Wouldnt kelly be 0.010101 period then? I mean 0.01 / (2*0.495) = 0.010101010...
Youre right... but this max win on changed chance has to be changed automatically to still meet kelly. Thats something hopefully will be set up for obvious reasons.
But what i mean is... lets assume all bets are 50%. Kelly 1% would only be the optimum for an investor when a player plays to full max profit. But most of the bids are way way lower than the amount needed for max profit. So the risks that 1% involves are never met for the single investor. He could easily use 10x kelly and still remain way under the 1% of his personal investment used in each bet since only a small portion of the house is used in each bet. The individual investor would have to set a way higher kelly to rally maximize his profits. Though of course he has to hope that its not often happening that someone really plays 1% max profit of the house since then 10% of his investment would be involved. Which is a high risk. I mean there should be a sweat spot for the real kelly value for an individual investor. And thats not 1% kelly. Its higher. I guess im not smart enough for calculating...
We can't make investors go up to 10x kelly on small bets because we we can't force users (players) to increase their bets if they bet small. A small bet is a small bet: every investor takes a share of the gains proportional to their investment; no bets are allowed which will exceed the 1% maximum optimum bet since any bets that exceed it will reduce long term gain.
No need to simulate, the math isn't that bad:
when you win, you win:
(1+f*b)*current_bankroll
where f is your kelly percentage, b is how much you win when you win (minus original bet, e.g., "1" for a 50:50 roll)
when you lose, you lose:
(1-f*a)*current_bankroll
where a is your wager.
is if you win p*N times, and lose q*N times (where p is the probability you win, q is the probability you lose, and N is the total number of games played), your ending money is:
endbankroll=(1+f*b)^(p*N) * (1-f*a)^(q*N) * starting bankroll
your percentage profit is then:
C=(1+f*b)^(p*N) * (1-f*a)^(q*N)
you can plug in any numbers you want for any type of roll, house edge, or kelly fraction and you will see what you get
the optimum f to maximize C is simply the derivative of C (or log of C since log is a monotonic function) respect to f set to 0.
optimum f = p/a-q/b
C is negative (that is, a losing strategy when)
p*log(1+f*b)+q*log(1-f*a) is negative, even if you have a positive house edge.
