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# Would bettors bust out if the house had no edge?

23 May 2011

Everyone knows why solid citizens always lose to casinos. It’s because the house has an edge or advantage in the games. Of course, “always” losing applies to players collectively. Individuals may have rounds, sessions, visits, and even extended periods when gambling is lucrative. The bosses bank on huge numbers of decisions for the edge to earn them a net profit.

The opposite is also true. Absent edge, both groups would break even in the long run. Players, again collectively. Particular bettors might prosper or wither; it’s the net of the wins and losses that would zero out. Enquiring minds accordingly want to know how much of the chance a player will finish in the hole is due to edge, how much to other factors, and what those other factors are.

Comparisons of several alternate bets, each configured normally and as they would be without edge, help answer these questions. To avoid extraneous influences that might distort the picture, consider several craps bets – all of which cost players \$30 on a seven: Buying a four or 10 for \$29 and Placing the five, six, eight, or nine for \$30. When the house has its usual advantage, four and 10 pay winners a net of \$57 (\$58 for the \$29 bet after the house subtracts \$1 vigorish from the \$30 proffered), five or nine pay \$42, and six or eight pay \$35 . With no edge or vigorish, the \$30 would pay \$60 on four and 10, \$45 on five and nine, and \$36 on six and eight.

Say you bet \$30 on just one of the numbers. Further, make believe your criterion for failure is exhausting a \$300 poke before the end of four hours – about 240 throws. When the house has its usual advantage, the probabilities you’ll have gone belly-up are: 41 percent on four or 10, 39 percent on five or nine, and 32 percent on six or eight. With no edge the probabilities drop from 41 to 36 percent on four or 10, 39 to 32 percent on five or nine, and 32 to 29 percent on six or eight. You may bust either way, but the edge makes doing so somewhat more likely.

Pretend you raise your loss tolerance to \$600. Subject to the house advantage, the likelihood you’ll be knocked for a loop within four hours is: 8 percent on four or 10, 6 percent on five or nine, and 4 percent on six or eight. If the house had no edge, the probabilities would change from 8 to 7 percent on four or 10, 6 to 5 percent on five or nine, and 4 to 3 percent on six or eight. Again, edge has a modest deleterious bearing. However, increasing your bankroll relative to your bet size – or, conversely and more in line with players’ thinking – decreasing your bet size relative to your bankroll – the impact is considerably more substantial. Sure, you’ll earn less money with lower wagers, but you’re also less apt to scrape the bottom of your fanny pack.

Another factor influencing the chance of busting out is the trade-off between the theoretical frequency of winning coups and the amounts of the payoffs. This is demonstrated by contrasting a \$30 bet on one number with an equivalent “\$32 across.” As an example, a \$30 six with the standard house advantage pays \$35 and is expected to hit an average of five times in 36 throws. The \$32 across is a combination bet involving \$5 on each of four, five, nine, and 10, with \$6 each on six and eight This is expected to hit a total of 24 times in 36 throws. With a house edge, six of these hits (on four or 10) pay \$9 and 18 (on five, six, eight, or nine) pay \$7; with no edge, six (on four or 10) would pay \$10, eight (or five or nine) \$7.50, and 10 (on six or eight) \$7.20.

To keep the \$2 discrepancy in bet size from biasing the results of a comparison, assume stakes of \$300 for the \$30 six and \$320 for the \$32 across. As above, your chance of cratering within four hours betting the \$30 six is 32 percent with and 29 percent without the edge. That of busting out on the \$32 across is 24 percent with the edge and 16 percent without it.

Keeping all other factors equal, the conclusions to be drawn from these results are as follows:

Edge raises the likelihood of busting out, although the impact on players during sessions of reasonable duration is small.
Overbetting a bankroll substantially increases the probability of going broke.
Increasing the rate while decreasing the amount of wins reduces the chance of a wipe-out, the influence being moderate.

This all has to do with volatility, a parameter of gambling largely ignored by casinos and players alike, both of whom think almost exclusively in terms of edge. Casinos perhaps justifiably, other then when the bosses panic because a player seems to be on a roll, because volatility effects are small relative to those of house advantage given the enormous numbers of decisions the joints typically experience. Players at their own peril because in the short term of a single session or casino visit, volatility swamps edge on changes in fortune, making it possible to win big. Or to lose a great deal more than the erosive action of the edge suggests. Here’s how the inimitable inkster, Sumner A Ingmark, described the situation:
Good gamblers can do many things, to regulate their bankroll swings.
Control, however, lets them choose the magnitude, not win or lose.