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Mathematics
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Applications of Intelligent Game and Reinforcement Learning
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Applications of Intelligent Game and Reinforcement Learning

A Special Issue of Mathematics (ISSN 2227-7390) belonging to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 30 April 2027 | Viewed by 6158

Editor

Dr. Yuxiang Sun

E-Mail Website
Guest Editor
School of Robotics and Automation, Nanjing University, Nanjing 210008, China
Interests: intelligent game simulation; multi-channel human–computer interaction (behavioral decision-making); intelligent decision-making (large models, reinforcement learning); human–computer collaborative decision-making

Special Issue Information

Dear Colleagues,

The convergence of intelligent game theory and reinforcement learning (RL) is revolutionizing decision-making systems across diverse fields. This Special Issue aims to showcase cutting-edge research and practical applications at this intersection. We invite submissions that explore how intelligent game models, powered by advanced RL algorithms, can solve complex, multi-agent problems in dynamic and competitive environments. Topics of interest include the integration of large language models for strategic reasoning, human–computer collaborative decision-making frameworks, multi-channel behavioral interaction analysis, and simulations for complex system management. We seek high-quality original research and review articles that demonstrate theoretical advances, novel methodologies, and innovative applications. The goal is to foster discussion on developing more adaptive, robust, and explainable intelligent systems for the future.

Dr. Yuxiang Sun
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • intelligent game theory
  • multi-agent reinforcement learning
  • decision-making systems
  • human–AI collaboration
  • large language models
  • behavioral simulation
  • Nash equilibrium
  • deep reinforcement learning
  • multi-agent systems
  • human–computer interaction

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Published Papers (4 papers)

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Research

24 pages, 10404 KB  
Article
Experience Extractor for Adaptive Tradeoff Between Exploration and Exploitation in Reinforcement Learning
by Zhi Yi, Zhongmin Wu, Yongming Xie, Ming Chen and Yinglong Dai
Mathematics 2026, 14(10), 1624; https://doi.org/10.3390/math14101624 - 11 May 2026
Viewed by 604
Abstract
In Reinforcement Learning (RL), the agent cannot distinguish between exploratory and exploitative experience. Not all sequential experiences contribute equally to the agent’s optimization, and the same experience holds different importance at different learning stages. We propose the Extractor for Adaptive Tradeoff Between Exploration [...] Read more.
In Reinforcement Learning (RL), the agent cannot distinguish between exploratory and exploitative experience. Not all sequential experiences contribute equally to the agent’s optimization, and the same experience holds different importance at different learning stages. We propose the Extractor for Adaptive Tradeoff Between Exploration and Exploitation (EATBEE), a task-oriented tool that tightly couples with the agent’s current knowledge and enables adaptive knowledge acquisition. We compare the originally sampled data with the task-driven data distribution to clearly illustrate their deviation. Then, we show how EATBEE identifies and extracts beneficial data for the agent. The monotonic improvement policy is theoretically validated under the assumption that the experience trajectory keeps a high degree of trajectory similarity after extraction. EATBEE serves as an independent module that can be seamlessly integrated with most existing RL algorithms. We substantiate the efficacy and practical applicability of the EATBEE method through experiments conducted in both discrete and continuous environments. Full article
(This article belongs to the Special Issue Applications of Intelligent Game and Reinforcement Learning)
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39 pages, 67440 KB  
Article
LLM-TOC: LLM-Driven Theory-of-Mind Adversarial Curriculum for Multi-Agent Generalization
by Chenxu Wang, Jiang Yuan, Tianqi Yu, Xinyue Jiang, Liuyu Xiang, Junge Zhang and Zhaofeng He
Mathematics 2026, 14(5), 915; https://doi.org/10.3390/math14050915 - 8 Mar 2026
Viewed by 1559
Abstract
Zero-shot generalization to out-of-distribution (OOD) teammates and opponents in multi-agent systems (MASs) remains a fundamental challenge for general-purpose AI, especially in open-ended interaction scenarios. Existing multi-agent reinforcement learning (MARL) paradigms, such as self-play and population-based training, often collapse to a limited subset of [...] Read more.
Zero-shot generalization to out-of-distribution (OOD) teammates and opponents in multi-agent systems (MASs) remains a fundamental challenge for general-purpose AI, especially in open-ended interaction scenarios. Existing multi-agent reinforcement learning (MARL) paradigms, such as self-play and population-based training, often collapse to a limited subset of Nash equilibria, leaving agents brittle when faced with semantically diverse, unseen behaviors. Recent approaches that invoke Large Language Models (LLMs) at run time can improve adaptability but introduce substantial latency and can become less reliable as task horizons grow; in contrast, LLM-assisted reward-shaping methods remain constrained by the inefficiency of the inner reinforcement-learning loop. To address these limitations, we propose LLM-TOC (LLM-Driven Theory-of-Mind Adversarial Curriculum), which casts generalization as a bi-level Stackelberg game: in the inner loop, a MARL agent (the follower) minimizes regret against a fixed population, while in the outer loop, an LLM serves as a semantic oracle that generates executable adversarial or cooperative strategies in a Turing-complete code space to maximize the agent’s regret. To cope with the absence of gradients in discrete code generation, we introduce Gradient Saliency Feedback, which transforms pixel-level value fluctuations into semantically meaningful causal cues to steer the LLM toward targeted strategy synthesis. We further provide motivating theoretical analysis via the PAC-Bayes framework, showing that LLM-TOC converges at rate O(1/K) and yields a tighter generalization error bound than parameter-space exploration under reasonable preconditions. Experiments on the Melting Pot benchmark demonstrate that, with expected cumulative collective return as the core zero-shot generalization metric, LLM-TOC consistently outperforms self-play baselines (IPPO and MAPPO) and the LLM-inference method Hypothetical Minds across all held-out test scenarios, reaching 75% to 85% of the upper-bound performance of Oracle PPO. Meanwhile, with the number of RL environment interaction steps to reach the target relative performance as the core efficiency metric, our framework reduces the total training computational cost by more than 60% compared with mainstream baselines. Full article
(This article belongs to the Special Issue Applications of Intelligent Game and Reinforcement Learning)
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28 pages, 3462 KB  
Article
Enhancing Multi-Agent Reinforcement Learning via Knowledge-Embedded Modular Framework for Online Basketball Games
by Junhyuk Kim, Jisun Park and Kyungeun Cho
Mathematics 2026, 14(3), 419; https://doi.org/10.3390/math14030419 - 25 Jan 2026
Cited by 1 | Viewed by 1722
Abstract
High sample complexity presents a major challenge in applying multi-agent reinforcement learning (MARL) to dynamic, high-dimensional sports such as basketball. To address this problem, we proposed the knowledge-embedded modular framework (KEMF), which partitions the environment into offense, defense, and loose-ball modules. Each module [...] Read more.
High sample complexity presents a major challenge in applying multi-agent reinforcement learning (MARL) to dynamic, high-dimensional sports such as basketball. To address this problem, we proposed the knowledge-embedded modular framework (KEMF), which partitions the environment into offense, defense, and loose-ball modules. Each module employs specialized policies and a knowledge-based observation layer enriched with basketball-specific metrics such as shooting success and defensive accuracy. These metrics are also incorporated into a dynamic and dense reward scheme that offers more direct and situation-specific feedback than sparse win/loss signals. We integrated these components into a multi-agent proximal policy optimization (MAPPO) algorithm to enhance training speed and improve sample efficiency. Evaluations using the commercial basketball game Freestyle indicate that KEMF outperformed previous methods in terms of the average points, winning rate, and overall training efficiency. An ablation study confirmed the synergistic effects of modularity, knowledge-embedded observations, and dense rewards. Moreover, a real-world deployment in 1457 live matches demonstrated the robustness of the framework, with trained agents achieving a 52.43% win rate against experienced human players. These results underscore the promise of the KEMF to enable efficient, adaptive, and strategically coherent MARL solutions in complex sporting environments. Full article
(This article belongs to the Special Issue Applications of Intelligent Game and Reinforcement Learning)
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23 pages, 7737 KB  
Article
Training Agents for Strategic Curling Through a Unified Reinforcement Learning Framework
by Yuseong Son, Jaeyoung Park and Byunghwan Jeon
Mathematics 2026, 14(3), 403; https://doi.org/10.3390/math14030403 - 23 Jan 2026
Viewed by 1469
Abstract
Curling presents a challenging continuous-control problem in which shot outcomes depend on long-horizon interactions between complex physical dynamics, strategic intent, and opponent responses. Despite recent progress in applying reinforcement learning (RL) to games and sports, curling lacks a unified environment that jointly supports [...] Read more.
Curling presents a challenging continuous-control problem in which shot outcomes depend on long-horizon interactions between complex physical dynamics, strategic intent, and opponent responses. Despite recent progress in applying reinforcement learning (RL) to games and sports, curling lacks a unified environment that jointly supports stable, rule-consistent simulation, structured state abstraction, and scalable agent training. To address this gap, we introduce a comprehensive learning framework for curling AI, consisting of a full-sized simulation environment, a task-aligned Markov decision process (MDP) formulation, and a two-phase training strategy designed for stable long-horizon optimization. First, we propose a novel MDP formulation that incorporates stone configuration, game context, and dynamic scoring factors, enabling an RL agent to reason simultaneously about physical feasibility and strategic desirability. Second, we present a two-phase curriculum learning procedure that significantly improves sample efficiency: Phase 1 trains the agent to master delivery mechanics by rewarding accurate placement around the tee line, while Phase 2 transitions to strategic learning with score-based rewards that encourage offensive and defensive planning. This staged training stabilizes policy learning and reduces the difficulty of direct exploration in the full curling action space. We integrate this MDP and training procedure into a unified Curling RL Framework, built upon a custom simulator designed for stability, reproducibility, and efficient RL training and a self-play mechanism tailored for strategic decision-making. Agent policies are optimized using Soft Actor–Critic (SAC), an entropy-regularized off-policy algorithm designed for continuous control. As a case study, we compare the learned agent’s shot patterns with elite match records from the men’s division of the Le Gruyère AOP European Curling Championships 2023, using 6512 extracted shot images. Experimental results demonstrate that the proposed framework learns diverse, human-like curling shots and outperforms ablated variants across both learning curves and head-to-head evaluations. Beyond curling, our framework provides a principled template for developing RL agents in physics-driven, strategy-intensive sports environments. Full article
(This article belongs to the Special Issue Applications of Intelligent Game and Reinforcement Learning)
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