The dark side of gambling maths: Creating slots illusions by design
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Gambling math underlies the games of chance in their commercial form, as these games are conceived on the basis of rigorous mathematical models.
The mathematical design of gambling games is indispensable: On the one hand, it is that design that dictates the characteristics of a game, makes the game functional, and guarantees the operator that they will always be in profit in the long run, no matter the players’ behavior, luck, or skills.
On the other hand, the players who have knowledge about the mathematical design of a game can play informed and use objective strategies in their play, which are based on the mathematical facts of the game. It appears that gambling math is the illuminating force that keeps gambling functional and fair; however there is a dark side of it that manifests in the most popular category of casino games, namely slots.
Catalin Barboianu, Mathematician and Philosopher of Science PhD
Each slot game is conceived by mathematical design that makes it unique, but the purpose of this design is not limited to functionality and identity. I take a look at the mathematical design of slots – which is not transparent as for the other casino games. It can produce illusions in the form of the so called near-misses, with implications for players’ behavior and decisions, many times harmful.
What is a near-miss?
In gambling, a near-miss is a losing outcome that looks close to being successful. It is a non-winning outcome differing in one or a few elements from a winning one.
Per this definition, we can see near-misses in every game of chance. If talking about blackjack, baccarat, craps, poker, lottery, or bingo, any near-miss is a chance near-miss, that is, that particular non-winning outcome occurs with the same probability as any other, including a winning one, due to the characteristics of the game and its randomness (Bărboianu, 2022). Things are different in slots:
In slots, a near-miss may occur on a payline, as well as below or above a payline. On a payline, an outcome holding for instance three award symbols and needing four such symbols to be winning is a near-miss. In slot games displaying more than one row of spots, if a missing winning symbol on a payline is displayed right above or below the spot where it was expected, this outcome is also a near-miss.
Such slots near-misses may occur by chance, but also as a result of the inner design of a slots game, as we will see further.
The mathematical design of slots
The characteristics of a slot game are described by a set of parameters that uniquely identify that game: the number of (unique) symbols used, the number of reels, the number of instances of each symbol on each reel (called the symbol weighting of that reel), and the arrangement of the symbols on each reel (Bărboianu, 2013). Each particular combination of symbols on a payline occurs with a probability given by those parameters.
Classical physical reels hold a limited number of stops (usually 22), including those in the second generation (electromechanical). With the third generation of slots the virtual reels have been introduced. A virtual reel can have unlimited number of stops, so the innovation was to make the actual symbol weighting on the virtual reel then map the virtual reel onto the physical reel (several instances of the same symbol on the virtual reel are mapped into one symbol on the physical reel). The RNG randomly pick a stop on each virtual reel, and the machine commands the physical reel to hold from spinning at the mapped (corresponding) stop (Turner & Horbay, 2004).
The parameters of a slot machine also give the value of the main statistical indicators that reflect the behavior of that machine in the long run, such as hit frequency of every winning combination (or the probability of that combination), the RTP (return to player percentage), and the volatility index. The parametric configuration of a slot machine and some of the statistical indicators are noted in the so-called PAR sheets (Probability Accounting Reports), which are internal documents of the producer.
The parametric configuration of a slot game reflects the goals of its producer and is obtained by at-order mathematical work performed by a game mathematician.
Secrecy about the mathematical design
Slots are the only casino games whose characteristics are not revealed entirely. Slot players can see the symbols of a machine, the number of reels, and its payout schedule; however, the rest of the parameters (as the symbol weighting of the reels) are hidden. In the absence of these data, one cannot estimate the probabilities of hitting the various winning combinations, nor some statistical indicators. The only indicator available for most of the slot games is the RTP. PAR sheets can be retrieved by legal interventions, through hard tedious process, depending on jurisdiction.
Gambling regulations are at the moment very permissive with this secrecy, which by the way appears as unjustified: On the one hand, it is unlikely that players would quit slots if they came to know how low are the probabilities of hitting prizes – take the example of lottery, where players continue to play against “astronomical” odds. On the other hand, no producer can actually claim to protect their slots design as a commercial secret against competition or stealing, as the infringer would self-eliminate from the market if caught; besides, in the wide diversity of slot games, originality is a must for the new startups (Bărboianu, 2014).
Still there may be a valid reason for this secrecy, pertaining to what the mathematical design of slots can do beyond functionality, namely to create illusions that work in the producers’ and operators’ benefit.
Creating artificial near-misses in slots
As game mathematicians can develop a certain parametric design for slots such that to meet certain requirements for the hit frequency of some winning combinations, they can also do that design to have a certain hit frequency for other combinations, including non-winning.
That is, the symbol weighting of the reels and the symbol arrangement can be manipulated within the design to have some non-winning combinations occurring more frequent than others or having certain symbols occurring more frequent than others on certain stops, on the payline or on other spots of the grid. The purpose of doing that is to produce frequent artificial (or false) near-misses as outcomes, which the player would perceive as chance near-misses.
There are two main techniques of creating artificial near-misses by mathematical design: the award symbol ratio and clustering.
Award symbol ratio technique
By choosing the right weightings for each reel, artificial near-misses can be made to occur frequently on a payline. For example, on a 3-reel machine, a high-award symbol (determining a high prize if occurring in three instances, one on each reel) is weighted in a high ratio on two of the reels and in a low one on the third.
Thus, the symbol will occur frequently on the reels where is more present, creating the illusion of a near-miss. This award symbol ratio technique applies to both physical and virtual reels.
The artificial near-miss can also occur above or below the payline. The display of the classical machines with one payline shows not only the stop of the reel on the payline, but also a portion of the reel neighboring it. In old classical physical-reel machines, such artificial near-misses adjacent to the payline can be produced trivially, by arranging the symbols on the reel strip such that an awarding symbol will be adjacent to a non-awarding one.
Thus, when a non-awarding symbol occurs in the payline on a reel, an awarding one is often visible in the display right above or below it. Hence for creating false near-misses adjacent to the payline, symbol arrangement on the reels is manipulated and not the weighting.
Clustering technique
In modern slots, creating artificial near-misses adjacent to the payline is done by clustering. The technique relies on the flexibility of the symbol weighting and arrangement on the virtual reels.
The idea is to put a stop with a high-award symbol on the virtual reel between several stops with low-award symbols or blanks, which are mapped into the corresponding positions on the physical reel or the virtual reel visible in the display. You will better understand how clustering works if explained through an example in a slots game.
Example of clustering
In the figure above is depicted a portion of a 72-stop virtual reel (showing 13 stops, with symbols defined in the legend). This virtual reel is mapped onto a physical reel with 22 stops, of which 11 stops are blanks. The portion of the visible reel holding the TD symbol (the highest award symbol) displays 1B…TD…3B. TD stands in virtual stop no. 29 and is surrounded by five blanks above and below it (Harrigan, 2008).
Each stop on the virtual reel has a relative frequency of occurrence in the payline of 1/72, hence each of the five blanks in stops 24 to 28 will occur with the same relative frequency for a total of five times. The five blanks 24 to 28 are adjacent to the TD, hence the TD on stop no. 29 will appear just below the payline on the visible reel with a relative frequency of 5/72. The TD on stop no. 29 will occur on the payline in average once in 72 spins, as frequently as on the visible reel. The five blanks on stops 30 to 34 will each occur on the payline with a relative frequency of 1/72, for a total of five times. The five blanks on stops 30 to 34 are adjacent to the TD, so the TD on stop no. 29 will appear above the payline with a relative frequency of 5/72.
If the outcome of the game were determined by the visible reel randomly, the near-misses for TD (chance near-misses) would have occurred with a relative frequency of 2/22, that is, once in 11 spins. However, due to the mapping and the clustering technique, those portions will actually occur on average 10 times out of 72 spins, that is, once is 7.2 spins. Although there are 11 blanks on the 22-stop physical reel, the chances the RNG will pick a blank (on the virtual reel) are much higher than 1 in 2. Therefore the blanks adjacent to (above and below) the high-award symbol are favored on the physical reel. When a player hits these blanks they have the illusion that they closely missed the high-award symbol, just because that symbol is physically close to the payline; but probabilistically it is not.
The near-miss effect and problematic gambling
Near missing is part of the excitement of gambling, however in slots it raises ethical issues. The main reason is that a near miss is not just a gaming event in the experience of a gambler, but it is followed by what psychologists call ‘the near-miss effect’, an emotional state that makes the gambler feeling that a win is on the way or due, which then influence their gambling behavior is the sense that it urges the gambler to play more.
The near-miss effect is considered by problem-gambling researchers a risk factor for a problematic gambling behavior and related to other risk factors, such as the illusion of control and other misconceptions and fallacies in the cognitive zone of gambling psychology (Griffiths, 1991).
These harmful effects of the near-miss obviously manifest also for a chance near-miss, however if we talk about artificial or engineered near-misses (as it is the case in slots), they fall within the ethical side of gambling, as increasing or multiplying a risk factor. The ethical issue of the artificial near-misses in slots adds and is related to the general issue of secrecy about the inner design of these games.
For playing informed – an important norm of responsible gambling – slots players should and have the right to know the information about the mathematical design of slots and how it relates to the artificial near-misses. Knowing a machine’s RTP and perhaps whether it is a low- or high-volatility one is not enough for playing informed. Knowing about the real probabilities of the occurrence of the winning combinations and how often you may expect to see artificial near-misses (or at least that they are real in most of the slot games) are far more important information for a responsible play (Bărboianu, 2022).
Conclusion
The mathematical design of any slot game is the blueprint of that game. Just modifying one parameter in this design gives a new slot game and this is the reason why there is such a wide diversity of slot games on the market. The mathematical design of a slot game responds to the requirements and goals of the producer, in what concerns the statistical indicators of the game, with direct implications for the profit of the operator in the long run.
Among such requirements there is also the producing of artificial near-misses, that is, outcomes that look like physically close to a win and don’t occur by chance, but as the result of the particular mathematical design of the game, pertaining to certain symbol weighting and arrangement on the reels. The purpose of these outcomes is of course to make the slot players spinning more the machines. Near-misses and their effects are important risk factors for problem gambling.
But if really there is a dark unethical side of slots design, who should we blame? Mathematics of gambling? Mathematics is universally applicable and for whatever purpose. Besides, without mathematics there would be no casino games at all. Game mathematicians? They just do their profession of applying the same innocent math to make a living of that. The slots producers themselves or the gambling regulations that allow such practices? The debate remains open.
References:
- Bărboianu, C. (2013). The mathematics of slots: Configurations, combinations, probabilities. Craiova: Infarom.
- Bărboianu, 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, Vol. 29, 1.
- Bărboianu, C. (2022). Understanding your game: A mathematician’s advice to rational and safe gambling. Târgu Jiu: PhilScience Press.
- Griffiths, M. (1991). Psychobiology of the near-miss in fruit machine gambling. The Journal of psychology, 125(3), 347-357.
- Harrigan, K. A. (2008). Slot machine structural characteristics: Creating near misses using high award symbol ratios. International Journal of Mental Health and Addiction, Vol. 6, 353-368.
- Turner, N. E., & Horbay, R. (2004). How do slot machines and other electronic gambling machines really work? Journal of Gambling Issues, Vol. 11.