% of your bankroll to that game. Similarly in a 1% edge game you should invest 1% of your bankroll.

The second thing is that this is a dynamic staking system - your stake is automatically increasing as your bankroll increases and decreasing as your bankroll decreases.

This is a simple explanation of a mathemetically complex model. But the combination of the dynamic changing in bankroll-at-risk and varying stake based on your edge has been shown mathemetically to be superior to all other forms of bankroll management - from gambling to investing (to anyone who has studied finance; the Markowitz portfolio management theory is based on the same fundamentals).

It seems an interesting strategy that allows maximum usage of bankroll for the maximum gain. The only thing is that the full-Kelly (as the above described method is known) is sometimes seen as super-aggressive bankroll management. It is common to reduce the absolute element of risk by choosing a half-Kelly or quarter-Kelly (whereby you take whatever the Kelly Criteria suggests you should invest and cut it in half, or in a quarter).

Does anyone have an opinions on this method? Is it superior or inferior to your own method? Did you already know about the Kelly Criteria and have i thus wasted a few minutes of your life??!!

Ste05

Good post, I have one query though.....
IMHO the most important point of this theory above would be how you calculate your % edge??
I think this has to be done scientifically, for example would you use such things as your BB/100 and extrapolate out from that an edge you feel you have over your opponents at that level, (then how does it translate as you move up in levels????)

On the other hand is it just a fanciful figure plucked out of the air, (using PT and notes) such as "well I think I have a good edge over these guys and so I'll give my self a 25% edge over the game. I feel this could be dangerous for some players who could use it as a way to talk themselves into games that they are not good enough for (or don't have the BR for!!), and they'll be able to explain and rationalise it to themselves by saying that they were exercising proper BR management (the Kelly Criteria!!!) but just got unlucky?????

Just my 0.02c

Having said that, you mentioned a caveat above, that this is only for profitable players and so the above obviously wouldn't apply to them.......

hotspur

I've never heard of the Kelly criteria being applied to poker before. Poker is very different from the other gambling spheres in which this system is applied, in which it determines an actual bet size depending on EV. It seems like an interesting application of it, but as Ste05 pointed out it is meant to require a far more precise evaluation of EV than is normally the case in poker game selection. But beyond that, there are problems -

Let's assume that we have perfect knowledge of our EV for every poker game on offer. Ought we to play the game where we have a 50% advantage with 50% of our bankroll or the game where we have a 20% advantage with 20% of our bankroll? The Full Kelly method has you betting 50% of your roll if you have a 50% advantage no problem because the maths say you should, but it rarely happens to my knowledge because it's just too volatile.

Most sports bettors will eschew this for betting a fraction of the amount suggested by the Kelly method, and others will use what's known as a Constant Kelly which is betting % according to the orignal bankroll not the emerging one.
Surely we should play the 50% advantage game but with less than 50% of our bankroll, the Half Kelly suggests 25% of our roll, the Quarter Kelly 12.5%.

But we are limited by our maximum buy in in NL hold em. So you can multitable to increase the amount of money you are wagering such that it equals the % of you bankroll that you are committing to according to the Kelly system. But as you beat the game for a while you will quickly be unable to commit the % size of your bankroll to the game that the system dictates that you should.

And here is where most players make their error. Let's assume no multitabling for simplicity. They want to maintain their % win rate according to the size of their bankroll. So they were making 20% (their +EV) per $200 buy in table, so they commited 20% (Full Kelly) or 10% (Half Kelly) of their roll to it (actually amost players arent this conservative), which would be $1000 or $2000. Now they beat this game for a little while and get up to $4000.

They can no longer play at these limits and still use the non-Constant Kelly system, because the % of their increasing bankroll that they would have to use (ie 20% or 10%) is disallowed as a buy in at these limits. So many players make the decision to move up in limits because they feel that they want to make their bankroll work for them and continue to win a % of it. Higher up they may find that they actually have an insignificant +EV or even a -EV, and they've screwed up.

So that's the problem with appying the Kelly system to poker, in order to continue to wager as a % of your increasing bankroll you have to lower your EV because you are limited in the amount you can bet at your chosen +EV game, unlike sports betting (within reason).


A more interesting application of the system might be at the micro level of individual hand bets, would you in a freeze out tourny bet 20% of your chipstack according to the fact that you considered your hand to be 20% ahead? Well you'd probably bet it all if you thought you were 20% ahead. But the Kelly system shows mathamatically that if you are 20% with cards to come then it is incorrect stack management to bet more than 20% of your stack. Stupid Kelly

I may sound like a degenerate gambler talking about these systems, I'm not, I'm just interested in the systems of gamblers cause they're all such nonsense And the book you're reading - every academic study I've ever read on stock market prediction shows that it's unpredictable. That's why the monkey with a pin wins the Sunday Independent share competition every year versus economist and experts (apart from 1 year where it came 2nd, and the winner kinda cheated by only picking 3 shares instead of the requested 5).

Hectorjelly

Great post, for some reason Bankroll management and related topics seem to inspire the craziest ideas in poker, so its good to hear a rational approach to the subject.