Is Volatility the Secret of Winning Big?

Basic Strategy for multi-deck blackjack dictates hitting rather than doubling on a two-card 11 against a dealer's ace. Both options favor the players. That is, the hand is projected to win more often than it loses, and to make money, either way. "The book" says to hit because doing so offers higher "expectation" than doubling. The reason, in a nutshell, is that the net number of positive results - wins minus losses - is over twice as great hitting than doubling.

Comparable conditions - hitting and doubling both profitable but the former more so - hold for six sets of hands. These are shown in the accompanying list. Also given is the theoretical earnings premium for hitting per $100 gross bet in six-deck games.

Situations when hitting has greater expectation than doubling although players are favored either way
player		dealer	extra profit per $100 initial bet
11	vs	ace	$2.19
9	vs	7	$5.97
8	vs	6	$2.20
8	vs	5	$5.60
A-2	vs	4	$3.57
A-3	vs	4	$1.48

Both alternatives are advantageous and the theoretical premium for hitting is small. So blackjack buffs may wonder whether to go ahead and double down in these situations despite the reduced expectation. Wouldn't the doubles cause greater bankroll swings, hand-by-hand? And, since chances of winning are good, wouldn't these fluctuations tend to be positive? As a result, although hitting should yield a greater return over extended trials, wouldn't the extra bets under favorable conditions help in the short haul, when the effect of volatility swamps that of edge?

Increased volatility through aggressive play is certainly rewarding when the cards fall beautifully into place. And it leads to disaster when the dealer is tearing up the tables. But, under normal circumstances, neither extreme prevails. Then what?

In fact, $5 bettors don't grab another $0.74 every time they hit or $0.63 whenever they double 11 against an ace, although these are the statistical expectations in six-deck games. They win or lose $5 or $10, or occasionally push. Volatility does dominate. And few players get enough of these hands in their lifetimes for the law of averages to zero in on the theoretical profit. So, after 100 instances of 11 versus ace with initial bets of $5 per round, nobody should assume they'll be precisely $73.69 up by hitting and $62.75 by doubling. Likewise, nobody should believe they'll win $500 or $1,000 by hitting or doubling, respectively, because they have an edge. What should individuals with finite bankrolls and limited time to gamble anticipate by hitting or doubling on these hands during typical sessions or casino visits?

A good way to gauge the normal bankroll swings associated with a bet is to perform a "risk of ruin" analysis to find the chances of reaching a specified profit level before losing the same amount. To illustrate the effect, consider 11 versus ace with a $5 bettor and a $50 profit or loss target. A solid citizen who hits has 96 percent chance of winning $50 before losing the same amount. A more aggressive punter who doubles down has a lower, although still enviable, 66 percent probability of success.

Another means of characterizing short-term bankroll swings uses the chance of winning or losing a specified amount within an arbitrary number of trials. Take 11 versus ace again, with a $5 bettor and 25 occurrences of the hand. Hitting, the chance is one in seven of winning $50, and only one in 200 of losing this much. Doubling, chances improve to over one in three of winning $50 in 25 trials, but also rise to one in five of losing the same sum.

Volatility is how small bets become big bucks at the tables and machines. But, when volatility is traded for even a small amount of edge, the price may be too high. Except, of course, on one of those days when you just can't do anything wrong. If only you knew which days they'd be. Sumner A Ingmark, the couplet king of the casino, cogently captured the crux of the conundrum thusly:

A gambling system's sorely vexed,
By ignorance of what comes next.