Retrieving the missing parametric configuration of a slot machine
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The so-called PAR sheets (Probability Accounting Reports), exposing the weighting of the reels, some of the probabilities associated with the winning combinations, and other statistical indicators, are kept secret by slots producers. Such secrecy is insubstantial (Barboianu, 2014). Game researchers can obtain them only through legal intervention. Since 2007, game researchers obtained the PAR sheets of some slot games through FIPPA (Freedom of Information and Protection of Privacy Act) in Canada (Harrigan & Dixon, 2009). Most of these PAR sheets became public after researchers studied them; others were also available before 2007 through other channels (see Wilson 2004a, 2004b, 2004c, 2004d, 2004e). Still, these are the PAR sheets of a miniscule part of all slot games on the market.

Mathematics provides us with some statistical methods of retrieving the missing data (number of stops of each reel, number of distinct symbols, and the distribution of these symbols on the reels  the latter is also called the weighting of the reels), which are essential for the numerical probability computations in slots.
The methods briefly described here (presented in detail in our book The Mathematics of Slots: Configurations, Combinations, Probabilities; see the Books section) can be applied in an organized professional environment in order to estimate and expose the parametric configuration of any slot game whose PAR sheet is missing.
The application of such methods involves volunteers committed to the collection of data through long-time observation and recording and a mathematical unit where these data are processed.

Statistical methods for estimating the parameters of the configuration of a slot machine

In the next sections we use the same denotations used in section The configuration of slot machines.

The raw approximation
This method is based on the well-known result from probability theory called the Bernoulli's Theorem, which states that in a sequence of independent experiments performed under identical conditions, the sequence of the relative frequencies of the occurrence of an event is convergent toward the probability of that event.
Applied to slots, that principle says that if N is the number of spins of a reel with t stops where we observe as an outcome a specific symbol S that is placed on c stops and n(N) is the number of occurrences of S after the N spins, then the sequence  is convergent toward the probability of occurrence of S, namely P(S) = c/t.
The ratio n / N is the relative frequency of occurrence of S. It follows that for large values of N, the relative frequency of occurrence of S approximates the probability of S occurring. The higher N, the more accurate this approximation. Obviously, the number of spins N must be large enough for obtaining good approximations of the ratios , and this is the main issue of this method. As theory does not provide us with tools for choosing N for a given error range, all we have is the principle the larger N, the better.
As one can notice, this method of approximation based on statistical observation is subject to errors coming from idealizations and various assumptions, and the error ranges are not even quantifiable. Given these issues, the best way to use this method is not for individual records, but cumulating progressively the records coming from several sources and refining the estimations in correlation with the increase in total number of spins N. This principle is also common for the odds calculators based on partial simulations, used for various games.
Note that the described method provides us with approximations of the ratios  (the basic probabilities) for each reel and not the parameters of the configuration individually ( and t). However, knowing the basic probabilities is enough for any probability computation for a slot game.
A more accurate approximation of the ratios  and even of  and t individually is still possible through statistical observation, using a method which can refine the raw estimations obtained through the previously described method. Such a method is briefly described in what follows.

Denominator-match method
Denote by  the number of occurrences of symbols  to  respectively after N spins of a reel. There is a slight correlation between the recorded values  for various large numbers of spins N. Based upon this correlation, we can refine the estimation of the ratios  obtained through the previous method and also find estimations for  and t, by recognizing a numerical pattern across some sequences of fractions representing the ratios between possible values for  and t.
The denominator-match method is based on the numerical analysis of the fractions  and on a five-step algorithm briefly explained below:
We write each fraction  as a chain of equal fractions, having numerators from 1 upward and denominators not necessarily integers, for every i from 1 to p.  Across the p chains of equal fractions obtained, we choose that of the minimal length (let m be the minimal length). Then, across the p chains of equal fractions, we extract m sequences of fractions (one fraction from each equality chain), having the denominators the nearest to the denominators from the minimal equality chain respectively. From the m sequences of fractions obtained, we choose one sequence of p fractions by applying progressively the following filtering criteria: having denominators as close to each other as possible, having the highest number of instances of the same denominator, and the repeating denominator with the largest share being an integer. As final step, we adjust the numerators of the final sequence of fractions, as follows: If the sum of the numerators lies between the minimum and maximum of the denominators, then we take the numerators as the symbol distribution on the reel () and their sum as the number of stops of the reel (t); if their sum does not lie in that interval, then through addition or subtraction, we distribute, proportionally with their values, the difference between their sum and the integer nearest to the mean of the minimal and maximal denominator, rounding the added/subtracted quantities to integers. For our resulting estimation, we take the adjusted numerators as the symbol distribution on the reel (), and the integer nearest to the mean of the minimal and maximal denominators as the number of stops of the reel (t).
This method provides us with the most probable number of stops t and associated symbol distribution  of a reel in a certain probability field; the error range of this approximation is quantifiable in terms of probability (Barboianu, 2013).

Practical application
Regarding the practical application of the methods through statistical observation, it is obviously an arduous task, since we have to watch and record spins in numbers of thousands. For online games, software can be developed to help in such an endeavour. For physical machines, it is far more difficult to watch and note down thousands of outcomes just for one reel of a machine, not to mention that the slots operator might not allow this action.
Infarom is launching the project project Probability Sheet for any Slot Game, dealing with collecting statistical data from slot players, using the data to estimate the parameters of the slot machine, refining the estimations with the newly collected data and computing the probabilities and other statistical indicators attached to the payout schedule of the slot machine, in order to provide the so-called PAR sheet of any slot game on the market.
We are in the phase of looking for collaborators and funding for this project. After the mathematical unit will be established, we will have a webpage dedicated to volunteer registration. Volunteers will be provided with the results of the game they watched and will have access to all the results obtained for other slot games.
Contact us with subject "slots data project" if you want to be part of our project.

References

Bărboianu, C. (2013). How to estimate the number of stops and the symbol distribution on a reel. In Infarom (Ed.), The Mathematics of Slots: Configurations, Combinations, Probabilities (pp. 46-63). Craiova: Infarom.

Barboianu, C. (2014). Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation. Journal of Gambling Issues, in press.

Harrigan, K.A., & Dixon, M. (2009). PAR Sheets, probabilities, and slot machine play: Implications for problem and non-problem gambling. Journal of Gambling Issues, 23, pp. 81-110.

Wilson, J. (2004a). PAR excellence: Improve your edge. Slot Tech Magazine, February 2004, pp.1623.

Wilson, J. (2004b). PAR excellance: Part 2. Slot Tech Magazine, March 2004, pp.1621.

Wilson, J. (2004c). PAR excellance: Part 3. Slot Tech Magazine, April 2004, pp. 2026.

Wilson, J. (2004d). PAR excellence - Improving you game - Part IV. Slot Tech Magazine, May 2004, pp. 2124.

Wilson, J. (2004e). PAR excellance: Part V: The end is here! Slot Tech Magazine, June 2004, pp. 2429.

This entry should be cited as:
Barboianu, C. (2014).

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