Playing is smart - How casinos choose payoffs

Ever wonder how casinos decide what to pay on various bets? Or why returns in some games are always the same while those in others not only vary among casinos but from one table or machine to the next within a single joint?

Payoffs are governed chiefly by chances of winning and the edge a casino gets on a bet. The practicality of betting and paying rounded-off values and, in some cases, the psychological trade-off of frequent small versus rare large returns are also factors.

Options available to the casinos depend on two conditions. First, whether probabilities are predetermined or can be set by the house. Second, whether the bets simply win or lose, or have multiple payoff levels as in games scored by poker rankings.

Predetermined probabilities and bets that either win or lose are simplest. These are instances where payoffs tend to be universal.

Double-zero roulette offers examples. The wheel has 38 identical grooves. Chance of winning on a single number therefore has to be one out of 38 or 1/38; chance of losing is the other 37/38. If a $1 bet paid $34, house advantage would be [(1/38) x $34 - (37/38) x $1]/$1 or 7.89 percent. With a payoff of $35, edge is [(1/38) x $35 - (37/38) x $1]/$1 or 5.26 percent. At $36 payoff, edge would be [(1/38) x $36 - (37/38) x $1]/$1 or 2.63 percent. For typical bet sizes and rounds per hour, 7.89 percent would chew players up so fast that nobody would play. At 2.63 percent, the house wouldn't average enough income to cover operating costs. It's impractical at roulette to pay anything but integral multiples of units bet, so 35-to-1 is the only sensible alternative.

Pretend you bet your $1 on a three-number "street" rather than a single spot. Chances are 3/38 of winning and 35/38 of losing. Were this to pay $10, edge would be [(3/38) x $10 - (35/38) x $1]/$1 or 13.16 percent. At $11, the formula yields 5.26 percent. For $12, you'd have a 2.63 percent advantage. The first of these would scare players off while the third would put the casino in the poorhouse. So 11-to-1 it is. Moreover, analogously to other combination roulette bets, 11-to-1 yields the same return betting $3 on the street or $1 on each of the three individual numbers.

In single-zero roulette, chances are 1/37 of winning and 36/37 of losing on a single spot. A payoff of 34-to-1 would yield a house advantage of 5.4 percent. At 35-to-1, it's 2.70 percent. And at 36-to-1, edge would be zero. A 5.4 percent edge defeats the purpose of offering single-zero to high rollers. Zero edge is a no-go. So casinos pay 35-to-1 and settle for 2.70 percent, taking less bite from a bigger pie than at double-zero tables.

Craps offers a host of illustrations. Say you Place the nine. You win on the four ways the dice total nine, and lose on the six ways they equal seven. So the chance of winning is four out of the ten relevant results, and that of losing is six out of 10. Were a $5 bet to pay $6, edge would be [(4/10) x $6 - (6/10) x $5]/$5 or 12 percent. At $7, edge equals 4 percent. An $8 payout would favor bettors by 4 percent. The 12 percent would kill the bet and the bosses can't give players an edge; this leaves $7.

Bets on the six have chances of 5/11 to win and 6/11 to lose. Some casinos offer a Big Six bet, booking wagers in any denomination and paying even money. Edge is [(5/11) x $1 - (6/11) x $1]/$1 or 9.09 percent. Little wonder this is considered a sucker bet. Were $5 to pay $6, edge would be [(5/11) x $6 - (6/11) x $5]/$5 or zero. Little wonder this bet isn't offered. Instead, Place bets on the six are made in multiple of $6. Edge works out to 9.09 percent at even money and 1.4 percent at 7-to-6. At 8-to-6, solid citizens would get 6.06 percent advantage. The casinos are pretty much boxed into the 7-to-6 payoff.

Payoffs are easy to specify in games where probabilities can't be adjusted and bets either win or lose. It's a matter of an edge both bettors and bosses can bear. So casino bean counters get the comfort conveyed by the doyen of the dithyramb, Sumner A. Ingmark:

Decisions are made with the least apprehension,
When choices comprise just a single dimension.