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Ever have one of those days when you never get ahead?

19 September 2011

Every true casino aficionado can recall instances of being ahead of a game then giving it all, often all-and-then-some, back. In such cases, it’s easy to recall precisely when discretion would have been the better part of starry-eyed optimism or wanton greed. Perhaps at the high point, or maybe at some amount down from the peak, such as a fall-off decided in advance when the rush of the action doesn’t cloud a person’s judgement. Both, of course, exemplify 20-20 hindsight since there’s no way to know how a fortune will fare at any particular juncture.

Still, quitting with a profit is a different matter than never getting ahead, cutting, and running. Losing the first round of a game then failing to climb into the win column can happen to the best of bettors in any session or visit. Contrary to what intuition may suggest, the chance this fate will befall has relatively little to do with house advantage or volatility. The factor that most strongly influences this aspect of gambling is the ratio of bankroll or loss limit to bet size.

As a base example, picture a hypothetical game with no edge. Pretend that a million patrons arrive with $100 bankrolls and make uniform $10 even-money bets. Being ahead at any time means having won at least $10.

Since this game has no edge, on the average, solid citizens’ wins and losses offset each other, so the bosses accordingly earn nothing. Assume players all quit as soon as they get ahead or exhaust their stakes. Then 909,090 (90.90 percent) would each leave $10 richer, pocketing a total of $9,090,900 in earnings along with their original stakes; the remaining 90,909 (9.09 percent) would each skulk home $100 poorer, sacrificing $9,090,900 overall. Net of wins minus losses is zero. The same proportions of winners and losers apply to 10-to-1 bankroll-to-bet ratios regardless of the actual sums, for instance $50 and $5 or $250 and $25. Alternate bankroll-to-bet ratios yield different results. At 5-to-1, for instance a $50 poke and $10 bets, the figures are 83.33 percent getting ahead by one bet then quitting and the complementary 16.67 percent never being in the black. At 20-to-1, say a $500 bankroll and $25 bets, the values are 95.24 percent stopping one bet to the good and.4.76 percent never seeing the light of day. The percentages cited for the million bettors can be interpreted as the probabilities for anyone gambling as indicated.

Single- and double-zero roulette afford instructive models to illustrate what happens in the real world. The two versions of the game have individual values of house advantage. Further, players can pick propositions having a range of payoffs other than 1-to-1, so they can tailor the volatility to their personal preferences for frequent small or occasional large returns.

At single-zero roulette, edge is 2.70 percent across the board. For bets on 12-number columns, which pay 2-to-1, “standard deviation” – a measure of volatility that can be pictured as the representative bankroll jump per spin – is 1.40 times the wager. At a 10-to-1 bankroll-to-bet ratio, the probability of never being one bet ahead is 10.38 percent; this is worse than the 9.09 percent of the idealized no-edge game, but not by much. Playing single-zero roulette with the same 10-to-1 bankroll-to-bet ratio but wagering on a single number, for a 35-to-1 payoff, leaves edge at 2.70 percent but boosts standard deviation to 5.84 times the bet. The effect is to trim the chance of always being in the red to 9.19 percent, only marginally greater than the 9.09 percent of the no-edge model. Analogous outcomes apply to other bankroll-to-bet ratios.

Giving the bosses yet greater edge has an impact but, again, it’s comparatively small. Double-zero roulette, which has a 5.26 percent edge, illustrates the phenomenon. Standard deviations for the column- and single-number wagers are slightly lower than for their single-zero counterparts, 1.39 and 5.76 times the wager, respectively. The probability of never being above water on the column wager with a 10-to-1 bankroll-to-bet ratio is 11.74 percent. For the single number wager, the chance is 9.24 percent. Both are worse than is true with the corresponding lower edge bets. The distinctions are not great, however, and the less so as volatility increases.

Sophisticated players may be aware that the volatility of their action goes up when they vary the sizes of their bets. And raising the volatility lowers the probability of never being ahead. The question may therefore follow as to whether they’d be wise to press their bets when they’re behind. Not only does the math suggest that the higher volatility improves their prospects for recovery, but intuition intimates that they’ll need fewer lucky hits to get over the top. The fallacy here is not only that bigger bets can lose as well as win, but pumping wagers reduces the bankroll-to-bet ratio, the negative effect of which is stronger than the positive impact of higher volatility. So this approach may succeed in any particular situation but wouldn’t be a sensible general rule. Here’s how the punter’s poet, Sumner A Ingmark, viewed this sort of trade-off:
Most poorly grounded theories lack,
The basis for a sound attack,
So each step up drops two steps back.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.