kelly criterion

[BillC]

Here is a text of the article. A new article that gives precise calculation and analysis of precise excess pot odds required is in the works. Comments welcome.

BillC

What Poker Players Can Learn from Blackjack Pros: Expectation isn't Everthing

Variance is the very stuff of gambling. It makes for the roller coaster that we ride, hopefully to riches. The average slope is your unit expection (EV), while the standard deviation (SD) is something like the average amount you go up or down. We use a random walk with upward drift model. If your slope is zero or negative, you will eventually crash. So let's stick to the positive EV situation; the rest is for losers. EV is the first thing to consider when evaluating a wager; variance is the second.

In University-level Mathematics and Statistics, two difficult concepts in courses are expectation (EV) and variance (they are difficult because the depend on the concept of random variable). These are fundamental concepts for games of chance and are worthy of serious study. Without a good understanding of these concepts, your thinking about poker will be incomplete. Now, EV has been discussed many times in the literature, while standard deviation SD (and its square, variance) gets less and usually vague attention. There have been several threads on the twoplustwo.com Poker Theory forum where the effect of variance (and utility) in poker has been batted around, with some arguing that its effect is neglible and should be ignored (e.g. http://forumserver.twoplustwo.com/sh...Number=1668690 and also see the March Magazine (vol. 1 no. 4) article Your Risk Level by Jared Lunsford ). While this last viewpoint is sometimes apt, it is not always, and the discussions have not taken place in a quantitatively rigorous framework. I will discuss variance in a non-technical way in this article, pointing out what it means to winning poker players. In part, this article is an explanation for poker players of the advanced math-oriented article Risk Formulas for Proportional Betting, see http://www.bjmath.com/bjmath/proport/riskpaper1.pdf. Specifically I will address

1. Variance and Bankroll requirements (macro stategy)
2. How standard deviation can be used to compare different games (macro strategy)
3. How variance affects decisions such as longshot draws (micro straegy)

An estimate for SD is given in Mason's book Gambling Theory and Other Topics. Even if you don't run your own numbers, it is important to get feel for the impact of SD.
A first thing to notice is that decision-making doesn't only involve expectation. If that were the case, we would run out and bet all our money on any wager with the slighest positive expecation, e.g. a slightly favorable lottery. The second thing to realize is that variance will affect your win rate. This is because negative swings can force you to lower your stakes. Variance is bad because it entails downward fluctuations that can put a signifigant dent in your bankrolls. But just how bad is it? According to economists and mathematicians, the answer to that question comes down to one's risk tolerance profile, or utility function. Often, it is indexed by specifying one's acceptable risk of ruin, or other nearly equivalent risk parameters such as the risk of ever being reduced to half one's original bankroll.

Poker vs. Blackjack: All of this has well understood or a long time by professional-level blackjack players (and this is my background), but the concepts are mostly only vaguely understood by poker players. Reasons for this are the relative difficulty in estimating EV in poker, the possible difficulty in moving around in stakes, and the fact that some players are overbankrolled for the games they are either playing, or able to play sufficiently profitably. Still, the theory is valid and can have practical effect in poker. We'll direct our comments mostly toward limit holdem games, though similar comments will apply elsewhere (e.g. tournaments, no-limit), and it would interesting to see the subject developed further. While blackjack is a higher variance game than poker (relative to EV), the downsides of variance in poker are manifested in different and perhaps in more insidious psychological ways. In blackjack you can know your EV and SD, but in poker you often have less good simulation and data and rely more on deductions based on experience and the nature of your opponents (i.e. you are partly guessing).

In blackjack, one can use risk-averse indices to vary from basic strategy depending on the count. This is done by using risk-averse indices, which are developed by considering the variance associated with various plays.

See http://www.bjmath.com/bjmath/kelly/kellyfaq.htm for a discussion of Kelly betting, Certainty Equivalence and Utility from a blackjack perspective. These topics form the theoretical basis for accounting for variance in all games of chance.


1. Bankroll: The bottom line of the mathematical theory, known as Kelly betting, is that your mathematically optimal bankroll for a game, with a given EV and variance for some fixed unit of time (say an hour of play) or fixed no. of hands (e.g. 50), is equal to the ratio

B=variance/EV

Your bankroll is optimal in the sense that if you choose games in this way, you will optimize the rate of bankroll growth. So if your hourly EV is 1 big bet and your variance is 100 big bets squared (so SD=10 big bets), then the optimal bankroll is 100 big bets. Using these numbers from now on as our benchmark, we see that a juicer but looser (or faster online) game might have an EV of 1.5 big bets, but a variance of 150 big bets, but will have the same optimal bankroll of 100 big bets. Thus a $1000 bankroll corresponds to an optimal $10 big bet for the benchmark game. It is of some importance to note that this ratio is independent of time unit. When your bankroll grows, you ramp up to a bigger game. If you have a bad enough run, you move down. Wagering thus results in the maximum (geometric) rate of bankroll growth. This pure Kelly bet is in a sense a "risk neutral" one -- betting any bigger will result in a lower rate of bankroll growth.

But this bankroll of only 100 big bets is too risky for most people. Also it is impractical since if you have a losing run, you must immediately move down in stakes. And theoretically, if you always played with 100 big bets, then you would have a risk of 50% of being reduced to one-half of your bankroll. So most opt for a more conservative approach and use a bankroll or 300 or 400 big bets as a standard bankroll. (This corresponds to betting 1/3 and 1/4 Kelly for those who know what this means; these Kelly fractions parameterize risk tolerance). This in line with what is recommended (by different reasoning) by Mason in Gambling Theory and Other Topics. The scaled-back wagering makes the ride smoother but results in a lower win rate relative to bankroll size. We lean toward the more conservative bankroll of about 400BBs for the benchmark, erring perhaps on the conservative side, taking into account the difficulties of poker EV estimation (and other infelicities) mentioned above. Some recommend even larger bankroll requirements, though this might depend on how rigorous one's definition of bankroll is. It is certainly not your "gambling bankroll" or "what you can afford to lose". Purists would insist that your bankroll is your net worth including the present value of future earnings, minus all expenses needed to support your desired life-style.

Some of you would view all this as silly because you have enough resources to withstand variance. But your bankroll can be too big for a given game. That is, if you have too much money for a particular game, then you are playing for fun (or whatever fetish), not for significant rate of bankroll growth. Bill Gates playing 4-8 holdem at the Bellagio is an example. Secondly, some of you complain that you can't readjust to say 300 big bets continously. Well, the situation is that the mathematical model is continuous, which is approximated by discrete reality. With online games, readjusting stakes can be done much more precisely, because of the wide range of stakes available and the ability to play games simultaneously at different stakes. Blackjack teams resize bets incrementally over time and usually not in the middle of a trip. What you can do, say, is to keep say 300 big bets as a lower bound and perhaps 400 BBs as an upper bound or just wait until your bankroll size justifes of bigger or smaller available game. A third problem is that you may not want to scale up to a bigger game if e.g. you cannot beat the bigger game for as much, or the game doesn't exist where you play. This is addressed by having a good way of comparing 2 different games, addressed below. A kind of EV ceiling can occur which is dependent on skill and game selection. The blackjack analog that the heat level increases as you move into higher stakes.

2. Comparing games. It turns out that a good measure of how good a game is is the ratio EV/SD, the expected value divided by the standard deviation (for some fixed amount of play, e.g. an hour or 100 hands). This ratio has been used by blackjack players for a long time. It corresponds the long term growth rate of your bankroll for the game in question. Notice the difference between this and the ratio used to compute bankroll above. This assumes that you are approximately following a fixed fractional-Kelly strategy such as playing with 300-400 BBs (assuming benchmark no.s). If you are overbanked for the game you are playing, then the SD plays a lesser role, a role that tends to zero as your bankroll goes to infinity. So if you are overbanked, you can tend to decrease the weight of variance and overweight expectation. This can be made more precise, but that's for another essay.

3. Longshot Draws

It is often said that high-variance plays should be avoided if the decision is close and bankroll is a concern. Let's make this precise using Utility Theory. The variance in a pure drawing situation, say drawing to a small set, or a gutshot, means that the pot really needs to be a bit bigger than what is dictated by pure expectation. But how much extra? We want our wagers to have positive utility, not just positve EV. We approximated the Certainty Equivalence break-even point (read on if you don't know what this is) and came up with the following felicitously simple but very good (under-) approximation for the excess pot odds needed due risk aversion, assuming you have a bankroll commensurate with your risk aversion. For our benchmark game and facing a bet of one big bet, the excess pot odds you need is (a bit more than)

d^2/200

where you odds of drawing out are 1:d, and the pot contains about d big bets. Thus a pure gutshot on the river needs about .7 extra big bets to call (ignoring implied odds). Drawing one card to a small set (d=23) requires almost 3 more big bets. If you are drawing to one card on the river, you need about 5 extra bets.
Of course if you are overbanked, these numbers diminish. But if you are underbanked they grow. For example if you normally want to have 400 BBs, but are playing with only 200 BBs (of course maybe you shouldn't be playing at all...), then the excess pot odds needed doubles! These numbers are analogous to risk-averse indices in blackjack. The accuracy of the approximation has been checked carefully.

A more general approximation formula for the excess pot odds needed is

d^2/2kB

where B is the bankroll and k is the Kelly fraction (typically 1/3 or 1/4). Notice that the k cancels out if you use the prescribed bankroll B=variance/(kEV). B should be expressed in the number of bets you need to call to draw to a winner. Thus if you are facing two big bets to draw, then the excess pot odds needed goes up by a factor of 2, and the excess amount money in the pot needed is quadrupled. The approximation depends on the bankroll being fairly large compared to the pot. A slightly better approximation is (d2+d)/2kB.

All this may seem like splitting hairs, but that is what poker theory is largely about. You split a lot of hairs and, well, you've got a lot of split hairs. What we have done is to quantify the vague notion we started this section with, and to put into numerical perspective how much variance matters The effect of variance is small, but it is there, just like risk-averse playing indices in blackjack. Serious players are always looking to push all the edges available.

©2005 by William Chin, All Rights Reserved