Are multi-line slot machines more than just extra boops and beeps?

Multi-line machines have become the devices of choice among many slot aficionados. In part, because solid citizens who bet on lots of lines get the impression they're winning all the time. Or, at least, until they notice their totals gradually fading.

One reason for the popularity is the high level of action, with sights and sounds proclaiming success even on combinations of wins and losses that result in net failures. Another factor is the opportunity for players having limited budgets to make small wagers -- as low as a penny per pull -- by risking the minimum on a single line; few do so, but that's another issue altogether.

More basic principals are afoot, however. To explore them, assume a hypothetical machine on which all individual lines have the same probabilities and returns, and therefore equal edges and volatilities. What happens when lines are played in combination?

Edge, the average fraction of each bet that the bosses earn through the fundamental math of the game, doesn't change. Play one line or 10 lines with 8 percent edge and the house chalks you up for 8 percent of your total bet. For example, whether you drop $1 on a single line or $0.10 on each of 10 lines at the same time, joint's theoretical profit is $0.08. Of course, if your alternatives are $1 on one line or $1 on each of 10 lines, 8 percent edge monetizes to $0.08 or $0.80, respectively.

Volatility, an indication of the magnitude of the bankroll swings typically associated with the game, works differently. Volatility is commonly measured with a quantity known as "standard deviation," which can be pictured as representative bankroll jumps up or down per coup. Volatility is not well appreciated in casino gambling. Over the statistically short haul of a session or casino visit, though, its impact on individual players usually swamps that of edge. Volatility is what lets fortunate gamblers overcome the edge. And it's what benches hapless bettors much faster than the gradual erosive effect of edge would suggest.

In contrast to edge, which stays the same, volatility per dollar bet decreases when identical lines are played in combination. For instance, at a machine with 8 percent edge, the probabilities and returns on each line might be such that standard deviation is 3.22 times the wager. So, when playing one line for $1, on the average, edge accounts for an $0.08 loss and standard deviation a $3.22 gain or loss. No matter how many lines are played, if the total bet is $1, $0.08 is attributable to edge. But, standard deviation is reduced to $2.28 playing two lines for $0.50 each, $1.61 with four lines for $0.25 each, $1.44 on five lines at $0.20 each, and so forth.

To be sure, increasing the number of lines without cutting the individual bets in proportion boosts the monetary effect of both edge and volatility. For two lines at $1 rather than $0.50 each, edge amounts to 2 x $0.08 or $0.16 and standard deviation to 2 x $2.28 or $4.56. Note that the latter is still less than the 2 x $3.22 or $6.44 standard deviation of one line at the same $2 total. Playing five lines at $1 each, $5 total up for grabs, edge is an average setback of 5 x $0.08 or $0.40 and standard deviation a characteristic bankroll jump of 5 x $1.44 or $7.20.

Lower volatility is bad news for those seeking a big score and willing to go bust for a shot at getting it. Not only do chances of jackpots shrink rapidly when they require multiple lines to be hit for the biggie simultaneously, but edge creeps up faster on the runs of good luck needed for a bankroll to reach any given earnings level. On the other hand, it's good news for folks seeking modest profits and long sessions on moderate stakes. Edge stays in there nibbling away at the totals but steep declines become less likely because wins, while small, are more frequent.

Take your pick. Single- versus multi-line play offers slot buffs another option for tailoring the casino experience to their personal preferences. As that intrepid inkslinger, Sumner A Ingmark, poetically prompted prudent punters:
Players who widely distribute their bets,
Rarely get rich but cut down on regrets.