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Best of Alan Krigman

Gaming Guru

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Drooling Dogs, Pecking Pigeons, and Sanguine Slot Zealots

22 February 2006

Pavlov used food to do it with dogs. Skinner used bells to do it with pigeons. Casinos use multi-line slot machines to do it with gamblers, which may or may not be the same as using bells to do it with pigeons. "It," in all three cases, being reinforcement for behavioral training and eliciting of desired responses.

What if you started playing a machine and hardly got any hits maybe you averaged one every five spins or less. You'd leave. Maybe go to another machine, a different casino, an activity other than gambling. More, say this happened on a few casino visits running. You might just skip your next planned jaunt to the jackpot joint. This isn't talking wins, although they'd be nice. It's talking hits. Getting at least something back in a fair fraction of rounds motivates players to keep betting and feeds their fantasies about this shaping up as their lucky day.

The quandary for casinos is, how to set up a slot machine with a hit rate that provides good reinforcement, yet also offers engaging payouts and still holds the money needed to cover the mortgage. The traditional problem is that these factors represent trade-offs. For instance, if hit rate gets to 50 percent, you'd only get your money back on each try and the casino would just break even. To offer big jackpots and decent intermediate returns at some established house advantage, the bosses have to lower the hit rate below 50 percent. You can see the dilemma.

This is the genius of the multi-line machines now swallowing the slot scene. They can offer hit rates well over 50 percent, while retaining attractive prize structures and keeping a lot of the solid citizens' hard-earned moolah at the end of the month.

To get an idea of how this is done, picture a highly simplified machine with two win-or-lose lines. More complex devices would work the same way. For a $1 bet, each line of the hypothetical machine has a 40 percent chance of hitting and returning $2.25 ($1.25 win and get the $1 back), offset by a 60 percent chance of losing. The house has a steep 10 percent edge, giving a payback of only 90 percent. For enquiring minds who want to know, this is found as 40 percent of $1.25 minus 60 percent of $1.00.

The accompanying table shows what can happen if both lines are statistically equivalent and you bet $1 on each. Entries in the "probability" columns are the chance of the combined event; these are the products of the probabilities of the two separately so, for example, win-lose is 40 percent times 60 percent, 0.4 x 0.6 = 0.24, or 24 percent. The theoretical net per hundred spins is the probability times the profit or loss multiplied by 100.

Effect on hit rate and casino edge of playing two lines together

result
probability of result
win or loss betting $1 per line
theoretical net for 100 spins
win-win
16%
+$2.50
+$40
win-lose
24%
+$0.25
+$ 6
lose-win
24%
+$0.25
+$ 6
lose-lose
36%
-$2.00
-$72

Sum the first three rows of the "probability" column and to get the overall hit rate on this machine as 64 percent. Up from 40 percent on one line. What about edge? It was 10 percent on each line by itself. Bet $200 in all and the casino chalks up $20. On two lines together, at $1 apiece, the handle on 100 spins would be $200. The "theoretical net" column shows you pick up a total of $52 and drop $72. The loss is still $20 on a $200 handle, 10 percent. So the casino has made the game tantalizing and gives you reinforcement, but hasn't sacrificed its take on the action.

Pretend you play across the board on a 10-line machine with a 90 percent return. Further, imagine that to show an imposing list of possible returns, each line has a paltry 20 percent hit rate. The chance you'll be told you've won and get something back on a spin, even if you actually lose by missing on nine lines and only recovering your bet on the 10th, is 9,999,998,976 out of 10 billion. So reinforcement acts two ways. It keeps you playing and encourages you to bet on more lines. Unless, of course, you've heeded this hoary hortation of the Helicon, Sumner A Ingmark:

Since appearances can fool you,
Don't let smoke and mirrors rule you.

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.