Every slot operator knows the overall average hold percentage of the slot floor. On a day-to-day basis, this does a nice job of letting you know how lucky you (or your customers) were on any given day. Unfortunately, this number is often aggregated over longer time periods and then used to discuss the “price” of your games. However, this aggregation of hold often provides irrelevant and misleading data. While the hold percentage on a single game is a relevant component (although not the entirety) of price, it becomes irrelevant when talking about the price of a group of games, or even the entire slot floor.
Introducing Power Sets
Power sets are an important concept in mathematics, originating in set theory, but used in many other branches of mathematics, including Real Analysis and Topology. According to Wikipedia, the “power set” (or “powerset”) of any set S, written P(S), P(S), (S) or 2S, is the set of all subsets of S, including the empty set and S itself.
When applied to gaming, we are considering combinations of configurations. Let’s consider a gaming example, looking at the power sets associated with three gaming machines: X, Y and Z. The power sets of these three gaming machines have a total of eight elements (See Figure 1). The diagram in Figure 2, provided by Wikipedia, shows a Hasse diagram of the power sets of the set of X, Y, Z.
Figure 1
Figure 2: Hasse Diagram of Power Set
Power sets can create enormous numbers of sets from groupings of gaming configurations. According to Wikipedia, if S is a finite set with |S| = n elements, then the power set of S contains |P(S)|=2n elements. Figure 3 shows how even 100 machines results in an utterly un-manageable number of power sets. This means that in the real world, we need to use approximate methods to handle the large numbers of sets.
Figure 3
Understanding the Hold
To apply the concept of power sets to explaining the concept of hold, we need to define two terms: the “maximum relevance of hold” and the “number of configurations at maximum relevance.”
The maximum relevance of hold is a function of a specific portion (percentage) of the floor. Given a fixed percentage of the floor, we take from the power set of the floor all collections of games that comprise this percentage. Then, for each collection of games taken, we determine (heuristically) how relevant hold percentage is to that group of games. Finally, we compute the maximum relevance taken over all collections of games selected. You will find that as we look at larger and more diverse groups of gaming devices, the relevance of hold diminishes. Suppose, for example, we begin with 1 percent of the floor. In this case, a mix of video poker, wide area progressives and traditional mechanical reel games that would not have much relevance of hold percentage. However, 1 percent of the floor may also consist of a single theme, in which case hold percentage would be quite relevant. As we move on to the entire floor (100 percent), where the only element of the power set of this size is the entire floor itself, the diverse mix of games’ overall hold percentage is not very relevant. We’ll discuss this in detail in a moment.
First, we need to define the number of configurations at maximum relevance. This is also a function of a specific portion (percentage) of the floor. The number of configurations is the number of distinct gaming configurations that exist within a set for the sets with maximum relevance. In our example above, at 1 percent, the maximum relevance occurred when we had a single theme within that portion of the floor—i.e., when the element of the power set at maximum relevance only has one game configuration. At 100 percent, the maximum relevance occurs at the only possible place—the entire floor itself. And because the entire floor has dozens, hundreds or thousands of game configurations, depending on how big the casino is, again, that maximum relevance is very low.
Factors Affecting Hold Relevance
Quite simply, game play is not directly related to hold percentage. This is one of the most common misperceptions about how gaming devices function. The idea is that by loosening the gaming (or reducing the hold), one can increase time on device. However, that’s not entirely true. As an extreme example, consider two games, Coin Toss and Lottery Ticket, that have the same hold percentage, namely 0 percent (all the money is returned to the customer). Now, keep in mind that the game experience is quite different for Coin Toss, where the player would, in any sequence of bets, expect to win and lose some games, and for Lottery Ticket, where the player would expect to lose continuously. If we increase the hold to 50 percent, the player playing Lottery Ticket, for example, would still expect to lose continuously—in effect, having exactly the same gaming experience.
Figure 4
This difference in games can be represented by a measure called “skewness.” In our two example games, Coin Toss is not skewed and Lottery Ticket is extremely skewed (see Figure 4). This brings us to understand that “games are complex mixtures of probabilities shown, which we refer to as pay tables. The way that the game is adjusted makes an enormous difference to the player experience. This effect is described as the skewness of the gaming device.”1
Hold is an Outcome
After exploring game play, we can see that hold is an outcome of the adjustments to the gaming floor, and there are clearly different ways of adjusting the gaming floor to affect a change in hold. The table in Figure 5 shows examples of some of the alterations that can be made to the gaming floor that may result in a change in hold.
Figure 5
In the end, it is not the hold of the game that determines the time on device; it is the game play. This game play can be modeled using Monti Carlo simulations,2 and these simulations show the effect the different game configurations have on the player experience. Having said that, in a practical world of gaming machines, the hold is very relevant for a single game configuration—but it must be accompanied by analysis of the game play.
The Customer
At the end of the day, customers play on gaming machines, and these players may be time constrained or wallet constrained. If the customer is wallet constrained (and in today’s economic environment, many are), then discussions about the value of the hold are completely irrelevant. The key metric is the amount the customer is prepared to spend each month or each trip, and we think that changing the game play to result in less time on device but the same spend may not be the desired outcome.
Furthermore, the customers’ play patterns can also change the effective hold percentage of the game. Considering questions such as “Does the customer play max bet?” or “Are they playing optimal video poker?” make player behavior a critical part of understanding the hold of the casino.
This effect gets amplified as the number of game configurations grows. To illustrate this, consider an example where a large group of players visits the property and all of them choose to play looser games. Clearly this change in player mix is important, but it would be analytically misleading to say that the hold of the property has changed—rather, it would be correct to say that the player mix has changed.
Thus, as the number of game configurations increases, the hold percentage metric often says more about the type of player that is in the building than it does about the type of games available on the floor. We summarize this line of thought in Figures 6 and 7.
Figure 6
Figure 7
Key Observations
At 1 percent of the floor, the maximum relevance of hold percentage occurs when we look at a single game configuration. In the case of a single configuration, hold percentage is more relevant, since it is a significant (albeit not singular) contributing factor to the customer experience.
At 100 percent of the floor, hold is essentially meaningless, as we are combining vastly different game types (such as video poker and video reel slots). We are also combining vastly different customer types. As we showed in a previous article,3 combining very different kinds of things can and does lead to misleading analysis.
Impact on the Server-Based World
As we saw above, the more configurations that are selected, the less relevant hold percentage becomes. Now let’s consider the server-based world, where each machine has access to a library with tens or even hundreds of games. At this point, even on a single machine, hold percentage is no longer relevant!
Yet, despite this fact, one can imagine that when Friday comes, high-level executives will still be demanding, “Raise the hold by 0.2 percent!” Since hold percentage is a poor measurement of price, trying to raise hold on the weekends is likely to be a fruitless endeavor. But what will fare worse is the customers’ perception. Whether operators choose to use server-based gaming as a way to raise hold on the weekends or not, they will nevertheless have the ability to do so—and our customers will eventually become aware of this. Will they decide that the casino is indeed changing the game on them? If so, will they lose interest in a game perceived as being “rigged”?
The Sacred Cow
We have often heard statements like “If we raised the hold percentage by 0.1 percent, we’d make an extra Coin In x 0.1%!” But as we have illustrated, these kinds of statements are misleading at best and may cost revenue at worst. The power sets and other illustrations provide counter examples of how to consider hold percentage and damage the sacred analytical cow!
Footnotes
1 CEM, May 2010, “Why Do I Need Math? Part V.” Cardno, Singh, Gewali.
2 CEM, May 2010, “Why Do I Need Math? Part V.” Cardno, Singh, Gewali.
3 CEM, October 2010, “Can Multi-Theme Slots Change Players?” Cardno, Singh, Thomas.
Andrew Cardno has more than 16 years of experience in business analytics, ranging from modeling health care drive times to casino gaming floor analytics. He often presents on the future of analytics across the world and has spent the last seven years living in the United States and working with corporations around the world. He can be reached at andrewcardno[at]yahoo.com.
Dr. Ralph Thomas is Vice President of Database Marketing for Seminole Hard Rock Gaming. During his years in the casino industry, Thomas has focused on maximizing profitability by applying statistical analysis to the company database. Previously, Thomas spent 15 years in academia, as both a student and a lecturer of mathematics. He can be reached at ralph.thomas[at]stofgaming.com.
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