“Optimal” Wagers on Penny Slots
Tags: gambling, modeling, simulations,
Categories: Mathematics, Modeling,
My girlfriend and I have been going about once a month to one of the two easily-accessible casinos in the NYC metro area (Resorts World Casino at Aqueduct Racetrack and Empire City Casino at Yonkers Raceway). But we never play table games. We’re not there to try to win big. We like the colors, sounds, and lights. We’re in it for the slots!
We only play low-denomination machines – mostly 1¢. Both of us like machines that provide the opportunity to win free spins. But we have different strategies. My girlfriend likes to find a machine she likes and place small bets until she gets a big win. I like to switch machines often, in search of one that’s hot. Our different strategies aren’t based on logic or a rigorous analysis of how these machines work. We just have different instincts.
So I decided I would do a more in depth study of how the betting strategy and choice of wager affects the expected winnings by running some simulations! I considered two versions of the same game (Cleopatra by IGT) that vary in volatility – basically the payout variance. Highly volatile machines pay out big but rarely, while low-volatility machines distribute small payouts more often but the jackpot is not as large. Low-volatility machines have become quite popular with casino owners in Vegas as the small payouts keep people hooked and playing for longer. When playing all 20 paylines of Cleopatra, it is a medium-to-low volatility game. When playing just one payline, it is a high volatility game.
Photo from Wizard of Odds
Three or more sphinx symbols trigger 15 free spins. It’s the perfect example of the type of game we’d play!
The Wizard of Odds and Casino Guru did a nice deconstruction of the machine, and the payouts one can expect for the 20 line and 1 line version of the game. Below are the 20 line odds.
| Win as x-times the bet interval | Share on RTP | Hits in 10 millions |
|---|---|---|
| <0x | 0% | 0 |
| <0.1x-0.2x) | 0% | 0 |
| <0.2x-0.5x) | 2.85% | 929482x |
| <0.5x-1x) | 4.68% | 740452x |
| <1x-2x) | 7.41% | 563289x |
| <2x-5x) | 26.11% | 1031867x |
| <5x-10x) | 9.5% | 149001x |
| <10x-20x) | 12.27% | 92050x |
| <2x-50x) | 18.23% | 58006x |
| <50x-100x) | 12.63% | 17450x |
| <100x-200x) | 4.74% | 3538x |
| <200x-500x) | 1.45% | 505x |
| <500x-1000x) | 0.12% | 20x |
The probability of ending up in each category is given by the value in the right-most column, divided by ten million simulations.
Some Results
All my code is available in this GitHub repository, and you can view the Jupyter notebook with the results here. Below is a histogram of our expected winnings in the 20 line game for our two different strategies (I switch more often than she does). Lori is more likely to lose her money faster, but she sometimes achieves bigger wins by playing more often. Full simulation details are in the Jupyter notebook.
I considered the best wager to place if you start with 40 dollars (my preferred maximum loss) and want at least a 10% chance of doubling your money. The optimal wager of 8 cents per bet emerges as the smallest one to give you this probability, as it reduces your risk of losing it all. The results below are for the 1 line game, and my switching strategy.
I also considered what happens when you adjust your wager in response to your winnings. Say you start with 100 dollars. What if you reduce your wager from 30 cents to 10 cents when you’re up by 30 dollars or more, and then readjust the wager to 30 cents if you’re back down to where you started? My theory was that this would improve my returns, as I reduce my chances of losing any winnings. On the 1 line game, I went from a 90.2% return to a 99.3% return using this strategy! I was less likely to lose all my money, as the weight of the probability distribution shifted towards the center.
All in all, it was fun to treat our trips to the casino as a case study for a math-related project. Let me know what you think!