Similarly, they could also leave the virtual casino which would result in a cash-out of the money that was currently on the machine. At the beginning of the game, participants could decide how much of the money on their account they wanted to load into the slot machine and which of the four machines they wanted to play on.
After a delay of 1 s, the wheels stopped spinning and displayed a combination out of 9 possible stimuli, all of which were pre-programmed in the game. The magnitude of the win was determined by the combination depicted; possible wins ranged from 5 Cents to 60 Euro the Jackpot win for a large bet.
Participants could look up the win table at any point in time by pressing an extra button on the machine. Fake wins have been found to reinforce the sense of winning in slot machine games, and were included here to identify whether these events play a role in characterizing impulsivity Jensen et al.
This type of trial outcome has been shown to enhance gambling motivation, to lead to physiological arousal and to activate reward-related brain areas Clark et al. For comparison across subjects and modeling purposes, we analyse performance over the trials only. Because the sequence of probabilities and reward levels were fixed across subjects, variability in performance could only result from the subject's own betting behavior and choices to engage in the DU option. The trace accounts for numerous relevant variables that may determine gambling behavior:.
Behavioral readouts are overlaid in different colors. Notably, the more impulsive subject showed more behavioral activation and risk-seeking behavior throughout the game, in particular during the more volatile phase. Two exemplary performance traces for subjects with different BIS scores. A Subject with BIS score below average B Subjects with BIS score above average Gray trace, performance over the course of the game in EUR.
Colored dots are overlayed to the performance trace and reflect events of interest in a particular trial. Notably, the more impulsive a subject i. In addition to computational modeling described below, we used classical multiple linear regression for analyzing the behavioral data and for evaluating the construct validity of computational models i.
To determine how much the variance of an estimated regression coefficient increased due to collinearity, we estimated the variance inflation factor VIF for each regressor. This paper is concerned with a proof of concept demonstration that generative modeling of gambling behavior can yield mechanistic descriptions of impulsivity in terms of individual beliefs and belief-to-response mappings. A generative model is a model which provides a joint probability distribution over all random variables involved e.
It specifies a forward mapping from hidden parameters and states to measurable observations. Such models allow the experimenter to infer upon the hidden states and parameters of an agent or subject engaged in a task. Critically, this class of models generate two things: sensory inputs and motor responses. Therefore, when specifying a generative model of gambling, one must consider what aspects of the sensory input administered and the behavioral responses observed are to be predicted by the model.
For example, does he treat near-misses similar to wins of any sort, and does he distinguish between true wins and fake wins? Secondly, how is a given belief transformed into a behavioral response or choice? In other words, generative models could be constructed for different combinations of perceptual and response variables.
In principle, finding the optimal model can be accomplished by means of Bayesian model selection BMS , which evaluates the relative plausibility of competing models in terms of the log evidence MacKay, and represents a principled trade-off between model fit and model complexity. However, a condition for BMS is that the competing models predict identical data.
This means that BMS can only proceed if both perceptual and response variables are identical. To deal with this issue, we implement a two-stage model selection in this paper. We then consider three different perceptual variables and four response variables; this results in 12 sensory-motor datasets. For any of these datasets, we can invert all five core models and select an optimal model using BMS.
In a second step, we can evaluate the relative goodness of these 12 selected models by assessing their construct validity against an external measure of impulsivity. To this end, we use an independent questionnaire-based measure of impulsivity the BIS and perform multiple regression analyses of individual parameter estimates on individual questionnaire scores, as described below.
Following this general overview of our modeling strategy, the following paragraphs will unpack these ideas and specify both the perceptual and response variables considered as well as the form of the generative models employed. Composition of perceptual and response variables for the computational modeling. Variables are binary and composed on a trial-by-trial basis for each of combinations shown. Wins are encoded as 1, losses as 0. A naturalistic paradigm like ours allows for numerous readouts of behavior, and thus, many possible response variables.
Here, we consider several combinations of readouts as candidate response variables. As our intention is to explain how impulsivity is manifested in a gambling paradigm, we use the BIS to inform the choice of response variables.
The reinforcement-learning and Bayesian models we consider below link the expression of the above responses to the agent's internal beliefs and their uncertainy. Simply speaking, we are modeling an agent for whom stronger beliefs of winning lead to increasingly risk-seeking behavior and, at the same time, result in an increasing frequency of erratic and sensation-seeking behavior in terms of CS and MS.
Critically, this probabilistic link between beliefs and actions is governed by a subject-specific parameter which, in some of the models described below, becomes a function of the agent's trial-wise uncertainty. As motivated in the Introduction, this paper adopts a Bayesian perspective on gambling.
The HGF represents a generic generative model of the sensory inputs an agent receives. It consists of hierarchically coupled Gaussian random walks, where this coupling is specified by subject-specific parameters. Different levels of the hierarchy encode a subject's estimates of different characteristics of environmental uncertainty. The first level, x 1 , follows the trajectory of the perceived variable in the environment, in the absence of perceptual noise.
The second level, x 2 , tracks the probability of trial outcomes over the course of the paradigm. The step-size of the random walk by x 2 depends on the highest level, x 3 , that tracks the global volatility of the environment. This three-level system underlies the player's belief-updating process during the game. To illustrate its general structure, let us assume that we track a quantity x 1 in our environment which evolves as a Gaussian random walk.
Let us now characterize the variance of this random walk as a function of a higher level, x 2 , which is itself a Gaussian random walk. We can continue this hierarchical coupling up to some n -th level:. In this model, the lowest level, x 1 , corresponds to the perceived variable e. The second level represents the evolution of the probability of trial outcomes over time. Critically, its variance depends on the third level which, in turn, represents the stability of the environment log-volatility.
In our context, this model describes how the player updates his beliefs about trial outcome probabilities under the influence of a higher belief of how these probabilities change in time i. By linking beliefs to choices via a response model, we can invert this model given measured responses; for details, see Mathys et al.
This model inversion allows for inference on subject-specific model parameters, and thus, on an individual's hierarchical belief trajectories and their associated uncertainties. Notably, the posterior estimates of subject-specific model parameters describe an individual's approximation to Bayes-optimal behavior.
In the HGF, model inversion rests on a variational approximation to full Bayesian learning which results in simple analytical belief update equations for a detailed derivation, see Mathys et al. Intuitively, one would imagine that a belief update occurs when an agent compares the predicted to the actual sensory input, calculates an error term, and then back-propagates this error up the hierarchy to adjust beliefs at all levels.
In the HGF, this occurs by passing back a prediction error that is weighted by precision inverse uncertainty. This precision term is proportional to the inverse step size of the Gaussian random walk on different levels. The equations above show that our network updates in a manner similar to RL, in which the model trains on the error signal between model predictions and observed data.
A key difference, however, is that the HGF not only provides estimates of states, but also of their uncertainty posterior variance or precision ; enabling a precision-weighting of prediction errors. This precision weighting means that prediction errors lead to greater updates the more precise less uncertain predictions are. The HGF thus takes into account estimates of uncertainty about the hidden hierarchically related processes which generate sensory inputs. The detailed update equations for precisions or uncertainties can be found in Mathys et al.
The HGF described above represents a Bayesian player who updates his beliefs about trial outcome probabilities at the 2nd level under the influence of a higher belief at the 3rd level of how these probabilities change in time, i. Collectively, these subject-specific model parameters describe the coupling of belief updates across levels and thus an individual approximation to Bayes-optimal behavior.
This function describes a sigmoidal mapping from the gambler's beliefs to his chosen action:. We consider four classes of response models that will be tested on each of the four aggregate response variables see section Response Variables.
Overview of the 5 core models tested. Model 1—4 are different types of the HGF that differ in the response model used, whereas Model 5 is a classical Rescorla-Wagner Model with a standard softmax response function.
Finally, Model 4 was inspired by a response model in Vossel et al. Here, we imagine an agent who is more sensation-seeking, erratic and risk-taking the higher his trial-wise uncertainty about winning probability. This uncertainty corresponds to the variance of a Bernoulli distribution at the first level of the perceptual model and takes a maximum value of 0. Using a scaling factor of 4 ensures that this argument enters the softmax appropriately, such that the maximum leads to the greatest probability of eliciting a response.
In particular, this includes RL models which have found application in some analyses of gambling tasks e. Having said this, we are not aware of any RL analyses of trial-wise data from casino slot machine gambling. The RW model is a trial-wise learning model, originally developed for estimating associative learning mechanisms in conditioning.
It is also frequently used in a reduced form, for example, for estimating on-line the probability of a trial-wise outcome; this is the form we use here. Updates are governed by prediction errors, scaled by a fixed learning rate:.
In this paper, we adopt a two-stage model selection procedure that evaluates different models with regard to two things: how well a model explains a given set of perceptual and response data features step 1—BMS , and for which of these different data features the parameters of an optimal model best predict an external measure of impulsivity step 2—construct validity. The best of the five core models for a given dataset is selected via Bayesian model comparison.
This rests on the log evidence, a principled index of a model's trade-off between fit and complexity MacKay, Critically, BMS implementations exist which can deal with heterogeneity across subjects and enable proper random effects group-level inference Stephan et al. Model selection stage 1. This optimal model then entered stage 2 construct validation. Model selection stage 2.
To determine the best model with respect to an external measure of impulsivity, we regressed individual BIS scores on model parameter estimates from the 12 models one for each pair of perceptual variable and response variable provided by model selection stage 1.
The winning model is picked using a BIC comparison across regression models, to account for differing model complexities. The approach we employ in the present analyses is that of approximating the log evidence by negative free energy. The free energy is an upper bound approximation to the agent's surprise about seeing the data and, in contrast to the log-evidence which is analytically intractable for all but the simplest models, can be computed as part of model inversion by means of variational Bayesian VB optimization see Mathys et al.
For further details on model comparison using free energy, please see Penny and Stephan et al. Having selected an optimal model for each of the 12 sets of data features, we can evaluate the models' construct validity, i. For this purpose, we use the independent questionnaire scores of impulsivity BIS and perform multiple regression analyses on the model parameter estimates. In this case of competing predictions based on multiple regression models, potential differences in model complexity due to differences in the number of generative model parameters and thus number of resulting regressors can be corrected using the BIC.
The significance of the ensuing best prediction is adjusted for multiple tests using Bonferroni correction. Regression of BIS scores on 4 major behavioral readouts. These selected 12 models then entered a second stage of model comparison, where we examined the construct validity of these models by testing how well their parameter estimates predicted the independent BIS scores. Summary of model comparison results across all 12 classes of 5 models each i.
The posterior expectation of model probability, obtained from a random effects Bayesian model selection procedure, is plotted on the y-axis. The perceptual variables span the x-axis; the response variables span the y-axis. RW, Rescorla—Wagner Model.
As the computational models vary in the number of free parameters e. Note that this model-based prediction of BIS scores is significant, even after Bonferroni correction. This study aimed to evaluate the utility of computational modeling in characterizing slot-machine gambling behavior under realistic conditions and establish construct validity in relation to standard questionnaire measures of impulsivity.
To this end, we created a naturalistic slot-machine paradigm to accrue realistic behavioral readouts from a group of healthy subjects and used a hierarchical Bayesian model of individual learning and decision-making to model the paradigm outputs. The task builds upon previous research using slot machine tasks to explore gambling e. Overall, we find that impulsivity as measured by the BIS score was significantly related to an exploration of these game features. The mechanistic model aims at formalizing how humans solve the task at hand on a computational level.
It relates potential beliefs and their evolution over time to behavioral choices. Variability across individuals within this process is captured by subject specific parameter estimates that can then be related to traits of the individual like impulsivity.
To unearth this hidden information, the models have to consider i what information of the game players are using in order to infer their chances of winning on a trial by trial basis perceptual variables , ii how they update their beliefs over time and express these beliefs through actions core models, each representing a particular combination of perceptual and response models , and iii which aspect of the observed responses should be used for estimating model parameters response variables.
As described above, we use a two-step procedure that combines initial BMS of the core model best explaining a given data set of perceptual and response variables with subsequent construct validation through multiple regression of parameter estimates from this selected core model on BIS scores.
The reason for this is simply that the perceptual variables differ much more from another than the response variables. With regard to the latter, the four response variables are nested in each other and are dominated by the frequent occurrences of BIs. By comparison, CS, and MS are less frequent and their addition to BI does not change the resulting response variable dramatically.
However, it should not be overlooked that these are in-sample predictions and the effect size estimates i. We will address this issue in future studies with larger samples which enable out-of-sample predictions. While this limited variance in a healthy population poses an even harder problem for statistical predictions than dealing with a highly variable population, there is no guarantee that the mechanisms highlighted by our model-based analyses will extrapolate to pathological gamblers.
Instead, it is possible that qualitatively different mechanisms operate during pathological as compared to recreational gambling. This would be signaled by a different outcome of our model comparisons and will be examined in future studies with patients. Notwithstanding these caveats, the present study is important because it suggests a novel two-step modeling procedure for slot machine gambling data, and it provides concrete suggestions of which data features in slot machine gambling may be most useful for future studies.
Several of the competing models also successfully predict the BIS The models differ solely in the response variables they predict. One potential cause for that could be that some of the behavioral readouts like CS were relatively sparse and contributed less to the individuals' variance in gambling behavior. Finally, a variant of model 1 which contains an additional free parameter compared to model 2 also significantly predicts BIS, but with a worse BIC score.
Interestingly, this is the only predictive model which rests on WLG learning from fake and true wins as perceptual variable and a constant decision noise that is independent of the current uncertainty. Thus, this model variant might capture a general bias toward reward-related processing and behavior. This is not in line with an earlier study Jensen et al. However, this previous study compared two slot-machines which not only differed in the number of fake wins, but also in the number of wheels 3 vs.
In our slot machine paradigm, wins and fake wins were of constant appearance; in addition subjects were informed about win magnitude after each trial, thus facilitating the distinction between real and fake wins. Altogether, our modeling results emphasize that uncertainty plays two important roles in gambling.
Second, the optimal response model captures a direct influence of this belief uncertainty on the individual's decision process in that decision noise is modulated by trial-wise uncertainty about winning probability. That is, the more uncertain a subject is whether he will win on the next trial the less his actions will be informed by his a priori beliefs, leading to seemingly more random behavior.
Encoding of uncertainty has previously been linked to an individual's impulsivity Averbeck et al. That is, for our particular paradigm and healthy volunteers, uncertainty about winning probability appears to be more strongly related to impulsivity than the prior belief about volatility.
Having said this, the accuracy of these questionnaire-based assessments suffers from a number of limitations. PG has a high co-morbidity with mood disorders and depression, both of which tend to overshadow gambling habits and their subsequent symptoms, and may thereby cause distorted self-reports Allcock and Grace, ; Black and Moyer, Further bias stems from patients lacking the requisite capacity for self-reflection Wilson and Dunn, It has thus been suggested that interactive, computer-based neuropsychological tests provide more reliable measures of impulsivity Kertzman et al.
Combining such tasks with a computational model of impulsivity in a naturalistic gambling setting may allow us to go even further. Four advantages of a computational approach to such problems are particularly worth mentioning.
First, computational models i can provide interpretations of trait like impulsivity by replacing the more descriptive nature of questionnaires with more mechanistic descriptions of how players update their beliefs during gambling and transform these into choices. In our case this is done by establishing a link between the individuals' uncertainty about winning and loosing and the resulting increase in more erratic and riskier responses. Furthermore, computational models can ii assess the degree of impulsiveness during actual gambling and without any need of potentially distorted self-reports, and iii they allow us to generate not only response traces observed in our subjects, but possible candidate response traces that reflect extreme cases of impulsive behavior.
Such traces could help to identify patterns in gambling data that earmark potential problem gamblers. This approach is therefore particularly interesting for prevention with respect to online gambling. Finally, iv the trial-wise traces of beliefs and uncertainties, inferred by a model, can serve to inform analyses of neurophysiological or fMRI data for examples using the HGF, see Iglesias et al.
The hierarchical Bayesian modeling approach presented here is capable of revealing cognitive mechanisms in gambling that are linked to traditionally defined impulsive traits of the individual. In particular, the gambling behavior of subjects, who are more impulsive, is best described by models that encode for greater uncertainty at various levels in their hierarchy, and show uncertainty-dependent coupling between beliefs about winning and subsequent decisions.
Our analyses provide a proof of concept that individual heterogeneity in gambling behavior can be quantified by computational models, enabling a mechanistic interpretation of individual gambling. Future research will have to assess the generalizability and practical utility of this approach in predicting disordered gambling behavior in various gambling settings such as online gambling. The Review Editor Dr.
Harriet Brown declares that, despite being affiliated to the same institution as author Prof. Klaas E. Stephen, the review process was handled objectively and no conflict of interest exists. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. We would like to thank Carolin Wolters for assisting in data collection and the volunteers who tested and provided much-valued feedback about the paradigm.
A sincere thank you to Christoph Mathys for very helpful guidance and feedback. Stephan ]. National Center for Biotechnology Information , U. Journal List Front Hum Neurosci v. Front Hum Neurosci. Published online Jul 3. Stephan 1, 2, 3, 4.
Frederike H. Author information Article notes Copyright and License information Disclaimer. Received Feb 20; Accepted May The use, distribution or reproduction in other forums is permitted, provided the original author s or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
This article has been cited by other articles in PMC. Abstract Impulsivity plays a key role in decision-making under uncertainty. Introduction Uncertainty is a fundamental aspect of human decision-making Bland and Schaefer, Table 1 Descriptive statistics. Open in a separate window. Slot-machine paradigm We designed a naturalistic behavioral paradigm to approximate the experience of true casino gambling by simulating a simple Electronic Gambling Machine EGM.
Figure 1. Figure 2. Data analysis In addition to computational modeling described below, we used classical multiple linear regression for analyzing the behavioral data and for evaluating the construct validity of computational models i.
Computational modeling Generic considerations This paper is concerned with a proof of concept demonstration that generative modeling of gambling behavior can yield mechanistic descriptions of impulsivity in terms of individual beliefs and belief-to-response mappings. Table 2 Composition of perceptual and response variables for the computational modeling.
Response variables A naturalistic paradigm like ours allows for numerous readouts of behavior, and thus, many possible response variables. Core models Hierarchical Gaussian Filtering As motivated in the Introduction, this paper adopts a Bayesian perspective on gambling.
Persistent state refers to passive features on some slot machines, some of which able to trigger bonus payouts or other special features if certain conditions are met over time by players on that machine. Roll-up is the process of dramatizing a win by playing sounds while the meters count up to the amount that has been won. Short pay refers to a partial payout made by a slot machine, which is less than the amount due to the player. This occurs if the coin hopper has been depleted as a result of making earlier payouts to players.
The remaining amount due to the player is either paid as a hand pay or an attendant will come and refill the machine. A scatter is a pay combination based on occurrences of a designated symbol landing anywhere on the reels, rather than falling in sequence on the same payline. A scatter pay usually requires a minimum of three symbols to land, and the machine may offer increased prizes or jackpots depending on the number that land.
Scatters are frequently used to trigger bonus games, such as free spins with the number of spins multiplying based on the number of scatter symbols that land. The scatter symbol usually cannot be matched using wilds, and some games may require the scatter symbols to appear on consecutive reels in order to pay.
On some multiway games, scatter symbols still pay in unused areas. Taste is a reference to the small amount often paid out to keep a player seated and continuously betting. Only rarely will machines fail to pay even the minimum out over the course of several pulls. Tilt is a term derived from electromechanical slot machines' " tilt switches ", which would make or break a circuit when they were tilted or otherwise tampered with that triggered an alarm.
While modern machines no longer have tilt switches, any kind of technical fault door switch in the wrong state, reel motor failure, out of paper, etc. A theoretical hold worksheet is a document provided by the manufacturer for every slot machine that indicates the theoretical percentage the machine should hold based on the amount paid in. The worksheet also indicates the reel strip settings, number of coins that may be played, the payout schedule, the number of reels and other information descriptive of the particular type of slot machine.
Volatility or variance refers to the measure of risk associated with playing a slot machine. A low-volatility slot machine has regular but smaller wins, while a high-variance slot machine has fewer but bigger wins. Weight count is an American term referring to the total value of coins or tokens removed from a slot machine's drop bucket or drop box for counting by the casino's hard count team through the use of a weigh scale.
Wild symbols substitute for most other symbols in the game similarly to a joker card , usually excluding scatter and jackpot symbols or offering a lower prize on non-natural combinations that include wilds. How jokers behave are dependent on the specific game and whether the player is in a bonus or free games mode.
Sometimes wild symbols may only appear on certain reels, or have a chance to "stack" across the entire reel. Each machine has a table that lists the number of credits the player will receive if the symbols listed on the pay table line up on the pay line of the machine. Some symbols are wild and can represent many, or all, of the other symbols to complete a winning line. Especially on older machines, the pay table is listed on the face of the machine, usually above and below the area containing the wheels.
On video slot machines, they are usually contained within a help menu, along with information on other features. Historically, all slot machines used revolving mechanical reels to display and determine results. Although the original slot machine used five reels, simpler, and therefore more reliable, three reel machines quickly became the standard. This limited the manufacturer's ability to offer large jackpots since even the rarest event had a likelihood of 0.
Although the number of symbols eventually increased to about 22, allowing 10, combinations,  this still limited jackpot sizes as well as the number of possible outcomes. In the s, however, slot machine manufacturers incorporated electronics into their products and programmed them to weight particular symbols. Thus the odds of losing symbols appearing on the payline became disproportionate to their actual frequency on the physical reel. A symbol would only appear once on the reel displayed to the player, but could, in fact, occupy several stops on the multiple reel.
In Inge Telnaes received a patent for a device titled, "Electronic Gaming Device Utilizing a Random Number Generator for Selecting the Reel Stop Positions" US Patent ,  which states: "It is important to make a machine that is perceived to present greater chances of payoff than it actually has within the legal limitations that games of chance must operate. With microprocessors now ubiquitous, the computers inside modern slot machines allow manufacturers to assign a different probability to every symbol on every reel.
To the player it might appear that a winning symbol was "so close", whereas in fact the probability is much lower. In the s in the U. These used a number of features to ensure the payout was controlled within the limits of the gambling legislation. As a coin was inserted into the machine, it could go either directly into the cashbox for the benefit of the owner or into a channel that formed the payout reservoir, with the microprocessor monitoring the number of coins in this channel.
The drums themselves were driven by stepper motors, controlled by the processor and with proximity sensors monitoring the position of the drums. A "look-up table" within the software allows the processor to know what symbols were being displayed on the drums to the gambler.
This allowed the system to control the level of payout by stopping the drums at positions it had determined. If the payout channel had filled up, the payout became more generous; if nearly empty, the payout became less so thus giving good control of the odds. Video slot machines do not use mechanical reels, instead of using graphical reels on a computerized display.
As there are no mechanical constraints on the design of video slot machines, games often use at least five reels, and may also use non-standard layouts. This greatly expands the number of possibilities: a machine can have 50 or more symbols on a reel, giving odds as high as million to 1 against — enough for even the largest jackpot. As there are so many combinations possible with five reels, manufacturers do not need to weight the payout symbols although some may still do so.
Instead, higher paying symbols will typically appear only once or twice on each reel, while more common symbols earning a more frequent payout will appear many times. Video slot machines usually make more extensive use of multimedia , and can feature more elaborate minigames as bonuses.
Modern cabinets typically use flat-panel displays , but cabinets using larger curved screens which can provide a more immersive experience for the player are not uncommon. Video slot machines typically encourage the player to play multiple "lines": rather than simply taking the middle of the three symbols displayed on each reel, a line could go from top left to the bottom right or any other pattern specified by the manufacturer.
As each symbol is equally likely, there is no difficulty for the manufacturer in allowing the player to take as many of the possible lines on offer as desire — the long-term return to the player will be the same. The difference for the player is that the more lines they play, the more likely they are to get paid on a given spin because they are betting more.
To avoid seeming as if the player's money is simply ebbing away whereas a payout of credits on a single-line machine would be bets and the player would feel they had made a substantial win, on a line machine, it would only be five bets and not seem as significant , manufacturers commonly offer bonus games, which can return many times their bet. The player is encouraged to keep playing to reach the bonus: even if he is losing, the bonus game could allow then to win back their losses.
All modern machines are designed using pseudorandom number generators "PRNGs" , which are constantly generating a sequence of simulated random numbers, at a rate of hundreds or perhaps thousands per second.
As soon as the "Play" button is pressed, the most recent random number is used to determine the result. This means that the result varies depending on exactly when the game is played. A fraction of a second earlier or later and the result would be different.
It is important that the machine contains a high-quality RNG implementation. Because all PRNGs must eventually repeat their number sequence  and, if the period is short or the PRNG is otherwise flawed, an advanced player may be able to "predict" the next result. Having access to the PRNG code and seed values, Ronald Dale Harris , a former slot machine programmer, discovered equations for specific gambling games like Keno that allowed him to predict what the next set of selected numbers would be based on the previous games played.
Most machines are designed to defeat this by generating numbers even when the machine is not being played so the player cannot tell where in the sequence they are, even if they know how the machine was programmed. This is known as the "theoretical payout percentage" or RTP, "return to player". The minimum theoretical payout percentage varies among jurisdictions and is typically established by law or regulation. The winning patterns on slot machines — the amounts they pay and the frequencies of those payouts — are carefully selected to yield a certain fraction of the money paid to the "house" the operator of the slot machine while returning the rest to the players during play.
Within some EGM development organizations this concept is referred to simply as "par". Play now! A slot machine's theoretical payout percentage is set at the factory when the software is written. Changing the payout percentage after a slot machine has been placed on the gaming floor requires a physical swap of the software or firmware , which is usually stored on an EPROM but may be loaded onto non-volatile random access memory NVRAM or even stored on CD-ROM or DVD , depending on the capabilities of the machine and the applicable regulations.
Based on current technology, this is a time-consuming process and as such is done infrequently. Other jurisdictions, including Nevada, randomly audit slot machines to ensure that they contain only approved software. Historically, many casinos, both online and offline, have been unwilling to publish individual game RTP figures, making it impossible for the player to know whether they are playing a "loose" or a "tight" game.
Since the turn of the century some information regarding these figures has started to come into the public domain either through various casinos releasing them—primarily this applies to online casinos—or through studies by independent gambling authorities. The return to player is not the only statistic that is of interest. The probabilities of every payout on the pay table is also critical.
For example, consider a hypothetical slot machine with a dozen different values on the pay table. However, the probabilities of getting all the payouts are zero except the largest one. Also, most people would not win anything, and having entries on the paytable that have a return of zero would be deceptive.
As these individual probabilities are closely guarded secrets, it is possible that the advertised machines with high return to player simply increase the probabilities of these jackpots. The added advantage is that these large jackpots increase the excitement of the other players. This game, in its original form, is obsolete, so these specific probabilities do not apply. He only published the odds after a fan of his sent him some information provided on a slot machine that was posted on a machine in the Netherlands.
The psychology of the machine design is quickly revealed. There are 13 possible payouts ranging from to 2, The payout comes every 8 plays. The payout comes every 33 plays, whereas the payout comes every plays.
Most players assume the likelihood increases proportionate to the payout. The one mid-size payout that is designed to give the player a thrill is the payout. It is programmed to occur an average of once every plays. The payout is high enough to create excitement, but not high enough that it makes it likely that the player will take their winnings and abandon the game.
In contrast the payout occurs only on average of once every 6, plays. The player who continues to feed the machine is likely to have several mid-size payouts, but unlikely to have a large payout. He quits after he is bored or has exhausted his bankroll. Despite their confidentiality, occasionally a PAR sheet is posted on a website.
They have limited value to the player, because usually a machine will have 8 to 12 different possible programs with varying payouts. In addition, slight variations of each machine e. The casino operator can choose which EPROM chip to install in any particular machine to select the payout desired.
The result is that there is not really such a thing as a high payback type of machine, since every machine potentially has multiple settings. Without revealing the proprietary information, he developed a program that would allow him to determine with usually less than a dozen plays on each machine which EPROM chip was installed. Then he did a survey of over machines in 70 different casinos in Las Vegas. He averaged the data, and assigned an average payback percentage to the machines in each casino.
The resultant list was widely publicized for marketing purposes especially by the Palms casino which had the top ranking. One reason that the slot machine is so profitable to a casino is that the player must play the high house edge and high payout wagers along with the low house edge and low payout wagers.
Other bets have a higher house edge, but the player is rewarded with a bigger win up to thirty times in craps. The player can choose what kind of wager he wants to make. A slot machine does not afford such an opportunity. Theoretically, the operator could make these probabilities available, or allow the player to choose which one so that the player is free to make a choice.
However, no operator has ever enacted this strategy. Different machines have different maximum payouts, but without knowing the odds of getting the jackpot, there is no rational way to differentiate. In many markets where central monitoring and control systems are used to link machines for auditing and security purposes, usually in wide area networks of multiple venues and thousands of machines, player return must usually be changed from a central computer rather than at each machine.
A range of percentages is set in the game software and selected remotely. In , the Nevada Gaming Commission began working with Las Vegas casinos on technology that would allow the casino's management to change the game, the odds, and the payouts remotely. The change cannot be done instantaneously, but only after the selected machine has been idle for at least four minutes. After the change is made, the machine must be locked to new players for four minutes and display an on-screen message informing potential players that a change is being made.
Some varieties of slot machines can be linked together in a setup sometimes known as a "community" game. The most basic form of this setup involves progressive jackpots that are shared between the bank of machines, but may include multiplayer bonuses and other features.
In some cases multiple machines are linked across multiple casinos. In these cases, the machines may be owned by the manufacturer, who is responsible for paying the jackpot. The casinos lease the machines rather than owning them outright. Casinos in New Jersey, Nevada, and South Dakota now offer multi-state progressive jackpots, which now offer bigger jackpot pools. Mechanical slot machines and their coin acceptors were sometimes susceptible to cheating devices and other scams.
One historical example involved spinning a coin with a short length of plastic wire. The weight and size of the coin would be accepted by the machine and credits would be granted. However, the spin created by the plastic wire would cause the coin to exit through the reject chute into the payout tray.
This particular scam has become obsolete due to improvements in newer slot machines. Another obsolete method of defeating slot machines was to use a light source to confuse the optical sensor used to count coins during payout. Modern slot machines are controlled by EPROM computer chips and, in large casinos, coin acceptors have become obsolete in favor of bill acceptors.
These machines and their bill acceptors are designed with advanced anti-cheating and anti-counterfeiting measures and are difficult to defraud. Early computerized slot machines were sometimes defrauded through the use of cheating devices, such as the "slider", "monkey paw", "lightwand" and "the tongue".
Malfunctioning electronic slot machines are capable of indicating jackpot winnings far in excess of those advertised. In the United States, the public and private availability of slot machines is highly regulated by state governments. Many states have established gaming control boards to regulate the possession and use of slot machines and other form of gaming. Nevada is the only state that has no significant restrictions against slot machines both for public and private use.
In New Jersey , slot machines are only allowed in hotel casinos operated in Atlantic City. Several states Indiana , Louisiana and Missouri allow slot machines as well as any casino-style gambling only on licensed riverboats or permanently anchored barges. Since Hurricane Katrina , Mississippi has removed the requirement that casinos on the Gulf Coast operate on barges and now allows them on land along the shoreline.
Delaware allows slot machines at three horse tracks; they are regulated by the state lottery commission. In Wisconsin, bars and taverns are allowed to have up to five machines. These machines usually allow a player to either take a payout, or gamble it on a double-or-nothing "side game".
The territory of Puerto Rico places significant restrictions on slot machine ownership, but the law is widely flouted and slot machines are common in bars and coffeeshops. In regards to tribal casinos located on Native American reservations , slot machines played against the house and operating independently from a centralized computer system are classified as "Class III" gaming by the Indian Gaming Regulatory Act IGRA , and sometimes promoted as "Vegas-style" slot machines.
As a workaround, some casinos may operate slot machines as "Class II" games—a category that includes games where players play exclusively against at least one other opponent and not the house, such as bingo or any related games such as pull-tabs.
In these cases, the reels are an entertainment display with a pre-determined outcome based on a centralized game played against other players. Some historical race wagering terminals operate in a similar manner, with the machines using slots as an entertainment display for outcomes paid using the parimutuel betting system, based on results of randomly-selected, previously-held horse races with the player able to view selected details about the race and adjust their picks before playing the credit, or otherwise use an auto-bet system.
Conversely, in Connecticut , Hawaii , Nebraska , South Carolina , and Tennessee , private ownership of any slot machine is completely prohibited. The remaining states allow slot machines of a certain age typically 25—30 years or slot machines manufactured before a specific date.
For a detailed list of state-by-state regulations on private slot machine ownership, see U. In essence, the term "lottery scheme" used in the code means slot machines, bingo and table games normally associated with a casino. These fall under the jurisdiction of the province or territory without reference to the federal government; in practice, all Canadian provinces operate gaming boards that oversee lotteries, casinos and video lottery terminals under their jurisdiction.
OLG piloted a classification system for slot machines at the Grand River Raceway developed by University of Waterloo professor Kevin Harrigan, as part of its PlaySmart initiative for responsible gambling. Inspired by nutrition labels on foods, they displayed metrics such as volatility and frequency of payouts.
In Australia "Poker Machines" or "pokies"  are officially termed "gaming machines". In Australia, gaming machines are a matter for state governments, so laws vary between states. Gaming machines are found in casinos approximately one in each major city , pubs and clubs in some states usually sports, social, or RSL clubs.
The first Australian state to legalize this style of gambling was New South Wales , when in they were made legal in all registered clubs in the state. There are suggestions that the proliferation of poker machines has led to increased levels of problem gambling ; however, the precise nature of this link is still open to research.
Australia ranks 8th in total number of gaming machines after Japan, U. This primarily is because gaming machines have been legal in the state of New South Wales since ; over time, the number of machines has grown to 97, at December , including the Australian Capital Territory. By way of comparison, the U.
State of Nevada, which legalised gaming including slots several decades before N. This new law also banned machines with an automatic play option. All gaming machines in Victoria have an information screen accessible to the user by pressing the "i key" button, showing the game rules, paytable, return to player percentage, and the top and bottom five combinations with their odds. These combinations are stated to be played on a minimum bet usually 1 credit per line, with 1 line or reel played, although some newer machines do not have an option to play 1 line; some machines may only allow maximum lines to be played , excluding feature wins.
Western Australia has the most restrictive regulations on electronic gaming machines in general, with the Crown Perth casino resort being the only venue allowed to operate them,  and banning slot machines with spinning reels entirely. This policy had an extensive political history, reaffirmed by the Royal Commission into Gambling: .
Poker machine playing is a mindless, repetitive and insidious form of gambling which has many undesirable features. It requires no thought, no skill or social contact. The odds are never about winning. Watching people playing the machines over long periods of time, the impressionistic evidence at least is that they are addictive to many people. Historically poker machines have been banned from Western Australia and we consider that, in the public interest, they should stay banned.
While Western Australian gaming machines are similar to the other states', they do not have spinning reels. Therefore different animations are used in place of the spinning reels in order to display each game result. Independent candidate Andrew Wilkie , an anti-pokies campaigner, was elected to the Australian House of Representatives seat of Denison at the federal election.
Wilkie was one of four crossbenchers who supported the Gillard Labor government following the hung parliament result. Wilkie immediately began forging ties with Xenophon as soon as it was apparent that he was elected. During the COVID pandemic of , every establishment in the country that facilitated poker machines was shut down, in an attempt to curb the spread of the virus. Bringing Australia's usage of poker machines effectively to zero. In Russia, "slot clubs" appeared quite late, only in Before , slot machines were only in casinos and small shops, but later slot clubs began appearing all over the country.
The most popular and numerous were "Vulcan " and "Taj Mahal". Since when gambling establishments were banned, almost all slot clubs disappeared and are found only in a specially authorized gambling zones. Slot machines are covered by the Gambling Act , which superseded the Gaming Act Slot machines in the U.
Casinos built under the provisions of the Act are allowed to house either up to twenty machines of categories B—D or any number of C—D machines. As defined by the Act, large casinos can have a maximum of one hundred and fifty machines in any combination of categories B—D subject to a machine-to-table ratio of ; small casinos can have a maximum of eighty machines in any combination of categories B—D subject to a machine-to-table ratio of Category A games were defined in preparation for the planned " Super Casinos ".
Despite a lengthy bidding process with Manchester being chosen as the single planned location, the development was cancelled soon after Gordon Brown became Prime Minister of the United Kingdom. As a result, there are no lawful Category A games in the U. Category B games are divided into subcategories. The differences between B1, B3 and B4 games are mainly the stake and prizes as defined in the above table.
Category B2 games — Fixed odds betting terminals FOBTs — have quite different stake and prize rules: FOBTs are mainly found in licensed betting shops , or bookmakers, usually in the form of electronic roulette. The games are based on a random number generator ; thus each game's probability of getting the jackpot is independent of any other game: probabilities are all equal. If a pseudorandom number generator is used instead of a truly random one, probabilities are not independent since each number is determined at least in part by the one generated before it.
Category C games are often referred to as fruit machines , one-armed bandits and AWP amusement with prize. Fruit machines are commonly found in pubs , clubs , and arcades. Machines commonly have three but can be found with four or five reels, each with 16—24 symbols printed around them. The reels are spun each play, from which the appearance of particular combinations of symbols result in payment of their associated winnings by the machine or alternatively initiation of a subgame.
These games often have many extra features, trails and subgames with opportunities to win money; usually more than can be won from just the payouts on the reel combinations. Fruit machines in the U. It is known for machines to pay out multiple jackpots, one after the other this is known as a streak or rave but each jackpot requires a new game to be played so as not to violate the law about the maximum payout on a single play.
Typically this involves the player only pressing the Start button for which a single credit is taken, regardless of whether this causes the reels to spin or not. Slot machines are a fairly new phenomenon and they can be found mostly in pachinko parlors and the adult sections of amusement arcades , known as game centers. The machines are regulated with integrated circuits , and have six different levels changing the odds of a Japanese slot machines are "beatable".
Parlor operators naturally set most machines to simply collect money, but intentionally place a few paying machines on the floor so that there will be at least someone winning, [ citation needed ] encouraging players on the losing machines to keep gambling, using the psychology of the gambler's fallacy.
For example, there must be three reels.
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|Betting on ms machine systems||By the betting on ms machine systems the New Jersey Alcoholic Beverages Commission ABC had approved the conversion for use in New Jersey arcades, the word was out and every other distributor began adding skill stops. Inspired by nutrition labels on foods, they displayed metrics such as volatility and frequency of payouts. Final bets 7, 8 and 9 cost three chips. The bayesian brain: the role of uncertainty in neural coding and computation. The only thing an attendant or floor person can give you is historical information.|
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|Betting on ms machine systems||The winnings are then paid betting on ms machine systems anyone who has placed a successful bet. Neural Netw. The games are based on a random number generator ; thus each betting on ms machine systems probability of getting the jackpot is independent of any other game: probabilities are all equal. Doing so requires sufficient building infrastructure, such as clearance beneath floors and behind walls, to allow for these many, many cable connections. Using model comparison, we compared a set of hierarchical Bayesian belief-updating models, i. As a result, there are no lawful Category A games in the U.|
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Most approaches let you place more bets than would otherwise be the case. While each software has its own process, there are some common elements. Typically, you that the strategy is fixed at the start. This means adding rules on when to bet and when to cash out. The betting bot will then follow these rules to the letter. In some cases, you add your own strategy. You might need to look for the bets to make too. In other cases, the automated betting software will do this for you. It is important that you understand at the beginning what you are going to get.
You will also want to see if it only covers Betfair or bookies too. Like most betting systems, there is no guarantee of profits. It all comes down to the strategy that you use. All that it can do is faithfully follow the strategy that you choose.
Next, we need to look at the benefits of doing this. Why should you consider going to an automated approach? Is it going to help you win more money? Or are there other benefits to be aware of? The fact is that there are a few different benefits to automated trading. The following are some of the main advantages you could get from doing this. Building a bot to use on Betfair might be easier than you think.
This post from Betfair explains how it can be done without any coding knowledge. In this way, you get automated betting software that is built to suit you. The starting point is with a system or strategy. You need to know how you are going to win money. It is then a question of building a bot to carry it out. Or paying someone to do the coding for you. So, this is a possible starting point even for a beginner.
Even if you can code, you need a betting strategy too. You need both elements — the bot and the strategy —to make this work. One without the other is no use to anyone. This is an approach for someone who is a Betfair expert above all. Perhaps you feel that you have gone as far as possible with manual trading.
Maybe a lack of time is now holding you back. With automated betting software, you can move on to the next level. You can also look at the following options for setting up a bot. These are pre-built tools that let you put your strategy into action right away. Take a look at the DIY options noted below. This tool promises to give you an easy approach to betting. You can automate all of your activity and just check out the results as suits you.
Let it run on its own as long as you want. Bet Angel lets you carry out complex trades effortlessly. You can set up strategies on different markets and add complex rules if needed. It is possible to use varying rules in different markets at the same time. Sharing your algorithms with other users can be done too. It also links to a spreadsheet, so that you can set your rules more easily. Therefore, you need to have some excel skills to use it to its potential.
There is a practise mode that comes in handy for newcomers. If you are new to automated betting then this is definitely advisable. You need to be comfortable with the functions before using real money. This is a recommended way to give it a try. If you want a longer period then there are various options. Here is another way of making your strategy easier to implement. Again, this is a tool that lets you automate your own betting system. You need to know how to trade before you start. There is a trial run feature on this automated betting software.
This means that you can use it with dummy money until you are comfortable. Different bots can be used on numerous sports. You can get automated betting on Betfair for football, horse racing, tennis, basketball and other sports. They offer a trial period offer for new members. This gives you 5 days of free access. If you subscribe for longer it works out cheaper. You can also use it on Betdaq or Matchbook. However, for many people using it on Betfair makes most sense.
You may not realise that Betfair also offers apps. One of them is this highly-rated automated betting software tool. The BetEngine gives you three downloading options. You can download the app with the database included. This is the right choice for newcomers looking for an easy start. The next choice is to download without the database. In this case, it is aimed at existing, experienced users.
Your final download choice is a user manual. This is well worth looking at, especially when you are just starting out. Choose from some different subscription options. It is a good idea to get started with the 7-day free trial. A more extensive range of features can be found on the Standard and Professional options. Perhaps you want your betting tool to do it all for you.
In that case, you want an automated betting system. This is more of an all-in-one package that you can start using on your behalf. You just need to set your betting limits and any other variables. It is the best choice if you want an easy life. This is one of the few automated betting systems to work with bookmakers too. Here we can see that this strategy has had winning runs of 30 or more wins in a row on 3 occasions. It has had winning runs of 5 in a row on occasions. And we can see a very smooth, consistent and stable change from short winning runs to longer winning runs.
We can see the Losing Sequences. This is very important in understanding the nature of our strategy and what kind of mental preparation we need. This strategy has lost 5 bets in a row on just 1 occasion, and it lost 4 bets in a row on 2 occasions and 3 bets in a row on 11 occasions. TSM is very useful in the way we can see how many times it has had losing runs and what kind of losing runs they were.
We can go further into this by looking at the cumulative drawdown tab. We can see where losing runs came close together and brought the bank lower. The worst case being -7 points drawdown reached on 4 occasions. This is again useful in planning our bank and being prepared for losing runs. Another tab I find very interesting is the Odds Frequency chart where we can see how often the odds come up in each odds range.
We can see this strategy is very heavily under 1. Even better, we can see the edge or yield for each odds range. And we can specify the odds ranges we want to see by editing the top right box. Now I can see that bets under 1. I see that the higher odds bets have the best yield. The 1. Next, and probably the most important feature of TSM is the ability to see which staking plan is the best for your betting strategy. We see straight away that Fixed Staking Target Profit Staking gets the best results with , points profit Cumulative Profit , but the betting bank does go bankrupt along the way.
So we should not use this staking plan with those settings. The best staking plan listed here with no bankruptcy is the Square Root plan with 1, points profit which beat level stakes profit of 97 points by a lot. The Square Root plan is a form of staking where we bet more when we are in profit and return to original stakes when in loss.
We can read more about it on the Staking Machine help file or website. Or you can see an in-depth analysis, demonstration and the best settings for this staking plan in The Staking Plans Book by Tom Whitaker. We can get a lot information from this screen, for example we can see that using the Square Root Staking plan the biggest stake used would have been 45 units and the smallest stake was 2 units.
We can see the average stake was Other important figures are the highest and lowest peaks of 2, and 83 points. The lowest peak in particular tells us how far down into our initial betting bank we went.
If we start with points, then a 83 point low means a loss was the worst we went into our starting bank. When we first click we might see a strange graph as the most successful yet bankrupt staking plan pushes the others out, so we can deselect the bankrupt plans. Then we can see much clearer which staking plans gave the smoothest profit ride. Hovering the mouse over each line will tell us which one it is.
From this we can see that Secure Staking the orange line seems to have the smoothest profit progression. So we would be interested in checking this staking plan out further. One important thing to note here is that each staking plan is on its default settings right now. That means there may be better settings that can be used to get even more profit out of the strategy. We can see that percentage staking now beats all the other staking plans and gets a result of over 7, points.
It is this kind of analysis that can show us very quickly why using one staking plan will be better than others. We get a profit of , points and the lowest trough was just That looks good. We can check this by doing a Monte Carlo Simulation.
This is where the data is shuffled so that the order of matches is randomized. This means we can check if the fixture list had been different, would we still have been ok and not gone bankrupt. It is a good guide for how safe the staking settings are. You can select how many times that you want to shuffle the data up to a maximum of , times. The more simulations you do, the longer it will take your computer to calculate it. Here we try 1, simulations.
Here we get a very important graph that gives us valuable information for selecting the correct stake size. Good news. The Percentage Staking plan did not go bankrupt in any of the simulations. And the average profit was , points profit. We can see a graph of this. This graph shows the number of times each profit level was reached. We can see that in most cases we will get between , and , points profit.
Even more importantly, we see that the Secure Staking plan that was previously our smoothest staking plan actually went bankrupt on 13 out of 1, simulations. That means that if the fixture list had been different, 13 times we would have gone bankrupt. That means that the secure staking plan with its current settings is not good for this strategy.
You can find the best staking plan for your Betaminic big data betting strategy by slowly increasing the stake size in the staking plan settings, checking it does not go bankrupt, checking the lowest trough is not too low and then checking Monte Carlo simulations to make sure it does not go bankrupt.
By repeating this process you can find the best staking plan for your strategies. One more important concept to think about is profit taking. When do you take out from the bank. Also, these results are based on letting the bank ride for the 7 years that this strategy has been going. In reality, we need decision points where we choose to take out some of the profit or to let it ride.
In The Staking Plans Book , the suggestion is to set profit taking at the doubling point. When the starting bank doubles, then take out the profit and reset the bank to points again. For research purposes this means we are constantly resetting the bank to and seeing how the strategy performs from different starting points. So it gives us a better way to see how the strategy copes in the bad runs. In reality, when the bank doubles, you can choose to let it ride or take some out.
It is a good moment to make a judgement about profit taking. Each time it reached points, it withdrew points and started the bank again. This means it doubled its bank 10 times! If you had not withdrawn the points and let it ride, it would be , points. The mini-graph then has much more meaning for us now. Each peak is where the bank reached and had a withdrawal. The gaps between the peaks are how long it takes to double the bank. We can see that sometimes there is a long gap where we had a bad run but we still finally got back to doubling the bank.
And there are some short gaps between peaks where we had a good run and doubled the bank quickly. This gives us an excellent visual guide of how much mental discipline we will need to follow this strategy with this staking plan. Look carefully at the long gaps and imagine if you will be able to wait and persevere through those bad runs so that you can enjoy the good runs.
We can see how the winning and losing runs might affect the strategy if the fixture list and order of wins and losses are different in the future. We can see again long gaps and short gaps between doubling the bank. This is a great visual guide for the nature of the strategy and also a key part of big data strategies is that you need a long term plan to take advantage of them and then to stick to the plan without changing things when a bad run comes.
The right staking plan applied to the right strategy can multiple and leverage profits immensely. You can try different staking plan settings to see if you can get better profits without going bankrupt and without having a lower trough. There are a wide variety of settings, and you can find the best general settings for each staking plan in The Staking Plans book where it has already gone through this process for you. It can give you a good starting point for settings to apply to other betting strategies.
A highly recommended general staking plan that is good for most betting strategies is the Secure Staking plan. Betting different amounts depending on the odds bracket. The best settings for the Secure Staking plan, which we shared in the previous article on staking here , are these. We can manipulate the data in other ways, too. For example, if when importing data, we use the opening odds instead of the closing odds, we can easily see which is better, to bet early on the opening odds, or to bet late on the closing odds.
You will find that overall, it does not affect the final outcome of profit, since some odds go up and some odds go down, as explained in one of our articles. The importance is to identify the value selections. Also, we can find the importance of getting the best odds. If you shopped around and managed to get just 1 tick higher on each bet e. Nearly double the profit just by getting slightly better odds.
This really shows the importance of trying to get the best odds every team, even by just 1 decimal point higher. On level stakes points profit is only improved to points, which shows how better odds impact profit more if the right staking plan is being used.
Using odds comparison sites like Bettingmetrics or always requesting one tick higher on Betfair is highly recommended. In conclusion, if we analyze our strategies and find the best staking plans for them, we can multiply our level stakes profits. But with more research on individual strategies, it is be possible to squeeze even more profit out of strategies.
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