The Kelly Criterion revisited

In my previous post about the Kelly Criterion for gambling and investing, I went through the mathematics to derive the Criterion. One of the models we discussed was this:

You are going to play an infinite series of games starting with a bankroll of B dollars. In each game, for each dollar bet, with probability p you will win W dollars and, with probability q (equal to 1-p) you will lose L dollars. In each game, you bet some fraction f of your current bankroll at the start of that game. The question is what should f be?

The solution given by the Kelly Criterion is the value of f that maximizes the geometric mean of your final bankroll (or equivalently the arithmetic mean of the logarithm of your final bankroll). For the model just given, that value of f is

f = (pW – qL) / WL

In the long term (an infinite series of bets), with probability one, this value of f minimizes the time required to reach any specified goal for the bankroll and will result in a larger final bankroll than any other strategy.

As we discussed, there is no controversy about the math underlying the Kelly Criterion, but there is a significant controversy about whether the f specified by the Kelly Criterion should actually be used in real life gambling or investing situations. In those situations, the number of games is finite, and the properties of infinite sequences might not be an appropriate guide to what you should do.

Some people are concerned that when the value of f corresponding to the Kelly Criterion is used, there is a large amount of volatility in the bankroll as the sequence of games progresses. For example, starting at any point in the infinite sequence, the probability is 1/3 that the bankroll will halve before it doubles. Many of these people would recommend betting Half Kelly (half the value corresponding to the Kelly Criterion), which gives about 75% of the growth rate but with significantly less volatility. (For example, the probability is 1/9 that the bankroll will half before it doubles.)

Other people would recommend betting more than the Kelly Criterion, which increases the volatility, but allows a larger possible gain (but with a low probability). For example, a value of f equal to one and a half times the Kelly Criterion also give about 75% of the growth rate, but with greater volatility, and a value of f equal to twice the Kelly Criterion gives a growth rate of 0 and even greater volatility.

To gain some insight into this situation, I made up a simple example, with

p = ½, W = 1, and L = .5

For these values

Kelly f = 0.5 and gain G = 1.06

Assume we are going to play only four games, and the initial bankroll B = 100.

We consider using four different values of f

f = 0.25, f = 0.5, f = 0.75, and f = 1.

For each of these values of f and each of the possibilities after four games, the final bankroll is given by the table.

Final Bankroll After Four Games

	Probability	1/2 Kelly	Kelly	3/2 Kelly	2 Kelly
		f = .25	f = 0.5	f = .75	f = 1.0
4 Wins, 0 Losses	0.06	244	506	937	1600
3 Wins, 1 Loss	0.25	171	235	335	400
2 Wins, 2 Losses	0.38	120	127	120	100
1 Win, 3 Losses	0.25	84	63	43	25
0 Wins, 4 Losses	0.06	59	32	15	6.25
Arithmetic Mean		128	160	194	281
Geometric Mean		105	106	105	100

The first column in the table enumerates the possible number of wins and losses after four games. The second column gives the probabilities of each of these outcomes.

The remaining columns give the final bankroll for each outcome for each value of f.

The last two rows of the table give the arithmetic and geometric means of the outcomes for each value of f.

The median and mode of the distributions for each value of f occurs when there are 2 wins and 2 losses. The median and modes are both largest (as is the geometric mean) for the value of f corresponding to the Kelly Criterion, (f = 0.5), but the arithmetic mean is largest for the value of f corresponding to twice the Kelly Criterion.

But suppose we removed all references to the Kelly Criterion from the table and just looked at the four possible ways to gamble. It certainly isn’t clear (to me at least) which is best.

For example, some people might want to bet their entire bankroll on each game (twice the Kelly Criterion) even though the most likely outcome is that they just break even, because there is some probability that they win a large amount of money (1600 dollars). Real people often make “bad” bets because of the small possibility of making a large amount of money. Think of state lotteries. (Insurance is also a bad bet because the insurance companies set the premiums so that they have an edge, but people nevertheless buy insurance because when they need it, they need it.)

I copied this quotation from the Web (http://www.bjmath.com/bjmath/kelly/mandk.htm)

If I maximize the expected square-root of wealth and you maximize expected log of wealth, then after 10 years you will be richer 90% of the time. But so what, because I will be much richer the remaining 10% of the time. After 20 years, you will be richer 99% of the time, but I will be fantastically richer the remaining 1% of the time.

That person is willing to take a large risk if there is a possibility of making a large amount of money

Some other people are not willing to take a significant risk and might want to bet the Half Kelly because it has the smallest possible loss of the four alternatives even though it has the smallest arithmetic mean and the smallest maximum gain.

Again, looking at the table, it seems to me that there are some people who would want to use each of the four ways to gamble (as well as possibly other ways).

I am not sure about what all of this says about the Kelly Criterion, except that it is mathematically interesting and a good benchmark to use to begin thinking about how to manage your money when you have an edge in gambling or investing.