The Hidden Power of Math: How Mathematics Makes Money for the Gaming Industry

Gaming is no longer only fun, since it now works as a multi-billion-dollar company across every platform. Every game, whether it’s a casino slot machine or a quick smartphone shooter, has a careful mathematical design that influences how the game functions.

For executives, investing in mathematicians, data scientists, and statisticians acts as a profit lever rather than a cost.

In this article, gaming includes mobile, PC, console, cloud, VR, online casinos, sports betting, and land-based casinos. The math behind these products involves probability theory, statistics, optimization, algorithm design, machine learning, and simulation.

The goal is to show how these methods influence five major sub-industries and how they drive value, manage risk, and support monetization. This framework matters to executives who focus on business outcomes and to technical teams who build the models that support them. Both groups gain a clearer view of how mathematics in gaming affects real results.

Overview of the Five Sub-Industries

The gaming ecosystem can be grouped into 5 sectors that apply mathematical models in different ways. These five categories provide the structure for how mathematics in gaming shows up across the industry.

1. Land-based casinos – rely on probability and payout modeling to control risk across table games and slots.
2. Online betting and sportsbook – depend on real-time odds calculation, pricing models, and predictive analytics.
3. Online casino and iGaming – use random number generation, volatility curves, and return modeling to shape player experience.
4. Mobile video gaming – leans on progression math, economy balancing, and user behavior modeling for monetization.
5. Console, PC, cloud, and live-service gaming – use optimization, simulation, and large-scale data systems to manage complex player ecosystems.

Land-Based Casinos

Casinos rely on structured math to keep games profitable while managing the swings that happen on the floor. These models guide how games are designed, priced, and monitored.

Key Mathematical Roles

1. House Edge and Expected Value

Expected value comes from calculating the weighted average of all possible outcomes for a bet. Casinos set payoff odds so this value stays positive for the house, which creates the margin known as the house edge. You can see this difference in games like American versus European roulette or when comparing common table game payouts.

2. Volatility and Risk (Variance and Standard Deviation)

Variance and standard deviation show how much player results swing above or below the expected value during short sessions. These measures help casinos understand streaks, cold runs, and irregular play patterns. Over time, these swings form a normal distribution that supports bankroll forecasts and long-term planning.

3. Player Behavior and Value Modeling (“Theo Win”)

Theo win estimates how much revenue a player generates by looking at their average bet size, hold percentage, and time on the floor. Casinos use this measure to segment players, identify high-value guests, and shape comp programs that improve retention.

Research from the Wharton Marketing Department on customer response modeling supports these predictive methods and shows how they guide decisions at scale, especially when paired with tools like Monte Carlo simulation in games.

4. Gambler’s Ruin / Risk of Ruin

Gambler’s ruin theory estimates the chance a player loses their entire bankroll before reaching a positive outcome. Casinos use this model to understand how bet size, game pace, and volatility affect how long a player stays active. These insights help set table limits, maximum bets, and risk thresholds that keep financial exposure predictable.

5. Game Design & Algorithmic Balancing

Casino games are built through models that keep the house profitable while still giving players a fair and enjoyable experience. Designers run large simulations to study how payouts behave across millions of plays. The Casino Mathematics Guide by IACSP explains how these models support game tuning and long-term balance.

• Payout tables built to maintain a steady margin for the house
• Side bets and progressives shaped by probability curves
• Monte Carlo simulations used to validate behavior across large trial sets

6. Combinatorics & Permutations

Combinatorics helps casinos map every possible outcome in games that rely on cards or dice. These counts show how blackjack hands or craps rolls distribute across thousands of plays. Designers use these probabilities to build side bets, including hard-way wagers in craps, that follow predictable mathematical patterns

7. Game Theory / Strategy Optimization

Skill-based games rely on models built from probability in gambling and decision theory to guide optimal play, such as the basic strategy charts used in blackjack. Poker and other competitive formats apply game-theoretic ideas to study how players balance risk, expected value, and long-term strategy.

These concepts are explained clearly in the mathematics and gambling analysis published by Betting Websites UK, which shows how equilibrium forms between players and the house.

8. Regulatory Compliance & Certification

Casinos must show that their random number generators produce fair results across long stretches of play. Regulators look for clear statistical testing and secure system design before they approve any game. The RNG fairness overview by Kenka Oscar explains how these checks protect both the casino and the players.

• Chi-squared and entropy tests confirm that results behave as expected
• Cryptographic RNGs reduce the chance of patterns or bias
• Seed controls keep the randomness source from being manipulated

9. Pricing “Comp” Programs & Incentives

Casinos study how comps, free play, and loyalty rewards affect a player’s long-term value. The goal is to offer these perks only when the expected profit from retention is higher than the cost of the incentive. Probability models based on play time, average bet size, and win or loss cycles help estimate the return on each reward.

10. Inventory & Floor Planning Optimization

Casinos use arrival patterns and queueing models to study how players move across the floor and where demand builds. These patterns shape machine placement, table spacing, and traffic flow to keep usage high throughout the day. The same models help forecast staffing needs and guide how resources are allocated during peak and slow periods.

• Player flow used to plan slot and table placement
• Layout adjustments improve machine uptime and engagement
• Staffing forecasts based on expected traffic patterns

Online Betting / Sportsbook

Sportsbooks depend on mathematical models built from probability theory that estimate the real chance of each outcome, with research showing that forecasting quantiles of outcome distributions is the central task of statistical models used in wagering markets which is the same probabilistic foundations that operators use when compiling odds as discussed by Dmochowski et al. in 2023.

A. Key Mathematical Roles

Studies of betting market efficiency by Spann and Skiera show that skilled bettors seek arbitrage opportunities created by mispriced odds across sportsbooks, prompting operators to adjust prices or hedge positions to eliminate inefficiencies and prevent unmanaged risk from accumulating over time.

Online Casino / iGaming

Online casinos run on mathematics in games because every outcome needs to feel fair while still behaving predictably over time. Slots, table games, and live dealer formats all rely on controlled randomness that keeps the experience steady for the operator and unpredictable for the player.

A. Key Mathematical Roles

1. RNG (Random Number Generators)

Online casinos use random number generators to create outcomes that cannot be predicted or influenced.

• Secure pseudorandom generators, including Mersenne Twister and cryptographic systems, produce the sequences that drive slots and virtual table games.
• Uniform distributions, clean entropy sources, and strong seeding keep results from repeating or forming patterns.
• The RNG and fairness in online casino guide by ICDST explains how these systems stay secure and why cryptographic integrity matters for digital games.

2. Return to Player (RTP) & House Edge Calibration

RTP sets the long-run return that players receive, such as 95 or 96 percent, and becomes the anchor for a game’s financial behavior. Designers adjust paytables, symbol weights, and hit frequencies so the final RTP matches the goal for the product. These choices shape both volatility and how often bonus features appear during play.

3. Volatility / Variance Modeling

Variance shows how wide the swings in payouts can be, which helps operators manage cash-flow risk. High-volatility games create rare but large wins, while low-volatility games produce smaller payouts at a steady pace. Sessions are tested through Monte Carlo simulations to see how results behave across millions of spins.

4. Combinatorics & Probability for Game Outcomes

iGaming products rely on combinatorics to calculate how reel symbols, bonus triggers, and jackpots line up over time. These probabilities guide how often features appear and how the game feels to the player. They also help determine the expected return of bonus rounds, free spins, and gamble features.

5. Simulation & Statistical Testing

Simulation work helps teams see how a game behaves once it is played millions of times, not just in theory. These models show whether payouts, bonus features, and jackpots line up with the design goals. Statistical testing then confirms that the game stays fair and random before regulators approve it.

• Monte Carlo simulations look at long-run behavior such as bonus frequency, volatility patterns, jackpot growth, and return stability.
• These runs reveal issues that do not appear in small samples, including pacing problems, clustering effects, or payout drift over time.
• Fairness checks such as chi-square tests and other statistical randomness tests are required during certification and auditing, as outlined in Gaming Laboratories International’s technical standards for RNG and game testing (GLI-19), which define the statistical methods used to validate randomness and long-term outcome stability.

Gaming Laboratories International (GLI) – GLI-19: Interactive Gaming Systems Gaming https://gaminglabs.com/standards/gli-19

6. Player Value Modeling & Retention

Online casinos study player behavior to predict which users are likely to become high-value players and how long they might stay active. These forecasts use segmentation, CLV modeling, and statistical patterns that show who returns, how often they bet, and how their spending changes over time.

A clear example is a player who wagers 50 dollars per session and returns five times a week, which creates a predictable value profile that guides retention planning.

• Models help identify “whales” by tracking visit frequency, average stake, and long-term trends
• Reward systems, loyalty tiers, and free-play offers are adjusted to match each segment’s expected value
• Retention forecasts show when to offer support, bonuses, or reminders to keep active players engaged

7. Game Optimization (Balancing)

Teams use Game balancing math to adjust values like RTP, hit rate, and bonus frequency so the game feels rewarding without harming long-term profitability.

Advanced methods such as: evolutionary search, bandit algorithms, and reinforcement learning test thousands of parameter combinations, a process explained in the N-Tuple Bandit Evolutionary Algorithm paper on arXiv, which shows how automatic tuning improves game outcomes over repeated trials.

These tools work alongside real A/B tests that compare payout structures and feature setups, helping designers find a version that keeps players engaged while meeting business goals.

8. Risk & Liability Management

Online casinos track how jackpots, progressive pools, and bonus features build financial pressure as more players join. Probability models help teams see when exposure grows too quickly or when reserves need extra support.

Some operators hedge large jackpots or use insurance partners to steady their balance sheet during rare but expensive payouts.

9. Compliance & Provably Fair Systems

Blockchain casinos use cryptographic proofs to let players check that each result came from a fair RNG. These proofs rely on hash functions and modular arithmetic, which reveal if an outcome was altered at any point. Regulators then review the math behind RTP, volatility, and fairness to confirm the game meets approved standards.

10. Marketing Analytics & Bonus Economics

Marketing teams study how bonuses influence actual betting patterns so they can target players who bring long-term value. Models track redemption behavior, wager size after claiming an offer, and how often bonus abuse occurs.

These insights guide terms like wagering requirements and max cash-out limits so promotions stay profitable without hurting the player experience.

Mobile Video Gaming (Free-to-Play)

Mathematics in mobile games shapes how players spend, how long they stay, and how the game feels moment to moment. These numbers guide monetization, player segmentation, and the balance between challenge and reward.

Teams rely on Video game analytics to watch these patterns closely and adjust the game as players move through it.

A. Key Mathematical Roles

1. Economy Design and Balancing

Free-to-play economies use clear currency flows that decide how rewards enter and leave the game. Designers set upgrade costs and reward levels with formulas that shape how fast players advance. These choices depend on Game balancing math, since small changes can shift the entire pace of play.

2. Player Segmentation and LTV / CLV Modeling

Studios group players by spending level and activity so they can forecast Customer lifetime value video games more accurately. These groups show which players return often, who spends the most, and who may drop off early. Predictive models help identify high-value users by looking at early behavior such as purchase timing and session length.

3. A/B Testing and Experimental Design

Teams run controlled tests to compare prices, rewards, or new features before releasing them widely. Each version measures clear outcomes such as purchase rate, play time, or session count. Multi-armed bandit tests then adjust which version a player sees based on what produces stronger results.

4. Churn Prediction and Retention Analytics

Studios estimate churn by watching for clear drops such as a 40 percent cut in session time, longer gaps between logins, or fewer actions per session. These patterns often show when a player is likely to quit within the next 3 to 7 days.

Teams then send targeted messages or offers based on that risk level to bring the player back before they leave for good.

5. Dynamic Pricing and In-Game Offers

Some games adjust prices or offers based on a player’s history, such as past purchases or time spent in key features. These adjustments follow clear rules so the game remains fair while still reacting to individual behavior. Reinforcement learning setups can decide which offer appears next, using the player’s past actions as the deciding factor.

6. Reward / Drop Mechanics (Loot Tables)

Loot systems rely on simple probability rules that shape how items appear during play.

• Drop rates decide how common or rare each item is inside a loot table.
• Expected value checks keep rewards fair while preventing whales from breaking the economy.
• Rarity tiers help control progression so players earn meaningful items at the right pace.

7. Engagement & Session Modeling

Studios track how players enter, play, and return to the game to understand overall activity.

• Models estimate session length, return chance, and play frequency using methods like Markov or Poisson analysis.
• These patterns help forecast DAU, MAU, and long-term retention curves.
• Clear models guide decisions on pacing, content drops, and feature timing.

8. AI / NPC Behavior

NPC behavior in mobile games relies on clear math rules that shape how characters move and make choices.

• Pathfinding methods like A* and Dijkstra’s guide NPCs around obstacles, choose efficient routes, and react when the map changes.
• Decision trees and behavior trees control simple actions such as chasing, hiding, or switching attack patterns based on player movement.
• Some games use reinforcement learning to adjust difficulty, changing how tough an enemy feels as players improve, as described in the MoldStud overview of math in game development.

9. Procedural Content Generation

Procedural generation uses math tools like Perlin noise and cellular automata to build levels, terrains, and missions on the fly. These methods reduce development time while creating worlds that feel fresh each time a player returns. Difficulty and resource placement stay predictable because the random output follows controlled statistical patterns.

10. Fraud Detection & Monetization Risk

Mobile games watch for fraud by looking for patterns that real players do not produce, such as rapid currency gains or identical actions repeated across many accounts. These checks use simple outlier rules that flag accounts when their behavior sits far outside the normal range.

Take Note: Once flagged, teams measure how the fraud affects revenue so they can block the activity and protect long-term value.

Console / PC / Cloud / Live-Service Video Gaming

Large-scale games rely on mathematics in gaming to handle graphics, AI behavior, monetization, and server operations. These systems support everything from AAA visuals to long-term balance in live-service worlds. VR and AR fit into this same group because they use the same math foundations for motion, rendering, and interaction.

A. Key Mathematical Roles

1. Graphics and Rendering

Modern graphics depend on linear algebra to move objects, rotate models, and control camera angles. Vectors, matrices, and quaternions handle these transformations so scenes stay smooth while players move through the world.

Techniques such as real-time ray tracing and physically based rendering simulate the physical behavior of light to produce realistic lighting, shadows, and reflections in modern 3D games, helping developers achieve cinematic visuals while maintaining performance.

Cross-Cutting Themes (Mathematics Across All Sub-Industries)

Math shows up in every corner of the gaming world, even when the products look completely different. Casinos, sportsbooks, iGaming, mobile games, and AAA titles all rely on the same core ideas to stay fair, balanced, and profitable. These shared themes show  regardless of which part of the industry you work in.

Simulation and Monte Carlo Methods

Simulations help teams see how a system behaves after thousands or millions of runs instead of guessing from small samples. Casinos use them to measure long-term risk, while game studios use them to test balance and catch rare problems. The method works everywhere because it reveals patterns normal play would never show.

Probability and Expected Value

Probability shapes outcomes across gaming, from sports odds and slot payouts to loot drops and upgrade paths. Expected value helps teams understand the long-run impact of repeated actions, whether it is a bet, a reward, or a player purchase. These ideas give every sector a simple way to predict outcomes from uncertain events.

Data Science and Machine Learning

Data science helps teams understand who their players are and how they behave over time. Machine learning adds another layer by spotting patterns that guide offers, pricing, and fraud checks. These tools appear everywhere because they turn raw activity into decisions that actually move the business.

Game Theory

Game theory explains how players react when they can influence each other or the market around them. It shows up in betting pools, in-game markets, competitive modes, and pricing systems. These models help keep systems balanced even when player behavior shifts quickly.

Optimization and Algorithm Design

Optimization helps teams tune values such as difficulty, economy flow, and server usage without breaking the experience. Good algorithms improve performance and keep costs under control. Every part of the industry depends on this kind of tuning to keep products running smoothly.

Statistical Testing and Regulatory Compliance

Statistical tests confirm that RNG systems behave fairly and follow the distributions they were built to match. Regulators rely on these checks in gambling, and game studios use the same methods for stability and performance testing. It is the easiest way to make sure outcomes behave the way the math says they should.

Business Value / Executive Take-Aways

Math helps companies understand what drives revenue, where risks sit, and how players act. The next section breaks down how that supports executive decisions.

Revenue Growth

Math gives teams the ability to shape pricing, odds, and in-game economies in ways that increase revenue without damaging player trust. Predictive models also help identify which players have strong long-term value and which segments respond to specific offers. These insights raise overall revenue while keeping acquisition and retention costs under control.

• Better pricing decisions based on real spending patterns
• Higher returns from improved LTV and churn forecasts
• Less wasted spend in marketing and live-ops

Risk Reduction and Liability Management

Risk models help companies avoid the kinds of rare events that can disrupt a quarter’s financial results. As odds and payouts shift, these models keep exposure steady and reduce swings in performance. That stability gives leadership cleaner forecasts and fewer unexpected losses.

Customer Segmentation and Retention

Segmentation makes it easier to understand who the high-value players are and how to keep them engaged. These models identify whales, steady spenders, and at-risk users long before behavior changes become obvious. When teams act on these signals early, retention improves and lifetime value grows.

Compliance and Trust

Math-backed fairness systems help companies prove that outcomes are legitimate, which protects both reputation and regulatory standing. Clear probability checks, verified RNGs, and audit trails reduce the risk of fines or trust issues. In markets that allow it, transparency tools like provably fair systems can set a company apart.

• Stronger compliance posture with fewer regulatory issues
• Greater player confidence through verified fairness
• Competitive advantage when transparency becomes a selling point

Operational Efficiency

Forecasting models help teams plan server capacity, live-ops pacing, and cash needs before problems appear. Bonus programs and promotions also become more efficient when their performance is predicted rather than guessed. This steadier approach reduces operational waste and supports smoother growth.

Innovation and Product Differentiation

Math supports features such as procedural generation, adaptive AI, and automated tuning that give modern games their depth. These systems shorten development cycles while keeping quality high. Companies that use these tools often move faster and create products that feel more polished and competitive.

Technical Summary / For Mathematicians & Engineers

Here you’ll find the core math concepts and engineering workflows that sit behind game design, operations, and large-scale platforms. The goal is to show how technical teams apply these tools in real settings rather than as abstract theory.

Key Mathematical Disciplines Used

• Probability theory and statistics for modeling variance, expected value, and long-run behavior
• Combinatorics and permutations for card outcomes, loot tables, and event trees
• Simulation methods, including Monte Carlo, for stress testing and system modeling
• Optimization, evolutionary search, and reinforcement learning for tuning game parameters
• Linear algebra, numerical analysis, and geometry for graphics, physics, and rendering
• Cryptography and secure RNG design for fairness and anti-cheat systems
• Data science and predictive modeling for segmentation, retention, and forecasting

How Teams Are Typically Organized

• Game mathematicians or “math ops” roles inside iGaming firms responsible for RTP, volatility, and payout structures
• Data science and analytics teams in mobile and live-service studios focused on LTV, churn, and player behavior
• Quant and risk groups in sportsbooks handling odds, liability, and hedging
• AI and ML research engineers in AAA and cloud-gaming teams building agents, pathfinding, and adaptive systems

Tooling and Workflow

• Simulation frameworks written in Python, R, or C++, often combined with custom internal tools
• Statistical and machine-learning libraries such as scikit-learn, TensorFlow, and PyTorch
• Game engine integrations in Unity and Unreal for physics, AI behavior, and procedural systems
• Data pipelines using streaming platforms, warehousing layers, and large-scale telemetry systems
• RNG certification workflows supported by cryptographic libraries and formal fairness testing

Potential Risks, Challenges, & Ethical Considerations

As math becomes central to game design and operations, new risks and responsibilities emerge. The points below outline the areas where teams must stay careful, from compliance to player fairness.

Mathematical Risk vs Real-World Player Behavior

Mathematical models can miss how real players behave, especially when emotions or unexpected choices drive decisions. Overfitting past data can create systems that fail when rare events or high rollers appear. These gaps remind teams that models support decisions but should never replace real observation.

Regulatory and Compliance Risk

Weak RNG design or incomplete testing can expose a company to regulatory penalties and serious reputation damage. Regulators are also increasing scrutiny of loot boxes and other mechanics that resemble gambling. Teams must show clear math behind drop rates and fairness to avoid compliance issues.

Ethical Issues

Math can be used to identify high-value or vulnerable players, which raises concerns about how these groups are targeted. Personalization can improve the player experience, but it can also blur the line between engagement and exploitation. Companies must balance revenue goals with fairness and social responsibility.

Technical Debt and Complexity

Systems that depend on advanced AI, large-scale simulations, or procedural generation tend to grow more complex over time, increasing the effort required for testing, validation, and long-term maintenance

Each layer of complexity adds cost to testing, validation, and long-term updates. Keeping these systems reliable requires ongoing investment, not just strong initial design.

Conclusion

Mathematics in gaming has become a core enabler of revenue growth, risk control, and innovation across both gambling and video games. It now functions as a strategic asset, giving teams clearer decisions, steadier systems, and products that stay competitive over time.

For executives, this means investing in math talent and building data science into real business strategy, while technical teams benefit from deeper collaboration across design, engineering, and analytics to create systems that are both rigorous and aligned with actual business needs.

If you need to learn how mathematics can help your gaming project, check my page on mathematical modeling.

References

Betting Websites UK. (n.d.). The role of mathematics in gambling and problem gambling. https://www.bettingwebsites.org.uk/articles/betting-psychology/role-of-mathematics-in-gambling-and-problem-gambling/

CU Independent. (n.d.). Random number generators and fairness in online gambling. https://www.cuindependent.com/

Dmochowski, J. P. (2020). A statistical theory of optimal decision-making in sports betting. Journal of Risk and Financial Management, 13(11), 259. https://doi.org/10.3390/jrfm13110259

Forrest, D., McHale, I. G., & McAuley, K. (2008). Detecting insider trading and match fixing in betting markets. Journal of the Royal Statistical Society: Series A (Statistics in Society), 171(2), 465–497. https://doi.org/10.1111/j.1467-985X.2008.00547.x

Gandar, J. M., Zuber, R. A., & Dare, W. H. (2004). Forecasting betting volume and liquidity in sports wagering markets. Journal of Forecasting, 23(2), 159–175. https://doi.org/10.1002/for.873

Hegarty, T., & Whelan, K. (2023). Estimating expected loss rates in betting markets: Why bets lose more on average than you are warned (Working Paper). University College Dublin. https://www.karlwhelan.com/Papers/Overround.pdf

Hing, N., Russell, A. M. T., & Browne, M. (2017). Risk factors for gambling problems among sports bettors. Journal of Gambling Studies, 33, 1281–1304. https://oasis.library.unlv.edu/cgi/viewcontent.cgi?article=1511&context=grrj

IACSP. (n.d.). Casino mathematics: Understanding casino math. International Association of Certified Surveillance Professionals. https://iacsp.org/casino-mathematics-understanding-casino-math/

ICDST. (2025). How math shapes fairness in online casinos. https://icdst.org/blog/index.php/2025/04/02/how-math-shapes-fairness-in-online-casinos/

Kenka Oscar. (n.d.). How mathematics shapes modern game design. https://www.kenkaoscar.fi/how-mathematics-shapes-modern-game-design/

Levitt, S. D. (2004). Why are gambling markets organised so differently from financial markets? The Economic Journal, 114(495), 223–246. https://academic.oup.com/ej/article/114/495/223/5086012

MoldStud. (n.d.). The role of mathematics in game development. https://moldstud.com/articles/p-the-role-of-mathematics-in-game-development

Nair, H. S. (2014). Intertemporal price discrimination with forward-looking consumers: Application to the US market for console video games (Working paper). Wharton School, University of Pennsylvania. https://marketing.wharton.upenn.edu/wp-content/uploads/2016/10/Paper-Nair-Harikesh-03-06-2014.pdf

NVIDIA. (n.d.). Introduction to real-time ray tracing. https://developer.nvidia.com/rtx/ray-tracing

Perols, J., Xie, Y., & Huang, A. (2013). Predicting customer churn in the gambling industry. Decision Support Systems, 56, 458–469. https://www.sciencedirect.com/science/article/abs/pii/S0148296312003530

Renz, J., Ge, X., & Thielscher, M. (2016). Physics-based simulation games. Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning. https://users.cecs.anu.edu.au/~jrenz/papers/renz-ge-psg16.pdf

Vlastakis, N., Dotsis, G., & Markellos, R. N. (2009). How efficient is the European football betting market? Journal of Forecasting, 28(5), 426–444. https://doi.org/10.1002/for.1091

Zhang, Y., Li, X., & Wang, J. (2022). Predicting player churn in free-to-play mobile games using machine learning. Applied Sciences, 12(6), 2795. https://doi.org/10.3390/app12062795

Zhang, Y., Li, X., & Wang, J. (2024). Consumer behavior and adoption patterns in cloud gaming services. Journal of Business Research, 175, 114620. https://www.sciencedirect.com/science/article/pii/S0148296324006301