Confidence levels for interval resizing

Posted by David D'Aquin on June 27, 1997 at 13:21:58: In Reply to: Long run formula for Kelly betting posted by David D'Aquin on June 27, 1997 at 08:41:49:

Ok, this will be a specific BJ example for the long run calculations. For interval resizing I'm going to simply give a generalization that goes with the procedure for finding the confidence levels for ideal Kelly resizing. This is based on resizing bets when 20 to 30% of the bank is won or lost. When resizing at fixed intervals, the number of hands required to reach any particular confidence level for winning is approximately 10% fewer than the number required for perfect Kelly betting. It takes fewer hands because the bets are resized less frequently. For interval resizing plans, for each increase in resize frequency you increase your rate of log growth and prolong the time horizon for reaching any confidence level of winning.

Since most people here have Don's book, I'll use an example from BJ Attack, p198, tbl 10.4, 6 decks, 5.5/6,S17, 1 to 8 spread. The average bet is 1.74 units, the standard deviation per hand is 2.91 units, win rate per hand is .0176 units. First let's see the number of bets for fixed bet sizes to get into the long run:

Long Run= [(2.91 * 3) / .0176]^2= 246,039 hands. For perfect Kelly:
Long Run= [(2.91 * 6) / .0176]^2= 984,155 hands.

Using a program I wrote to find confidence levels when resizing at fixed intervals, I used a 479 unit bank and resized when 100 units were won or lost. The number of hands required to reach a 99.87% certaintly of winning was just under 900,000, about 10% less than that required for perfect Kelly.

The same procedure can be used to find other confidence levels. Suppose you wanted to know how many hands were required to have an 84.13% confidence level of winning when resizing at intervals with the same starting bank. (84.13% is confidence within 1 standard deviation). In this case you'd find the number of hands required to overcome 2 standard deviations for fixed bet sizes, and then reduce it by 10%. It came out almost exactly perfect in all cases I tested. So for this game, using a 479 unit bank and resizing when 100 units are won or lost, the number of hands required to have an 84.13% confidence of winning would be:

84.13% confidence= {[(2.91 * 2) / .0176]^2} * .9 = 98,415.42

That's the medicine you have to take to be a Kelly bettor. The pluses are reduce chance of catastrophic loss (ruin) and maximaztion of log growth. The drawback is the prolonged time required to get into the long run.

I should point out also that these calculations only work for the left side of the distribution. I have some ideas that work for win projections for the right side but haven't found the relationships to fixed bet sizing that make them easy.

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