Kelly with non-binary outcomes

[Gonski]

Can anyone provide me with the formula for expected value (as opposed to expected growth) for n bets?

Additionally, does anyone have any literature on applying Kelly to non-binary outcomes? Preferably for multiple simultaneous, non-binary outcomes?

[Ganchrow]

Can anyone provide me with the formula for expected value (as opposed to expected growth) for n bets?

Since EV is additive across multiple bets (as is the log of expected growth), it's all rather straightforward:

If the EV for the i^th bet is given by EV_i, then the total EV across all n bets would be simple be:

Where if we assume m > n different possible outcomes such that X_i;j is the outcome of bet i given outcome j, and p_j is the probability of the occurrence of outcome j then for a given i such that 1 ≤ i ≤n:

Putting it all together then gives us:

}

Additionally, does anyone have any literature on applying Kelly to non-binary outcomes? Preferably for multiple simultaneous, non-binary outcomes?

Perhaps this post (attached spreadsheet) of Simultaneous Event Kelly (Mut Ex) with changing prices might be of some use at getting you started?

[Pancho sanza]

How does one adjust a group of 5 team teasers that all have the same leg in them.

Example:


WinProb20 %Payout5.0Bankroll50000

Say I have 4 teasers, all with the same win prob & payout as above.

Full Kelly says bet $1000 on each teaser.

However given that 1 team that is unique to all the teasers, this would increase risk somewhat.

Given that, how do I adjust the bet sizes?

[Gonski]

Thanks very much for your reply Ganchrow.
I think I have it.
If I use an example you used in a previous blog:
For a player betting at an edge of 5% and odds of -200, the proper Kelly stake is 10%. Over 100 bets, he has an expected return of 64.7% with a 36.7% probability of not turning a profit and a 3.4% probability of losing two-thirds or more of his stake.

For a player betting at the same 5% edge but at odds of +400, were he to bet the 10% stake of the -200 player, while he’d have the identical 64.7% expectation, he’d have a 73.5% probability of no profit, while his probability of losing two-thirds or more of his stake would be 55.8%.

(If no formulae are visible below, then I don't know how to copy formulae drafted in Word to the Blog, any suggestions welcome.) Please see attacment for full copy of reply.

Then: 12EV=i=1nj=1mpj*Xi;j'>
Where: 12pj=n!j!n-j!* pj1-pn-j'>
12Xi;j=1+O-1* k%j* 1-k%n-j-1'>

And: n = number of bets
j = outcome or number of wins
O = Decimal Odds
k% = % of bankroll bet
p = probability of winning = ( E +1 ) / O
m = n +1
Then I exactly match your results – thank you!

Where I was getting stuck was that I kept thinking that win sequence had to have an impact on EV. What dawned on me is that it is only relevant if you bet a set dollar (as opposed to %) amount
12>1n*B'> OR you start factoring in the time value of money (which is only relevant over longer periods of time or when interest rates are very high).

Can I confirm that the formula for Expected Growth (EG) is:
12EG=(1+O-1*k%)n*p* (1-k%)n1-p-1'>
And that the Expected Growth for the above two examples is:
EGa = 28.98%
EGb = -71.57%