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Playing It Smart: How to size up your bankroll26 February 2009
Winning in a repetitive bet milieu like a casino game often takes a big enough bankroll to weather the cold streaks. It's a matter of probabilities, not absolutes. The question isn't what multiple of your average bet is needed to play a certain game. Rather, how big a bankroll will yield some percentage confidence of surviving the normal downswings of the game with a given betting strategy.
Few solid citizens appreciate the influence of bankroll size. Fewer, yet, realize how funding needs are altered with considerations such as average bet, edge, volatility, and number of coups they hope to experience during a session or visit.
Each game, in some cases each way of playing, is unique. But the general effects of the various elements can be broadly applied, qualitatively if not quantitatively. Blackjack affords a useful paradigm for examining the diverse implications.
To start, picture three ways to play following Basic Strategy. You bet exactly or an average of $10 per round, giving the casino 0.5 percent edge. Your "expectation" is a theoretical loss of $0.05 per coup. A risk of ruin analysis shows what stake yields 90 percent confidence you won't go bust in some number of rounds.
One way would be to bet a flat $10 on a single spot. You'd need at least $304, $436, and $628 for 250, 500, and 1,000 rounds.
Pretend you vary your bets from $5 to $50 based on hunches or a popular but statistically irrelevant rule such as pressing after every win. A typical betting profile that left you at $10 on the average might multiply the overall volatility in the game by a factor of 1.32. Edge stays the same. Minimum bankrolls would then rise to $399, $570, and $818 for the 250, 500, and 1,000 rounds.
Another alternative would be flat-betting two spots at $5 each. You're still risking $10 per round at 0.5 percent so there's no impact on theoretical loss. But volatility relative to $10 flat on one spot falls by a factor of 0.924 and bankroll needs become $255, $367, and $530 for 250, 500, and 1,000 rounds respectively.
Make believe you count cards and have an edge of 0.5 percent the converse of what the bosses earn from Basic Strategy bettors. You procure the edge mainly by varying your bet according to the proportions of high and low cards remaining to be dealt.
For comparative purposes, say you're guided by the count but still distribute bets from $5 to $50 at the same frequencies as in the random-pressing Basic Strategy case. You again bet $10 on the average and have a volatility increase factor of 1.32. Now, though, you have 0.5 percent edge. You get 90 percent confidence of survival with $379, $530, and $738 for 250, 500, and 1,000 rounds, respectively. Higher volatility demands more bankroll than flat-betting Basic Strategy. But having the edge makes it less than betting Basic Strategy arbitrarily at equivalent rates.
Double the bets for the card counter, and the average goes to $20 per hand while the volatility factor remains at 1.32. Bankroll requirements are then twice those for the $10 average: $758, $1,061, and $1,477 for 250, 500, and 1,000 rounds, respectively. If you play two spots using the $5 to $50 spread on each hand, an average of $10 each or $20 per round, the volatility factor drops to 1.32 x 0.942 = 1.25. Bankroll requirements decrease to $713, $997, and $1,387 for 250, 500, and 1,000 rounds, respectively.
Look at the trends. Bankroll minima vary in direct proportion to average bet size, grow with volatility, and shrink as edge gets less negative or more positive. Over the statistically small numbers of coups in single sessions or visits, volatility changes bear more strongly on survival than do those of edge. Counting cards at blackjack boosts volatility to gain edge, mandating more moolah for the same confidence in session endurance than does following basic strategy. This counter-intuitive effect typifies the peril of long-term answers to short-term questions. Here's how the poet, Sumner A Ingmark, saw such surprising situations:
Like waves that all boats drop or lift,
In steps that mask protracted drift.
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