Kelly Explanation (Corrected)

Editorial Comment: The post below corrects three typos that occurred in my original post. Two errors were inconsequential (spelling), but the third might have caused some confusion: The original post stated "KF<2.0 You go broke faster", while this corrected post states: "KF>2.0 You go broke faster" (> means "greater than"; < means "less than"). Sorry for any misunderstanding. I look forward to your comments, corrections or suggestions. Thanks...Freddie

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KELLY EXPLANATION

The Kelly Criterion was originally developed for communications theory. It can be extended to gambling, and used to determine the bet size that will maximize the LONG TERM growth rate (compound growth rate) of your bankroll. The original Kelly formula assumed that you either win or lose your bet. In this case:

Kelly Bet = Expectation x Bankroll

(note: Expectation is the expected $win per $bet)

Blackjack is more complex than a win-or-lose bet (due to doubling, splitting, insurance, naturals etc.). For games with complex expectations, and when the bet size is small relative to the bankroll, Kelly can be defined as follows:

Kelly Bet = Expectation x Bankroll / Variance

(note: Variance is Standard Deviation squared)

(note: Variance and Expectation are defined relative to initial bet for a hand)

If a player strictly follows Kelly, they will resize their bet each time their bankroll or expectation changes. If that player has exact information about their bankroll and expectation, and if they follow Kelly exactly, then they will never go broke and they will maximize their long term compound growth rate.

However, under casino conditions it is not possible to strictly follow Kelly. First, the expectation we think we have is only an estimate, and our actual expectation is probably lower than we believe, due to counting, playing and betting errors. In addition, most casinos do not allow perfectly sized bets (for example $36.41). And finally, we sometimes make deliberate errors in the interest of camouflage.

These are some of the reasons that the Kelly criterion should be only a starting point for determining your ideal bet size. In addition it is important to understand how your growth rate and its variability relate to your bet size. This can be seen by examining the effect of Kelly Fraction (KF), which is defined as the size of the actual bet as a fraction of the full Kelly Bet.

KF=0.0 You are betting nothing (therefore you don't win anything)

KF=0.5 Bankroll grows at 75% of optimal - Low Variability

KF=1.0 Optimal bankroll growth - Intermediate Variability

KF=1.5 Bankroll grows at 75% of optimal - High Variability

KF=2.0 Bankroll eventually goes to zero (you go broke)

KF>2.0 You go broke faster

Whenever the Kelly fraction is greater than one, two undesirable things happen: growth rate goes down and variability goes up. So there is absolutely no benefit in using a Kelly Fraction greater than one, and it makes a lot of sense to keep the Kelly fraction significantly lower than that. My understanding is that professional gamblers typically use 0.3 as their Kelly Fraction. At this betting level, their bankroll will grow at about 50% of the optimal rate, variability will be relatively low, and there will be an allowance for the inevitable errors that occur in casino play.