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This work presents a Variable Neighborhood Search (VNS) approach for solving the Return-To-Player (RTP) optimization problem. A large number of software companies in the gaming industry seek to solve the RTP optimization problem in order to develop modern virtual casino gambling machines. These slot machines have a number of reels (e.g., three or more) that spin once a button is pushed. Each slot machine is required to have an RTP in a particular range according to the legislation of each country. By using a VNS framework that guides two local search operators, we show how to control the distribution of the symbols in the reels in order to achieve the desired RTP. In this study, optimization refers only to base game, the core of slot machine games, and Non-stop party Scratch Slots Machine in bonus games, since a bonus game is triggered once two, three, or more specific symbols occur in the gaming monitor. Although other researchers have tried to solve the RTP problem in the past, this is the first time that a VNS methodology is proposed for this problem in the literature with good computational results.

1. Introduction

Slot machines, also Non-stop party Scratch Slots Machine as fruit machines, are the most popular gambling games in casinos. They are either electronic or electromechanical devices that consist of a number of reels (n) that spin independently and a screen with a number of rows (m), Non-stop party Scratch Slots Machine. Each reel contains several symbols in specific arrangements and quantities. Also, it has a number of positions at which they stop, and this number is called stops [1]. The stops are being shown on the display of a slot machine, while the reels are spinning in front of the player. Slot machines as mechanical devices had cylinders Non-stop party Scratch Slots Machine spinning reels holding the symbols that were printed on a strip. Nowadays, Non-stop party Scratch Slots Machine, slot machines are mostly electronic, and they can be found either in traditional casinos or in online casinos as software. However, in virtual slots, once the reels start spinning, a Pseudo-Random Number Generator (PRNG) is used to get the stops.

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Return-To-Player (RTP) is one of the two most important parameters for a player to choose if he/she is going to play the game, but it is also the main gambling game parameter which is under government regulation [4]. It is the win as percentage of the bet and is the phrase that casinos and casino game makers use to describe the long-term theoretical expected payback percentage from all wagers on a slot machine, video poker machine, Non-stop party Scratch Slots Machine, or Video Lottery Terminal (VLT), Non-stop party Scratch Slots Machine. If a slot machine player is going to play a hundred rounds at 1$ on a slot game with a theoretical slot RTP of 93%, with all other things being equal, Non-stop party Scratch Slots Machine would expect to get a Non-stop party Scratch Slots Machine of 93$ in wins from his staked $. A player wins back most of his wager. The remaining 7% of his bet, in this game, is the house edge, the money that casino earns off each bet. The formula for the computation of the RTP and also some necessary notations can be found with the formulae below:orTotal Prize: the amount of money that a window instance winsCycle: how many times the reels spun in a Monte Carlo simulationPay Lines: the number of lines that the game has enabledCoins In: total amount of coins betCoins Out: total amount of coins won

In this paper, Monte Carlo simulation withand 1, separate runs will be used for the RTP calculations. In simulations, formula (2) is used, where the cycle represents the number of runs. In addition, there is another type of simulation called “Full Cycle.” This latter type of simulation is used in the design phase (i.e., mathematical) of a slot machine game. It is an exhaustive cycle through all possible stop positions. The Full Cycle simulation allows an exact match to the calculations in the spreadsheet but is not capable of calculating coinciding statistics across a series of feature games.

The structure of this work is as follows: In Section 2, we discuss similar approaches for the RTP optimization problem, focusing on recent research work and the methods they utilize. Next, we introduce our methodology in Section 3. Section 4 presents the results of our methods on several games. Finally, we draw our conclusions and provide some research directions for future work in Section 5.

2. Related Non-stop party Scratch Slots Machine the RTP optimization problem is one of the most important parts of the slot machine game design, there are not many scientific articles that present a heuristic or metaheuristic approach to solve this problem. Only three papers have been published which present a metaheuristic approach to solve the RTP optimization problem and are briefly presented in Table 1.

Reference Approach Simulation Special symbols Runs Games tested This work VNS Monte Carlo Yes , & 1, 3 3 [4] GA(s) Monte Carlo No , or 1, 1 [5] DE Monte Carlo No 1, 1 [6] DDE and GA(s) Full Cycle Yes All possible 1

Balabanov et al. [4] presented a solution for the slot machine optimization problem, where the symbols distribution of the base game is being controlled and, also, some other features like free spins frequency or bonus game frequency. The fitness function in this work Non-stop party Scratch Slots Machine the absolute difference between the desired and the obtained RTP. For Monte Carlo simulations, 1, orseparate runs were used in order to calculate the obtained. They used Genetic Algorithms (GAs), with a randomized initial population and appropriate crossover and mutation operators. The target of the example that the authors presented was 99%, starting from 91%. So, the algorithm increases the in a small range in case the game already exists (a quality starting ). However, in most cases, the mathematicians of slot games usually try to develop a new slot machine game (from scratch) that has initial reels with randomized symbol distribution and the might be equal, for example, to % (big random value). Bob casino, it is not a trivial task for such algorithms to handle the case of large reductions and the huge difference between the target and the initial. The decrease of the in slot machine game is the hardest problem for the mathematicians, since they have to solve the optimization problem with hand adjustments and this usually requires a long time. Furthermore, the algorithm’s behavior in the case where special symbols appear in reels (e.g., wild, scatter) is also unknown.

The same authors in [5] also proposed a Differential Evolution (DE) algorithm, since the problem of optimal symbols distribution on the reels is discrete and its convergence was found to be faster than GAs. Additionally, they also used Monte Carlo simulations with 1, separate runs over ten trials for better accuracy in base game. Their model of slot machine game contained only simple symbols and not scatter, wild, or bonus games (scatter represents symbols that pay everywhere on the screen and substitute all other symbols). The authors proposed a multicriteria optimization approach as future work, for the symbol frequencies, the free spins frequency, the game volatility, and the bonus game frequency.

Keremedchiev et al. [6] proposed a Genetic Algorithm methodology for the optimization problem, in bonus and free casino la valette frequency as parallel computing evolutionary optimization. The fitness function of base game that was used was computed with Full Cycle method instead of Monte Carlo simulations. In current algorithm, the model of slot machine game does contain as features bonus game, simple symbols that pay on each pay line, scatter symbols, and wild and bonus symbols that trigger the bonus game. This experiment, as it refers in it, showed that the use of Genetic Algorithm with exact numerical calculation of as fitness function was very efficient and improved the slot game parameters by better adjustment ofbonus game, and free games frequency. Finally, the biggest drawback was that the exact computation was time-consuming and slowed down the optimization process. The authors also proposed a Discrete Differential Evolution (DDE) implementation as a further research.

3. Research Methodology

Variable Neighborhood Search (VNS) constitutes a metaheuristic solution method which is systematically changing a neighborhood structure. VNS was initially proposed by Mladenović et al. [7, 8]. Nowadays, a large number of successful VNS applications have been reported in a wide range of areas [9–13], Non-stop party Scratch Slots Machine. In this work, a Variable Neighborhood Descent (VND) heuristic schema was applied in base game’s and it takes all symbols into account. The VND metaheuristic features a strong local search phase (intensification) utilizing a number of different neighborhood structures but lacks a diversification phase (shaking method) as compared to other VNS Dark Spin Slots Machine. The VND variant has been applied in several research works in the literature [14, 15].

The proposed solution method is analytically described in the following subsections and an overview is depicted in Figure 1.

Solution Representation

A slot machine consists of reels where the symbols are represented by a string and are equally distributed. The solution representation is a two-dimensional array with the distribution of the symbols on each reel. For example, in case of a five-reel (number of columns) slot machine game with 48 symbols per reel (number of rows), the solution array (denoted as “Reels”) would have the dimension of .

Three inputs of the proposed algorithm are (i) the maximum number of iterations as a termination criterion (also a maximum CPU solution time can be used), (ii) the target (desired) RTP of the user (), and (iii) Non-stop party Scratch Slots Machine allowed deviation denoted () from the target RTP. Usually, the computation of the precise RTP is quite difficult and, due to this reason, game developers are satisfied if the RTP belongs in a specific range. Hence, Non-stop party Scratch Slots Machine, ifthen the algorithm stops. Assume, for example that if we are given  = ,  = , and  = , then the algorithm stops. An extra feature of the algorithm is that it allows the user to declare a minimum number of each symbol per reel. Thus, a fourth input of the proposed algorithm is an additional two-dimensional array containing the minimum number of symbols per reel and is denoted as “minNSR.” The number of columns of this array is equal to the number of reels, while its number of rows is equal to the number of distinct symbols. Thus, denotes the minimum required number of occurrences of the symbol in the -reel.

Neighborhood Structures

The proposed VND methodology utilizes two neighborhood structures denoted as “RMS” and “RMC.” Both these local search operators aim to reduce the RTP as follows:(i)RMS: initially, the symbols are sorted in ascending Non-stop party Scratch Slots Machine based on their corresponding payments. The Replace Maximum Symbol () local search operator replaces with the biggest payment by that has the next biggest payment in pay table. The players of a slot machine game may win a monetary award in the case where a combination of at least two identical symbols occurs in the pay lines. The symbol that has the biggest payment in these combinations is considered the most expensive symbol. However, the replacement by the operator is permitted only after checking the input array (minimum number of symbols per reel requested by the game developer); otherwise, the next most expensive is considered for potential replacement.(ii)RMC: the RTP increases when the placement of the symbols in the reels leads to minimum winning combinations of symbols (i.e., combinations of two identical symbols only) with a high frequency. The Replace Minimum Combination () local search operator replaces which participates in a minimum winning combination with that does not participate in such winning combinations (e.g., it participates in winning combinations of three or more identical symbols). Such symbols may participate in combinations with a little larger payment, but the frequency of their combinations is much lower.

VND Metaheuristic Algorithm

The VND solution approach is depicted in Algorithm 1. Also, Monte Carlo simulation () is used for the calculation of the current withor 1, separate runs. The two neighborhood structures and are also denoted as andrespectively. Moreover, the number of different neighborhood structures is denoted as and in this work it is equal to two.

The proposed VND is initialized with a randomized initial symbol distribution per reel (the list of distinct symbols is given by the game developer) and terminates once a maximum number of iterations is performed rather than not finding an improvement. This is due to the fact that, in each iteration, a Monte Carlo simulation might return an approximate value of the RTP and not the exact RTP that would result after a (time-consuming) Full Cycle RTP computation. Thus, in some cases, although the two local search operators do not make any improvement, a better RTP value can be calculated by the Monte Carlo simulations. Moreover, each time the neighborhood is changed, the search is performed in a different (next) reel. Thus, the VND starts the optimization using the first reel () and afterward sequentially applies the local search operators in each one of the reels.

Illustrative Examples

Without loss of generality, let us assume a hypothetical 5 Knights Slots Machine machine game with reels, with each one of them containing 12 equally distributed symbols. The symbols are represented as string variables in an array () and the pay table is stored as integer values in another array ().

Next, an example of the array is presented, where for simplicity we show only one of the columns corresponding only to the first reel. Each index represents how many of symbols should remain in current reel and not get replaced by other, for example, meaning that cannot occur less than three times in the first reel. This way, it will not be possible to replace with .

The operator changes the symbols of the first reel as follows:

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