PLAYING IT SMART by ALAN KRIGMAN

Do chance-payoff shifts affect your session prospects?

Say you bet $10 on a win-or-lose proposition that pays $20 with a 32.75 percent chance of success. The house would have 1.75 percent edge; its theoretical fee for booking the bet would therefore be 1.75 percent of $10 or 17.5 cents. Volatility, as measured by "standard deviation" (picture it as the average jump your bankroll takes up or down in any round) is $14.08.

Instead, pretend you bet $20.35, win or lose, with a strong 67.25 percent chance of collecting $9.65. The house would have 0.86 percent edge. But you're wagering $20.35 so the bosses take a theoretical fee equal to 0.86 percent of $20.35 or (surprise!) 17.5 cents. And the standard deviation would be $14.08.

As a third option, imagine you venture $14.25, up or down, on a 50 percent shot at garnering $13.90. You've undoubtedly guessed you'd average a 17.5-cent loss on this bet over the long haul and your bankroll steps would have a standard deviation of $14.08.

In all three instances, edge and volatility are the same. Yet, your gut may tell you the gambles differ. True, these bet sizes, payoffs, and probabilities are hypothetical -- fabricated so edge and standard deviation are equal for the alternate cases.

However, the trade-offs are typical of those you'd make covering varying numbers of spots at roulette, deciding between Placing and Laying numbers at craps, or choosing one or more lines at a multi-line slot machine. So, enquiring minds will want to know if and how sessions based on such bets will have divergent profiles.

The distinctions among these bets are in what the statisticians call "skewness" or "skew." Relative to the amount at risk, positive skew means a small chance of a big win, negative skew implies a high likelihood of a small gross, and zero skew is an even chance of a comparably-sized profit. For the examples cited, skew is +0.74 for the 32.75 percent $10.00/$20.00 wager, -0.74 for the 67.25 percent $20.35/$9.65 bet, and 0.00 for the 50 percent $14.25/$13.90 case.

If you're planning to make a bet or two then quit, you can easily envision which approach best suits your personal preferences. But what about their impact on sessions of reasonable duration?

A goal many players share is to double their money or bust out trying. Suppose you buy-in for $400 with this objective. Computer simulation shows your chance of success to be the same for the three bets, about 33 percent. Skew doesn't affect this outcome.

Other solid citizens focus on time, not earnings. They want extended action without downswings that send them to the lockers.

Assume, as an illustration, this is your criterion. Further, posit that you start with a $400 stash and hope to be in the ring for at least 500 rounds. Computer simulation indicates that sustainability is equivalent with all three types of bets. Wagers with the edge and standard deviation as stated afford 73 percent chance of at least this long a session regardless of skew.

Results of the simulations for win goal and sustainability offer a point of philosophical interest. Simulation is a "brute force" approach. It doesn't entail deriving and solving equations. Instead, it involves programming a computer to act in the virtual domain like an analogous situation in the corporeal world.

True formulaic approaches are far more "elegant" but they often include simplifying assumptions. And if the assumptions aren't valid -- well, that's why some of the early Mars missions crashed instead of landed gently on the planet.

For the three bets and the two exit criteria discussed here, simulation yielded the same conclusions as equations based on laws of probability which don't explicitly account for skew. This vindicates the theoretically-minded. And it should give pause to the pure pragmatists. Or, as that renown rhymer of revelatory arithmetic, Sumner A Ingmark, recognized when he wrote:

Understanding the physical is often quite quizzical,
'Till a probe analytical uncovers what's critical.

(c) 2005, ICON/Information Concepts Inc

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