Slot machines are electronic gambling devices offering a variety of games, commonly found in casinos and some bars. They are popular and highly profitable due to their relatively low operating costs compared to games like roulette, blackjack, or poker. A typical slot machine consists of three or five reels that spin when a button is pressed. The game pays out based on the patterns of symbols displayed on the screen after the reels stop. This paper focuses on the distribution of symbols on the reels.
Each reel in a slot machine contains ordered symbols, often represented by numbers in mathematical models. The order in which these symbols appear on the reel follows a discrete probability distribution. Based on this distribution, various winning combinations can emerge. Mathematicians who design slot machines must select a discrete probability distribution that achieves a desired Return to Player (RTP). RTP is calculated by dividing the money won by the money lost, then multiplying by 100. This value is critical for slot machine vendors, as it is a key parameter regulated by government authorities. RTP represents the expected value of the game. While volatility is another important parameter in slot machines, it is not discussed in this study as it is less frequently regulated by governments. The method proposed for optimizing the symbol distribution can be a valuable tool for mathematicians developing new slot games.
This section reviews the basic concepts of slot machines, particularly the game reels.
Slot machines operate based on the concept of spinning reels. Initially, slot reels were mechanical, and players manually pulled a handle to spin the reels. Today, most slot machines are computerized, with virtual reels controlled by a Random Number Generator (RNG) [1]. While five independent reels are most common, variations with three or more than five reels exist. When the player presses a button, the reels begin to spin, stopping one by one. The player's winnings are determined by the combination of symbols displayed on the screen. Each game has a paytable, which is usually displayed separately for the player to review. Some symbols appear more frequently than others. Less frequent symbols form winning combinations less often but yield higher payouts.
A key characteristic of each slot machine is its RTP%. This value is calculated by dividing the amount of money won by the amount of money lost, then multiplying by 100. RTP can vary from 80% in some areas like Las Vegas to up to 98% in certain EU member states. For example, fruit machines in the UK are required by law to meet a minimum payout percentage within a set period [2]. Typically, RTP values exceed 90%. To achieve the desired RTP, mathematicians and game designers collaborate to populate the game reels with symbols in an optimized discrete distribution [3].
Genetic Algorithms (GAs) are search heuristics inspired by the process of natural selection [4][5]. These algorithms are commonly used to explore potential solutions in a given solution space. Through inheritance (crossover), mutation, and selection techniques, GAs can guide candidate solutions toward an optimal outcome. GAs are population-based algorithms, where each solution represents an individual within the population. Each individual has a set of properties that are subject to mutation or crossover. Traditionally, properties are represented as binary sequences (0s and 1s), though other encodings, such as binary trees, can also be used.
Optimization begins with a randomly generated population of individuals. The process is iterative, with each iteration referred to as a generation. For each individual in the generation, a fitness value is calculated, representing the quality of the solution. The most fit individuals are selected based on a defined rule, and they are recombined (through crossover and mutation) to form a new generation. The algorithm continues until it reaches a maximum number of generations or achieves a desired fitness level.
To apply GAs, two components must be defined: 1) the genetic representation of the solution space and 2) an appropriate fitness function to evaluate solutions. Once these elements are established, the GA can proceed with population initialization and iterative improvement through selection, crossover, mutation, and evaluation.
In the proposed model, each individual in the population represents a set of symbols distributed across the slot machine's reels. Each symbol is represented as a single integer, with both the symbol and its position on the reel being significant. The solution space is finite and discrete, with each position on each reel corresponding to one integer from a list of possible symbols.
Population initialization is typically done randomly, but in this model, it is based on an initial configuration of the reels, with random noise added to the configuration [6]. The population size is determined experimentally and can range from a few individuals to several thousand.
During the selection process, individuals are chosen based on their fitness values. Some selection methods prioritize the best individuals, while others select a random subset of the population. The fitness function depends on the problem at hand and measures the quality of the solution. In this model, the fitness function is defined as the absolute difference between the desired RTP and the RTP obtained from the current symbol distribution. Monte Carlo simulations are used to estimate the RTP based on 100,000 or 1,000,000 runs of the game.
Elitism is applied, ensuring that the best individuals survive to the next generation.
Crossover involves selecting pairs of parent individuals and recombining their attributes using a single-point cut. This process creates child individuals. Future research may explore the possibility of using more than two parents for crossover. After crossover, mutation occurs by randomly selecting and replacing symbols in the child individuals.
The algorithm terminates either after a set number of generations or upon manual intervention. The final solution produced by the GA is an integer vector representing the optimal distribution of symbols on the reels. For example, in a slot game with five reels and 63 possible symbols per reel, the final solution would be a 5x63 integer vector [6].
Experiments were conducted using a non-commercial slot machine consisting of five reels, each displaying three symbols. The slot machine had a specific paytable and combination lines, and the results of the simulations were analyzed.
