IDEAS home Printed from https://ideas.repec.org/a/gam/jstats/v5y2022i3p47-818d890803.html
   My bibliography  Save this article

Deriving the Optimal Strategy for the Two Dice Pig Game via Reinforcement Learning

Author

Listed:
  • Tian Zhu

    (Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA)

  • Merry H. Ma

    (Stony Brook School, 1 Chapman Pkwy, Stony Brook, NY 11790, USA)

Abstract

Games of chance have historically played a critical role in the development and teaching of probability theory and game theory, and, in the modern age, computer programming and reinforcement learning. In this paper, we derive the optimal strategy for playing the two-dice game Pig, both the standard version and its variant with doubles, coined “Double-Trouble”, using certain fundamental concepts of reinforcement learning, especially the Markov decision process and dynamic programming. We further compare the newly derived optimal strategy to other popular play strategies in terms of the winning chances and the order of play. In particular, we compare to the popular “hold at n” strategy, which is considered to be close to the optimal strategy, especially for the best n, for each type of Pig Game. For the standard two-player, two-dice, sequential Pig Game examined here, we found that “hold at 23” is the best choice, with the average winning chance against the optimal strategy being 0.4747. For the “Double-Trouble” version, we found that the “hold at 18” is the best choice, with the average winning chance against the optimal strategy being 0.4733. Furthermore, time in terms of turns to play each type of game is also examined for practical purposes. For optimal vs. optimal or optimal vs. the best “hold at n” strategy, we found that the average number of turns is 19, 23, and 24 for one-die Pig, standard two-dice Pig, and the “Double-Trouble” two-dice Pig games, respectively. We hope our work will inspire students of all ages to invest in the field of reinforcement learning, which is crucial for the development of artificial intelligence and robotics and, subsequently, for the future of humanity.

Suggested Citation

  • Tian Zhu & Merry H. Ma, 2022. "Deriving the Optimal Strategy for the Two Dice Pig Game via Reinforcement Learning," Stats, MDPI, vol. 5(3), pages 1-14, August.
  • Handle: RePEc:gam:jstats:v:5:y:2022:i:3:p:47-818:d:890803
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2571-905X/5/3/47/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2571-905X/5/3/47/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. John Jumper & Richard Evans & Alexander Pritzel & Tim Green & Michael Figurnov & Olaf Ronneberger & Kathryn Tunyasuvunakool & Russ Bates & Augustin Žídek & Anna Potapenko & Alex Bridgland & Clemens Me, 2021. "Highly accurate protein structure prediction with AlphaFold," Nature, Nature, vol. 596(7873), pages 583-589, August.
    2. Alan J. Brokaw & Thomas E. Merz, 2004. "Active Learning with Monty Hall in a Game Theory Class," The Journal of Economic Education, Taylor & Francis Journals, vol. 35(3), pages 259-268, July.
    3. David Silver & Aja Huang & Chris J. Maddison & Arthur Guez & Laurent Sifre & George van den Driessche & Julian Schrittwieser & Ioannis Antonoglou & Veda Panneershelvam & Marc Lanctot & Sander Dieleman, 2016. "Mastering the game of Go with deep neural networks and tree search," Nature, Nature, vol. 529(7587), pages 484-489, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bingren Chen, 2023. "Point Cloud Registration via Heuristic Reward Reinforcement Learning," Stats, MDPI, vol. 6(1), pages 1-11, February.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Patrick Bryant & Gabriele Pozzati & Wensi Zhu & Aditi Shenoy & Petras Kundrotas & Arne Elofsson, 2022. "Predicting the structure of large protein complexes using AlphaFold and Monte Carlo tree search," Nature Communications, Nature, vol. 13(1), pages 1-14, December.
    2. Zhenchong Mo & Lin Gong & Mingren Zhu & Junde Lan, 2024. "The Generative Generic-Field Design Method Based on Design Cognition and Knowledge Reasoning," Sustainability, MDPI, vol. 16(22), pages 1-34, November.
    3. Sun-Ting Tsai & Eric Fields & Yijia Xu & En-Jui Kuo & Pratyush Tiwary, 2022. "Path sampling of recurrent neural networks by incorporating known physics," Nature Communications, Nature, vol. 13(1), pages 1-10, December.
    4. Cui, Tianxiang & Du, Nanjiang & Yang, Xiaoying & Ding, Shusheng, 2024. "Multi-period portfolio optimization using a deep reinforcement learning hyper-heuristic approach," Technological Forecasting and Social Change, Elsevier, vol. 198(C).
    5. Min Yan & Can Huang & Peter Bienstman & Peter Tino & Wei Lin & Jie Sun, 2024. "Emerging opportunities and challenges for the future of reservoir computing," Nature Communications, Nature, vol. 15(1), pages 1-18, December.
    6. Ye Yuan & Lei Chen & Kexu Song & Miaomiao Cheng & Ling Fang & Lingfei Kong & Lanlan Yu & Ruonan Wang & Zhendong Fu & Minmin Sun & Qian Wang & Chengjun Cui & Haojue Wang & Jiuyang He & Xiaonan Wang & Y, 2024. "Stable peptide-assembled nanozyme mimicking dual antifungal actions," Nature Communications, Nature, vol. 15(1), pages 1-17, December.
    7. Ivica Odorčić & Mohamed Belal Hamed & Sam Lismont & Lucía Chávez-Gutiérrez & Rouslan G. Efremov, 2024. "Apo and Aβ46-bound γ-secretase structures provide insights into amyloid-β processing by the APH-1B isoform," Nature Communications, Nature, vol. 15(1), pages 1-14, December.
    8. Léon Faure & Bastien Mollet & Wolfram Liebermeister & Jean-Loup Faulon, 2023. "A neural-mechanistic hybrid approach improving the predictive power of genome-scale metabolic models," Nature Communications, Nature, vol. 14(1), pages 1-14, December.
    9. Xiaoyue Li & John M. Mulvey, 2023. "Optimal Portfolio Execution in a Regime-switching Market with Non-linear Impact Costs: Combining Dynamic Program and Neural Network," Papers 2306.08809, arXiv.org.
    10. Pedro Afonso Fernandes, 2024. "Forecasting with Neuro-Dynamic Programming," Papers 2404.03737, arXiv.org.
    11. Stella Vitt & Simone Prinz & Martin Eisinger & Ulrich Ermler & Wolfgang Buckel, 2022. "Purification and structural characterization of the Na+-translocating ferredoxin: NAD+ reductase (Rnf) complex of Clostridium tetanomorphum," Nature Communications, Nature, vol. 13(1), pages 1-11, December.
    12. Pierre Azoulay & Joshua Krieger & Abhishek Nagaraj, 2024. "Old Moats for New Models: Openness, Control, and Competition in Generative AI," NBER Chapters, in: Entrepreneurship and Innovation Policy and the Economy, volume 4, National Bureau of Economic Research, Inc.
    13. Riya Shah & Thomas C. Panagiotou & Gregory B. Cole & Trevor F. Moraes & Brigitte D. Lavoie & Christopher A. McCulloch & Andrew Wilde, 2024. "The DIAPH3 linker specifies a β-actin network that maintains RhoA and Myosin-II at the cytokinetic furrow," Nature Communications, Nature, vol. 15(1), pages 1-17, December.
    14. Yashan Yang & Qianqian Shao & Mingcheng Guo & Lin Han & Xinyue Zhao & Aohan Wang & Xiangyun Li & Bo Wang & Ji-An Pan & Zhenguo Chen & Andrei Fokine & Lei Sun & Qianglin Fang, 2024. "Capsid structure of bacteriophage ΦKZ provides insights into assembly and stabilization of jumbo phages," Nature Communications, Nature, vol. 15(1), pages 1-12, December.
    15. Anthony C. Bishop & Glorisé Torres-Montalvo & Sravya Kotaru & Kyle Mimun & A. Joshua Wand, 2023. "Robust automated backbone triple resonance NMR assignments of proteins using Bayesian-based simulated annealing," Nature Communications, Nature, vol. 14(1), pages 1-15, December.
    16. Bret M. Boyd & Ian James & Kevin P. Johnson & Robert B. Weiss & Sarah E. Bush & Dale H. Clayton & Colin Dale, 2024. "Stochasticity, determinism, and contingency shape genome evolution of endosymbiotic bacteria," Nature Communications, Nature, vol. 15(1), pages 1-14, December.
    17. Nathan Companez & Aldeida Aleti, 2016. "Can Monte-Carlo Tree Search learn to sacrifice?," Journal of Heuristics, Springer, vol. 22(6), pages 783-813, December.
    18. Jun-Yu Si & Yuan-Mei Chen & Ye-Hui Sun & Meng-Xue Gu & Mei-Ling Huang & Lu-Lu Shi & Xiao Yu & Xiao Yang & Qing Xiong & Cheng-Bao Ma & Peng Liu & Zheng-Li Shi & Huan Yan, 2024. "Sarbecovirus RBD indels and specific residues dictating multi-species ACE2 adaptiveness," Nature Communications, Nature, vol. 15(1), pages 1-15, December.
    19. Yuchen Zhang & Wei Yang, 2022. "Breakthrough invention and problem complexity: Evidence from a quasi‐experiment," Strategic Management Journal, Wiley Blackwell, vol. 43(12), pages 2510-2544, December.
    20. Deyun Qiu & Jinxin V. Pei & James E. O. Rosling & Vandana Thathy & Dongdi Li & Yi Xue & John D. Tanner & Jocelyn Sietsma Penington & Yi Tong Vincent Aw & Jessica Yi Han Aw & Guoyue Xu & Abhai K. Tripa, 2022. "A G358S mutation in the Plasmodium falciparum Na+ pump PfATP4 confers clinically-relevant resistance to cipargamin," Nature Communications, Nature, vol. 13(1), pages 1-18, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jstats:v:5:y:2022:i:3:p:47-818:d:890803. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.