Slot game development guide: Everything you need to know

Slot game development stands at the forefront of the evolving game industry and provides a thrilling fusion of technology, entertainment and opportunities. The creation of attractive and innovative slot games is extremely important as Digital Landscape continues to redefine the leisure experience. These virtual on e-arm bandits not only attract players around the world, but also play a role as the foundation for online casinos.

This guide delves into the art and science of slot game development and shows a very important role in forming a modern game ecosystem. Understanding this process from concept to execution can unleash developers and lovers as well.

What is slot game development?

Slot game development is a complex process of creating digital slot machines for online casinos and game platforms. The core includes the design, programming, and implementation of virtual slot games that mimic the mechanisms and excitement of the traditional slot machines in the real store casino.

Developers focus on various important factors, such as graphics, sound design, user interface, and gameplay mechanics, and promise attractive and immersive players. The central concept is to produce random results in a way that reflects the unpredictable possibility of a physical slot machine. This is usually realized using a random number generator (RNG) to ensure fairness and fairness.

In addition, slot game development incorporates a wide range of themes, from ancient civilizations to nea r-futuristic science fiction settings, and supports the diverse preferences of players. In addition, by incorporating functions such as free spins, bonus rounds, mult i-pliers, etc., we will enhance entertainment and provide more victory opportunities for players. In short, slot game development combines stat e-o f-th e-art technology and creative design, providing players a dynamic and attractive game experience from comfortable devices.

Important features in slot game development

In the area of ​​slot game development, some important features need to pay close attention to creating outstanding game experiences. These elements form the charm and engagement of the game:

Graphic and visual design: impressive visuals are the most important. Hig h-quality graphics, dynamic animation, and attractive artwork will bring life into the game and immerse the players in a beautiful environment.

Sound design and effect: Hearing aspects are also important. The attractive sound effect complemented by the attractive soundtracks enhances the overall experience and amplifies the excitement of spin and victory.

Theme and story: The attractive theme enhances the immersive player. The wel l-developed themes, such as exploration of ancient civilization, adventures to the universe, and traveling to the fantasy world, resonate with the player's heart.

The layout of the payline and the bet options has a significant effect on game play. Developers must carefully balance the number of payline and enable a variety of betting options that can handle a wide range of players.

Bonuses and special features: By incorporating bonuses such as free spins, mult i-pliers, and interactive mini games, you add excitement and potential reward layers to enhance the player's retention and fun.

Random number Generator (RNG): The robust RNG ensures fairness by randomly producing the results of each spin and reproduces the unpredictable possibility of the conventional slot machine.

User interface (UI): Intuitive and friendly interfaces are essential for seamless navigation and fun players.

Compatibility and accessibility: Ensuring various devices and platform compatibility, such as desktops, mobile, and tablets, allows more users to access games.

Security and Fair Play: The introduction of robust security measures and transparent game play mechanism will build a trusting relationship with players and guarantee a fair and secure game environment.

Slot game developers pay close attention to these important factors, not only draw players's interest, but also create an attractive and unforgettable game experience that makes you want to play many times. You can do it.

Slot Game Development Step Bay Step Process

Slot game development has a systematic process, guaranteeing a seamless transition from concept to a completely functional game. Here, I will explain each stage in detail:

Conceptualization: The beginning of a process in which creative ideas are shaped. Developers brainstorms theme, functions and mechanisms, taking into account the target user preferences. At this stage, the basics of the entire project are built.

Design: This stage focuses on creating visual and aural elements. Graphic artists create eye-catching visuals, while sound engineers create compelling audio effects. The game layout and user interface (UI) are designed to be intuitive to use.

Development: The conceptualized ideas are converted into code. Programmers use a variety of technologies and languages ​​to bring the game to life, integrating features such as reels, paylines, and bonus features.

Testing: Rigorous testing is paramount to identify and fix bugs, glitches, and performance issues. Our Quality Assurance (QA) team conducts extensive playtesting to ensure the game works flawlessly across different devices and platforms.

Balancing and Optimization: This stage involves fine-tuning game mechanics, such as adjusting payout percentages, to ensure an optimal player experience.

Regulatory Compliance: Developers must adhere to industry regulations and standards to ensure their games meet legal requirements for fair play and responsible gambling.

Integration with Gaming Platforms: The game is integrated into the online casino platform to ensure seamless compatibility and functionality within the larger gaming ecosystem.

Deployment: After rigorous testing and compliance checks, the game is ready for release. It is available for players to enjoy on the online casino platform.

Post-release support and updates: Continuous monitoring and updates are important to address any issues that may arise after launch. This stage also involves implementing player feedback and introducing new features to keep the game fresh and engaging.

Taking these steps allows developers to create polished, high-quality slot games that provide players with an immersive gaming experience.

Benefits of Slot Game Development

Revenue Potential

Slot game development is a significant source of income in the gaming industry. The global demand for compelling slot games continues to increase, giving developers the opportunity to tap into a lucrative market.

The engaging gameplay and compelling features inherent to slot games drive high player engagement. The thrill of spinning the reels and the possibility of hitting a jackpot keep players entertained and immersed in the game.

Brand Recognition

Creating outstanding slot games can greatly enhance the brand awareness of developers. A successful slot game establishes developers as a reliable source of attractive entertainment, and builds trust and trust in the fierce competition game industry. This improvement in recognition leads to lon g-term success and faithful player base.

Technology used for slot game development

Programming language

• JavaScript
• Html5
• CSS3
• Node. js
• Graphics and rendering:
• WebGL
• OpenGL

Game development framework

Serverside operation

Networking and multiplayer function

• Networking library/API

Integration of i n-ap p-i n-app and payments

• Payment gateway API

AI integration:

• Machine learning for personalized game experiences

Blockchain technology:

• Safe transactions with smart contracts and proof fair gaming

AI and blockchain provide additional functions for personalized experiences and safe and transparent gameplay.

Reason for choosing Gamesdapp as a slot game development company

As a leading company for casino games, Game Dap offers outstanding options in slot game development. Game dap has a proven track record in the production of hig h-performance hig h-performance games and combines stat e-o f-th e-art technology and creative innovation. Our experienced development teams are good at utilizing the latest trends and technology, and will surely provide attractive game experiences. In addition, Gamesdapp considers customer satisfaction first and emphasizes seamless communication, strict delivery date, and cos t-effectiveness. With this quality and dedication to custome r-centricism, Game Dap has become a reliable partner for those who seek to p-class slot game development services.

In conclusion, slot game development is a dynamic fusion of creativity and technology, leading to the prosperity of the game industry. With great care of design, functions, and user experiences, developers can create attractive games that catch a wide range of users. An thoughtful development not only ensures the player's engagement, but also opens a door to the possibility of great revenue. It is clear that investing in slot games's thoughtful productions succeed in the game industry and are the most important in forming impactful existence.

Efficient Monte Carlo method for mult i-dimensional modeling of slot machines and jackpots

At present, entertainment is one of the largest industries and continues to expand. In this study, we deal with issues that are important for each casino and gambling clubs that estimate the ratio of the prize for the jackpot. Solving this problem leads to calculation of mult i-dimensional integration. For this purpose, the most powerful stochastic sem i-carlo promot, especially the lattice series, digital affiliate, Halton series, SOBOL series, and Latin super cube sampling are adopted. These techniques have significantly improved than the classic Monte Carlo method. After accurately calculating the integrated integration, it shows a method of calculating the expected value of the real consolidated prize, taking into account the time distribution when a player of a different number of people bet. In addition, we propose a solution for hig h-dimensional problems. All proposals are verified by calculation experiments using actual data. The results obtained in this study can be applied not only in gambling but also in many fields, such as finance and ecology.

62p30; 65c05; 68w20; 91a60

1. Introduction

The gambling industry plays an important role in modern life. It is one of the most profitable businesses in the world, and is expected to exceed $ 640 billion in 2027. In recent years, there have been more opportunities for everyone who wants such fun. The most popular game consoles are slot machines, also known as "fruit machines" and "on e-arm bandit" [5]. However, due to competition, the payback rate (Return rate to players, RTP) reaches 98% [6]. For all casinos and gambling clubs, it is essential to plan an expenditure in an accurate way to achieve both competitiveness and profits.

1. 1. General framework

In this paper, we consider a gambling club system with slot machines [7]. All revenues are formed by players' bets. The majority of expenditures are made up of direct "payline" wins. Here, bonuses and standalone jackpots are also included. Then comes the linked progressive jackpot (hereafter called jackpot). The size of the jackpot is determined by the size of the bet, and therefore is a deterministic expenditure. However, the consolation prize is not. Its upper limit is the same as the jackpot or a part of it. However, its size is a random variable, since its specific size depends on the player who won the jackpot and the amount of his bet. The aim of this paper is to propose a robust method to calculate its expected value. For completeness, the other parts of the costs concern drinks and food in the casino, staff salaries, equipment, and housing.

The consolation money is distributed to all players except the player who hit the jackpot. Everyone gets a share in proportion to the amount of their bet. So, basically, if E[X] is the expected bet share of the winner, then 1 - E[X] is the expected size (in percentage) of the consolation prize. How bets are made is explained in the next subsection.

1. 2. Bet Collection

First, the player deposits cash into the machine. This is done either directly using a bill validator or by the dealer using an "attendant" electronic key. Then, before selecting a game, the player chooses how many credits to bet per spin and sets the denomination of credits (the winnings of 1 credit). After that, the game is selected. From the game settings, the number of lines can be selected. Roughly speaking, a bet of 30 lines and 30 credits means playing 30 games at the same time, with a bet of 1 credit each game.

The first slot machines had real mechanical reels, but modern slot machines have virtual reels. When the player presses the button, the reels spin. When the reels stop spinning and the same symbols line up on the active lines (generally a pattern, not a straight line), a payline wins. Regardless of the outcome, the bet is collected on each spin.

All games are characterized by volatility. Volatility is measured on a discrete scale between 1 and 5. There is no accurate formula, but it is necessary to interpret that the larger the number, the higher the winner of the winning. Games have various RTP level settings [3, 10, 11, 12], often fluctuates between 88%and 96%, but usually higher than 91-92%. All RTP levels are associated with different reels. The RTP level is preset by the owner of the machine and cannot be changed on the go.

1. 3. Get jackpot

For all bets, the amount to configure the jackpot prize is set. The winning of the Jackpot is won by one player. As described earlier, the winning probability of each player is proportional to the bet amount. The process up to the jackpot is visualized as follows. On the screen, a tube along the outer periphery appears (of course, a rectangular shape). The tube is divided so that each player can assign a tube segment with a proportional length. Then, the ball begins to circulate while slowing down the tube. Jackpot winner is determined by the segment where the ball stopped. Other players receive the money.

The main new nature of this study is that the problem of the expected value of the alimony has been r e-defined into a multidimensional integral evaluation problem, an algorithm that uses advanced probability approaches and numerical calculation techniques. It was designed. The configuration of this paper is as follows. In the following Section 2, we will introduce the integral expression model, the point conversion algorithm, and the probability method used for integral calculation. In Section 3, a detailed presentation of the obtained results and its thorough explanation are provided. In Section 4, a special case study using actual data is considered, indicating a method of deriving the expected value of the Real Labor Award. This article is connected by section 5.

2. Algorithms and methods

Before introducing the model, we will reveal another constraint that must be paid attention. No matter how small a bed is, the relevant probability does not fall below 0. 5 %. This automatically suggests that the upper limit of the probability of one player is 100-0. 5 ( n-1) %. Then, L: = 0. 005 and U: = 0. 995 are defined. Obviously, this setting is effective for the maximum N ≤ 200 players.

Define the probability of an I (I = 1, ..., n) to acquire a jackpot as X I. The following discussions show the following:

∑ i = 1 n X I = 1 L ≤ x I ≤ u-L -n for I = 1, ..., n.

The formula (1) suggests that the point should be drawn from the N-dimension of Simplex. However, it is more complicated to meet the inequality (2). In order to deal with this, we will introduce two approaches. However, in order to explain them better, the expected value operator first.

2. 1. Integration expression

Let's explain our purpose again to seek the expected prize for comfort. If the first player gets a jackpot, the (relative) size (relative) of comfort is 1-x 1. However, the probability that the first player gets a jackpot is just X 1. Of course, this applies to all players. Therefore, in the case of n-players, the size of the consolidation award (CP) obtained in the percentage is as follows.

E [c p] = P [winning the first player] ∗ Size [C P .. + + P [win the last player] ∗ size [C P

In this case, the expected value is as follows. E [c p] = ∫ x ∈ n-1 ∫ ∫ i = 1 n x = ( 1-x i) d x n d x n-1 ⋯ d x 1, Here, δ n-1 is a standard ( n-1) -Simplex converted according to (2).

However, it can be reduced by one dimension of integration (4) using x n = 1-x 1-x 2- ... - x n-1 (1).

E [c p] = ∑ i = 1 n-1 x I ( 1-x I) + 1 -∑ i = 1 n-1 x I ∑ i = 1 n-1 x I. If d: = n-1, the integral is as follows.

E [c p] = ∫ ∫ x ∈ v ¯ ∫ ∫ = ∫ = = = = = = = = = = = = ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = 1 d x ∑ ∑ = = = = 、 、 、 、 、 、 、 、 、 、 、

Here, V ¯ D is a space between the standard ( d-1) -SIMPLEX and the coordinates of the RD, and is converted again according to (2).

To make it easier to understand, write a integral with each extreme for the first value of D:

∫ L U X 1 ( 1-x 1) + ( 1-x 1) x 1 d x 1,

∫ l ( u-l) ∫ l ( u-x 1) x 1 ( 1-x 1) + x 2 ( 1-x 2) + ( 1-x 1-x 2) (x 1 + x 2) d x 2 2 d x 1,

∫ l ( u-2 l) ∫ l ( u-l -x 1) ∫ l ( u-x 1-x 2) x 1 ( 1-x 1) + x 2 ( 1-x 2) + x 3 (x 3 ( 1 - x 3) + ( 1-x 1-x 2-x 3) (x 1 + x 2 + x 3) D x 3 d x 2 d x 1,

∫ l u- ( k-1) l ∫ l u- ( k-2) l-x 1 ∫ l u -∑ ∑ ∑ ∑ k k ∑ ∑ ∑ = = = ( 1-x i) + 1 -∑ i = 1 K x I ∑ I = 1 k x I d x k ⋯ d x 2 d x 1.

Fig. 1 is the integral of d = 1 and d = 2.

In the next section, we explain the drawing algorithm for both points (4) and (6). From now on, let the total number of points be C ≫ 1.

2. 2. Drawing algorithm for points (4)

This method is simple. First, draw C D-dimensional points x ^ D uniformly distributed in a D-dimensional hypercube. Next, sort the coordinates of each point independently of the other points, and name the new points x 〜 D. Then, map x 〜 D to a point x 〜 D + 2 in the ( D + 2 )-dimensional hypercube as follows:

x 〜 D + 2 ( i ) : = x 〜 D ( i - 1 ) for i = 2 , ... , D + 1 x 〜 D + 2 ( 1 ) : = 0 , x 〜 D + 2 ( D + 2 ) : = 1 .

Now, we can be sure that the coordinates of each point in x 〜 D + 2 are sorted and lie on the line [ 0 , 1 ] that contains the boundaries 0 and 1. If we take the difference between two adjacent coordinates and define a new coordinate in this way, we arrive at the point x ^ N in the ( D + 1 )-dimensional hypercube.

x ^ N ( i ) : = x 〜 D + 2 ( i + 1 ) - x 〜 D + 2 ( i ) . The point x ^ N certainly belongs to the standard D-simplex. In this way, (1) is satisfied. To satisfy (2), we apply a linear transformation to x^N: xN : = x^N ( 1 - N - L ) + L , where the operation is element-wise. Thus, the point xN (7) satisfies both (1) and (2) and belongs to Δ D.

2. 3. Algorithm for drawing points in (6)

This algorithm is also not complicated. First, draw C D-dimensional points x^D uniformly distributed in a D-dimensional hypercube. Now, we linearly scale the point coordinates as follows:

xD(i) : = x^D(i) - U - ( D - i + 1 ) L - ∑j = 1 i - 1 x^D(j) + L , where, of course, for i = 1, the sum is equal to 0. The points x D (8) satisfy ∑ i = 1 D x i ≤ U and x i ≥ L f o r i = 1 , ... , D . Hence, they truly belong to V ¯ D and satisfy the requirements used to evaluate (6).

Of course, to do so, we can also use the former algorithm and truncate the last coordinate of x N (7). This operation is in fact the projection of x N onto the D-dimensional hyperplane O x 1 x 2 ... x D .

It is worth mentioning that we actually use the first algorithm, which truncates the last coordinate, since the second algorithm does not allow us to distribute the points in an optimal way.

2. 4. Monte Carlo Algorithm

Monte Carlo methods[13] are useful when deterministic methods fail[14, 15, 16]. Monte Carlo methods have applications in many areas, such as valuing financial derivatives[17] and playing slot machines[18] and reconstructing the reels[19]. Here we describe some basic Monte Carlo methods.

Plain Monte Carlo is the oldest and probably the most used Monte Carlo (MC) method for solving multidimensional integrals[14]. The MC quadrature formula lies in the probabilistic interpretation of the integral:

I [ f ] = ∫ Ω f ( x ) p ( x ) d x . Let the random variable θ = f ( ξ ) be: E θ = ∫ Ω f ( x ) p ( x ) d x ,

where ξ 1 , ξ 2 , ... , ξ N are independent observations of ξ with probability density function p ( x ), and θ 1 = f ( ξ 1 ) , ... , θ N = f ( ξ N ). Then, a simple MC approach for integral I is defined as follows [14]:

θ ¯ N = 1 N ∑ i = 1 N θ i .

Latin Hypercube Sampling (LHS) is a type of stratified sampling (SS) [14]. For SS, we need to divide [ 0 , 1 ] d into M d discontinuous subregions (each with a volume of 1 M d ) and sample one point in each region. It has been proven in [15] that the variance of SS does not exceed that of ordinary random sampling. LHS is a highly researched topic [20, 21, 22, 23].

By definition, Quasi-Monte Carlo (QMC) methods are based on quasi-random sequences that are constructed to minimize the deviation from uniformity, called the discrepancy [24, 25].

Let x i = ( x i ( 1 ) , x i ( 2 ) , ... , x i ( s ) ) , n = ... a 3 ( n ) , a 2 ( n ) , a 1 ( n ) , n > 0 , n ∈ Z .

The radical inverse sequence is defined by the following equation [27]: n = ∑ i = 0 ∞ a i + 1 ( n ) b i , ϕ b ( n ) = ∑ i = 0 ∞ a i + 1 ( n ) b - ( i + 1 ) and the discrepancy satisfies: D N * = O log N N . The Van der Corput sequence [28] is obtained when b = 2.

The Halton sequence [29, 30] is defined by the following equation. s n ( k ) = ∑ i = 0 ∞ σ i + 1 ( k ) a i + 1 ( k ) ( n ) b k - ( i + 1 ) ,

where ( b 1 , b 2 , ... , b s ) ≡ ( 2 , 3 , 5 , ... , p s ) , p i is the i-th prime number, and is the set of permutations of σ i ( k ) , i ≥ 1 : ( 0 , 1 , 2 , ... , p k - 1 ) .

The Sobol sequence [17, 31, 32, 33, 34] is defined by the following formula. It is defined by x k ∈ σ ¯ i ( k ) , k = 0 , 1 , 2 , ... .

where σ ¯ i ( k ) , i ≥ 1 : the set of permutations for 2 k , k = 0 , 1 , 2 , ... successors of the Van der Corput sequence, where v i , i = 1 , ... , s is the set of direction numbers [34].

To scramble Halton sequences, we used permutations of radical inverse coefficients obtained by applying the inverse radix operation to all possible coefficient values. This algorithm is proposed in [35].

To scramble Sobol sequences, we used random linear scrambling combined with random digital shifts. This algorithm is proposed in [36].

When the integrand is sufficiently regular, lattice sequences with special arrangements with less discrepancies generally outperform basic MC methods. Sloan and Kachoyan [37], Niederreiter [38], Hua and Wang [39], Wang and Hickernell [40], and Sloan and Joe [41] provide comprehensive descriptions of the theory of lattice sequences.

We construct lattice sequences with optimal generating vectors using special algorithms [42, 43, 44, 45] for rank-1 lattice rules with prime points and rank-1 lattice rules with product weights of 2 30 points.

Niederreiter [27, 46] introduced a special family of digital ( t , m , s )-nets on F b . These nets are obtained from rational functions over finite fields [47, 48, 49]. We use a special kind of digital sequence, namely interlaced digital sequence [50, 51, 52, 53]. It is a special class of digital sequences, a concept similar to lattice sequences, but based on linear algebra over finite fields [54].

Here, we use the generator matrix for the implementation of the 21201-dimensional Sobol' sequence from [55] for digital sequences, and the generator matrix for the interlaced Sobol' sequence with interlace factor d = 2 for interlaced digital sequences.

3. Results and Discussion

This section gives an overview of the results obtained by the methods mentioned above. C represents the total number of points used in the integration.

For 10-dimensional integration (Table 1), the best approach is the lattice sequence. It achieves the smallest error in 7 out of 21 cases. The other suitable method is the interlaced digital sequence, which produces the five smallest errors, but they are obtained for large values ​​of C and are an order of magnitude better than the errors produced by the other methods.

Regarding the 2 0-dimensional integral (Table 2), the best approach is the smallest digital sequence with five errors. An appropriate approach when C is large is a scramble version of Halton and Sobol sequence, but the best error is almost the same size as other errors.

Figure 2 shows the results of 1 0-dimensional and 2 0-dimensional integration.

Considering the 3 0-dimensional integral (Table 3), the scrambled Harton sequence stands out with 10 minimum errors. If the C is large, only the interlace digital sequence has the same consequences.

The same situation is also found in the 4 0-dimensional integral (Table 4), and the best method is a scrambled Harton sequence and a sobol array, respectively. The scrambled Harton series has achieved an error 1E-9 in both integral.

The results are shown in FIG.

The best approach to the 5 0-dimensional integral (Table 5) is the Scramble Harton series with five minimum errors. If the value of C is the maximum, the digital array gives the best result. It is also worth noting that the former has achieved an error 1E-9.

In the last test 6 0-dimensional integral (Table 6), the most accurate approaches were scrambled Sobolt Squances and Latin Hyper Cube Sampling, respectively. LHS is suitable for small C values, while scrambled, sobolor sequence and lattice sequence are effective when C is large. Most methods have achieved an error in 1E-9 order, but looking at the convergence of the auda, ​​it is clear that this phenomenon is a pure luck in the rough Monte Carlo approach.

The results of 5 0-dimensional and 6 0-dimensional integration were plotted in Fig. 4.

Many conclusions are derived. First, even for larg e-scale issues, the achievement absolute error is very small. Second, lattice series and digital series are effective when the dimensions are small, and the scramble Halton series and the SOBOL series are suitable for a large number of dimensions. However, it is not always the best method. < SPAN> Regarding the 2 0-dimensional integral (Table 2), the best approach is the smallest digital sequence with five errors. An appropriate approach when C is large is a scramble version of Halton and Sobol sequence, but the best error is almost the same size as other errors.

Figure 2 shows the results of 1 0-dimensional and 2 0-dimensional integration.

Considering the 3 0-dimensional integral (Table 3), the scrambled Harton sequence stands out with 10 minimum errors. If the C is large, only the interlace digital sequence has the same consequences.

The same situation is also found in the 4 0-dimensional integral (Table 4), and the best method is a scrambled Harton sequence and a sobol array, respectively. The scrambled Harton series has achieved an error 1E-9 in both integral.

The results are shown in FIG.

The best approach to the 5 0-dimensional integral (Table 5) is the Scramble Harton series with five minimum errors. If the value of C is the maximum, the digital array gives the best result. It is also worth noting that the former has achieved an error 1E-9.

In the last test 6 0-dimensional integral (Table 6), the most accurate approaches were scrambled Sobolt Squances and Latin Hyper Cube Sampling, respectively. LHS is suitable for small C values, while scrambled, sobolor sequence and lattice sequence are effective when C is large. Most methods have achieved an error in 1E-9 order, but looking at the convergence of the auda, ​​it is clear that this phenomenon is a pure luck in the rough Monte Carlo approach.

The results of 5 0-dimensional and 6 0-dimensional integration were plotted in Fig. 4.

Many conclusions are derived. First, even for larg e-scale issues, the achievement absolute error is very small. Second, lattice series and digital series are effective when the dimensions are small, and the scramble Halton series and the SOBOL series are suitable for a large number of dimensions. However, it is not always the best method. Regarding the 2 0-dimensional integral (Table 2), the best approach is the smallest digital sequence with five errors. An appropriate approach when C is large is a scramble version of Halton and Sobol sequence, but the best error is almost the same size as other errors.

Figure 2 shows the results of 1 0-dimensional and 2 0-dimensional integration.

Considering the 3 0-dimensional integral (Table 3), the scrambled Harton sequence stands out with 10 minimum errors. If the C is large, only the interlace digital sequence has the same consequences.

The same situation is also found in the 4 0-dimensional integral (Table 4), and the best method is a scrambled Harton sequence and a sobol array, respectively. The scrambled Harton series has achieved an error 1E-9 in both integral.

The results are shown in FIG.

The best approach to the 5 0-dimensional integral (Table 5) is the Scramble Harton series with five minimum errors. If the value of C is the maximum, the digital array gives the best result. It is also worth noting that the former has achieved an error 1E-9.

In the last test 6 0-dimensional integral (Table 6), the most accurate approaches were scrambled Sobolt Squances and Latin Hyper Cube Sampling, respectively. LHS is suitable for small C values, while scrambled, sobolor sequence and lattice sequence are effective when C is large. Most methods have achieved an error in 1E-9 order, but looking at the convergence of the auda, ​​it is clear that this phenomenon is a pure luck in the rough Monte Carlo approach.

The results of 5 0-dimensional and 6 0-dimensional integration were plotted in Fig. 4. Many conclusions are derived. First, even for larg e-scale issues, the achievement absolute error is very small. Second, lattice series and digital series are effective when the dimensions are small, and the scramble Halton series and the SOBOL series are suitable for a large number of dimensions. However, it is not always the best method.< a , b , c >.

There is a reason for this result. The integral function is a smooth multiple polymorphism. Some adaptive methods are advantageous for irregular and no n-smooth functions, such as, for example, Latin super cube sampling, but this is not the case. Thus, the scrambled Harton series and the lattice series with good generated vectors are recommended for accurate evaluation of the integral of this study.

Please note a few precautions about the implementation of the Stock Stick Algorithm. First of all, two digital sequences (regardless of the presence or absence of inter racing) are implemented in C ++ to improve their performance, but other methods and remaining operations are implemented in Matlab ®. 。 Second, it is difficult to hold 30 mult i-dimensional points in the RAM in the 'Double' type with an 8-byte coordinates in the RAM. Even in the 1 0-dimensional point, the necessary storage reaches 80GIB (jibovite). But after all, it is not necessary. We have implemented a simple deformation of the Mapreduce model. It evaluates the integral with two 10 chunks, preserves the results, and finally averaging all approximates. Some stochastic methods are suitable for parallel implementation, but other sem i-sequences are not effective because they depend on the elements before the sequence.

4. Actual case study< a , b , c >= < 2.006 , 1.014 , − 0.9962 >This section applies algorithm described in the valgus prize value for actual data from the Bulgarian gambling club.

It also reminds that D-dimension integral F (D) represents the expected value of the comfort award (CP) in games with n = d + 1 players. In a game with a player, (S) he gets a jackpot and becomes C P = 0. Here, the results of d = 1, ..., and 63 are plotted in FIG. < SPAN> There is a reason for this result. The integral function is a smooth multiple polymorphism. Some adaptive methods are advantageous for irregular and no n-smooth functions, such as, for example, Latin super cube sampling, but this is not the case. Thus, the scrambled Harton series and the lattice series with good generated vectors are recommended for accurate evaluation of the integral of this study.

Please note a few precautions about the implementation of the Stock Stick Algorithm. First of all, two digital sequences (regardless of the presence or absence of inter racing) are implemented in C ++ to improve their performance, but other methods and remaining operations are implemented in Matlab ®. 。 Second, it is difficult to hold 30 mult i-dimensional points in the RAM in the 'Double' type with an 8-byte coordinates in the RAM. Even in the 1 0-dimensional point, the necessary storage reaches 80GIB (jibovite). But after all, it is not necessary. We have implemented a simple deformation of the Mapreduce model. It evaluates the integral in two 10 chunks, preserves the results, and finally averaging all the approximation values ​​again. Some stochastic methods are suitable for parallel implementation, but other sem i-sequences are not effective because they depend on the elements before the sequence.

4. Actual case study

This section applies algorithm described in the valgus prize value for actual data from the Bulgarian gambling club.

It also reminds that D-dimension integral F (D) represents the expected value of the comfort award (CP) in games with n = d + 1 players. In a game with a player, (S) he gets a jackpot and becomes C P = 0. Here, the results of d = 1, ..., and 63 are plotted in FIG. There is a reason for this result. The integral function is a smooth multiple polymorphism. Some adaptive methods are advantageous for irregular and no n-smooth functions, such as, for example, Latin super cube sampling, but this is not the case. Thus, the scrambled Harton series and the lattice series with good generated vectors are recommended for accurate evaluation of the integral of this study.

Please note a few precautions about the implementation of the Stock Stick Algorithm. First of all, two digital sequences (regardless of the presence or absence of inter racing) are implemented in C ++ to improve their performance, but other methods and remaining operations are implemented in Matlab ®. 。 Second, it is difficult to hold 30 mult i-dimensional points in the RAM in the 'Double' type with an 8-byte coordinates in the RAM. Even in the 1 0-dimensional point, the necessary storage reaches 80GIB (jibovite). But after all, it is not necessary. We have implemented a simple deformation of the Mapreduce model. It evaluates the integral with two 10 chunks, preserves the results, and finally averaging all approximates. Some stochastic methods are suitable for parallel implementation, but other sem i-sequences are not effective because they depend on the elements before the sequence.

4. Actual case study

This section applies algorithm described in the valgus prize value for actual data from the Bulgarian gambling club.

It also reminds that D-dimension integral F (D) represents the expected value of the comfort award (CP) in games with n = d + 1 players. In a game with a player, (S) he gets a jackpot and becomes C P = 0. Here, the results of d = 1, ..., and 63 are plotted in FIG.

The value of f (d) indicates the prior to the jackpot, which is expected to be paid by the casino as a d + one player always plays. Of course, it doesn't. Especially in small gambling clubs, most of the players are small. Furthermore, the number of visitors is day and night periodic. The distribution is binometric or polymer. Figure 6 is an example of two gambling clubs in a mediu m-scale city in Bulgaria in 2017. Each club has 32 slot machines, but the first club seemed to play more than 25 people and 16 or more visitors in the second club.

The X axis shows the number of staff being played, and the y axis shows a relative time. Here, the real comfort prize F '(D) is calculated. This is defined as a CP expectation value when at least one person, at least one person, plays at least one person, at least one person, in consideration of the defined distribution. In order to calculate the confidence interval (CI), the standard deviation must be calculated using the standard formula (9).

E [f ′ (d)] = ∑ i = 1 d W i 1, ..., 24,

Here, w, i = 1, ..., d is a relative weight, and ∑ i = 1 d W i = 1 must be for all D.

Finally, assume the regularity of d = 1, ..., 24 real CP f '(d), and plot the expected value and CI in FIG. Is shown).

The real CP of the second gambling club has a large standard deviation, and therefore the CI is wide. See Fig. 5). On the other hand, the lower the number of players, the lower the expected CP expected.

The overall conclusion is that 99%of the reliability (Z *≈ 2. 57), the reality awards that have been realized will not be more than 1%from their expected value. be. This is the case when a jackpot hits less than two years (n = 500 days) every day. The value of

At the end of this section, we briefly consider large casinos. With a large number of machines and a low or negative skew in the time distribution of simultaneous plays, the effective expected value of CP will be high, close to 1. Of course, in this case the jackpot will be hit more frequently, but the absolute size of the jackpot and CP will be much larger than in the corresponding small gambling club. Therefore, the accurate calculation of the expected value of CP is of utmost importance even for large casinos.

Here, the calculation of integrals in hundreds of dimensions becomes a problem. The approach described in the paper is robust and can be scaled to large problems, but all methods become slow when the number of dimensions becomes huge. It is a legitimate question whether the CP value f ( D ) in large dimensions can be extrapolated from the CP value in lower dimensions that has already been calculated.	Our investigations have shown that f ( D ) can be approximated by the functional form of the Michaelis-Menten saturation curve (10), see Figure 5:
CP	g ( D ; p ) = a D b + D + c ,
where the parameter p =	Recalling that f ( 0 ) = 0, we fit the model (10) only to the first 10 values ​​of f ( D ), D = 0 , ... , 9. The fitted value of p is called the nonlinear least squares estimator and is denoted as p ˇ. Let us also define the least squares error function as Φ ( p ) = ∑ D = 0 9 f ( D ) - g ( D ; p ) 2 .
SS	The fit is indeed good, since the step norm δ p k = 8. 24163 e - 5, the first-order optimality measure ∇ Φ ( p ˇ ) ∞ = 2. 35 e - 9, and the error function Φ ( p ˇ ) = 4. 24941 e - 8 are very small. Moreover, the variance of the residuals σ 〜 2 = 2. 0614 e - 5 and the mean squared error σ ^ = 7. 7914 e - 5 are very small, and the coefficient of determination is practically R 2 = 1. 0000.
All parameters p are statistically significant, and their fit values ​​are p ˇ = .	Finally, we evaluate g ( p ˇ ; D ) for D = 0 , ... , 63, and plot the absolute error ϵ = f ( D ) - g ( D ; p ˇ ) in Figure 9.
The error magnitude is 1e-4, but the fit was only done for the first 10 values. If all known values ​​were fitted, the error would be negligibly small. This shows that (10) can be used to extrapolate the true consolation prize to larger values ​​of D with acceptable error and low computational cost.	5. Conclusion
CI	This new experimental study solved the problem of determining the expected value of the Jackpot's percentage of the prize. This is extremely important for each casino or a gambling club for a strict budget plan, regardless of its scale. This problem is formulated as a multidimensional integral evaluation. There are several quas i-advanced quas i-carlo methods, especially Sobol, Harton sequence, lattice with interlaces, and Latin Hyper Cube Samplings. It has been demonstrated that all of these have excellent performance compared to the basic Monte Carlo method.

Another new factor in this paper is the formula of the expected value of the real consumption award, taking into account the time distribution of different players. Ultimately, we propose an approach that responds to a very high dimension by inserting the already calculated results.

1. The proposed algorithms can calculate not only the linked jackpots, but also the wide area jackpot (executed over multiple casinos machines). Another possible way to further develop this study is to optimize a stochastic approach to execution time, accuracy, and number of dimensions.
2. The results obtained in this survey may be available in various fields of life. It may play an important role in estimating the SOBOL sensitivity index of a larg e-scale pollution model. Furthermore, such knowledge is frequently used in weighing finance to evaluate and calibrate mult i-dimensional financial derivatives. Finally, this result will be useful for other scientists to calculate demands in a variety of knowledge.
3. Disclaimer: This paper must not be understood as advice to play slot games or not. The purpose is to propose an algorithm for practical and applied scientific purposes.
4. Author's contribution
5. All authors contributed to the same. Concept, S. G., methodology, V. T., Software, S. G., verification, V. T., formal analysis, formal analysis, S. G., S. G., S. G., V. T., Writing-Editing, Editing, V. T., V. T., V. T. All authors are , I read and agree with the published manuscript.
6. Funding < Span> This new experimental study solved the problem of determining the expected value of the Jackpot's percentage of the prize. This is extremely important for each casino or a gambling club for a strict budget plan, regardless of its scale. This problem is formulated as a multidimensional integral evaluation. There are several quas i-advanced quas i-carlo methods, especially Sobol, Harton sequence, lattice with interlaces, and Latin Hyper Cube Samplings. It has been demonstrated that all of these have excellent performance compared to the basic Monte Carlo method.
7. Another new factor in this paper is the formula of the expected value of the real consumption award, taking into account the time distribution of different players. Ultimately, we propose an approach that responds to a very high dimension by inserting the already calculated results.
8. The proposed algorithms can calculate not only the linked jackpots, but also the wide area jackpot (executed over multiple casinos machines). Another possible way to further develop this study is to optimize a stochastic approach to execution time, accuracy, and number of dimensions.
9. The results obtained in this survey may be available in various fields of life. It may play an important role in estimating the SOBOL sensitivity index of a larg e-scale pollution model. Furthermore, such knowledge is frequently used in weighing finance to evaluate and calibrate mult i-dimensional financial derivatives. Finally, this result will be useful for other scientists to calculate demands in a variety of knowledge.
10. Disclaimer: This paper must not be understood as advice to play slot games or not. The purpose is to propose an algorithm for practical and applied scientific purposes.
11. Author's contribution
12. All authors contributed to the same. Concept, S. G., methodology, V. T., Software, S. G., verification, V. T., formal analysis, formal analysis, S. G., S. G., S. G., V. T., Writing-Editing, Editing, V. T., V. T., V. T. All authors are , I read and agree with the published manuscript.
13. Funds providing this new experimental study solved the problem of determining the expected value of the Jackpot's percentage of the prize. This is extremely important for each casino or a gambling club for a strict budget plan, regardless of its scale. This problem is formulated as a multidimensional integral evaluation. There are several quas i-advanced quas i-carlo methods, especially Sobol, Harton sequence, lattice with interlaces, and Latin Hyper Cube Samplings. It has been demonstrated that all of these have excellent performance compared to the basic Monte Carlo method.
14. Another new factor in this paper is the formula of the expected value of the real consumption award, taking into account the time distribution of different players. Ultimately, we propose an approach that responds to a very high dimension by inserting the already calculated results.
15. The proposed algorithms can calculate not only the linked jackpots, but also the wide area jackpot (executed over multiple casinos machines). Another possible way to further develop this study is to optimize a stochastic approach to execution time, accuracy, and number of dimensions.
16. The results obtained in this survey may be available in various fields of life. It may play an important role in estimating the SOBOL sensitivity index of a larg e-scale pollution model. Furthermore, such knowledge is frequently used in weighing finance to evaluate and calibrate mult i-dimensional financial derivatives. Finally, this result will be useful for other scientists to calculate demands in a variety of knowledge.
17. Disclaimer: This paper must not be understood as advice to play slot games or not. The purpose is to propose an algorithm for practical and applied scientific purposes.
18. Author's contribution
19. All authors contributed to the same. Conceptualization, S. G., methodology, software, Software, S. G., V. T., formal analysis, formal analysis, S. G., S. G., S. G., Resource, V. T., Written-Editing, Editing, V. T., V. T., V. T., V. T. All authors , I read and agree with the published manuscript.
20. Funding for funds
21. Venelin Todorov is supported by the Bulgarian National Science Fund under the bilateral project KP-06-Russia/17 "New Highly Efficient Stochastic Simulation Methods and Applications" and the Bulgarian National Science Fund under the project KP-06-N52/5 "Efficient methods for modeling, optimization and decision making". Slavi Georgiev is supported by the Bulgarian National Science Fund under the project KP-06-M32/2-17. 12. 2019 "Advanced probabilistic and deterministic approaches to large-scale problems in computational mathematics", the National Program "Young Scientists and Postdoctoral Fellows-2" - Bulgarian Academy of Sciences, Ruse University Scientific Research Fund under the project FNSE-03.
22. Limitations on data use
23. There are limitations on the use of these data.
24. Conflicts of interest
25. The authors declare that they have no conflicts of interest. The funders were not involved in the design of the study, the collection, analysis and interpretation of data, the writing of the manuscript or the decision to publish the results.
26. Abbreviations
27. The following abbreviations are used in this manuscript:
28. RTP
29. Return to Player
30. Consolation Award
31. LHS_τLatin Hypercube Sampling
32. Stratified Sampling
33. (Q)MC
34. (Quasi-)Monte Carlo
35. (F)CBC
36. (Fast)Component-by-Component
37. Confidence Interval
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