[bztips]

Can someone point me to an explanation of exactly how "partial" Kelly differs from "full" Kelly? What exactly is the function that's being maximized with "partial" Kelly?

[Ganchrow]

Can someone point me to an explanation of exactly how "partial" Kelly differs from "full" Kelly? What exactly is the function that's being maximized with "partial" Kelly?

Here's the Kelly Utility function I use in C/C++:

inline double DLLEXPORT KUTIL(double dResult, double dKellyMult) {
	// Kelly Utility for dResult > 0 , dKellyMult > 0
	// dKellyMult = 1 → full Kelly,
	// dKellyMult = ½ → half Kelly,
	// dKellyMult = ¼ → quarter Kelly, etc.
	if(dResult <= 0|| dKellyMult <= 0 ) {
		return -DBL_MAX;
	} else if(abs(dKellyMult - 1) <= FLT_EPSILON) {
		return log(dResult);
	} else {
                double dKellyExp = 1 - 1 / dKellyMult;
                return ( (pow(dResult, dKellyExp) - 1) / dKellyExp);
	}
}

To economists, it's what's known as the "isoelastic utility function" with "constant relative risk aversion" = 1 dKellyMult .

[bztips]

Thanks.
I'll have to dust off my old notes on CES utility and Arrow risk aversion.

[Wrecktangle]

Partial Kelly is the betting public's reaction to wanting to use Kelly to get close to the "optimal" bankroll fraction given that they are really too lazy or too unknowledgeable to calculate the risk involved with a bet. Unlike Blackjack where you can exactly calculate your Kelly win fraction given conditionals (i.e. deck count), in sports (unless you calculate all the conditionals in your sport and your win% under each condition - injuries, weather, rest, coaching trends, etc, and hope your betting series is not serially correlated - actually you can derive a Kelly fraction given you know your correlation factor), you do not know your true win fraction.

So unfortunately unlike Full Kelly, there is no published theoretical base for fractional Kelly and unless you are willing to do the work, it tends to be the bastard child of the betting public who are too lazy or too dumb to drill out their true risk by calculating all the conditionals on a sport.

[djiddish98]

Partial Kelly is the betting public's reaction to wanting to use Kelly to get close to the "optimal" bankroll fraction given that they are really too lazy or too unknowledgeable to calculate the risk involved with a bet. Unlike Blackjack where you can exactly calculate your Kelly win fraction given conditionals (i.e. deck count), in sports (unless you calculate all the conditionals in your sport and your win% under each condition - injuries, weather, rest, coaching trends, etc, and hope your betting series is not serially correlated - actually you can derive a Kelly fraction given you know your correlation factor), you do not know your true win fraction.

So unfortunately unlike Full Kelly, there is no published theoretical base for fractional Kelly and unless you are willing to do the work, it tends to be the bastard child of the betting public who are too lazy or too dumb to drill out their true risk by calculating all the conditionals on a sport.

Isn't partial kelly the acceptance of the fact that your edge in sports betting is a lot less firm than your edge in blackjack?

What is the proposed alternative? (betting a % of your bankroll to win each time?)

Doesn't the market take all the conditions you mentioned above into account? Hypothetically, If you had 3 buckets of bets, where the edge compared to the closing line is 2,3 and 4% over the long term, would you not recognize these edges and bet each the same?

Thanks.

[Optional]

Can someone point me to an explanation of exactly how "partial" Kelly differs from "full" Kelly? What exactly is the function that's being maximized with "partial" Kelly?

A very complete paper on that question here:

http://www.edwardothorp.com/sitebuil...lationsNew.pdf

[bztips]

Thanks Op, that paper is very interesting, but it assumes you already know exactly what fractional Kelly is and how it's derived. Fortunately, it cites another paper (Mgmt Science, Nov 1992) that goes step by step through the definition and analysis of full vs. fractional.

Having (briefly) perused that paper, I must say that I think Wrecktangle is being a bit harsh on the "betting public". His claim that "there is no theoretical base" for fractional Kelly is simply not true -- it all depends on your level of risk aversion. Straight log utility has "low" risk aversion, and if one prefers more "security" at the expense of "growth", then fractional Kelly is exactly what you want.

[Wrecktangle]

Bz, you only have one mathematically provably optimal betting fraction for a situation. And it is the so called "Full" Kelly Criterion. Dr. Kelly actually found the mathematical proof and published it. That means EVERYTHING else is sub-optimal.

Non-mathematicians seem to not understand the enormity of this. There are very few useful things in life that come under this situation.

[bztips]

Wreck, full Kelly is optimal for maximizing expected growth, I get that. But whether you choose to believe it or not, pure maximizing of expected growth may or may not be a bettor's only goal.

Read the paper by MacLean, Ziemba and Blazenko that I cited -- if I have a security goal as well (which can be defined in different ways eg., protection against losing more than x% of my wealth at any point in time), then fractional Kelly provides the appropriate strategy for any such growth-security tradeoff profile.

[Ganchrow]

Bz, you only have one mathematically provably optimal betting fraction for a situation. And it is the so called "Full" Kelly Criterion. Dr. Kelly actually found the mathematical proof and published it. That means EVERYTHING else is sub-optimal.

Non-mathematicians seem to not understand the enormity of this. There are very few useful things in life that come under this situation.

In the economics of preferences, optimality is a relative term.

What's optimal for me (a butterscotch and banana lover), may very well be suboptimal for you (a chocolate and vanilla lover).

In Kelly's original paper, A New Interpretation of Information Rate, Kelly explores the expected growth maximizing methodology that would later become known as the "Kelly Criterion". The question of suitability for any particular investor was never actually established. Rather, Kelly made the a priori assumption that expected growth maximization (and expected growth maximization alone) was the only conceivable logical goal of all sensible investment strategies.

John Kelly was not an economist ... he was however, an apples and cinnamon lover and he never seemed to quite figure out what the hell it was with all those damned bananas, butterscotch candies, and chocolate and vanilla ice cream sandwiches (or to reference the dearly departed Kurt Vonnegut, Jr. "[who] wrote a story about flying-saucer people who visited Earth. Two things about the United States really bewildered them. 'What is it,' they wanted to know, 'about blow jobs and golf?'")

So the point is that while full Kelly is necessarily the optimal investment strategy for all market participants whose preferences happen to align themselves with those of Kelly's "Everyman Bettor", there is in reality absolutely nothing sacrosanct about expected growth maximization (at least not any more so than there is about a golden ratio of chocolate to vanilla swirls).

In an earlier post in this thread I mentioned that Kelly utility belonged to a class of utility function known as "isoelastic". These exhibit a property called "constant relative risk aversion" (or "CRRA") which, mathematically stated, means that the Utility function U(x) is a particular solution to the second-order differential equation:

x * d² dx² U(x) + ρ * d dx U(x) = 0

Where constant ρ > 0, is known as the coefficient of relative risk aversion.

The intuition behind CRRA is straightforward.

In the parlance of sports betting, if two CRRA bettors were described with identical ρ's, then irrespective of relative bankroll size, both bettors would choose to stake the same percentage of their respective bankrolls on any given wager.

It's this virtually axiomatic (if implicit) assumption of CRRA that is the impetus behind players and betting advisers (spanning what seems nearly a full cross-section of betting skill and temperament) structuring wagers as percentages of bankroll without regard to absolute bankroll size. Note that this is only economically "correct" (from the standpoint of traditional expected utility maximization) for bettors whose preferences exhibit CRRA.

So at this point let's see how the broader class of n-Kelly Utility (i.e., Kelly Utility across all positive Kelly multipliers) relates to CRRA preferences. We have the standard Kelly Utility function (for Kelly multiplier κ = 1):

U(x) = log(x)

As well as "fractional" or "multiple" Kelly for κ > 0, ≠ 1:

U(x) = (x^1-1/κ - 1 )/( 1 - 1/κ)

(Note: the subtraction of one from the above numerator is simply a convenience that ensures utility of 0 for x = 1. As it's simply a translation of the utility curve by a constant, this term is often omitted in the literature, its presence or absence having no real effect on results. For the sake of anal retentivity, I always tend to include it.)

(Also note: As κ → 0, U(x) → 0 ∀ x. This represents, "total risk aversion", where a bettor will refuse all risk without regard to potential reward. By contrast, as κ → +∞, U(x) → x. This represents "risk neutrality", where a bettor will make his allocation decisions based solely on expectation without any regard to risk. The net effect of this would be that such a bettor would always wager his entire bankroll on the single bet with the highest edge. Hence, risk neutrality and total risk aversion are in fact subsets of n-Kelly utility.)

As it's simply a translation of the utility curve by a constant, this term is often omitted in the literature, its presence or absence having no real effect on results. For the sake of anal retentivity, I always tend to include it.)


Despite the different looking equations, one will note that the full Kelly utility function is simply "fractional" Kelly
utility taken as κ → 1.

The first derivative of Kelly Utility is given by:

U'(x) = x^-1/κ

And the second derivative:

U''(x) = -1/κ * x^{-1/κ - 1}

So plugging these in to the CRRA differential equation and solving for κ:

x * d² dx² U(x) + ρ * d dx U(x) = 0
x ρ * -1 / κ * (x^{-1/κ - 1}) + x^-1/κ = 0
κ = 1 ρ (for x≠ 0)

So what we see is that the so-called "Kelly multiplier" is simply the reciprocal of the coefficient of relative risk aversion ... implying that as a bettor's Kelly multiplier decreases he will become (relatively) more risk averse.

So once we allows ourselves to get past the presumed primacy of expected growth maximization, we see that the optimal allocation strategy for bettors with various Kelly multipliers, is nothing more and nothing less than the argmax of expected utility for bettors exhibiting CRRA with varying RRA coefficients.

This allows Kelly to prescribe behavior across a wide array of risk preferences, provided those preferences exhibit CRRA.

Now at this point I'm going to preempt anyone itching to argue with me that CRRA preferences are not valid in general for all classes of investors (or indeed for any class of investor) by simply stating:

CRRA preferences are not valid in general for all classes of investors (or indeed for any class of investor).

In fact, anyone who attempted to claim that CRRA preferences accurately described all "reasonable" classes of betting behavior would be guilty of precisely the same "sin" as the Right Honorable Dr. Kelly.

What we see is that Kelly only considered market operators with CRRA preferences who have a relative risk coefficient/Kelly multiplier of unity. Now "fractional" Kelly utility expands on full Kelly to include the entire class of CRRA bettors with nonnegative relative risk aversion coefficients, and while a fairly broad generalization of full Kelly, it by no means should be presumed to reflect the entirety of the universe of "reasonable" risk preferences.

Recall that the Taylor expansion of a function U(x) about x=a is given by:

[nbtable][tr][td]U(x) = [/td] [td] [/td] [td] U⁽ⁱ⁾(a) * (x-a)ⁱ/ i ! [/td][/tr] [/nbtable]

where U⁽ⁱ⁾(a) corresponds to the i^th derivative of U wrt x evaluated at x=a.

Taking the first 3 terms of the Taylor expansion around x=1 (recall the above mentioned convenience of subtracting 1 from the numerator of the partial Kelly utility function -- who's laughing now?) gives us:

U(x) ≈ (x-1) - (x-1)² 2κ

And taking the expectation for Δ Bankroll → 0 we find:

E[U] ≈ μ - σ² 2κ

which should be familiar to anyone who's taken (and not forgotten) undergraduate financial economics as the Markowitz utility function.

So what we see is that not only is n-Kelly a generalization of full Kelly, but also (for small deviations in bankroll) is an asymptotic generalization of simple quadratic expected utility at the Markowitz-efficient frontier.

The problem comes when considering the higher order moments (technically about the origin) of the results distribution.

Because n-Kelly includes but a single descriptive parameter (i.e., the "Kelly multiplier"), it's unable to fully specify trade-offs between more than 2 distribution moments. So while n-Kelly can account for any relative utility/disutility trade-off between expectation and variance, the coefficient for the third moment is "locked in" (at κ+1 6κ² , it so happens), and so on and so forth for increasingly higher order moments.

In the end that's what the set of n-Kelly utility functions comes down to: a single κ parameter that specifies a predefined functional trade-off between the economic "good" that are betting edge and the further odd-numbered moments about the origin, and the economic "bad" that are (approximately) variance and the further even-numbered moments about the origin.

So is n-Kelly the end-all be-all of allocation utility? Clearly not. But at the very least it is a natural and logical extension of the parameter-less full-Kelly utility function. And while full-Kelly utility maximization does "happen" to imply expected growth maximization, there's nothing that should lead us to construe it as somehow "more optimal" across all possible sets of risk preferences. I love butterscotch ... do you?

[bztips]

woo-hoo!!! I am going to proudly take Ganchrow's elegant (yet still hard to understand!) explanation as an endorsement of (or at least consistent with) what I said -- full Kelly is only optimal if all you care about is expected growth; plus the other assumptions he mentioned -- you have isoelastic utility with constant relative risk aversion. It's not the end-all and be-all as some apparently believe.

ps. I happen to love butterscotch and vanilla; chocolate not so much.

[Data]

I just want to scream out of all of my meager 160 IQ that this was very humbling. Welcome back, Ganchrow. BTW, I am full of my own invention, Lemon Grey Goose + Hpnotiq.

[Optional]

I'd been thinking a bit like Wrecktangle, that only full Kelly was optimal.

Now I feel a bit better about my .4 Kelly multiplier.

[Wrecktangle]

Very nice work, Ganch. Unfortunately, in order to use any kind of Kelly (and I remain convinced that "Full" Kelly is optimal as I prefer to drive down the size of my Bankroll (BR) at Full, rather than using a larger one at fractional given the inherent changeability of sport) you must still drill out your conditionals in order to nail down that win%. This drives you to getting into Bayesian stat and fully checked out dbs with those conditionals.

All this gets the average to advanced punter back to unit betting (I always advise using the Century system as it will get you thru a season). I've stated on more than one occasion that Kelly, or even fractional Kelly, while it might be optimal or close, cannot be used if you do not understand your true win% in a given situation. This remains an area in which most folks simply won't spend the time or have the skills to do.

So folks, using Kelly is a little like using nitroglycerin to build tunnels thru mountains. Black powder (unit betting) will do the job, but much slower, Nitro will be faster if you know exactly what you are doing and have a steady hand, otherwise you can lose your BR life very quickly.

[djiddish98]

All this gets the average to advanced punter back to unit betting (I always advise using the Century system as it will get you thru a season).

So do you risk the same amount on a game A with +200 odds and game B with +100 odds?

Or is this more applicable to spreads?

[u21c3f6]

Can someone point me to an explanation of exactly how "partial" Kelly differs from "full" Kelly? What exactly is the function that's being maximized with "partial" Kelly?

Here is how I look at it and this is also based on many years of use.

Kelly is without a doubt the mathematically proven optimal wager for bankroll growth assuming (and this can be a big assumption) that you know the actual edge of your wager. Early on I used full-Kelly. However, being relatively conservative, I did not like the volatilty that my bankroll had using full-Kelly. In researching money management further, it seemed to me that half-Kelly might be the way to go. The comparison goes like this:

Using full-Kelly, there is a 1/3 chance that your bankroll would fall in half before it doubled. Using half kelly, you still have 75% of the growth rate as full-Kelly but with half-Kelly there is only a 1/9 chance that your bankroll will fall in half before it doubles.

Having used both, I am personally much more comfortable using half-Kelly. I prefer to watch my bankroll increase in size by a nice small zig-zag pattern than the more extreme ups and downs of full-Kelly. Once you double your bankroll, you will be making wagers equivalent to full-Kelly of your original bankroll only now with a greater degree of safety.

One other thing I also do is to segregate different types of wagers as well as different odds to determine my edge. For myself, I find that different types of wagers and different odds have a different edge which is critical for the Kelly calculation. If I was to lump all of this into one data pool I could come up with one composite edge but that is not as optimal as knowing that I have a higher or lower edge based on the type of wager and the odds and then sizing my wagers accordingly.

Joe.

[Wrecktangle]

So do you risk the same amount on a game A with +200 odds and game B with +100 odds?

Or is this more applicable to spreads?

Well yes, if your perceived advantage (overlay) is the same in both cases. A good advantage player is capable of seeing an average 1-2% advantage over the long term, thus the real origin of the Century System; at its core it's a full to 1/2 Kelly system using a realistic "rule of thumb" win% for the long term win advantage a serious non-modeler typically achieves.

The best modelers can achieve a higher win%, but only with the best dbs with as many conditionals identified as possible.

So, for the average punter, the century slows the bleeding so that the player can last an entire season before his bankroll is dried up.

[u21c3f6]

Well yes, if your perceived advantage (overlay) is the same in both cases.

I think this needs to be clarified. If you have the same % edge for a +200 and a +100 wager, you would wager half as much on the +200 as you would on the +100. When you use (have) the same % edge across various odds you essentially size your wager to win the same amount (edge/odds).

As I posted above, this is one of the reasons that I segregate my types of wagers as well as the odds ranges to ensure I am using a more accurate edge % for my sizing calculation.

Joe.

[djiddish98]

I think this needs to be clarified. If you have the same % edge for a +200 and a +100 wager, you would wager half as much on the +200 as you would on the +100. When you use (have) the same % edge across various odds you essentially size your wager to win the same amount (edge/odds). As I posted above, this is one of the reasons that I segregate my types of wagers as well as the odds ranges to ensure I am using a more accurate edge % for my sizing calculation. Joe.

This is what I was getting at - what you should bet depends on both odds and edge, not just edge.

Thanks Joe.