How does volatility affect your chances of gambling success?

Casino games are biased toward the bosses. Not because of the odds against winning particular bets, but by the offset between those odds and the payoff for beating them. Except in the occasional cases where the offset is implemented as a fee or commission, players rarely notice the edge when they’re ahead or falsely attribute their poor performance to it when they’re behind.

Use double-zero roulette as an example. The house advantage on most bets in this game is an edge equal to 5.26 percent of the total wager. If $10 is at risk, say $1 on each of 10 spots or $10 on a single number, this comes to 5.26 percent of $10 – or 52.6 cents. The 52.6 cents is essentially invisible for two main reasons. 1) It’s not paid or deducted from winnings directly but buried in the 28-to-10 versus 26-to-10 or the 37-to-1 versus 350-to-10 odds versus payoff offsets for the respective bets. 2) It’s eclipsed by what’s actually exchanged, $10 loss and $26 or $350 gain for the two cases. Also, a 5.26 percent edge accounts for one bet every 19 coups. Few players know how often they expect to win or lose in 19 spins, let alone whether their bankrolls are short the amount this would represent.

Were casino gambling just a matter of letting the house pinch pennies from players’ fanny packs, it wouldn’t be any fun. Profits can be earned despite the edge. And losses can be incurred greatly exceeding what the edge implies. These situations arise because the jumps in fortune that occur whenever a decision is won or lost jumps are usually large relative to the edge – $350 or $10 compared to less than $0.53 when a solid citizen wins or loses a $10 straight-up roulette bet.

These jumps give games and propositions characteristic volatilities. Math mavens measure volatility with “standard deviation,” which can be pictured as the representative bankroll change per coup. Standard deviation is $15.83 for $1 on each of 10 spots and $57.62 for $10 on one spot.

Volatility is bidirectional, such that the big ups and downs are self-cancelling over time. Edge is unidirectional so the impact of its small spin-by-spin disadvantage grows gradually but relentlessly. In the short run, volatility therefore swamps edge and leads to those rewarding or disproportionately expensive sessions. Over the long haul, edge eventually becomes dominant and costs gamblers – as a group – close to what’s predicted by its action on multitudes of wagers.

The two strategies cited for betting $10 per spin at double-zero roulette illustrate situations where players face the same edge but different volatilities. They can accordingly be used to show how low or high volatility can affect the chances of ecstasy or agony during reasonable time frames.

Pretend an individual’s primary motivation is to earn $500 before depleting a $200 stake, regardless of how long it takes. A “risk of ruin” analysis based on edge and standard deviation gives the chances of success as 7.4 percent with $1 on each of ten spots and 26.3 percent with $10 on a single number. For doubling a $200 stake, they’d be 30 and 48 percent, respectively. In either instance with boom-or-bust as the sole criterion, higher volatility has better portents.

Instead, make believe a person’s chief objective is time at the table. Say the player would be satisfied getting in five hours, betting $10 per spin on a $200 stake. This period would amount to about 200 spins. Betting $1 on each of 10 spots, the risk of ruin analysis gives 46 percent chance of being in action – showing a surplus or fighting a deficit under $200 – after this period. With the entire $10 on a single number, the probability is 17 percent. Were the time target trimmed to three hours, roughly 120 spins, the figures would be 64 and 22 percent, respectively. With the survival criterion, lower volatility is more apt to be fruitful.

Of course, enquiring minds want to know the toll edge takes on these strategies. Were the game to have no edge, volatility would become irrelevant with respect to the strict win goal criterion; the chance of earning $500 before losing $200 would be 29 percent, and that of doubling $200 before losing the same amount would be 50 percent. With no edge, though, volatility still affects the chances of survival. The probability of lasting for five hours would be 63 percent betting $1 on each of 10 spots and 19 percent betting $10 straight-up. For three hours, the prospects would improve to 75 and 25 percent, respectively.

More conservative bet-to-bankroll sizing would help, too. If table minima, social pressure, desire for a comp to the all-you-can-eat buffet, or your ego keeps you at $10 per spin, this would mean playing with a bigger stake. All the probabilities examined would increase. But so would your pain, self esteem, and significant other’s criticism if you had more action or came closer to winning an amount you considered worthwhile, but ended up busting out for more than you could really afford. As that moolah-minded muse, Sumner A Ingmark, mumbled:

Losing gamblers often wonder, as they watch themselves go under,
Were the bosses set on plunder, or did they, themselves, just blunder?