Not by Edge Alone: Ranking Bets by Risk of Ruin

Casino bets are usually evaluated in terms of house advantage or edge. By this standard, baccarat is superior to roulette; the former has bets with edge as low as 1.17 percent, while the best you can do in the latter is about 2.6 percent. Likewise, place bets on six or eight in craps are preferred over those on four or 10 -- the house getting 1.5 and 6.66 percent, respectively.

Edge is a fine touchstone for casinos since it gives a reliable estimate of earnings after millions of wagers. And edge is also almost universally used in evaluating bets for players.

But house advantage is a weak criterion from the bettors' perspective. Solid citizens rarely make enough wagers in a year, let alone a day or session, for edge to be significant. Further, it ignores factors like odds and payout. A 30-to-1 longshot paying 30-to-1 and a 50-50 bet paying even money both have zero edge. Such elements affect players' ability to outride normal downswings but don't matter to casinos over the long term.

Say you start at a craps table with a $250 stake. Your goal is to make as much as you can in three hours, quitting early only if you lose the $250. You're considering three options: (a) dropping $5 on the pass line and taking $10 in odds, (b) placing the five and nine for $5 each, (c) betting $5 on the field.

In three hours, "expected" losses due to edge are roughly $8.50 on pass, $40 on the five and nine, and $100 on the field. Solely based on edge, pass is best, five and nine next, and field a distant late-running sucker bet. Conventional wisdom.

But nobody's planning to lose the expected amounts. Everybody's counting on fluctuation, hoping their bankrolls will let them withstand the cold spells so they can reap the hot streaks.

The fluctuations per roll of the dice are $8.22 for the line bets and $5.27 each for the place and field bets. Moreover, the fraction of the $250 bankroll at risk on each roll is not the same for the three choices. On the line, the average bet is $11.67 so the bankroll is $250/$11.67 or 21.4 units. On the place bets, $10 is at risk so the bankroll is $250/10 or 25 units. On the field, $5 is at risk so the bankroll is $250/$5 or 50 units. Considering all these terms, at once, quickly boggles the mind.

Fortunately, bets can be characterized using a single parameter that accounts for all these factors in terms meaningful to players. That parameter is "risk of ruin" -- the chance of going broke with any given stake -- following a session of reasonable length, making any particular bet. This isn't quite the same as the chance of dropping to a loss limit during such a session, then ending at some other point, lower or higher. However, the end-of-session figures are suitable for comparing bets.

Risk of ruin after three hours -- found using the values for edge, bankroll, and fluctuation -- is 6.1 percent for the line bet, 1.8 percent for the five/nine place bets, and 6.7 percent for the field bets. On this basis, the place bets are best, with the line and field running neck-and-neck for second and third. All offer reasonably good chances of surviving the session.

Changing the conditions can greatly alter the picture. For instance, playing for six hours on the $250, risk of ruin is 15 percent on the line, 12 percent on the place bets, and 36 percent on the field. For three hours with $100 bankrolls, risk of ruin becomes 28 percent on the line, 27 percent on the place bets, and 50 percent on the field. In both cases, place and line bets are roughly equivalent -- place still being better -- while the field is by far the worst; in addition to the rankings, the figures show the dangers of normal downswings and overbetting a bankroll.

Risk of ruin is not a simple calculation. But edge isn't easy to determine, either, until you know how. Or know someone who does. As the pragmatic poet, Sumner A Ingmark, said: