Optimal Bet Sizing Question

[TK]

Pardon me if this is too easy for all the smart folks on here, but could use a bit of guidance.

Assume that I have a bet avaiable which has the following payoff structure:

For the "bet", I need to risk $2.70,

And there are only two outcomes:

Win: I receive $2.30
Lose: I lose the $2.70 that I put at risk

Probablity of winning this bet over the long run is 60%, but for any single bet this can vary between 50% and 70% (i.e. 90% of all possible outcomes fall within this range - can asusme normally distributed if it helps)

(So, it follows that the probability of losing this bet over the long run is 40%, but again, for any single bet, this can vary between 30% and 50%)

My starting bankroll is $1000

I can place this bet over and over again (sequentially, not simultanesouly). Obviously, this bet is +EV, but the outcome has some variance. My goal is to maximize the growth of my bankroll, and not go bust.

I've attempted to simulate, but I am sure there is a closed "solution". In this case, I can be somewhat confident of the probability of winning, but full Kelly is not appropriate due to the variance of the outcome. Can guess at a fraction, but if possible would like to be more scientific.

How would I size this bet - What % of my bankroll do I put on this bet?

Any guidance would be much appreciated.

Thanks,

[u21c3f6]

I am going to assume that you have no way of knowing when a wager may fall into the 50 or 70 % range but that your overall win % is 60%.

All Kelly cares about for optimum bankroll growth is that at some point in the future you will achieve the edge and win rate that you plug into the formula. In this case full-Kelly would be approx 13% of bankroll. As long as you achieve a 60% win rate at some point at 2.3/2.7, you will optimize your bankroll by wagering 13.04% (to be more exact) of your bankroll per wager.

Now, because of variance and/or the violent swings in bankroll that full-Kelly may produce, you may want to consider a fractional Kelly. IMO there is no one right answer to this. I personally use half-Kelly and it serves me well. My bankroll has less violent swings and I don't have to be overly concerned that my edge and/or win % is exact. The biggest problem IMO in using Kelly is over-estimating your inputs as overbetting using Kelly can and will decimate your bankroll.

Joe.

[TK]

I am going to assume that you have no way of knowing when a wager may fall into the 50 or 70 % range but that your overall win % is 60%.

Correct. I can predict that, over the long run, this bet will hit at 60%.

Interesting, and intuitively it makes sense. Kelly would only care if I could not predict the long term win rate. My bet size is automatically adjusted for the next bet to make up for the variance with each individual event.

I've left out some of the finer points to this bet (e.g. sometimes the payoff is 2.5/2.5 or 2.2/2.8, but I have enough to go on now).

Many Thanks.

[Blax0r]

I am going to assume that you have no way of knowing when a wager may fall into the 50 or 70 % range but that your overall win % is 60%.

All Kelly cares about for optimum bankroll growth is that at some point in the future you will achieve the edge and win rate that you plug into the formula. In this case full-Kelly would be approx 13% of bankroll. As long as you achieve a 60% win rate at some point at 2.3/2.7, you will optimize your bankroll by wagering 13.04% (to be more exact) of your bankroll per wager.

Now, because of variance and/or the violent swings in bankroll that full-Kelly may produce, you may want to consider a fractional Kelly. IMO there is no one right answer to this. I personally use half-Kelly and it serves me well. My bankroll has less violent swings and I don't have to be overly concerned that my edge and/or win % is exact. The biggest problem IMO in using Kelly is over-estimating your inputs as overbetting using Kelly can and will decimate your bankroll.

Joe.

I'm with Joe; you can just use your expected win% when calculating your kelly stake. If you get some more info on the distribution of win% (which would probably be tough though), then just be sure to re-calculate your E[w%].