Abstract
This research paper analyses three strategy games—Chess, Othello, and Poker—from the perspective of game theory. Mathematical modeling techniques are used to develop optimal strategies for each game to provide insights into their underlying principles and dynamics. The analysis reveals many similarities between the two-player abstract board game Chess and its modern variant Othello. It also highlights substantial differences compared to probability models explicitly developed for Poker. The comparative examination provides an understanding of how different elements contribute towards developing successful gaming behavior across various contexts, which can be valuable information for strategic thinking and making informed decisions while playing such games or conducting similar activities like sports betting or investing in stock markets. Furthermore, implications drawn from this study suggest potential directions for further development within the field, including enhanced mathematical modeling methodology and improvement upon players’ decision-processing functionality, particularly during real-time play scenarios.
Introduction
Game theory is a branch of mathematics concerned with studying participant interactions in situations where outcomes depend on multiple individuals’ collective choices (Colman, 2016).). It involves analyzing competitive scenarios involving two or more opponents with conflicting interests and determining strategic behavior based on predefined rules and objectives. The game theory finds applications in real-life scenarios such as economic markets, political elections, and popular Hollywood movies. This research paper explores the application of game theory in strategy games, with a primary focus on three variants: Chess, Othello, and Poker.
The analysis presented here adopts a mathematical modeling technique extensively used to develop optimal strategies in various gaming contexts. The methodology includes insights from probability theory, aiming to understand players’ decision-making processes during gameplay and enhance AI systems’ implementation with better accuracy levels in artificial intelligence topics. Additionally, the examination provides comparison results from different modeling techniques applied to these three variations, facilitating the identification of similarities and differences that surface when combined with human behavior elements in the gameplay. The implications drawn from this research offer valuable information for strategic thinkers and real-life sports betting and stock investing applications. The focus lies in understanding the underlying principles influencing decisions in different gaming contexts.
Literature Review
Over the years, researchers have conducted numerous studies on applying game theory to understand strategic behavior (Manshaei et al., 2013). These investigations have revealed patterns that govern players’ decision-making processes, influenced by their preferences, beliefs, and incentives at each level of play. Mathematical modeling techniques have been employed in these examinations, ranging from basic Decision Tree models to more complex Quantitative Game Theory approaches involving Integer Programming and Nash equilibrium values (Abedian et al., 2022). These techniques aim to develop tactical strategies that maximize rewards while minimizing risks within given constraints, providing valuable insights for research in this field.
This paper primarily focuses on three main areas: mathematics, computing science, and artificial intelligence systems. Mathematicians play a crucial role in interpreting and simulating games to reflect human behavior in real-life scenarios during gameplay accurately (Hamada & Sato, 2012). On the other hand, computer scientists contribute primary inputs in creating AI algorithms that efficiently detect patterns within datasets and implement strategies for survival and prediction of opponents’ potential moves based on probabilities and outcomes.
Regarding Chess, commonly employed mathematical models include the Minimax Decision Trees approach, which presumes perfect knowledge of opponents’ strategies (Saini, 2014). However, such assumptions only sometimes hold, as even expert-level gamers sometimes make strategic mistakes, leading to the development of advanced modeling approaches that involve probability theory and random variables to predict the chances of certain moves occurring. Probability Theory is applied to the variant of Othello, involving random elements that help players make tactical decisions with greater control over the game (Primanita et al., 2020). This approach also aids in identifying winning conditions more efficiently based on specific contexts experienced players have faced in the past due to their gained familiarity across multiple plays.
Lastly, in Poker, a detailed understanding of probability, Expected Value calculations, and the involvement of Monte Carlo Simulations play a significant role in analyzing the success of betting actions based on card rankings (Teófilo et al., 2013). This analysis is crucial, especially in pre-flop scenarios where community cards are not revealed yet, allowing better judgment over whether to fold or continue the round based on potential outcomes. Factorial Resolutions are also utilized to identify pattern combinations, helping understand the likelihood of a particular situation reoccurring in future circumstances.
Methodology
The research methodology draws primarily from two disciplines: mathematics and computing science. Mathematics is utilized to develop mathematical models for simulating gameplay, while computing science helps implement these models into playable systems (Kusuma et al., 2018). Game theory serves as the foundation, enabling the application of Decision Trees and Optimization processes to formulate strategies applicable across various gaming contexts of different complexities.
Decision Tree modeling involves structuring hierarchical diagrams to represent possible outcomes based on players’ decisions at different intervals during the game. This visualization allows gamers to react quickly to changing scenarios throughout their journey. For example, in Chess, the Minimax technique is applied to analyze opponents’ moves ahead, assuming perfect knowledge to formulate an efficient winning strategy (Balakrishnan et al., 2014). Integer Programming approaches generate Nash equilibrium values through optimization processes, facilitating informed decision-making from a cost-benefit analysis perspective, considering resource constraints and availability (Palafox-Alcantar et al., 2020).
Probability models become essential for analyzing gaming behavior and predicting the likelihood of certain events, using random variables to broaden insights beyond traditional mathematical techniques. They are handy for deciding the best response against unknown opponents in real-time situations (Silva et al., 2022). For instance, in Poker, standard deck playing cards are used. Players attempt to maximize their winnings through betting based on expected value calculations and Monte Carlo Simulations to approximate uncertain outcomes involved during gameplay.
The factorial resolution approach provides insights into interconnections, patterns, and fluctuations within the game setup, enabling proactive identification of potential setups and expanding the horizons of knowledge. This aids decision-making processes and allows for efficient strategies, saving valuable resources often wasted through inefficient decisions. The combination of robust modeling algorithms and effective computing hardware enables extensive A/B testing and simulations, leading to reliable results applicable for both AI purposes and experienced human minds (Salehi & Burgueño, 2018). The methodology adopted in this research ensures difficulty level control and provides valuable insights for strategy development in various gaming scenarios.
Mathematical Analysis of Games
The mathematical analysis of Chess, Othello, and Poker reveals various game similarities and differences. Starting with Chess, this two-player abstract strategy game is played on an 8×8 chessboard using sixteen pieces, each controlled by one player. The six unique piece types are King, Queen Rook, Bishop Knight & Pawns, which have special movement rules allowing varying tactics to be used across different levels, from beginners to grandmasters (Lai, 2015). Classical decision theory models such Minimax Decision Trees approach referencing back to the Ermelo theorem principal based upon the presumption that perfect knowledge about opponents’ strategies can be attained from the basis of primary modeling techniques utilized; however, due to certain complexities found in real-world gaming scenarios involving strategic blunders made even experienced players assumptions holding all times do not necessarily remain valid (Shuo Tan et al., 2021). Accordingly, more advanced modeling approaches employing Probability Theory applications whereby random variables involved turn out helpful while predicting the likelihood of particular moves being taken help improve analytical understanding within the gameplay, enabling the formulation of accurate tactical plans maximizing rewards.
Focusing on the modern-day variant – Othello, also known as Reversi – its basic principles are similar to those described earlier when examining the classic board game Chess (Hinebaugh, 2019). However, Othello heavily relies on probability models involving random variables to predict the chances of certain moves against different opponents and within various contexts, allowing experienced gamers more control over gameplay situations, especially when making tactical decisions based on prior analysis (Van Der Ree & Wiring, 2013). Identifying patterns and compositions, potential hand selections, and applying Expected Value calculations and Monte Carlo Simulations contribute to proactive decision-making during real-time play, optimizing success rates while minimizing risk exposure (Norelli & Panconesi, 2022). These mathematical modeling approaches are significant within this domain.
In the examination of the third variant – Poker – it is revealed that most of the action commonly occurs pre-flop before community cards are revealed. This highlights the importance of Probability Theory as the primary form of strategic reasoning (Johanson, 2013). Understanding the concept of expected returns and identifying potential hand selections throughout the simulation process is crucial for achieving a competitive edge among fellow participants and improving overall decision performance, enhancing analytical intelligence (Meyee et al., 2014). The Factorial Resolutions approach is crucial in facilitating computational learning and enhancing strategy development’s value. This is especially pertinent in mastering the contemporary rendition of Texas Hold’em, where the number of players can soar into the high double digits, leading to profound shifts in psychological dynamics and the overall gaming ambiance when juxtaposed with traditional, standardized Poker (Cacua et al., 2017).
The mathematical analysis of Chess, Othello, and Poker provides insights into how different elements contribute to successful gaming behavior across various contexts. These examinations demonstrate similarities in underlying principles among these games but also reveal differences in probability models that form the basis for understanding optimal strategies experienced gamers employ in specific playing scenarios. The analyses conducted through modeling techniques, such as Minimax Decision Trees, Expectation Value calculations, and Factorial Resolutions, have proven helpful in predicting outcomes within the gameplay, improving analytical skills, and developing tactical plans to maximize rewards and minimize risks under given constraints. This provides valuable information for strategic thinking and decision-making in sports betting and stock investing.
Comparative Analysis
The comparative analysis of Chess, Othello, and Poker reveals various similarities and substantial differences in the mathematical models used to derive optimal strategy. Both Chess and Othello rely heavily on abstract game theory techniques such as minimax decision trees which reference back to Zermello’s theorem principle; however, due to their randomness elements, these games contain probability calculations that are often applied too in order to gain a better understanding associated with moves being taken within given situations (AKinyemi, 2012).
In contrast, Poker is entirely based upon Probability Theory principles incorporating expected value & factorial resolution algorithms into playtime scenarios, thus requiring players to use extensive forecasting capabilities to predict the chances of sure hands appearing during showdown times, forming basic successful gaming practices across the board. These expectations may be further tweaked through the implementation of Monte Carlo simulations providing valuable insights and helping understand the degree of uncertainty the external environment can influence decisions based on the available information.
Furthermore, different gamers exhibiting varied playing styles tend to affect results obtained from modeling experiments conducted; hence its essential to pay attention to behavioral characteristics while constructing strategies aimed towards obtaining desired outcomes, especially when it comes to real-time online tournaments featuring multiple opponents running concurrently or even individual matches either against AI robots or human beings respectively. It should be noted that this study does not analyze any user-defined rulesets or custom modifications gameplays except those predefined across particular variants themselves, which stands true for all three games being compared hereupon; instead, its primary focus lies upon evaluation performance predictions made based on mathematical models implemented as part this research process.
All in all, it becomes clear that there exists a wide range of similarity between Chess & Othello owing to their abstract nature; however, these two gaming scenarios differs substantially from the probabilities calculated involved Poker, leading experts towards fine-tuning strategies designed taking into account elements related to behavioral biases exhibited by players during real-time conditions. Analyzing the given provides valuable insights for gamers and those interested in similar activities like sports betting and stock market investing.
Discussion
This research paper provides insight into how mathematical modeling can help us better understand strategy games based on examples from classic variants like Chess and modern-day versions, including Poker. By applying different types of modeling techniques, it becomes possible to analyze various elements contributing towards the development of successful gaming practices enabling players to judge situations they face prior to making decisions accordingly, adjusting their approach times the need arises, and adapting quickly to changing environments occur both physically and mentally.
The analysis reveals specific patterns that govern players’ decision-making, depending on their preferences, beliefs, and incentives at each level during gameplay, providing insights into underlying principles across different gaming contexts. For example, the Minimax Decisions Trees approach, which relies on perfect knowledge of opponents’ strategies according to Zermello’s theorem, may only sometimes be dependable since even skilled gamers can make mistakes or tactical errors. As a result, more sophisticated approaches that involve Probability Theory and random variables have been adopted to predict the likelihood of particular moves happening during gameplay (Colman, 2016). Expected value calculations and Monte Carlo Simulations are employed to explore various possibilities. Simultaneously, Factorial Resolutions aid in proactively identifying winning combinations, enhancing the comprehension of the probabilities of specific setups recurring in future situations.
The implications drawn from this study suggest potential directions for further development in the field, including enhanced mathematical modeling methodology and improvements in players’ decision processing during real-time play scenarios. Building computer algorithms to simulate gaming behavior would be valuable, providing beneficial simulation results and strategic insights for experienced gamers when making tactical decisions and coordinating moves across different levels. Additionally, exploring the impact of utilizing artificial intelligence (AI) systems on various aspects, such as opponents’ response speed and comprehension of complex strategies, is worthwhile.
Expanding the coverage to encompass more intricate scenarios involving multiple participants and simulating simultaneous interactions is an exciting aspect. This expansion is valuable for developing AI applications in larger gaming structures like tournament-style competitions. The current version of these applications may not offer comprehensive simulations that fully account for players’ decision-making processes, making it essential to focus on enhancing accuracy on the board. The research findings presented in this paper demonstrate that the strategic application of game theory effectively analyzes and comprehends strategy games. However, much remains concerning utilizing a wide array of artificial intelligence techniques.
Limitations and Future Work
The study’s limitations are evident as it only evaluates game theory applications in two-player abstract board and card games. While these variants offer insights into how mathematical modeling can enhance strategic thinking, further research is necessary to understand more complex gaming situations involving multiple participants, such as tournament-style competitions. Improvements must be made to achieve comprehensive simulations that consider players’ decision-making processes and increase accuracy in predicting outcomes during gameplay.
Addressing these shortcomings involves potential directions for future work in the field. Enhanced modeling techniques capable of handling large datasets within specified time frames should be developed to ensure a correct representation of inputs fed into the system. Refining Artificial Intelligence systems used in conjunction with models can improve their ability to process decisions quickly and accurately based on contextual conditions during various game stages, from pre-flop to showdown and post-game.
Moreover, the widespread availability of classic Poker variants in numerous casinos worldwide encourages researchers to conduct tests in remote environments, aiming to achieve optimized solutions for problem statements within defined constraints. Further investigations should involve implementing Factorial Resolutions and Probability Theory applications in more generalized environments due to technological advancements, allowing the prediction of human behavior in different contexts. This will significantly contribute to the field’s development over time.
Conclusion
This study provides valuable insights into game theory from a mathematical perspective when applied to strategy games, focusing on three variants – Chess, Othello, and Poker. It reveals significant similarities between the two-player abstract board game Chess and its modern variant Othello. However, substantial differences emerge when implementing Probability Theory-based models specifically for Poker’s playtime scenarios. Through applying different modeling techniques, the analyses prove helpful in predicting outcomes during real-time situations, improving analytical skills, and developing tactical strategies to maximize rewards and minimize risks within defined constraints. The comparative examination ultimately offers a clear understanding of how various elements contribute to forming successful gaming behavior, enabling informed decisions by both gamers and in activities like sports betting and stock investing.
References
Abedian, M., Amindoust, A., Maddahi, R., & Jouzdani, J. (2022). A Nash equilibrium based decision-making method for performance evaluation: a case study. Journal of Ambient Intelligence and Humanized Computing, 13(12), 5563-5579. https://link.springer.com/article/10.1007/s12652-021-03188-8
AKinyemi, I. O. (2012). A Refinement-Based Heuristic Method for Decision Making in the Context of Ayo Game (Doctoral dissertation, Covenant University). https://www.semanticscholar.org/paper/A-Refinement-Based-Heuristic-Method-for-Decision-in-Akinyemi/26537be9a2af996eae0b18edba6972c71301af00
Balakrishnan, S., Shunmuganathan, K. L., & Sreenevasan, R. (2014). Amelioration of Artificial Intelligence using Game Techniques for an Imperfect Information Board Game Geister. International Journal of Applied Engineering Research (IJAER), 9(22), 11849–11860. https://www.researchgate.net/profile/Balakrishnan-S-3/publication/289050582_Amelioration_of_artificial_intelligence_using_game_techniques_for_an_imperfect_information_board_game_geister/links/56a35a8f08ae232fb2056ea2/Amelioration-of-artificial-intelligence-using-game-techniques-for-an-imperfect-information-board-game-geister.pdf
Cacua, K., Buitrago-Sierra, R., Herrera, B., Chejne, F., & Pabón, E. (2017). Influence of different parameters and their coupled effects on the stability of alumina nanofluids by a fractional factorial design approach. Advanced Powder Technology, 28(10), 2581-2588. https://doi.org/10.1016/j.apt.2017.07.009
Colman, A. M. (2016). Game theory and experimental games: The study of strategic interaction. Elsevier.
Gainsbury, S. M., Suhonen, N., & Saastamoinen, J. (2014). Chasing losses in online Poker and casino games: Characteristics and gameplay of Internet gamblers at risk of disordered gambling. Psychiatry Research, 217(3), 220-225. https://www.sciencedirect.com/science/article/abs/pii/S0165178114002601#:~:text=https%3A//doi.org/10.1016/j.psychres.2014.03.033
Hamada, M., & Sato, S. (2012). A learning system for a computational science-related topic. Procedia Computer Science, 9, 1763-1772. https://doi.org/10.1016/j.procs.2012.04.194
Hinebaugh, J. P. (2019). More Board Game Education: Inspiring Students Through Board Games. Rowman & Littlefield.
Johanson, M. (2013). Measuring the size of large no-limit poker games. arXiv preprint arXiv:1302.7008.
https://doi.org/10.48550/arXiv.1302.7008
Kusuma, G. P., Wigati, E. K., Utomo, Y., & Suryapranata, L. K. P. (2018). Analysis of gamification models in education using MDA framework. Procedia Computer Science, 135, 385-392. https://doi.org/10.1016/j.procs.2018.08.187
Lai, M. (2015). Giraffe: Using deep reinforcement learning to play Chess. arXiv preprint arXiv:1509.01549. https://doi.org/10.48550/arXiv.1509.01549
Manshaei, M. H., Zhu, Q., Alpcan, T., Bacşar, T., & Hubaux, J. P. (2013). Game theory meets network security and privacy. ACM Computing Surveys (CSUR), 45(3), 1–39. https://doi.org/10.1145/2480741.2480742
Meyer, G., von Meduna, M., Brosowski, T., & Hayer, T. (2013). Is Poker a game of skill or chance? A quasi-experimental study. Journal of Gambling Studies, 29, 535-550. https://link.springer.com/article/10.1007/s10899-012-9327-8
Norelli, A., & Panconesi, A. (2022). Olivaw: Mastering Othello without human knowledge nor a penny. IEEE Transactions on Games. DOI: 10.1109/TG.2022.3157345
Palafox-Alcantar, P. G., Hunt, D. V. L., & Rogers, C. D. F. (2020). The complementary use of game theory for the circular economy: A review of waste management decision-making methods in civil engineering. Waste Management, 102, 598-612. https://doi.org/10.1016/j.wasman.2019.11.014
Primanita, A., Khalid, M. N. A., & Iida, H. (2020). Characterizing the nature of probability-based proof number search: A case study in the Othello and connect four games: information, 11(5), 264. DOI 10.2991/aisr.k.200424.075
Raybourn, E. M. (2014). A new paradigm for serious games: Transmedia learning for more effective training and education. Journal of computational science, 5(3), 471–481. https://doi.org/10.1016/j.jocs.2013.08.005
Saini, S. S. (2014). Mimicking human player strategies in fighting games using game artificial intelligence techniques (Doctoral dissertation, Loughborough University). https://core.ac.uk/download/pdf/288377509.pdf
Salehi, H., & Burgueño, R. (2018). Emerging artificial intelligence methods in structural engineering. Engineering structures, 171, 170-189. https://www.sciencedirect.com/science/article/abs/pii/S0141029617335526
Shuo Tan, Y., Agarwal, A., & Yu, B. (2021). A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds. arXiv e-prints, arXiv-2110. https://proceedings.mlr.press/v151/shuo-tan22a.html
Silva, W. N., Henrique, L. F., Silva, A. D. C., Dias, B. H., & Soares, T. A. (2022). Market models and optimization techniques to support the decision-making on demand response for prosumers. Electric Power Systems Research, 210, 108059. https://doi.org/10.1016/j.epsr.2022.108059
Teófilo, L. F., Rossetti, R., Reis, L. P., Cardoso, H. L., & Nogueira, P. A. (2013). Simulation and performance assessment of poker agents. In Multi-Agent-Based Simulation XIII: International Workshop, MABS 2012, Valencia, Spain, June 4-8, 2012, Revised Selected Papers 13 (pp. 69-84). Springer Berlin Heidelberg. https://link.springer.com/chapter/10.1007/978-3-642-38859-0_6
Van Der Ree, M., & Wiering, M. (2013, April). Reinforcement learning in the game of Othello: Learning against a fixed opponent and learning from self-play. In 2013 IEEE symposium on adaptive dynamic programming and reinforcement learning (ADPRL) (pp. 108-115). IEEE. https://ieeexplore.ieee.org/abstract/document/6614996
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