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# What look like small changes in games can have big monetary effects

12 March 2012

Perhaps you’ve noticed marginal differences among nominally similar casino games. Well, caveat emptor – buyer beware. Minor alterations may cause major changes in house advantage, and what seem like attractive features for bettors could well be boons for the bosses.

Single- and double-zero roulette offer a classic example. Wheels have 37 grooves on the former and 38 on the latter . So the probabilities of the ball ending in any given position are 1/37 (2.7027 percent) and 1/38 (2.6316 percent), respectively. Does the 0.0711 percent offset matter much?

Make believe you want to bet on a three-number row. Payoffs are 11-to-1 in either configuration. The single-zero version has three ways to win and 34 to lose so you’re facing a house edge of (3/37) x 11 - (34/37) x 1 = -2.7027 percent. The double-zero version also has three ways to win but 35 to lose so edge is (3/38) x 11 - (35/38) x 1 = -5.2632 percent. If, over a period of time, you risk a total of \$1,000 on three-number rows, the house will rate your contribution to its coffers as 0.027027 x \$1,000 = \$27.03 in one case and 0.052632 x \$1,000 = \$52.63 in the other.

Were everything else equal, you’d be better off at single-zero tables. But, everything else isn’t equal. Minimum wagers are generally higher at these games. This makes sense for the casino. Were you to bet \$10 per spin, the house would figure your action as worth an average of 0.02703 x \$10 or 27.03 cents at single-zero roulette, compared with a heftier 0.05263 x \$10 or 52.63 cents at a double-zero table. Boosting the minimum for the privilege of fighting a lower edge raises the bosses’ earnings; for instance, a \$25 bet earns them an average of 0.02703 x \$25 or 67.57 cents.

Increasing your exposure to get the lesser edge can make your stake more vulnerable to the normal downswings of a session. Pretend you have a \$500 bankroll and risk \$10 per spin on one three-number row at double-zero roulette. You’d have roughly 81 percent chance of still being in the fray after 180 rounds, about four hours. A \$500 poke with \$25 per spin on a row at a single-zero table has only a 32 percent prospect of flouting the fickle finger of fate for this long.

Sometimes, differences between games are deceptive and solid citizens have to be relatively sophisticated to understand the trade-offs. Say two jacks-or-better video poker machines have the return schedules shown in the accompanying table. Game 1 pays more for full houses and up, less for triplets, and the same for all other outcomes. At first glance, the richer high-end payouts make Game 1 look more tempting. Considering overall return percentage, though, it’s worse. Trips hit far more often than full boats and above. The influence of payouts on return percentage of triplets therefore swamps that of returns at the higher levels. When the dust settles, Game 1 returns 94.4 percent while Game 2 is more liberal at 98.4 percent.

Returns for alternate jacks-or-better games
```Hand               Returns
Game 1   Game 2
Royal             1000      800
Straight Flush     100       50
Full House           9        8
Flush                6        6
Straight             4        4
Triplets             2        3
Two Pair             2        2
Jacks or Better      1        1
```

Blackjack offers another illustration. One-deck games in which blackjacks pay 6-to-5 (1.2-to-1) rather than the standard 3-to-2 (1.5-to-1) are now in the casino repertoire. They appeal to players who’ve heard that mixing fewer decks cuts house advantage. This is true when the rest of the parameters of the game are identical. And it holds for everyone – card counters, Basic Strategy buffs, and the hoi polloi. A primary reason is that the probability of a blackjack rises as the number of decks falls. For “N” decks, it’s 2 x (N x 4)/(N x 52) x (N x 16)/((N x 52)-1). This equals 4.7451 percent with eight decks, 4.7489 percent with six, and 4.8265 percent with one.

Players like to get blackjacks. They win immediately and receive bonuses. Few realize the impact of the bonus on house edge. A blackjack paying 1.5-to-1 lowers edge in the game by 0.5 times the probability it will occur. That’s 0.5 x 4.8265 = 2.4133 percent for one deck, a gain over 0.5 x 4.7489 = 2.3745 percent for six and 0.5 x 4.7489 = 2.3726 percent for eight. But with one deck where blackjacks pay 1.2-to-1, edge drops by a mere 0.2 x 4.8265 = 0.9653 percent. The benefit is 2.4133 - 0.9653 = 1.448 percent less than in an otherwise equivalent 3-2 one-deck game. Basic Strategy can keep edge below 0.5 percent at an eight-deck table with the usual rules. If a 3-2 one-deck game had zero edge, house advantage would soar to 1.448 percent when blackjacks paid 6-to-5. Failure to appreciate this phenomenon can accordingly be costly to the unwary.

Of course, gamblers can hit it big, lose their shirts, or finish between these extremes regardless of what or how they play. Still, credit is due to Louis Pasteur who said, “Luck favors the well prepared.” Praise is also owed to the illustrious inkslinger, Sumner A Ingmark, who intoned:

Examine minor variations,
Cause change may promise celebrations,