Quote:
Originally Posted by Thremp
I'm having trouble managing to implement Kelly Criterion for wagers that are correlated such as Basic Strategy Teasers. If I bet Kelly units... How would I allocate my bankroll from ranging from 2-6 teamers each weekend?
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There's no simple closed-form solution to this problem.
In the case of 3 candidates, with win probabilities p<sub>1</sub>, p<sub>2</sub>, and p<sub>3</sub>, and teasers that payout at decimal odds of o<sub>2</sub> for 2-team parlays and o<sub>3</sub> for 3-team parlays, utility as a function of the weightings of the three 2-team teasers (x<sub>12</sub>, x<sub>13</sub>, x<sub>23</sub>) and the one 3-team teasers would look like this:
U(x<sub>12</sub>,x<sub>13</sub>,x<sub>23</sub>,x<sub>123</sub>) =
p<sub>1</sub>*p<sub>2</sub>*p<sub>3</sub>*log(1+(o<sub>2</sub>-1)*(x<sub>12</sub>+x<sub>13</sub>+x<sub>23</sub>) + (o<sub>3</sub>-1)*x<sub>123</sub>) +
p<sub>1</sub>*p<sub>2</sub>*(1-p<sub>3</sub>)*log(1+(o<sub>2</sub>-1)*x<sub>12</sub> - x<sub>13</sub> - x<sub>23</sub> - x<sub>123</sub>) +
p<sub>1</sub>*p<sub>3</sub>*(1-p<sub>2</sub>)*log(1+(o<sub>2</sub>-1)*x<sub>13</sub> - x<sub>12</sub> - x<sub>23</sub> - x<sub>123</sub>) +
p<sub>2</sub>*p<sub>3</sub>*(1-p<sub>1</sub>)*log(1+(o<sub>2</sub>-1)*x<sub>23</sub> - x<sub>13</sub> - x<sub>12</sub> - x<sub>123</sub>) +
[p<sub>1</sub>*(1-p<sub>2</sub>)*(1-p<sub>3</sub>) + p<sub>2</sub>*(1-p<sub>1</sub>)*(1-p<sub>3</sub>) + p<sub>3</sub>*(1-p<sub>1</sub>)*(1-p<sub>2</sub>) + (1-p<sub>1</sub>)*(1-p<sub>2</sub>)*(1-p<sub>3</sub>)]
*log(1-x<sub>12</sub> - x<sub>13</sub> - x<sub>23</sub> - x<sub>123</sub>)
U is then of course maximized with respect to (x<sub>12</sub>, x<sub>13</sub>, x<sub>23</sub>, x<sub>123</sub>) ≥ 0 subject to the budget constraint of x<sub>12</sub> + x<sub>13</sub> + x<sub>23</sub> + x<sub>123</sub> ≤ 1. Similar logic would be used given a larger number of underlyings and larger teaser sizes.
It bears mentioning that this process is identical to that which would be used to determine any simultaneous bet staking with fixed parlay odds and the weightings on 1-team parlays constrained to zero.
As I said, there's no simple closed-form solution to the problem, but a numeric solution can be easily obtained computationally using an optimization package such as Excel Solver.
Using Excel Solver and assuming a 72% win probability for each teaser leg and payout odds for 2-6 team teasers of +100, +180, +300, +465, +750, respectively, one finds that the full-Kelly stake for each of the fifteen 2-team teasers would be 0.5666% of bankroll, the full-Kelly stake for the one 6-team teaser would be 1.3224% of bankroll, with all other full-Kelly teaser stakes at zero. For fifth-Kelly, the 2-team and 6-team stakes would be 0.1160% and 0.2322%, respectively, with all other stakes zero. Results were identical (and
much more quickly obtained) using both MINOS and IPOPT, two considerably more robust (and freely available) nonlinear optimization engines.
Kelly Criterion for Correlated Wagers
Linear Mode
