Using maths to parlay a small edge into billions

PUBLISHED : Sunday, 21 August, 2011, 12:00am
UPDATED : Sunday, 21 August, 2011, 12:00am


How did Stanley Ho Hung-sun, Steve Wynn and all the other casino kingpins get so rich? You may think it's because their casinos have overwhelming odds against the gamblers. In fact, the casino's edge is tiny. You may think the gaming business generates high returns with each transaction. In fact, some games produce a return of only 0.5 per cent to 2.7 per cent on average in each round of play. Any other business could not thrive on such puny returns per transaction. The gambling moguls get rich from three concepts: advantage, volatility and the scale of operations.

It's well known that the scale of Macau's casino industry is huge: more than 5,000 gaming tables operating 24 hours a day, 365 days a year. Last year, it raked in HK$182 billion. In the first half of this year alone, revenue from Macau's casinos has already exceeded what Las Vegas casinos made last year.

Less widely known is the fact is that in all casino games, the house advantage over the gambler per round of play is small: 2.7 per cent for roulette (using a wheel with 37 numbers) and about 0.5 per cent for blackjack. So for a US$100 bet, the gambler will lose on average US$2.70 in roulette and 50 cents in blackjack. The casino achieves this small advantage because of the rules of the game. For example, in roulette with 37 numbers, 1 to 36 plus 0, the gambler has 18 chances of winning if he bets on even numbers but 19 chances of losing when the ball falls on the odd numbers or zero. Thus the rules ensure he has a 49 per cent chance of winning but a 51 per cent chance of losing. In blackjack, the gambler will lose a hand if his cards total up to more than 21 even if later the dealer's hand also exceeds 21.

Small as the casino's advantage is, it brings in billions thanks to the mathematical concept least familiar to players: volatility. The casino business is probably the only one in the world based on the statistical concept of volatility: the wins and losses of a game as many thousands of rounds are played. Given its scale of operations, a casino can estimate how much money it will make over a period in any game not only from its advantage in the game, but also from its volatility.

Say a gambler makes only one bet of US$100 on either odd or even numbers in roulette: he has a 49 per cent chance of winning - not a fair game for sure, but the odds are good enough for many to take a chance. Few punters, however, stop at one round. Because of volatility, as he plays more rounds, betting US$100 a round, his 49 per cent chance of winning decreases to 17 per cent after 1,000 rounds, and 1.7 per cent after 5,000 rounds. After 10,000 rounds, he's certain to lose. So despite the small return per transaction, with thousands of tables on which tens of thousands of gamblers are playing hundreds of thousands of rounds, volatility virtually prints money for owners.

The effect of volatility, the random wins and losses as the gambler plays many rounds, also gives him the illusion that he can beat the casino. A gambler may have a short-term winning streak, but in the aggregate, volatility will always pay the casino and make its owners billionaires.

Can volatility favour the gambler? Only in certain games in which the outcome of the next round of play is influenced by previous rounds. If he can remember the outcome of the rounds that have been played, the gambler has an advantage in betting on future rounds. Card counting in blackjack is the classic case. This involves calculating the odds of the cards yet to be played by counting those already played.

Several teams from the Massachusetts Institute of Technology used card counting to win in Las Vegas in the 1990s. Such a technique can achieve a statistical gain of 1 per cent in each round, beating the dealer's advantage of 0.5 per cent without card counting. But the casinos soon plugged this hole: the cards are now shuffled after every round.

Stock markets are often likened to casinos. Given current market volatility, is the investor better off playing blackjack in Macau than betting on stocks? The stock market as measured by the S&P 500 Index has a long-term annual gain of 9 per cent and volatility of about 15 per cent. In blackjack, the player loses an average of 0.5 per cent with a volatility of 112 per cent. On the face of it, the market is a better bet, a positive gain with less volatility.

But here's another perspective: casino games have clear rules in which expected value and volatility can be calculated. Future performance is the same as past performance. In roulette, there is a 49 per cent chance of winning if only one round is played. This chance does not change whether there's a debt crisis in Greece or the US. Is any punter certain that he has a 49 per cent chance of making money in the stock market?

In recent weeks, the Hang Seng and S&P 500 dropped more than 5 per cent in a day. In the stock market, past performance is no guarantee of future performance. Of course, the market eventually corrects itself for long-term investors, whereas the roulette player will eventually lose for sure.

As in the financial industry, the gaming industry has scam artists. Last month a seven-man gang in Macau swapped the card-shuffling machine in a casino with one that read the cards inside the machine. The information was transmitted to a gang member outside the casino who telephoned an accomplice at the table on which way to bet. They won US$24 million in one day, but were arrested soon after.

In gambling there are no sure wins - unless, of course, you own the casino.

Tom Yam, a Hong Kong-based management consultant, holds a doctorate in electrical engineering and an MBA from the Wharton School of the University of Pennsylvania. He has worked at AT&T, Ernst & Young and IBM