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Best of Alan Krigman

Gaming Guru

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Playing It Smart: How flexible are the return schedules for various games?

3 August 2009

Many casino games have return schedules with payoffs contingent on the result of a round. Video poker and other slot machines are classic cases. Some table games now also have this feature integral to the primary action as well as on bonus side-bets.

On window-type slots, designers are free to select chances of particular outcomes as well as their associated payoffs to achieve desired performance. In video poker and table games where outcomes hinge on elements such as cards, dice, or numbered ping-pong balls, chances are generally predetermined and winning combinations with their payoffs are left to be ascertained. The goal is to find pairs of values that make the game attractive to players while letting the house earn a profit over the long term.

The probabilities must add up to 100 percent, of course. And hit rate (the frequency at which players get anything at all) can't exceed 50 percent or the house won't have an edge. But within general and game-specific constraints, there's a fair degree of flexibility. Common practice is to start by specifying a theoretical overall return percentage and minimum acceptable hit rate, requiring that chances tend to fall as returns rise, and recognizing factors inherent in a game that can't be changed.

It's not always mathematically possible to meet every objective on a wish list simultaneously. For instance, probabilities or payoffs that seem desirable may have to be raised or lowered, target hit rate may have to be cut, or solid citizens may have to be given different returns than the bosses might prefer.

Lots of calculations are involved. The standard technique is called "linear programming." This once entailed a huge effort. But it's now built into the spreadsheet software available on most computers so it's quick and easy... assuming you know how.

A benefit of computerization is that the math mavens don't have so much sweat equity in the arithmetic they're satisfied once they find a set of answers that actually "work." They can run through multiple configurations then pick-and-choose.

As a simplified example, picture a three-window slot machine. An "x" or an "o" can appear in any position. The chance of an x is the same for all windows. Assume the designer wants the overall return to be 90 percent, and the hit rate to be over 35 percent.

The computer is to determine the probability of an x, along with payoffs for three x's, two x's and an o in any order, and one x and two o's in any order, that satisfy these conditions (three o's lose). The first of the accompanying tables shows three solutions that meet the criteria. Many others are also possible.

Combinations of probabilities and payoffs for simplified slot machine that yield different hit rates at 90 percent return

As a second illustration, say a casino would like higher than usual payoffs for royals and straight flushes at jacks-or-better video poker. They'll sacrifice returns for lesser hands, but want to keep the return at 98.5 percent. The probabilities and hit rate are fixed. Only the payoffs are adjustable. The second of the accompanying tables shows a few of the numerous solutions a computer might grind out once the spreadsheet is set up. Note that payoffs for straight flushes could be rounded-off (perhaps to 625, 350, and 250) with little impact on return percentage.

Are you apt to see any creative alternatives like these at your friendly neighborhood casino any time soon or ever? The poet, Sumner A Ingmark, anticipated just this question when he wrote:

To be slyer than the average fox,
You must start to think outside the box,
And prepare yourself for jolting shocks.

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.