GGB is committed to providing updated news and analysis on our weekly news site,

Doing the Math

The creativity of game design goes hand in hand with what keeps players at the game—the program math

Doing the Math

Fifty years ago, the math of the slot machine was a simple proposition. Three reels, 64 stops per reel, and no special features outside of perhaps a wild symbol. The functions of game design and program math were usually relegated to one person.

The virtual reel system that came in the 1980s changed everything, of course, with an unlimited number of virtual stops at the game designer’s fingertips. Still, things remained relatively simple, at least until Australian-style multi-line games and bonus features debuted in the mid-1990s. With more possible outcomes to every spin, program math gained an importance it did not previously have.

These days, mathematicians are essential to game design. They take creative concepts that present a multitude of outcomes, and features from hold-and-re-spin to wheel bonuses to multiple progressive jackpots, and weave them into game programs that keep players playing while making money for the casinos.

“Math is critical,” says Michael Brennan, chief product officer for slot supplier Bluberi. “In the simplest terms, the old adage of ‘the art gets the player to stop and try your game, but the math keeps them there (and keeps them coming back)’ really does ring true today more than ever.”

“Game math is the foundation of the game,” notes Gerard Crosby, senior vice president and chief games product officer for Konami Gaming. “The game designer works closely with the mathematician creating a compelling math model, then works with a small team of artists, engineers and composers, developing the game vision, ensuring the player chases are transparent and fun to play.”

Other suppliers sing a similar tune when it comes to the importance of math in the game design equation. “The combination of math and art is extremely important to any slot product,” says Ainsworth’s Andrew DuBose, a game designer who also holds an M.S. in mathematics. “We are always trying to meet the players’ needs and expectations during game play.”

Ainsworth Lead Game Designer Terry Daly, a 30-year veteran of the craft, digs down deeper, noting that math is what governs the game style. “A gambler-style game will have a low win frequency, but higher pays,” he says. “An entertainment-style game will have more frequent bonuses, but lower pays. It’s important to determine what you are trying to accomplish with each game. You are really trying to reach two audiences. There are players who have more time than money and other players who have more money than time. We need to appeal to both on casino floors.”

“Konami’s math is targeted toward the gambler player type, and game themes tend toward the styles sought out by that type of gambler,” says Ian Arrowsmith, senior director of game development for Konami Gaming.

Michael Mastropietro, senior vice president of game development for Light & Wonder, says, “To achieve successful game design, mathematicians, game companies and designers must embrace the idea that math is the soul of the game. The experience of the game and the excitement behind each moment is driven by math. When you mesh these two elements together—creativity and mathematical function—you achieve a balance that not only is functional for the casino but is also excitable for the player.”

Anthony Baerlocher, vice president, innovation and mechanical reels for IGT, puts it succinctly: “To me, the math is the personality of a game,” he says. “Different people are attracted to different types of games and game math. The math itself is complicated, but it is a science. It is true, it is accurate. But there is an infinite number of permutations on how to get to a 95 percent payout. That’s where the creative process comes in—building the pay loading, the frequencies, how much goes into a bonus that creates that game’s unique personality.”

Signature Style

Those “unique personalities” of games have allowed each supplier to develop a signature style—game characteristics that players recognize as the product of a particular manufacturer.

“We do think we’re evolving a signature Bluberi approach,” comments Brennan. “Can we take the final game math and simply re-present key events and mechanics to the player to add variety, depth, and re-playability without affecting the core math pillars?

“For example, if you spin a wheel to win a multiplier, why not, for a less frequent subset of this occurrence, present the multiplier first then recreate the wheel spin experience? This is a method we used to great success in Devil’s Lock and are repeating with upcoming releases.”

“Ainsworth’s high-denom math models have been some of Ainsworth’s signature game styles,” says Daly. “That’s a much different math style than a traditional three-reel game or a five-reel low-denom game. It’s all about appealing to different players.”

“All mathematicians have what we’d consider a style or fingerprint to the designs we put out,” says L&W’s Mastropietro. “This signature fingerprint is woven consistently throughout studio outputs.”

Tweaking the math while keeping a signature style is particularly important in the case of recurring game themes. “When it comes to building game families, you typically have math models that stay consistent and then iterate the creative,” says Mark DeDeaux, general manager and senior vice president, slots for AGS. “As an example, with the game Rakin’ Bacon!, we have certain standards within the game to make sure they are always present. When we make sequels, there are certain items we never want to depart too far away from, like the base mechanics, because players have certain expectations of the game.”

“For game families, we try to keep jackpot and feature hit rates the same,” says Ainsworth’s DuBose. “On all our three-reel games the jackpot frequency will be similar. The math changes may be more subtle in line pays or adjusting other frequencies of events that players will want to chase.”

“Thinking back to many of our game families,” says Mastropietro at L&W, “you’ll see various iterations of the game, and with each new addition to the family, there’s something that makes it more compelling for the player. We look to take the fun parts of the game that really draw in the attention span and excitement and give them a plus-up.

“For example, our Dancing Drums began with the feature play of drums growing up to six stacked, but when we released Dancing Drums Explosion, we thought, ‘OK, how can we one-up this for the player?’ and that’s where we decided the drums now would grow to stack eight high.”

One theme that has maintained its signature style throughout the past 25 years and scores of individual titles is the Wheel of Fortune franchise from IGT.

“There are certain functionalities that are combined with the wheel and frequency of the wheel that have fulfilled play expectations over so many years,” says Dubravka Burda, senior vice president, global studios for IGT. “We evolved the math over the years, but the functionality, the color, and the parts in the game are very consistent. This is something that is definitely IGT’s signature.”

The Basic Process

“You are really trying to reach two audiences. There are players who have more time than money and other players who have more money than time. We need to appeal to both on casino floors.” —Terry Daly, Lead Game Designer, Ainsworth

As more and more diverse game features have been added, game design has become more complex over the years, as all those features and outcomes must be woven into an effective math program. However, the basic process of game development has remained as a joint effort of creative design and mathematics.

“Essentially the whole process hasn’t changed at all in the fact that you’re aiming to create an experience that is going to get the player excited to pursue a win and want to play the game each time they visit a casino,” says Ainsworth’s Daly. “What has changed is the technology. The math is much more complex. We can really drill down into how players interact with the game.

“Additionally, the math modeling is much more flexible. In older games, especially old mechanical games, you could get very limited in what you were able to execute. To put it simply, the power of microprocessors today is infinitely more powerful than anything that was available in the past. The ability to use graphics to reflect the math has obvious changed significantly over the years. You can do things today that would have been unfathomable 20 years ago.”

“Great game ideas come in all different shapes and forms,” says

DeDeaux at AGS. “Sometimes it’s iterating off of mechanics we are familiar with in the market and building math models that accentuate the game. It starts with the concept of a game and simultaneously, graphics, art and math come together to complement it. We identify what we want to achieve and have ongoing development and iterations of it to get there.”

“Concept first, then we assign a math model,” explains Martin Blais, mathematician manager for Bluberi. “And it all has to gel, following our cornerstone design principle of ensuring everything is ‘where it matters.’ Big animations should coincide with significant math contributions, for example.”

DuBose at Ainsworth comments that effective game design is a process that can take many forms. “In the recently released Grand Legacy and Royal Legacy, we thought we had a good game theme and wanted to incorporate some ideas like larger multipliers and the chance to win multiple jackpots on a single spin,” he says. “We sat on it for bit until we were able to come up with the right math for the game.”

“Konami’s R&D works closely with product management to determine the business and market needs with respect to all the different types of games required,” says Arrowsmith. “The game designer usually has an idea of the type of game they want to make. They then work with both math and art to come up with an initial concept, with the math and art complementing each other, to produce an awesome game.”

What has changed in this process is the complexity of the games themselves, according to Konami’s Crosby. “The games have become more complex as more and more exciting new free game and bonus features are included,” Crosby says, “and thus, product technical documents have grown to be the size of a large novel, and simulations of the math models have become a critical part of the math design process.

“Also, the design of the math model is much more collaborative between mathematician and game designer to ensure the best playing experience is achieved. Wizard Strike is a perfect example of this, where mathematician and designer worked hand-in-hand to produce this compelling model.”

“Games used to be pretty simple, but they’ve grown significantly in complexity,” says Blais at Bluberi. “The role of a mathematician has also evolved, where today a strong mathematician needs to be versatile—he or she has to know some programming, for example.”

“Back in the day when it was three reel, one-line stepper, it was here’s a math idea first,” says IGT’s Baerlocher. “That’s inverted in today’s world. It’s really more about coming up with the concept and then making sure that the math fits. And then what we see happening is the math actually adapts to the original concept. If we find it doesn’t play the way it was originally envisioned, we make modifications so that the concept and the math work hand in hand.”

“Game math is fairly finite,” says DeDeaux at AGS. “You may have a game concept that is creative in the types of bonuses you want to do, but adapting the game math to the game is challenging, and takes a lot of creativity and finesse to have the exact outcome mathematically in the bonus, features, and base game. It requires a lot of art, science, and being creative in solving challenges.”

Into the Future

Those challenges will no doubt increase as the creative teams at the slot suppliers come up with more innovative concepts and new ways to play.

“One thing that is certain is that games will become more complex in the future,” says DeDeaux at AGS. “As an industry, we will face the challenge of entertaining and capturing the younger demographic. When you look at how tech has evolved, and the types of entertainment experiences offered on cellphones, we will need to think about how we can compete with future technology, in general, to continue moving forward as an industry.”

IGT’s Baerlocher sees the challenges of game design evolving as more information becomes available on player preferences. “The big thing I see changing in the industry is the availability of data,” he says. “It used to be

really hard to get any type of information out of the casino industry on how games were performing, but in the past four or five years, operators are being more open to sharing data because they realize that by giving the suppliers some insights, they’re going to get a better product.

“With information like the Eilers report and ReelMetrics that shows which games are the top earners, you can spend more time studying those games and learning from them… using data to help our creativity.”

“I like to use the analogy that building a game is like building a great (music) album,” says DeDeaux at AGS. “You need lyrics, your music, and all the different elements to create one. But it’s the combination of all these things working harmoniously together that makes it great.

“Math is like the heartbeat of a game, similar to how the rhythm is the heartbeat of a song—and it is critical to the overall success of the game.”

Frank Legato is editor of Global Gaming Business magazine. He has been writing on gaming topics since 1984, when he launched and served as editor of Casino Gaming magazine. Legato, a nationally recognized expert on slot machines, has served as editor and reporter for a variety of gaming publications, including Public Gaming, IGWB, Casino Journal, Casino Player, Strictly Slots and Atlantic City Insider. He has an B.A. in journalism and an M.A. in communications from Duquesne University in Pittsburgh, PA. He is the author of the books, How To Win Millions Playing Slot Machines... Or Lose Trying, and Atlantic City: In Living Color.  

    Recent Feature Articles

  • Funding the Future

    Gary Ellis’ vision of a cashless casino ecosystem with Marker Trax and Koin

  • Age of the ETG

    Electronic table games have grown from simple automated roulette machines into a genre that is steeped in innovation.

  • Online in Ontario

    Stakeholders deem Ontario a success, but also a work in progress.

  • Mixing It Up

    Developing slot floor strategies for emerging markets.

  • Gaming & Diversity: Staying the Course

    DEI has encountered big resistance of late. Here’s how gaming companies continue to build a fairer workplace.