Utility function for fraction kelly

[Charlie Mosby]

Hi this is just a tiny theoretical question that I've always had. Thanks in advance for anyone who will read this.


Betting at full Kelly is optimal on many different levels for the log utility U(x) = Log(x) where x is the bankroll (wealth).

I wonder if betting at fraction kelly corresponds to (maximizing the expectation of) a different kind of utility function, or just simply
U'(x) = Log(kx), where for example k = 1/2 is half kelly and k = 1/3 one-third kelly.

That is, betting full kelly achieves the fastest growth rate and asymptotic optimality etc for the Log utility.
Is betting fraction kelly achieving similar things for some other kind of utility function?


I'm asking this because I think either there's a typo in this page or I just totally got it wrong:

http://www.bjrnet.com/articles/kellyfaq.htm


Under Q3, this pages shows a U(x) that in fact doesn't converge to Log(x) when k --> 1 (full kelly).
I think the author meant to use something like the following but made a mistake:
g(x) == [x^(1/k -1) - 1] / (1/k - 1)

This g(x) --> Log(x) when k --> 1 can be viewed as a way of defining the (natural) logarithmic function.

However, with the mathematically consistent (I believe) g(x) just above, one will arrive at something entirely different in maximizing the expectation (regardless taking the 2nd order approximation or not).
The optimal wager size of this g(x) is not the usual k * ev / var.

[Three]

when k--> 1 (full kelly) I think the author meant to use something like the following but made a mistake:
g(x) == [x^(1/k -1) - 1] / (1/k - 1)

I am not sure what they meant but I can tell you your equation is wrong. Full kelly k =1. that makes your radical 1/1 - 1 = 0. An exponent of 0 equals 1. So you have (1-1)/(1/1-1) or 0/0. First off you can't divide by 0. Second, if you divide by anything other than 0 you get 0 as your answer. Obviously the equation in question 3 and your equation can't be used with k=1 (full kelly).

The result of switching the sign of the radical and the denominator (what you suggest). Is to move the exponent when properly expressed as its absolute value of the exponent in the numerator or denominator to the denominator or numerator respectively and changing the sign of the result. Is that what you intended to accomplish when you decided the changes were necessary?

[Nyne]

I'm not so sure that equation is wrong. Different people have different utility functions, and as I recall, the utility function that corresponds to fractional Kelly is in fact formulated quite differently from the full Kelly utility function. I can't remember off the top of my head if what is in that link was it or not, but I don't remember having any disagreement with that article either. I'll try to remember to check my notes when I'm back home in a few days.

[Charlie Mosby]

(1-1)/(1/1-1) or 0/0. ...your equation can't be used with k=1 (full kelly).

Is that what you intended to accomplish when you decided the changes were necessary?

What I intend to accomplish is to find a utility function that
(1) gives the 'desired' optimal wager proportion k * ev/ var (perhaps under 2nd order approximation)
and
(2) converges to Log(x) when k --> 1


It has been bugging me a bit that there are plenty of material elaborating how proportional betting W = f * B makes sense (wager W proportional to your bankroll B and advantage), how Kelly betting is optimal (among the proportional scheme) with the factor f = ev/var, but then continue down the road we make a jump in the math saying that

"oh in reality full kelly is mathematically too risky (RoR = 13.5%) and at the same time most people cannot handle it emotionally"

so we simply

"reduce the factor f by directly multiplying it with a fraction so that we have k * ev/ var"

without a proper derivation, without going back to talk about how this k * ev/ var corresponds to some kind of optimal wager size for some kind of utility function.

(sure we have that exponential formula indicating what the new RoR is with the kelly fraction, but that's a different part of the story)



I quoted that FAQ webpage from bjrnet, but actually it was also one of the FAQ pages on Standford Wong's bj21 (now removed)......as I recall.

I made the 'changes' since that formula under Q3 immediately reminds of that particular way of defining logarithm function, the g(x) I used.

As for the 0 divided by 0 issue, I know for a fact that this limit of g(x) exists and it is the Log(x) as we know it (for any x > 0).
But yeah there are some details I should go back and double check.

[Charlie Mosby]

... as I recall, the utility function that corresponds to fractional Kelly is in fact formulated quite differently from the full Kelly utility function. I can't remember off the top of my head if what is in that link was it or not,....

Do you remember where you read about it? Is it some web discussion thread or maybe a book?

...but I don't remember having any disagreement with that article either. I'll try to remember to check my notes when I'm back home in a few days.

Thanks.

Yeah this is not so much about agreeing/disagreeing with what was written by Red Taylor, but about knowing how the k * ev/var can be derived instead of just multiplied adhoc.

As of now Log(kx) seems to me a fairly reasonable candidate.

[Nyne]

Well given a utility function u(x), you can write out the expected utility of the final bankroll after making a given bet with betting fraction f, which will look something like p_win*u(b+f*b*payoff_odds)-p_loss*u(b-f*b) for a simple win/loss 2 outcome bet, but you could write the formula for any number of possible outcomes. The optimal bet is the one that maximizes the expected utility, so you just take the derivative and find out what number in [0,1) makes the derivative of the expected utility equal to zero. The trick is figuring out your personal utility function. In truth, it is very common for your bets to be restricted by something other than maximizing your utility (particularly table limits, but also constraints on session win/loss due to heat concerns, etc.) but its always right to take whichever action maximizes your expected utility absent other concerns. But, your question is what utility function accurately describes a fractional Kelly bettor, and I can't answer that right now...

[bigplayer]

"oh in reality full kelly is mathematically too risky (RoR = 13.5%) and at the same time most people cannot handle it emotionally"

If you chop risk by 83% by going from Full Kelly to Half Kelly but only chop EV by 50% that's a pretty good deal regardless of emotions.

[ohbehave]

In the FAQ referenced in the OP, under Q5: What is the optimal bet size, where does the 0.144% figure come from for k=.3? And what would that value be for k=.1?

[Nyne]

In the FAQ referenced in the OP, under Q5: What is the optimal bet size, where does the 0.144% figure come from for k=.3? And what would that value be for k=.1?

The 0.144% number comes from simply plugging the values given into the formula given above in Q5.

akB/u = (0.6%*(T-0.4))*0.3*B/(5/4) = (0.6%*0.3*4/5)*B*(T-0.4) = 0.144%*B*(T-0.4)

k=0.1 would make the percentage 0.048% instead of 0.144%, but I would question the reasoning for choosing such a low Kelly fraction. Unless the bankroll is huge, it is probably wise to bet more aggressively. Somewhere in the 0.25-0.5 range is probably where most people should be, in my opinion.

[Charlie Mosby]

If you chop risk by 83% by going from Full Kelly to Half Kelly but only chop EV by 50% that's a pretty good deal regardless of emotions.

oh thanks for reminding me this : )

yeah it's like RoR reduced to 1/6 with ev reduced to 1/2.


I'm not questioning using fraction kelly instead of full kelly, just wondering in what sense is fraction kelly is optimal mathematically.

[Charlie Mosby]

Unless the bankroll is huge, it is probably wise to bet more aggressively. Somewhere in the 0.25-0.5 range is probably where most people should be, in my opinion.

yeah I personally will probably keep using 1/3 or 1/4 kelly for a long time....maybe drop to 1/6 when my bankroll is like 500k?


How would/have you guys gradually decrease the kelly fraction in correspondence with bankroll size?

This surely depends mostly on personal risk aversion, but I'm really curious how more experienced AP look at it.



I mean, mathematically one MUST chose diminishing RoR each time when resizing the bets after the bankroll has grown a substantial proportion.
This is to avoid the cumulative RoR going up, as detailed in some threads here and there and some front page articles in bj21.

Either one can stay at the same betting level (not resize as bankroll grows), or choose the ramp that goes with a new RoR that is slightly lower than the current one.
Ideally one would want the RoR for each 'period' goes to zero asymptotically.

[Three]

Either one can stay at the same betting level (not resize as bankroll grows), or choose the ramp that goes with a new RoR that is slightly lower than the current one.

Your current ROR changes with your changing BR. The correct statement is, you can either keep your bets the same and enjoy a decreased ROR as your BR grows or resize to have a higher EV by raising your ROR toward the original ROR. Don's statement was dead on but I think it caused some confusion because you didn't seem to understand it.

[ohbehave]

The 0.144% number comes from simply plugging the values given into the formula given above in Q5.

akB/u = (0.6%*(T-0.4))*0.3*B/(5/4) = (0.6%*0.3*4/5)*B*(T-0.4) = 0.144%*B*(T-0.4)

k=0.1 would make the percentage 0.048% instead of 0.144%, but I would question the reasoning for choosing such a low Kelly fraction. Unless the bankroll is huge, it is probably wise to bet more aggressively. Somewhere in the 0.25-0.5 range is probably where most people should be, in my opinion.

Not sure what I was trying to do at 3am.

If net worth is used to determine my bankroll as suggested in the article I get a figure that is 6-7 times larger than the figure I currently use. I would necessarily want a very low Kelly factor. I have CVCX/Data to get optimal figures, I was only interested to see where I would stand as far as Kelly factor is concerned with regard to this article.