The peril of faster action at the casino, and how to forestall it

Some casino aficionados find leisurely action tedious, if not boring, and prefer games that proceed at a rapid clip. Many, however, fail to recognize the cost inherent in a high decision rate.

Players’ fate follows from the joint influence of edge and volatility characteristic of every wager.

• Edge, a small fraction of each bet, arises because players earn less for each win than the odds fought to achieve it; casinos therefore reap more than they sow netted over many rounds and players collectively lose. Money edge represents flows only one way, from bettors to bosses.

• Volatility describes bankroll jumps as players win or lose decisions. The size may be either a fraction or a multiple of each bet. It’s a fraction, typically large, of wagers more likely to win than lose but which pay less than what’s at risk. It equals or exceeds the amount up for grabs on wagers that pay at or over 1-to-1 but are more apt to lose than win. Volatility is bidirectional; the money moves either way depending on whether a bet flies or flops. In a sense, volatility accounts for luck – the good kind soaring and the bad sinking.

On a single round basis, volatility overwhelms edge. Say, for instance, you make a Place bet on the nine at craps. The payoff is 7-to-5 for odds against winning of 7.5-to-5. With a $5 bet, one “standard deviation” – a quantity the math mavens use to gauge volatility – is $5.88. Edge has a monetary value of $0.20. Essentially nobody notices they’ve been shortchanged by $0.20 when they rack up their $7 profit or lament their $5 loss.

Bankroll erosion caused by edge accumulates steadily as play continues. The impact of volatility on a fortune also grows as action proceeds. However, the effect of volatility can be up or down so standard deviation represents a range within which outcomes can be expected to occur. Further, because the jumps up and down tend to be somewhat self-canceling, the extent of the range is not proportional to the number of decisions as with edge but is moderated. The impact of volatility accordingly grows more slowly than that of edge and eventually gets overtaken.

After 25 decisions on the nine, edge is equivalent to 25 x $0.20 or $5; after 100 decisions, it’s 100 x $0.20 or $20, after 2,500 decisions it’s 2,500 x $0.20 or $500, and after 10,000 decisions it’s 10,000 x $0.20 or $2,000. The cumulative effect of one standard deviation defines a range expected to cover 68 percent of all cases. The range attributable to volatility alone, as measured by one standard deviation, is between minus and plus 5 x $5.88 or $29.40 after 25 decisions, 10 x $5.88 or $58.80 after 10 decisions, 50 x $5.88 or $294.00 after 2,500 decisions, and 100 x $5.88 or $588.00 after 10,000 decisions. The point at which the expected range for the combined influences of edge and volatility tops off at break-even in 68 percent of all cases is 865 bets. The ranges for other emblematic numbers of resolutions are shown in the accompanying table.

Effects of edge and volatility for various numbers of resolved $5 Place bets on nine at craps

No of       Effect of   Effect of            Combined
decisions   edge ($)    volatility($)        effect ($)
25          -5.00       -29.40 to +29.40     -34.40 to +24.40
100         -20.00      -58.80 to +58.80     -78.80 to + 38.80
2,500       -500.00     -294.00 to +294.00   -794.00 to - 206.00
10,000      -2,000.00   -588.00 to +588.00   -2,588.00 to - 1,412.00

The table shows that casino patrons can win when the volatility of the games after the number of rounds played has a monetary value greater than that of the edge for the same action. The table also highlights the dark underbelly of volatility. If the net volatility swing is down they can lose far more than the edge would suggest.

As the pace of a game increases, more bets are resolved during sessions of any duration. Losses due to edge therefore catch up with possible gains due to volatility, and may eventually become large enough to virtually preclude an overall profit.

In principle making bets with higher volatility can forestall the drop-dead situation. But this approach can be perilous because volatility can works in both directions and can send solid citizens to the showers as well as keep them ahead of the steamroller. Playing with the lowest possible edge through choice of games or strategies is one a more effective way to circumvent this condition. As an illustration, for a $5 bet at blackjack played according to strict Basic Strategy, edge only represents a penalty of 2.5 cents while one standard deviation is $5.65. With these parameters, the crossover is beyond 51,000 rounds. For reference, at a table with two other players betting one hand each, you’ll average about 100 rounds per hour. At this rate, 51,000 rounds is 510 hours. At three hours per session, this will be 170 sessions. Not all that many, in the grand scheme of things. An observation that did not escape the poet, Sumner A Ingmark, as is evident from his couplet:

The dominance of edge is sequent,
For those whose play is fast and frequent.