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Why Betting Progressions Work When Experts Say They Can't19 April 1999
Casino table game players frequently debate progressive versus flat betting. Should they start low and follow any of a host of progressions, or begin higher and stay constant?
All tried and true gambling gurus guarantee that systems to raise or lower bets don't affect house advantage or edge when odds and returns are uniform from round to round. Casino balance sheets prove this over extended periods, showing "holds" remarkably close to the gross amounts wagered multiplied by the edge. If, during a year, players "place" the nine at craps for a total of $100 million, the hold on this action will be within a whisper of $4 million -- the edge on this bet being 4 percent -- independent of how the individual wagers are sized, shaped, or sequenced.
Players, however, know from experience that betting systems do affect performance. Some solid citizens believe the secret of success is "playing with the house's money;" they press their bets after rounds in which they win, attempting to escalate their earnings. Others adhere to the theory that long series of losses are rare; they raise their bets after rounds in which they're beaten, attempting to recover when the cold streak ultimately breaks. A third group takes the slow-but-sure flat bet approach.
Both viewpoints are right. Betting strategies don't matter... and they do. How so? Bets not only have a built-in edge favoring the house, but also an inherent volatility which can go either way. In the long run, for the casino's multitude of wagers, edge is overwhelming. In the short run, for an individual's action during a session or casino visit, volatility predominates. If it didn't, everybody would succomb to edge, nobody would ever win, and even free all-you-can-eat buffets wouldn't lure folks to the casinos.
On any selected wager, flat betting minimizes volatility. Progressions raise it. The details of the impact on a game depend to some degree on how players vary their bets. But the overall effects of increasing volatility are common to all strategies.
Say two craps players bet on the nine. Steady Stu always bets $10. Jumping Jill starts at $5, goes to $10 when it wins, presses to $25 if it hits again, then keeps at this level. Stu averages $10 with a fluctuation or "standard deviation" which works out to $11.76 per round. Jill's at $5 60 percent of the time, $10 another 24 percent, and $25 the remaining 16 percent. Her average bet is $9.40; her standard deviation comes to $13.86 per round.
Consider edge alone after 1,000 decisions. Stu's expectation is to lose 4 percent of $10,000 or $400. Jill's expectation is to lose 4 percent of $9,400 or $376. But volatility, which dominates far beyond 1,000 trials, paints a different picture.
For typical short-term play, think about status after 1,000 rounds. Stu has 14 percent likelihood of showing a profit and can be 90 percent confident of being from $213 ahead to $1,013 behind. Jill, with a smaller average bet, has 20 percent chance of showing a profit and can be 90 percent confident of being from $347 ahead to $1,099 behind. Of Jill's $134 better projected zenith, $24 is attributable to edge and $110 by volatility. The $86 worse projected nadir also represents $110 contributed by volatility, offset by $24 from edge. In both cases, the magnitude of the maximum loss exceeds that of the corresponding maximum profit by twice what the house expects to draw off through edge.
When does edge begin to overshadow volatility? Stu can go 59,000 rounds before edge causes his maximum theoretical profit to peak, and 235,000 rounds before his chance of being ahead falls below 5 percent. Jill, with higher volatility, can go 82,000 rounds before her maximum theoretical profit starts to fall and 327,000 rounds before her chance of being ahead is under 5 percent.
Raising volatility with progressive bets boosts the importance of fluctuation relative to edge. For equivalent amounts at risk lucky players get higher profits while the less fortunate suffer greater and faster losses. As Sumner A Ingmark, whose poetic edge is often buried in volatile versification, wrote:
When usages are retroactive.
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