How Long Is The Long Run? Part 1
Surprisingly, the length of time has little to do with it
by John Robison
Video poker experts frequently talk about a magical length of time known as the long run, but rarely if ever do they define how long the long run is. Playing a slot or video poker machine is, statistically speaking, an exercise in Random Sampling with Replacement. Statisticians use Random Sampling frequently (choosing people to participate in surveys, for example). We can use the formulas that statisticians have developed for Random Sampling to help us define the long run.
People usually describe the long run in terms of a length of time, like six months or a year. But time is not the proper measurement to use to express the long run for a slot or video poker machine, or for a slot or video poker player. Time, in fact, has little to do with the long run.
Consider a slot machine. We calculate its actual payback by dividing the total amount of money players have won on the machine by the total amount of money they have wagered on the machine. And we know that as we approach the long run, the result of that formula gets closer and closer to the machine’s long-term payback—the payback calculated by dividing the total amount of money that is won from all the winning combinations on the machine by the product of the total number of combinations multiplied by the number of coins played.
When the machine is not being played, is anything happening that affects the result of the actual payback formula? No, neither the amount of money paid by the machine nor the amount of money played on the machine is changing. The clock is ticking, but the machine is not getting any closer to the long run. The long run cannot be expressed as a length of time.
The only way to affect the result of the actual payback formula is to play the machine. The only measurement of the long run that is valid, therefore, is the number of spins played. We have to express the long run as a number of spins.
Using formulas developed for Random Sampling, slot manufacturers calculate a Volatility Index for each machine. Using the Volatility Index, they then calculate Confidence Intervals for the actual payback of the machine after a certain number of plays. An example is this interval from IGT Program PS0043 for a Double Diamond slot machine: After 1,000 spins, we can be 90 percent confident that the machine will have paid back between 57.35 percent and 122.86 percent of the money wagered on it. That’s a very large range. One thousand spins is not long at all in the life of a slot machine.
The word “confidence” is in “Confidence Interval” because we have to choose how confident we want to be that the actual payback will fall within the interval. The only way we can be 100 percent confident is to include all possible outcomes, no matter how unlikely. A 100 percent Confidence Interval for a slot machine after 1,000 spins would have to include 1,000 losing spins and 1,000 jackpots. I’d love to get 1,000 jackpots in a row (I could retire!), but it’s not likely to happen. Nevertheless, the only way to be 100 percent sure that the payback will fall in the interval is to include 1,000 jackpots in the interval.
We are more interested in what is likely to happen than in every possibility of what could happen, so we’ll use a confidence level of 90 percent, which is what most jurisdictions use in the calculation. (New Jersey uses a confidence level of 99 percent.)
Slot managers and slot regulators use the Confidence Intervals to verify that machines are working properly. They compare the amount of money a machine has paid back to its players with the amount predicted by the Confidence Interval for the amount of play the machine has received. If the actual payback falls outside the interval, they inspect the machine to make sure it has the correct payback program and that it is operating properly. It’s possible that this machine could be one of the 10 percent of machines whose paybacks do not fall in the interval (after all, we’re only 90 percent sure that the payback will fall in the interval). It’s also possible that someone has tampered with the machine.
I’ve never seen Confidence Intervals calculated for video poker machines. But slot managers do, however, know how much a paytable can pay back with optimal play and how much it pays back using the strategy a typical player uses. For example, a 9/6 Jacks machine can pay back as much as 99.5 percent, but most players play at only a 95 percent level. There’s nothing wrong with the machines. The problem is with the players. They’re not playing the optimal strategy.
Four things determine how much you win at video poker: the paytable, your strategy, the number of hands you play, and randomness (luck, if you will). The Confidence Intervals give you a way to quantify how big an effect luck has on your results. Using the Confidence Intervals, you can see whether it’s worthwhile for you to learn a more complicated—and mathematically correct—strategy. It’s possible that, for the number of hands you’ll play in your lifetime, randomness will still have a greater effect on your results than making, say, penalty card adjustments.
You can also use the Confidence Intervals to compare different paytables and see how many hands it takes before the effect of the paytable change has a greater effect on your results than randomness.
Finally, you can use the Confidence Intervals to determine how long the long run is. As the number of hands played increases, the size of the interval decreases. You can never totally eliminate the effect of randomness, but its effect diminishes as you play more hands. Say you decide that the long run is reached when the size of the interval falls below one percentage point. You can check the Confidence Intervals to see how many hands it takes for the size of the interval to fall below one percentage point.
At that point I have to admit that the Confidence Interval formula is not entirely correct for small (less than 1,000,000, say) hands. It’s close enough for our purposes, though. And, as I was told when I questioned why a marketing department in which I worked would complain about the many problems in their research numbers yet still base marketing decisions on them, they’re the only numbers we have.
Next month, we’ll look at Confidence Intervals for some video poker paytables to see how long the long run is.