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1. ## Simple closed form solution for unconstrained Simultaneous bet Kelly staking

I've for some time been a bit embarrassed by my rather egregiousness overcomplication of the mathematical curiosity that is the unconstrained simultaneous independent event Kelly solution, which I detailed over 3 years ago in this thread.

So to rectify in brief:

Given N independent events, x1, x2, ..., xN, with corresponding single bet Kelly stakes of κ1, κ2, ..., κN, the unconstrained Kelly solution (for any Kelly multiplier > 0) consists of the 2N-1 parlays such that the wager on a given parlay comprised of all events in set S would be:

[nbtable] [tr] [td] [/td] [td] κi[/td] [td]   [/td] [td]   [/td] [td] × [/td] [td]   [/td] [td] [/td] [td] (1-κi) [/td] [/tr] [/nbtable]

So given, for example, events A, B, C, D, and E, with corresponding single-bet Kelly stakes of κA, κB, κC, κD, and κE, then the Kelly stake for the 1-team parlay consisting of only bet A would be:
κA * (1-κB) * (1-κC) * (1-κD) * (1-κE)

While the Kelly stake for the 3-team parlay consisting of bets A, B, and C would be:
κA * κB * κC * (1-κD) * (1-κE)
Much simpler, no?

Now if only I had used this logic in my old-school JavaScript Kelly calculator, it would run a hell of a lot faster. Well, c'est la vie.

If anyone's interested in the C-code for this (and or/a DLL linkable from Excel) let me know and I'll post it here. (Although as I've said, this is really more of a curiosity than anything else.)

SBR Founder Join Date: 8/28/2005

2. ganch, i'll give 450 more SBR pts if you can deliver that halves/quarter .5 pt calculator you've been working forever and a day on
Points Awarded:
 Ganchrow gave trixtrix 649 SBR Point(s) for this post.

3. Originally Posted by trixtrix
ganch, i'll give 450 more SBR pts if you can deliver that halves/quarter .5 pt calculator you've been working forever and a day on
That's sweet of you, but as I'm no longer in SBR's employ I'll have to pass for the time being.

SBR Founder Join Date: 8/28/2005

4. So, Ganch. Are you back?

...as in the semi-Mod position you were "once" in?

In any regard, welcome back. We really, really missed you.

5. Whoops, I guess we posted about the same time...

...anyway, glad to see you back regardless.

6. Originally Posted by Flying Dutchman
So, Ganch. Are you back?

...as in the semi-Mod position you were "once" in?

In any regard, welcome back. We really, really missed you.
Nope ... just wasting time that would be better spent working ...

SBR Founder Join Date: 8/28/2005

7. ganch 4 mod!!11

pls don't desert us again

8. Dude....you know your stuff !!!!!!!!!!!!

9. Originally Posted by Ganchrow
I've for some time been a bit embarrassed by my rather egregiousness overcomplication of the mathematical curiosity that is the unconstrained simultaneous independent event Kelly solution.
No need to be embarrassed sir. People here will let you know when your egregerious overcomplications are worthy of a red face, this is not one of those times.

Hope all is well, glad to see you back at SBR.

10. Ganch, we are busy gathering all your posts into a little red book which we will title: Sayings of Chairman Ganchrow. And then when we venture into Players Talk, we will hold it out in front of us to keep JJ's flesh eaters at bay.

11. Ganch should start his own forum and charge \$\$\$ to join

12. yes!!!!!!!!!!!!!!!!

read many of your posts from the time long ago sir. didn't even read this thread yet. just count me among the extremely excited that you may have possibly potentially returned.

13. Originally Posted by statnerds
yes!!!!!!!!!!!!!!!!

read many of your posts from the time long ago sir. didn't even read this thread yet. just count me among the extremely excited that you may have possibly potentially returned.
lol

14. I signed up to this forum just to say that the work behind this post by Ganchrow is incredible.

I've being trying to find a closed form solution for independent, simultaneous events (but without the parlay option) for sometime now. I got my first class in maths at uni and i'm certainly no slouch, but this problem is so messy and unpleasant to navigate that i'm not suprised it has eluded you for 3 years in a closed form.

I would really like to know how you solved this. You can see that it has most of the desired properties, although I would have thought that if exactly one individual Kelly stake is 1 - for a certain event - then the all the parlay Kelly stakes would be zero except for a single parlay of size 1 for the certain event. But as you've shown this wouldn't then maximise growth.

I've tried multiple times to work through a similar problem (excluding parlays), for a small number of independent, simultaneous events, but there are pages full of terms and I just never had the patience to go through it all. Every time I try it again I seem to forget just how many damn terms there are. It's fairly straight forward computationally and so I put it off and never thought I would see a closed form solution...

It's just so elegant and simple, I can't believe out of all the mess something so simple comes out the other side. Well I just wanted to say that although I didn't get to solve this one myself, it's very gratifying to finally see it.

One last thing, if it's possible to post the method you used to solve this problem then I would really appreciate it, as I havn't been able to see anything other than by brute force only...

Really great work!

15. Originally Posted by Peeig
Ganch should start his own forum and charge \$\$\$ to join
I would be there in a minute. WB Ganch

Hopefully this gets posted as I am on Post Review......

SBR
Bash 2012
Attendee 8/17/2012

16. I managed to find a method used to solve this problem in the paper http://www.afaanz.org/research/AFAANZ%200676.pdf

They introduce a lot of notation, which is very natural and helps to break down the otherwise enormous mess of the situation into a compact, succinct structure that makes the theory much more approachable.

If your method is better than this Ganchrow I would really like to see it, but their approach seems to be quite powerful and comprehensive and definately worth a look.

17. THANK YOU! This is awesome for a stats-deficient person like myself.

18. ok, a really dumb question. In Ganchrow's post, when he talks about a "parlay", does he literally mean it in the usual sense where you must win all events in the parlay in order to cash? Or is he simply referring to n simultaneous independent wagers which are won or lost individually?

19. Originally Posted by Ganchrow
κN, the unconstrained Kelly solution (for any Kelly multiplier > 0) consists of the 2N-1 parlays such that the wager on a given parlay comprised of all events in set S would be:
[nbtable] [tr] [td] [/td] [td] κi[/td] [td]   [/td] [td]   [/td] [td] × [/td] [td]   [/td] [td] [/td] [td] (1-κi) [/td] [/tr] [/nbtable]
OMFG

7of9 called. She said that was the second most beautiful thing she has ever seen.

20. Originally Posted by Chris_B
I managed to find a method used to solve this problem in the paper http://www.afaanz.org/research/AFAANZ%200676.pdf

They introduce a lot of notation, which is very natural and helps to break down the otherwise enormous mess of the situation into a compact, succinct structure that makes the theory much more approachable.

If your method is better than this Ganchrow I would really like to see it, but their approach seems to be quite powerful and comprehensive and definately worth a look.
It appears to be the same solution.

I'm not sure why they're so concerned about fixed % vig in that paper since that doesn't actually exist.

21. i dont get it...whats the purpose for the kelly calculator & can it give me the best value when i bet on a team with multiple pointspreads to find out which spread gives me the highest value on my wager

22. Am I missing something here?

Suppose you have a series of simultaneous independent bets where you have odds of 2 and the real probability of winning is 80% (unrealistic, i know).
In the case of a single bet the Kelly criterion states that you should bet 60% of your bankroll.

My expectation would be, that as the amount of possible bets approaches infinity, the total amount of your bankroll waged should go up and ultimatly reach all of your bankroll. However, I find the opposite to be the case if you follow the solution given above.

1 bet - 1 bet of 60% of BR
2 bets - 2 bets of 24% = 48% of BR
3 bets - 3 bets of 9.6% = 28.8% of BR

My guess would be that my expectation of what should be happening is wrong, but what's wrong with it?

23. ...

24. I attempted to quote the |-| symbol in your formulas. I couldn't capture it. This is not kappa I'm referring to. What is the mathematical terminology for the |-| symbol used in your formulas? The parallel vertical bars connected by a single horizontal line at the top. Thanks. --

25. @bookiebust: It's the product of a sequence: See http://en.wikipedia.org/wiki/Multipl...al_Pi_notation

26. Originally Posted by Dash2in1
Am I missing something here?

Suppose you have a series of simultaneous independent bets where you have odds of 2 and the real probability of winning is 80% (unrealistic, i know).
In the case of a single bet the Kelly criterion states that you should bet 60% of your bankroll.

My expectation would be, that as the amount of possible bets approaches infinity, the total amount of your bankroll waged should go up and ultimatly reach all of your bankroll. However, I find the opposite to be the case if you follow the solution given above.

1 bet - 1 bet of 60% of BR
2 bets - 2 bets of 24% = 48% of BR
3 bets - 3 bets of 9.6% = 28.8% of BR

My guess would be that my expectation of what should be happening is wrong, but what's wrong with it?
You need to go back and read Ganch's original post/comments that he linked to above. I (and probably most others) have long had a fundamental misunderstanding, I think, of the "simplification" provided here.

Long and short: there's nothing wrong with your calculations per se, but the rub is that Ganch's formulas are optimal assuming you bet not only the individual events ("1-team parlay" in his terminology) but ALSO all of the multiple-team parlays!! So in your example of 2 simultaneous independent events, you would need to bet .6*(1-.6)=.24 on each single, but also .6*.6=.36 on the two-team parlay, giving a total bet of 84% of payroll.

Sort of a depressing realization, really, since in the most common situations people are not going to be betting all the different parlay combinations if they have a handful of simultaneous individual plays. Looking on the bright side, you're really going conservative (from a Kelly standpoint) by not betting the multi-team parlays, which is probably a good thing. Of course, you're probably also giving up a good amount of your expected bankroll growth.