Authors’ Note: In this fourth article of our 12 part series on customer centric gaming floors, we will examine game volatility and its use as an analytic tool in the casino of the future. Please note these articles are meant to stimulate thought and that we are using some deliberately provocative metaphors and examples which should be taken with a grain of salt.


Slot machine volatility typically refers to the amount of risk in a specific game: a game with infrequent large payouts is considered very volatile while a game with frequent small payouts is less volatile.

But as its very name connotes, volatility fluctuates often and can be influenced by a number of different factors. For instance, a normally less volatile game can see its volatility increase dramatically if it is linked to an extremely high jackpot. Similarly, on an individual game basis, a player can change a slot’s volatility simply by betting more or less lines on the game, impacting the size and chance of a payout.

All this means careful consideration must be paid to the pay table as player choice can dramatically impact volatility. For example, in a classic game like Double Diamond with no progressive payout, the amount of the payout is correlated with the amount gambled, and the prize for a two-coin bet is double the prize for a one-coin bet. But in Wheel of Fortune and similar games, the size of the payout can be vastly different for a maximum three-coin bet vs a one-coin bet as it takes a maximum bet to qualify for the top prize.

So to get a true measure of game volatility, understanding the player experience for individual players is critical. The player experience is a mixture of both the volatility of the game and the volatility of their betting patterns. In other words, players have a direct impact on the risk they are taking in the game and the nature of their gaming experience.

CALCULATED EFFECT

There are some complex calculations involved in deriving slot machine volatility. To start, let’s look at calculation and simulation—the two most common ways of determining the hold percentage on machine games.

Calculation:This method involves determining the hold percentage based on probabilities shown in the pay table. This approach is very accurate and given the correct mathematics is a great way of looking at many games. The essence of the method is to build out complex formulas to show all the possible outcomes from a game.

Simulation: In the real world of game design, we have extremely complex sets of pay tables that make up the gaming experience. It turns out that we can avoid the hard work of calculating the probabilities and just run hundreds or thousands (or even hundreds of thousands) of simulations and measure the outcome. This method comes into its own when we want to vary the inputs (such as pay lines or bet amount) to determine the outcome from a sequence of games. This simulation model is often called a Monte Carlo method.  It represents a broad class of events that are normally undertaken with computers that crank out large numbers of simulations.

Monte Carlo can be extended to cover quite complex behavior as we can vary the input parameters. For example, we can change the bet amount in response to wallet constraints or in response to sequences of wins or losses. This ability to change the input parameters allows us to simulate very complex behaviors that become extremely hard to model using calculations.

Stepping past the basic calculation of hold percentage outcome, we can add layers of complexity to the problem by looking at calculations of metrics such as volatility. If we think about the volatility of the game and include the changes in customer behavior, we introduce enough complexity that simulations are the best (and possibly the only practical) way of deriving the numbers.

GONE FISHING

The fisherman’s analogy also helps to explain some of the behavioral aspects of the gambling experience. In recreational fishing, the fisherman invests time (and money) in the hope of catching a fish. In fact, the time and money usually invested in fishing is nothing like the return that is received. However, the moment of catching a fish makes it all worthwhile; this moment in time is much like a winning moment on a gaming machine.

Furthermore, once the fisherman catches a fish, they can go fishing more frequently, fish with more rods and otherwise change their behavior to impact how many big fish they will catch. This change in behavior will probably result in the fisherman catching more fish; in other words, by changing his behavior the fisherman is able to impact the frequency of unlikely events. Players can do the same by betting on many lines or by changing their bet in response to a winning event.

In part two of this series that ran in March Slot Management & Marketing, we introduced the concept of outcome versus optimization metrics. The canonical outcome metric in gaming is theoretical win.  Theoretical win can also be illustrated using the fishing analogy. Consider this example: when looking at the amount of fish caught, if a fisherman catches a fish, on average, every six hours, then the expected value of a catch is 1/6 of a fish per hour. However, when we look at the fishing experience it is dramatically different—essentially the fisherman either catches a fish or they do not, there is no 1/6 of a fish. The fishing experience is defined by the fisherman’s experience of catching or not catching a fish. Furthermore, when this fisherman catches a fish they may think they are on a run and change their behavior; they may stay longer, throw in another line or pay a quick return trip the next day.

Meanwhile, in the gambling experience there is one huge difference: the players can immediately change their bet by, for example, doubling down. Following the fisherman’s analogy, we could think of this as using the first fish caught as bait, hoping that a bigger fish will be caught.

This is not the only example of theoretical win analysis falling short in the face of player experience calculations. While it remains a central metric to gaming, theoretical win has many flaws when it’s applied to the player experience. In short, while the casino can calculate the expected value of the betting event, the player is only playing for winning events and they measure their results on their wallet.

STREAKY MATH

When perceived through the prism of gaming experience, a unique transformation occurs to volatility—it becomes the aspect of the game the player equates with luck. Furthermore, the player’s response to winning (or losing) events dramatically impacts how this “luck” impacts them.

To fully understand this we need to look at the streakiness of the gaming experience. Winning five times in a row on a game, when looked at from a single starting point, is very unlikely; however the mathematics is both simple and deceptive.  The probability of winning five times in a row is equal to the probability of winning once to the power of five. For example, if the probability of winning one game is 0.5, then the probability of winning five games in a row is 0.5 * 0.5 * 0.5 * 0.5 * 0.5, which equals approximately 3 percent or one chance in 32. At face value, it sounds highly improbable. So when a player says, “Wow, I won a game five times in a row!” it seems like something very unlikely has happened. However, this 3 percent chance of winning five games in a row is actually about eight times higher when looking at sequences of 20 games in a row.

Now let’s take a look at winning streaks among these possible outcomes. The calculations in Table 1 (The Streakiness of 20 Games) are based on a python program written by the authors. The data is based on 20 sequential games, which gives 2^20, or 1,048,576, possible outcomes for the win/lose game. Keeping the probability of a win 50/50 (coin toss), a straight count of the number of outcomes gives the probability. As you can see from the table, almost 80 percent of the time players will win three games in a row when they play 20 games; and one in four times they will win five times in a row. This “streakiness” result shows that players are swimming in “unlikely” events when they play.  To imagine this, picture the rush of winning five hands of blackjack in a row.

And the story does not stop there; we can imagine the player changing their bet in response to these runs of lucky (or unlucky) events. Let’s consider the example of a player betting up when they win. This combined with the streakiness of winning sequences will result in periods when the player has what seem to be massively lucky or unlucky events. The player has changed the apparent volatility of the gaming experience.

To explain this using the fisherman’s analogy, players are like fishermen who change their fishing behavior in response to catching (or not catching) fish. This change in behavior means that in order for us to understand the player experience, we need to look not at the game, but how the player reacts to the gaming experience. The calculation of true volatility involves some deep computational methods that will be covered in future articles in this series.

BRINGING IT ALL TOGETHER

This article highlights how the player experience is a combination of both the game they are playing and their reaction to that gaming experience. We show that players are like fishermen who change their fishing behavior in response to catching or not catching a fish. Understanding this change in behavior is essential to understanding the gaming experience instead of just relying on the flat mathematics of the game itself. Furthermore, we have shown that streakiness of games has a dramatic impact on how the player experiences the game. Armed with this deeper understanding of true volatility, the challenge is now to go out and apply this knowledge to optimize your gaming floors.