Oy! Sorry, I have to do it my way. Let x == fraction of unit bet on Red , Let y == fraction of unit bet on R-14 , Let z == fraction of unit bet on R-16. x + y + z == 1 E == (4/37)(35z+x-y) + (3/37)(35y+x-z) + (16/37)(x-y-z) + (14/37)(-1) Now comes the hard part: 0 == 4(35z+x-y)/(1+k*(35z+x-y)) + 3(35y+x-z)/(1+k*(35y+x-z)) + 16(x-y-z)/(1+k*(x-y-z)) - 14/(1-k) Solve for k eek The quick-and-dirty solution for k (and probably a serious underestimate in this case) is k1 == (4(35z+x-y) + 3(35y+x-z) + 16(x-y-z) - 14) / (4(35z+x-y)^2 + 3(35y+x-z)^2 + 16(x-y-z)^2 + 14) Instead I strongly recommend a numerical equation cruncher like Excel's Solver routine.
I managed to crash my machine twice, because I was doing so many simulations, but I think I've resolved the discrepancy. Your top result (maximizing the F(), numerically): {R-16: 8%, R-14: 5%, Red: 11%} My top result (Monte-Carlo, utility function is the average log(endingBankroll)): {R-16: 8%, R-14: 5%, Red: 11%} (same as yours) My top result (Monte-Carlo, utility function is the median log(endingBankroll)): {R-16: 8%, R-14: 8%, Red: 6%} Kut2k2, you appear to be optimizing for the absolute profit, so your result would be different.
Here's an amusing thing I found: if you add the option to bet on black and/or green, the solver gives almost the same solution, but puts 0.6% on green and ups red to 11.4%. It appears that green, even with a slightly negative expectation, is such a nice hedge that you're better off including a tiny bit and betting more. This isn't supper surprising I suppose - there are analogous results in portfolio theory where including negative expectation, negatively correlated "assets" actually improves things. Black is so horribly negative expectation you could never use it that way though.
Did you arrive at 1/5 to 1/10 Kelly analytically, or is it just a gut feel? Do you use discrete or continuous Kelly?
The Kelly equation I posted indicates the total betting fraction (k) that grows a given asset allocation {x,y,z} the fastest in this roulette game. So it doesn't find the optimal allocation {x_opt, y_opt, z_opt}. x_opt, y_opt and z_opt are the values that maximize k*E.
It's based on monte carlo equity curves with various kelly fraction bets. 1/5 to 1/10 was the range where they started looking "acceptable" to my eye - no stomach churning drawdowns. Most of the stuff I do has discrete outcomes, so I use discrete Kelly (or the general log/maximization form thereof when there are weird correlations like in this case).
No. Reread the definitions. They're based on a unit bet. To put it another way, x == R/(R+F+S), and so on.
Kelly == 0.1963808 Expectation == 1.81572481 k*E == 0.35657349 There is overbetting here (betting fraction == 22% > k) but the proportions are very good.