Simultaneous-bet Kelly staking -- the simplest case

ganchrow · Jan 26, 2007

In case anyone's interested, following is the closed-form solution for simultaneous bet Kelly staking, given the simplest case where the single-bet Kelly stakes for each simultaneous bet are equivalent, all bets are uncorrelated, and the only bound on wagers is the size of the bankroll.

I've yet to work out the closed-form solution for the general case of correlation and differing single-bet Kelly stakes (if it even exists). That would obviously be considerably more difficult, and probably better left calculated by an optimizer.

If anyone's really interested in seeing the proof (not that I expect that), I could probably write it up. I've also created a very simple example spreadsheet as proof of concept. It will work for up to 255 bets that meet the above criteria.

<hr>

Given n uncorrelated binary bets, we define the Kelly-optimal allocation as the set of weightings for each of the 2n-1 n-or-fewer-team parlays (where a single bet is considered a 1-team parlay) that can be created from the n-single bets, which maximizes the expected logarithm of the bankroll.

Let oi = decimal odds on the ith bet,
Let pi = win probability of the ith bet,
Let ki = ith single-bet Kelly stake = pi + (1 - pi)/(1 - oi),

If ki = kj for all i,j on the interval [1,n],

then the Kelly-optimal weighting of each and every m-team parlay (as a percentage of the total bankroll), Knm, is given by:

Code:

       n
 K<sub>n</sub><sup>m</sup> = [SIZE=6]∑[/SIZE] combin(n-m, m-i) * k<sup>1+n-m</sup> * (-1)<sup>m-i</sup>
       i=n-m+1

<hr>
Cross-posted at SBR Forum here.

FatTony · Jan 26, 2007

my cat's breath smells like cat food.:howdy:

LogansRun · Jan 26, 2007

ganch, I'd love to see the proof!

There is a program out there {bet optimizer by Teppo Salonen} that calculates the Kelly bet sizes for multiples but it has a limitation of only 7 events. Can you publish the generalized formula or at least give me some idea of how to proceed with it, excel programming wise? I don't recall if this program allows correlations.

ganchrow · Jan 26, 2007

Simultaneous-bet Kelly staking -- the next case

This is the more general case where we relax the constraint that all single-bet Kelly stakes need to be equal. For the sake of sanity, the weightings are defined recursively.

<hr>

Given n uncorrelated binary bets, we define the Kelly-optimal allocation as the set of weightings for each of the 2n-1 n-or-fewer-team parlays (where a single bet is considered a 1-team parlay) that can be created from the n-single bets, which maximizes the expected logarithm of the bankroll.

Let oi = decimal odds on the ith bet,
Let pi = win probability of the ith bet,
Let ki = ith single-bet Kelly stake = pi + (1 - pi)/(1 - oi),

Define κ(n,m,{B}) as the sum of the Kelly optimal weights for all m-team parlays made up of all bets included the set {B}, then

Code:

κ(n,m,{[B]B[/B]}) = [SIZE="6"]∏[/SIZE] [FONT=Symbol]k[/FONT]<sub>i</sub>                       (for n = m)
           i Є {[B]B[/B]}

                       n 
κ(n,m,{[B]B[/B]}) = [SIZE="6"]∏[/SIZE] [FONT=Symbol]k[/FONT]<sub>i</sub>  -  [SIZE="6"]∑[/SIZE] κ(n,i,{[B]B[/B]})     (for n > m)
           i Є {[B]B[/B]}   i=m+1

Example:

Code:

[INDENT]given:
[FONT=Symbol]k[/FONT]<sub>1</sub> = 1%
[FONT=Symbol]k[/FONT]<sub>2</sub> = 2%
[FONT=Symbol]k[/FONT]<sub>3</sub> = 3%
[FONT=Symbol]k[/FONT]<sub>4</sub> = 4%
[FONT=Symbol]k[/FONT]<sub>5</sub> = 5%

κ(5,5,{[B]1,2,3,4,5[/B]})  = (weighting of the 5-team parlay as % of bankroll)
  = [FONT=Symbol]k[/FONT]<sub>1</sub>*[FONT=Symbol]k[/FONT]<sub>2</sub>*[FONT=Symbol]k[/FONT]<sub>3</sub>*[FONT=Symbol]k[/FONT]<sub>4</sub>*[FONT=Symbol]k[/FONT]<sub>5</sub> 
  = 1%*2%*3%*4%*5%
  = 0.0000012%

κ(5,4,{[B]1,2,3,4[/B]})  = (weighting of all 4-team parlays consisting of bets {1,2,3,4,5} as % of bankroll)
  = [FONT=Symbol]k[/FONT]<sub>1</sub>*[FONT=Symbol]k[/FONT]<sub>2</sub>*[FONT=Symbol]k[/FONT]<sub>3</sub>*[FONT=Symbol]k[/FONT]<sub>4</sub> - κ(5,5,{[B]1,2,3,4[/B]})
  = 1%*2%*3%*4% - κ(5,5,{[B]1,2,3,4,5[/B]})
  = 0.0000228%

κ(5,4,{[B]1,2,3,5[/B]})  = (weighting of the 4-team parlays consisting of bets {1,2,3,5} as % of bankroll)
  = [FONT=Symbol]k[/FONT]<sub>1</sub>*[FONT=Symbol]k[/FONT]<sub>2</sub>*[FONT=Symbol]k[/FONT]<sub>3</sub>*[FONT=Symbol]k[/FONT]<sub>5</sub> - κ(5,5,{[B]1,2,3,5[/B]})
  = 1%*2%*3%*5% - κ(5,5,{[B]1,2,3,4,5[/B]})
  = 0.0000288%

κ(5,3,{[B]1,2,3[/B]})  = (weighting of the 3-team parlay consisting of bets {1,2,3} as % of bankroll)
  = [FONT=Symbol]k[/FONT]<sub>1</sub>*[FONT=Symbol]k[/FONT]<sub>2</sub>*[FONT=Symbol]k[/FONT]<sub>3</sub> - κ(5,4,{[B]1,2,3[/B]}) - κ(5,5,{[B]1,2,3[/B]})
  = 1%*2%*3% - κ(5,4,{[B]1,2,3,4[/B]}) - κ(5,4,{[B]1,2,3,5[/B]}) - κ(5,5,{[B]1,2,3,4,5[/B]})
  = 0.00054720%

etc.
[/INDENT]

ganchrow · Jan 27, 2007

LogansRun said:
ganch, I'd love to see the proof!

There is a program out there {bet optimizer by Teppo Salonen} that calculates the Kelly bet sizes for multiples but it has a limitation of only 7 events. Can you publish the generalized formula or at least give me some idea of how to proceed with it, excel programming wise? I don't recall if this program allows correlations.

No, it does not appear to support correlationed bets.

trixtrix · Jan 27, 2007

ganchrow said:
This is the more general case where we relax the constraint that all single-bet Kelly stakes need to be equal. For the sake of sanity, the weightings are defined recursively.

<hr>
Given n uncorrelated binary bets, we define the Kelly-optimal allocation as the set of weightings for each of the 2n-1 n-or-fewer-team parlays (where a single bet is considered a 1-team parlay) that can be created from the n-single bets, which maximizes the expected logarithm of the bankroll.

Let oi = decimal odds on the ith bet,
Let pi = win probability of the ith bet,
Let ki = ith single-bet Kelly stake = pi + (1 - pi)/(1 - oi),

Define κ(n,m,{B}) as the sum of the Kelly optimal weights for all m-team parlays made up of all bets included the set {B}, then

Code:

κ(n,m,{[B]B[/B]}) = [SIZE=6]∏[/SIZE] [FONT=Symbol]k[/FONT]i (for n = m) i Є {[B]B[/B]} n κ(n,m,{[B]B[/B]}) = [SIZE=6]∏[/SIZE] [FONT=Symbol]k[/FONT]i - [SIZE=6]∑[/SIZE] κ(n,i,{[B]B[/B]}) (for n > m) i Є {[B]B[/B]} i=m+1

Example:

Code:

[INDENT]given: [FONT=Symbol]k[/FONT]1 = 1% [FONT=Symbol]k[/FONT]2 = 2% [FONT=Symbol]k[/FONT]3 = 3% [FONT=Symbol]k[/FONT]4 = 4% [FONT=Symbol]k[/FONT]5 = 5% κ(5,5,{[B]1,2,3,4,5[/B]}) = (weighting of the 5-team parlay as % of bankroll) = [FONT=Symbol]k[/FONT]1*[FONT=Symbol]k[/FONT]2*[FONT=Symbol]k[/FONT]3*[FONT=Symbol]k[/FONT]4*[FONT=Symbol]k[/FONT]5 = 1%*2%*3%*4%*5% = 0.0000012% κ(5,4,{[B]1,2,3,4[/B]}) = (weighting of all 4-team parlays consisting of bets {1,2,3,4,5} as % of bankroll) = [FONT=Symbol]k[/FONT]1*[FONT=Symbol]k[/FONT]2*[FONT=Symbol]k[/FONT]3*[FONT=Symbol]k[/FONT]4 - κ(5,5,{[B]1,2,3,4[/B]}) = 1%*2%*3%*4% - κ(5,5,{[B]1,2,3,4,5[/B]}) = 0.0000228% κ(5,4,{[B]1,2,3,5[/B]}) = (weighting of the 4-team parlays consisting of bets {1,2,3,5} as % of bankroll) = [FONT=Symbol]k[/FONT]1*[FONT=Symbol]k[/FONT]2*[FONT=Symbol]k[/FONT]3*[FONT=Symbol]k[/FONT]5 - κ(5,5,{[B]1,2,3,5[/B]}) = 1%*2%*3%*5% - κ(5,5,{[B]1,2,3,4,5[/B]}) = 0.0000288% κ(5,3,{[B]1,2,3[/B]}) = (weighting of the 3-team parlay consisting of bets {1,2,3} as % of bankroll) = [FONT=Symbol]k[/FONT]1*[FONT=Symbol]k[/FONT]2*[FONT=Symbol]k[/FONT]3 - κ(5,4,{[B]1,2,3[/B]}) - κ(5,5,{[B]1,2,3[/B]}) = 1%*2%*3% - κ(5,4,{[B]1,2,3,4[/B]}) - κ(5,4,{[B]1,2,3,5[/B]}) - κ(5,5,{[B]1,2,3,4,5[/B]}) = 0.00054720% etc. [/INDENT]