I've already qualified this. If the goal is to maximize the return with no regard to risk, and if you are willing to destroy hundreds of accounts in this game, then betting 100% is indeed the right strategy. For example, there is 1 in 791 chance that you hit R16 three times in a row, which would pay about $43 million. So, even if you lose everything in the other 790 trials (which would be 790 * 1K = $790K), you still come out well ahead. If, instead, you are treating your account as the only one that you have, the optimal strategy should be much more conservative. As my last simulation shows, the best risk-adjusted strategy would be betting 1% on Red. We can argue about what constitutes risk and how to measure it, but it looks undeniable that the "optimal strategy" is a function of the risk tolerance.
Please, prove it. Red-16 1-spin scenario has the odds stacked against it, 89.2% chances of losing your bet. You have to believe in Santa to make that *one* bet. But if you do, then just bet whatever you are comfortable losing. This not a matter of long-term profit maximization. One last comment as I have other things to do this week-end: whet makes you think that the odds of hitting Red-16 are best on the 1st spin than on the 2nd? Assuming no win, your 1st bet is $80, the second one $73, the 3rd one $67, etc. - this makes absolutely no sense, since all these spins have the same win probability. Stop the dogma, and start thinking.
Kelly prescribes betting 8% of the bankroll on Red-16. 8% of $1000 is $80. With that single $80 bet, you have an 89.2% probability of losing $80, and a 10.8% probability of making $2800 (80 * 35). Therefore, the expectancy of that single bet is: 0.892 * (-80) + 0.108 * (+2800) = $231. QED. There is nothing wrong with Kelly, either when using it with a single bet or the multiple ones. It's just that it aims to maximize the geometric rate of capital growth, which is not what most traders seek.
in this case , i would recomend the simple martingale progression.. as with 23 reds on the table the chance of hitting red is 60+% which is enough to overcome the house edge.... ergo bet 1 unit on red and on each loss do the martingale...
Hey, Visaria, can you point me to the way of calculating the relationship between Kelly and the standard deviation? What I am looking for is a way to maximize the (growthRate/stdev ) ratio. That is, the question is, by how much do I need to reduce Kelly so that I can maximize the quantity which I call the "risk-adjusted Kelly"? Thanks.
Nonlinear has already given an explanation which you need to read and understand. The brief answer is that you have fallen into some layman's fallacy. True story to illustrate: a couple of years ago, i was dating this young, very sexy girl who, unlike me, did not have any money and was not trained in the dark art of mathematics (which you refer to as dogma lol). I had that day received from an online casino a particularly juicy offer of a 50% cashback on any losses up to £1000. Between rounds (wink, wink), i explained the offer to her and asked her what she thought about it. She said she wouldn't be interested in participating since a) she couldn't afford to lose (even though i explained that she could bet a small amount that she was comfortable with rather than a bet of £2000) and more importantly b) because it was a ONE OFF deal, the casino would not give her another chance if she lost on the offer. Clearly she thought there was only value if there was a string of similar offers, she just couldn't grasp that there was value in just doing one offer.
The general relationship between an increase in bet size and variance is if you double your bet size, the variance increases by a factor of root 2 and vice versa. Hope that helps in what you are looking for.