[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?

[tacomax]

I've just paged Ganchrow for you.

[Thremp]

Originally Posted by tacomax

I've just paged Ganchrow for you.

Heh. Maybe I should've just PMed him.

I've looked into it a shade. Would be interested in any ideas for approximations as well. I came up with some rough figures, but would atleast like a second opinion before I move forward with my bastardized math.

[bigboydan]

Originally Posted by tacomax

I've just paged Ganchrow for you.

I was gonna say I would have if you didn't already do it.

[Ganchrow]

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?

There's no simple closed-form solution to this problem.

In the case of 3 candidates, with win probabilities p₁, p₂, and p₃, and teasers that payout at decimal odds of o₂ for 2-team parlays and o₃ for 3-team parlays, utility as a function of the weightings of the three 2-team teasers (x₁₂, x₁₃, x₂₃) and the one 3-team teasers would look like this:

U(x₁₂,x₁₃,x₂₃,x₁₂₃) =
p₁*p₂*p₃*log(1+(o₂-1)*(x₁₂+x₁₃+x₂₃) + (o₃-1)*x₁₂₃) +
p₁*p₂*(1-p₃)*log(1+(o₂-1)*x₁₂ - x₁₃ - x₂₃ - x₁₂₃) +
p₁*p₃*(1-p₂)*log(1+(o₂-1)*x₁₃ - x₁₂ - x₂₃ - x₁₂₃) +
p₂*p₃*(1-p₁)*log(1+(o₂-1)*x₂₃ - x₁₃ - x₁₂ - x₁₂₃) +
[p₁*(1-p₂)*(1-p₃) + p₂*(1-p₁)*(1-p₃) + p₃*(1-p₁)*(1-p₂) + (1-p₁)*(1-p₂)*(1-p₃)]
*log(1-x₁₂ - x₁₃ - x₂₃ - x₁₂₃)

U is then of course maximized with respect to (x₁₂, x₁₃, x₂₃, x₁₂₃) ≥ 0 subject to the budget constraint of x₁₂ + x₁₃ + x₂₃ + x₁₂₃ ≤ 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.

[rjp]

Originally Posted by Ganchrow

Results were identical (and much more quickly obtained) using both MINOS and IPOPT, two considerably more robust (and freely available) nonlinear optimization engines.

Just how fast do these run? I'm using a library with built in optimization algorithms that seem to converge quickly, and I only run into trouble if I've got a large number of bets such that it takes a long time to build the function to maximize.

Lets say 10 simultaneous wagers... how long does it take to maximize the growth function?

[Ganchrow]

Originally Posted by rjp

Just how fast do these run? I'm using a library with built in optimization algorithms that seem to converge quickly, and I only run into trouble if I've got a large number of bets such that it takes a long time to build the function to maximize.

Lets say 10 simultaneous wagers... how long does it take to maximize the growth function?

Obviously, convergence time will will depend on among other things, the degree of correlation, but in general it will handle 10 simultaneous events in a under a few seconds.

11 or 12 wagers is where it tends to really slow down, to the point where I can't even get it to run with the available memory (without making numerous dubious approximations in the utility curvce) for 13 or more simultaneous events.

[rjp]

Interesting. Any idea what they're using to perform the optimization?

The algorithms I was using to maximize the growth function are terrible when I try to apply something other than an even money bet, so I'm now working on a solution using simulated annealing to see if I have more success.

[DrunkenLullaby]

Originally Posted by Thremp

How would I allocate my bankroll from ranging from 2-6 teamers each weekend?

I would start by allocating $0 to the Jazzette family of books.

[Ganchrow]

Originally Posted by rjp

Interesting. Any idea what they're using to perform the optimization?

The algorithms I was using to maximize the growth function are terrible when I try to apply something other than an even money bet, so I'm now working on a solution using simulated annealing to see if I have more success.

The IPOPT algorithm uses "primal-dual path-following interior point methods".

"For problems with nonlinear constraints, MINOS uses a sparse SLC algorithm (a projected Lagrangian method, related to Robinson's method). It solves a sequence of sub problems in which the constraints are linearized and the objective is an augmented Lagrangian (involving all nonlinear functions). Convergence is rapid near a solution."

[bigboydan]

Originally Posted by DrunkenLullaby

I would start by allocating $0 to the Jazzette family of books.

Ain't that the truth.

Most people don't even like to talk about who accepts correlated parlays on the forums in fear they might ruin it for them.

[RickySteve]

The original post is talking about related simultaneous wagers, not correlated parlays.