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Best of Alan Krigman

Gaming Guru

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Playing it smart - How casinos choose payoffs

18 June 2007

Many gambling propositions either win, lose, or push, and have probabilities that can't be adjusted within the constraints of the game. Double-zero roulette offers examples. A wager on a 12-number column has a probability of 12 out of 38, roughly 31.6 percent, because of the design of the wheel. Casinos have little choice as to what they pay for bets like this. Given the probability, setting the payoff determines the edge. And there's a narrow range of edges simultaneously acceptable to the bosses and the bettors. Adding the practical requisite that returns be in whole numbers, often only one payoff will do the job.

Games in which winning rounds can earn varying amounts have far more flexibility. Slots are in this category, as are table games based on poker where for instance a winning flush is ranked higher and accordingly pays more than a pair. The basic rules of these games usually establish classes of hands which are winners. And, although in some cases decision strategies influence the probabilities of various final results, chances typically shift only slightly among players who show a modicum of common sense.

Jacks-or-better video poker affords an illustration. The ideal probability of winning one or another final hand is shown in the accompanying table. The "full-pay 1" column gives returns for the elusive "9-6" game (9-for-1 on a full house and 6-for-1 on a flush), where average player payback is essentially 100 percent.

Possible alternate return schedules for hands in jacks-or-better video poker, all averaging 100 percent player payback

Handprobabilityfull-pay 1full-pay 2full-pay 3
Royal0.0025%8003,15071
Str Flush0.0105%501115
Quads0.2364%25913
Full House1.1520%9810
Flush1.0800%669
Straight1.1350%445
Triplets7.4490%333
Two pair12.9400%222
High pair21.4500%111

But, what if a casino tried to attract more action with a jackpot exceeding 800-for-1 ($1,000 for $1.25)? The house can tweak returns all along the line, raising that on a Royal while cutting it on other winners, yet leaving the overall payback as-is. One set meeting this standard is shown in the "full-pay 2" column.

Casinos can also go in the other direction. They might figure that loading the returns into the jackpot will make a few solid citizens exceptionally happy, but will knock too many others for a loop so fast they won't enjoy their gambling experience and will go elsewhere next time. The jackpot can be lowered and returns for lesser-ranked hands increased in a manner that gives patrons more gambling time on their bankrolls, while still leaving the overall return percentage the same. The "full-pay 3" column gives a possible return schedule that achieves this goal.

Actually, it would be relatively simple to design a machine that lets players pick from among several alternate sets of payoffs, depending on whether they preferred to go for a big score or a long session. They could all give the house the same average earnings potential. The games wouldn't have to be full-pay, as in the example. The house could set any desired return percentage and calculate multiple payoff schedules to fit.

There's no great mystery to the math here, although it can get tedious and is vulnerable to errors if you do it by hand. The approach is to multiply the predetermined chance of each hand by the payout to get the average return at that level. Then add up the average returns to get overall payback percentage.

If you're good with Excel spreadsheet software, you can work backwards. Set up a matrix with the known probabilities for each hand, insert formulas to multiply what's in cells representing probabilities by the payoffs, and sum the results. Then use "solver" to find alternate sets of payoffs that make the sum the overall payback percentage any specified value. Of course, if everyone with a computer learns it's this easy, casinos won't pay me those big bucks to do it for them. And I might be forced back to seeking that missing link in the system I'm developing to beat the house. As the immortal muse, Sumner A. Ingmark, muttered:

Here's a puzzle, and you may have come across it: If
in casinos, mathematics reigns as causative,
Why won't betting double negatives go positive?

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.