by Dr Catalin Barboianu PhD | Date of Publishing: 16 May 2023
The game of slots is as popular among casino games as it is hard to “deconstruct” and understand in depth. If one wants to analyse a slot game statistically to reveal its parameters, they have no at-hand means for doing that, just because its producers keep the game’s design secret. We can see in the pay window the payout odds of the game, but we don’t even know how many stops each reel has and how many symbols of each kind are placed on each reel; in the absence of this information, one cannot know the probability of each winning combination and thus cannot further compute other statistical indicators either.
And yet one statistical indicator is displayed on many slot machines. It concerns the return to player (RTP), or payback percentage. Producers reveal that information either voluntarily or constrained by the local gambling regulations.
In this article, we shall see what the RTP is, how this information (when available) so often misinterpreted by players, and how such misinterpretations impact non-problematic gambling.
Table of Contents
Before defining RTP, we should know what the expected value of a bet is. For a bet in any game of chance, the expected value of that bet (EV) is the following sum of products:
EV = probability of winning x profit if you win + probability of loosing x the negative profit (loss) if you loose. The possible profit and loss are calculated using the payout odds that the game offers for that bet. In order for the bet to be profitable for the house, it is necessary for the EV to be negative. By the game’s design, the producer of the game chooses the payout schedule such that this condition to be met.
Thus the EV is a statistical indicator computed from the player’s perspective and depends on the payout odds and the structural design of the game. The latter element is that giving the probabilities of winning and losing.
The EV is a statistical average, and this is the way it should be interpreted in the real world of gambling. For instance, an EV of –1.5 cents for a 1$ wager should read as ‘by playing this bet continuously, you are expected to lose on average 1.5 cents at every dollar bet over the long run’. The EV can also be expressed as a percentage of the original stake; in our example, EV = –1.5%.
Although it looks like an arithmetical average, it is not. At most, it is a weighted mean, where the weights are the probabilities. Given that probabilities are involved in the equation of EV, the interpretation of the value of the EV as measuring a rate of the loss (or profit) relative to stakes can only be as a limit, so it cannot be applied to finite contexts. As such, ‘the long run’ actually means ‘infinite play’, and the mathematical expectation is not a real account of the play over short periods or low number of games. The longer the number of games in which we play the bet, the more accurately the EV reflects the real overall rate of loss or gain. This is the meaning of statistical average.
The house edge (HE) of a bet is defined as the opposite (in sign plus or minus) of the expected value, expressed as a percentage: HE = –EV. It is a statistical indicator calculated from the house’s perspective and reflects the ratio from the players’ total wagers going to the house as a profit.
When you press the play button in slots, you place a bet like in any other game of chance. Your bet is that a winning combination will occur on one or more paylines. If this happens, you are paid back your stake (credits bet) multiplied by the payout odds of that combination. This bet has its own expected value, per the definition in the previous section.
In general, if C1, C2, K, Cn are the winning combinations of symbols, p1, p2, k, pn are respectively the probabilities for these combinations to occur on a payline, o1, o2, k, on are respectively the payout multipliers, and c is the credit wagered, then
EV = p1*(o1 .c-c)+p2*(o2.c-c)+K+pn*(on .c-c)-(1-p1-p2-K-pn)*c
or, as a ratio of the initial credit:
EV (%) = p1*(o1-1)+p2*(o2-1)+K+pn*(on-1)-(1-p1-p2-K-pn) = -HE.
Obviously, for being able to concretely compute the EV and HE, we should know the probabilities pi, however in general the only parameters from the above formulas that we can know are the payout odds oi. Only the game producers have access to all the parametric information.
The return to player (RTP), also known as the payback percentage, is defined as
RTP = 1 – HE = p1*o1+p2*o2+K+pn*on
As the RTP is often the only mathematical information available about a game of slots (if itself available), it becomes very important as information characterises the game. There are three aspects of the importance of the RTP:
Those players who don’t understand the mathematical definition of the RTP may be influenced by its word description. As such, they perceive the “return” as a predicted gain that will add to their bankroll, from which other bets may be wagered with the hope of reaching a big hit (higher prizes or jackpot). In both its abstract form and its application in the real game, the RTP actually reflects the house edge (remember that RTP = 1 – HE), hence ultimately, the RTP reflects an average loss and not any gain.
The “positive” perception players may have about the RTP is also fuelled by its values which usually range between 80 – 97%.
Such values of this indicator close to 100% are usually perceived as advantageous from the player’s perspective. This is also a possible reason for exposure by the producers. However, the RTP is one minus the game’s house edge, and as long as the house edge remains positive, slots look like any game of chance concerning this principle. An RTP of 97% means a house edge of 3%, higher than the house edge of European roulette and far higher than the house edge of blackjack, for instance.
Then, we should take the RTP to express house edge (what the house takes from us over the long run) rather than any actual return.
As we explained, the RTP is a statistical average due to the involvement of probability. Therefore, mathematically it only makes sense at infinity, while in the real play (which is finite) we can only refer to it as a limit to approach over the long run.
Applying the RTP over definite play intervals is fallacious based on this principle.
Some players believe that the RTP can be applied for their playing session at a slots machine, whatever long; that the machine will pay back that percentage to them from the moment they start the session until they leave the chair, or the machine will pay it back to all players playing that machine over one day or one week.
However, the payback is not materialised over a given time or number of plays.
The RTP of a game should not be interpreted as the return for a given player from that player’s bets, but cumulatively, that is, the return from all players’ bets to all players over the long run. Or, interpreting the RTP for just one player, it is the return from that player’s investment to that player if they play that game an infinite number of times.
The payback that a slot machine returns to the players is not distributed by any rule over the entire series of plays of that machine and in particular it is not uniformly distributed. This happens because the outcomes of the game are randomly chosen by the random number generator (RNG). In conditions of randomness, any winning combination will occur as frequently as its probability indicates, but as a statistical average and not literally – that is, if the probability of a combination is 1/1,000, this does not mean that the combination will occur about once at every 1,000 spins. It could occur twice in one hundred spins and then once in the next three thousand spins. We know that the greater the number of plays (in order of tens of thousands or more), the more accurately the relative frequency of that combination approximates its probability. This nature of probability is also reflected in the non-uniform distribution of the RTP.
Some players get angry when someone hits a big prize on a machine they were just playing. They say they would have been the lucky winner if they had stayed there; they also say that it is not worth returning to that machine since it has just paid. None of such arguments is valid. First, every spin is independent of any other, and the outcomes are randomly chosen by the RNG, which generates hundreds or thousands of numbers every second; by the time the reels are spinning, the game is already over, as the RNG has already selected the stops; moreover, the RNG is that it is always working, even when you’re not playing. If the machine has just paid, this does not mean that it won’t pay again over the next spins because the RTP, even constant, applies to an indefinite number of spins.
Similarly, choosing a machine just because it has not paid anything or has paid very low for a long period before is not a rational argument for the same reason.
Such false beliefs are a form of the Gambler’s Fallacy relative to RTP – players affected by this fallacy see randomness to order and not a disorder and have the false belief that the next outcomes are influenced by previous outcomes for the statistical constant (probability, EV, or RTP) to be reflected in the actual series of games.
When RTP information is displayed on the showcard of a machine, that message needs to be elaborated as we explained the RTP in the previous sections.
Even though many such messages contain the attribute “average” for the payback, its mere presence does not guarantee that the player understands it is about a statistical average and what that means.
Researchers in problem gambling conducted many studies related to the misinterpretations of the RTP and RTP messages among slots players. They found that the classical message of the sort “this machine has an average payback of (say) 90%” can be meant by non-informed players in various incorrect ways, among which are:
Such meanings have nothing to do with the concept of average, either arithmetical or statistical. Yet we cannot say that such RTP messages are misleading but too simplistic to understand correctly.
Other RTP messages, while still simplistic, can turn out to be misleading. It is about those messages on machines having different RTPs depending on the amount staked. The variation comes from the extra bonuses and different payout odds the game offers for higher stakes.
Studies have shown that players may come to the correct interpretation of such a message, which is “the higher the stake, the higher the payout”. However, some come to interpret it as “the higher the stake, the greater chances of winning”, which is false and has nothing to do with the nature of the RTP. Whatever the stake, the winning probability is the same. However, the payout may be higher, which is reflected in an increase of the RTP. Of course, such misinterpretation is again favoured by the lack of information to correctly understand the term ‘average’.
All the mentioned misinterpretations of the RTP impact the problem-gambling and strategic aspects of using that information. The misinterpretations were confirmed by empirical research in the psychology of problem gambling and qualified as risk factors for problematic gambling behaviour.
To avoid them, slot players must not only be informed about the formal mathematical description of the RTP, but also have an adequate understanding of the concepts of probability, statistical average, and statistical independence, which has the potential to correct most of the classical misconceptions and fallacies specific to gambling. Regarding the RTP, the briefest recommendation is to see it as reflecting the house edge and not an actual return or gain and to remember that it is an abstract notion (as is probability) that cannot be applied to short or medium play sessions.
Bărboianu, C. (2022). Understanding Your Game: A Mathematician’s Advice for Rational and Safe Gambling. Târgu Jiu: PhilScience Press.
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, 1-23.
Beresford, K., & Blaszczynski, A. (2020). Return-to-player percentage in gaming machines: Impact of informative materials on player understanding. Journal of Gambling Studies, 36(1), 51-67.
Collins, D., Green, S., d’Ardenne, J., & Wardle, H. (2014). Understanding of Return to Player messages. In conference Harm Minimisation: Investigating Gaming Machines in Licensed Betting Offices, London (Vol. 10).
Harrigan, K. A. (2007). Slot machine structural characteristics: Distorted player views of payback percentages. Journal of Gambling Issues, Vol. 20, 215-234.
Source of the images: Collins, D., Green, S., d’Ardenne, J., & Wardle, H. (2014). Understanding of Return to Player messages. In conference Harm Minimisation: Investigating Gaming Machines in Licensed Betting Offices, London (Vol. 10).
Author: Dr Catalin Barboianu PhD
Catalin Barboianu PhD is a games mathematician and problem-gambling researcher. He authored ten books on gambling mathematics and several academic articles in the field of problem gambling and philosophy of science. Catalin is a science writer and consultant for the mathematical aspects of gambling for the gaming industry and problem-gambling institutions. Read More
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