Casino Enterprise Management

Authors' Note: In this article, we look at some real-world examples of multigame player information. Multigame player information is nonexistent in most gaming systems, and this complete lack of information makes a mockery of almost any theoretical win-based marketing program. This article is a case study of how advanced data matching makes this data easy to handle, relevant and actionable. Using examples from the Silverton Casino, we show how the correct theoretical win is quite different to the box average, with specific examples of how players are incorrectly compensated.

Once again, we find ourselves talking about the calculation of theoretical win. As discussed in earlier installments of this series, the fisherman’s analogy is an illustration of how the theoretical win calculation is a misrepresentation of a gaming experience. Essentially, if we catch a fish every six trips, then mathematically we would expect to catch one-sixth of a fish each trip. But, of course, in reality a fisherman only catches a whole fish. (See the February 2013 issue of CEM for more details on this analogy.¹)

Similarly, theoretical win is the “expected value” of the result of a gaming experience. This calculation is based on the probability of a win times the value of the win. The challenge is that most slot systems do not track the game performance; they track the asset performance. A solution to this problem involves applying computerized problem-solving to provide the corrected theoretical win per game (and thus by customer). As promised in previous articles in this series, we will now investigate in depth how computerized methods can be applied to data problems.

Problem-Solving
First, let’s consider these two broad categories of mathematical problems: those that can be solved by formulas and those that lend themselves only to computerized problem-solving. Fermat’s Last Theorem is a canonical example of the difference between computer-solved problems and formula-solved problems.

Mathematical Problem-Solving
Any student of geometry is very familiar with the 3-4-5 right triangle. And anyone who has forgotten their geometry can easily calculate that 32+42=52. However, the exponent 2 is quite unique in this regard. Centuries ago, Pierre de Fermat proposed the following statement, later called Fermat’s Last Theorem: Given an integer n>2, there are no positive integers a, b and c that satisfy an + bn = cn. So, for example, there do not exist positive integers a, b and c such that a3 +b3 = c3.

For more than 300 years, this mathematical problem remained unsolved, despite Fermat’s tantalizing note indicating that he had a proof, but it was too large to fit in the margin of the book he was reading at the time he made the conjecture. In 1995, 358 years after the problem was first formulated, Andrew Wiles of Princeton University finally published the first confirmed solution to this problem. It was the longest unsolved mathematics problem in history, and it was described as one of the most difficult problems in mathematics. Its solution required the development of many new areas of mathematics. Wiles himself spent decades pursuing his childhood dream of solving the most famous of math problems.

Computerized Problem-Solving
While there exists a wonderful, exact mathematical solution to Fermat’s Last Theorem, computers allow us to tackle these kinds of problems in a far simpler way with nearly the same outcome. And it doesn't have to take 358 years.

In the world of computerized problem-solving, there are two broad fields: traditional computer science and artificial intelligence. An informal definition of artificial intelligence is the computer science associated with all problems too hard to solve exactly with traditional methods. The word exactly is very important here, as it eliminates the opportunity to use approximations.

We can solve Fermat’s Last Theorem for a huge number of integers using computers. First, we need all combinations of a,b and c and then all values of n. Let's start with n=3, and say we only consider values of a, b and c in the set of (1,2). (See Figure 1.)

Figure 1 shows eight possibilities for the values of a, b and c. Filling out the matrix for n=3, we see in Figure 2 that in all cases, Fermat’s Last Theorem is correct so far.

Generalizing this a little, we see that the number of rows is

23 = 8. More generally, if we wanted to consider a million positive integers, we would need to consider 1,000,0003 cases—that’s 1 quintillion (10 with 18 zeros)! Adding to this calculation, if we wanted to consider all values of n up to 1 million, we would need our computer to consider 1 septillion cases (10 with 24 zeros). Now, this is a tremendous number of cases and could take approximately 31 million years to work through them all.²

However, using another approach, we could approximate. Approximation and guess work are fantastic, and computers are in fact very good at this kind of problem—it just often takes a far more sophisticated approach.

This leads us to the four kinds of problems as described in Figure 3. The approximate solution method is called a heuristic method.

Heuristic “Solution” to Fermat’s Last Theorem
Now, unless we are prepared to wait for a very long time, it is impossible to brute-force calculate a solution to Fermat’s Last Theorem to any reasonable size of a, b or c. This is where approximate computerized methods come into their own. To illustrate this, we can try a lot of random values of a, b, c and n. With, say, 100 billion random tries and zero successes, we can conclude that Fermat’s Last Theorem is, at least, highly unlikely to have an exception. To those unfamiliar with computer science, this may seem complex, but this is, in fact, a solution that almost all computer programmers could tackle with relative ease.

How does this relate to gaming? Quite simply, the gaming industry is loaded with data and complex problems. Furthermore, some of these problems have no precise solution, but heuristic solutions can be applied. Here is where the multigame player data enters.

Multigame Player Data
As we have discussed previously (see Part 1 of this series in the January 2013 issue of CEM⁴), the challenge with getting the correct data is that it simply does not exist in most deployed gaming systems. The reason that the player data is not matched with the detailed gaming data is twofold. First, the detailed gaming data is low quality, and second, the playertracking systems are separate from the slot accounting systems. Given this near complete lack of precise player data at the game level and a clear need to have this data for accurate marketing, we have to dig into some complex computer science to provide the tools we need in the modern world of big data. The core complex tool that is needed is heuristic data matching.

As a quick example, a machine may contain two games, one with 2 percent hold and the other with 10 percent hold. In the player tracking system, this machine will simply report the average of the two games, which is 6 percent hold. Thus, the theoretical win reported for all customers of this game is wrong—and by a significant amount.

The heuristic data matching in this situation is significantly more complex than the heuristic “solution” to Fermat’s Last Theorem. To begin with, we have a record of each customer on each slot machine for each day from our player tracking system. From our slot accounting systems, we have a record of each game on each slot machine for each day. While we won’t go into details about the technique, data matching applies a heuristic algorithm to take the data from the player tracking system (which is missing the play on each game of each machine) and combine it with the data from the slot accounting system (which is missing the play of each player on each machine) to provide with a high level of certainty the actual numbers for every player, every day, on every machine and for every game.

This allows us to know the exact hold percentage of every dollar spent by each player. This means that, for example, no longer will we assign a 6 percent hold to players of a machine containing a 2 percent hold game and a 10 percent hold game. We will have enough information to determine who played the 2 percent game and who played the 10 percent game. In short, this gives us the ability to measure theoretical win accurately for each customer. We call this CorrectTheo©.

Real-World Example
We have applied this method of heuristic data-matching in the real world, with results that clearly show the need to know not only what machines your customers are playing, but also what games they are playing on these machines. Let’s take a look at this real-world data, which has been obfuscated but shows the general trend.

In Figure 4, customers are placed on the 16 squares. The heat map shows the density of customers, as well as the following:

1)         All squares have approximately the same CorrectTheo win (this is shown by the top number in each of the quadrants).
2)         The second number is the interesting number, as it shows the theoretical win.
3)         Customers with high theoretical win are on the top.
4)         Customers with high CorrectTheo are on the right.

The method of making the 16 squares is called the quartal method, and customers are scattered across the graph in 3-D space to ensure that the customers contained in each square have the same total theoretical win (although the number of customers varies). The heat map shows the density of customers, though for obfuscation purposes, the number of customers is left out of the graphic. (Refer to the June 2009 issue of CEM³ for more information on this method.)

We can see a comparison of two quadrants of Figure 4 and the top left and the bottom right, in Figure 5.

As you can see, there is a remarkable difference between customers who are within 1 percent of the box value (asset) versus customers who have a CorrectTheo that is 1.8 times as large as the box (asset) value.

Now consider two customers, Norman (Normal) and Ursula (Unskilled). Our current theoretical win-based marketing program is correct for Norman. However, we can almost double our rewards to Ursula; essentially, she is playing a game like keno. Now consider the casino. Armed with this knowledge, the casino’s message to Ursula can be different and her rewards can near double. To this Ursula will, in all likelihood, respond with additional trips.

Bringing It All Together
The gaming industry is a business of selling mathematical model-based experiences to customers. Sometimes there is some very hard work involved in preparing the data, and in this article, we have gone to quite some lengths to introduce heuristic methods and how they can solve even the most difficult problems, albeit approximately. After all this hard computer science, the results are fun, drive profit and, at the end of the day, are irresistible. Furthermore, given that these methods are now available, in our opinion, it is fiscally irresponsible to build marketing programs—with huge reinvestment costs—based on fundamentally flawed mathematics. The correct numbers are available now, and it’s time to start using them.

Footnotes
1 Available at https://casinoenterprisemanagement.com/articles/february-2013/where%E....
2 Without optimization of method and with computers running in the GigaHertz range.
3 Singh, A.K., & Cardno, A. (June 2009). “Market Basket Analysis, Part VI: The Quartal Graph is Worth Several Thousand Word.” CEM. Available online at https://casinoenterprisemanagement.com/articles/june-2009/market-bask....
4 Available at https://casinoenterprisemanagement.com/issues/january-2013-volume-11-....

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@yahoo.com.

Dr. Ralph Thomas is Vice President of Strategic Analytics and Database Marketing for Seminole 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@stofgaming.com.