Static Game Research Articles

A Nash equilibrium against gun control

This work constructs a non-cooperative, static game of gun control between the citizen and a pacifistic society characterised by law enforcement imperfection, by which the retention of firearms and the certitude of punishment against all crimes emerge both as a strict Nash equilibrium, in pure strategies, and as a strict dominant strategy equilibrium. The reason is that ratified by the Second Amendment to the American Constitution, discerning the necessity of a militia to the individual and societal security of a free state, by which the right of the people to keep and bear arms cannot be infringed.

Artificial Intelligence Usage in Forming Computer Games Space

The computer games industry is growing rapidly today, and the number of users of these games (players) is increasing. All of this has an impact on modern society. The development of new computer games involves constant evolution in such areas as technological innovations, the development of game genres, and analysis of market trends in the entertainment sector. The development of artificial intelligence and its application in the development of computer games contribute to the dynamic change of the landscape and expansion of the space and genres of games. The purpose of the article is to study and analyse the state of the modern computer games industry, taking into account the main genres and trends in the use of artificial intelligence in their development and the formation of the corresponding game space, to assess technological innovations, to analyse the market and to forecast the future development of the game industry. The research methods are the main methodological approaches and technological means of artificial intelligence for the development of computer games and the formation of the corresponding game space. Such methods include, in particular, systematic and comparative analyses to identify the features of computer games of various genres and types, the method of expert assessments, which involves the analysis of literary sources and information resources, interviews and surveys of game industry experts. The novelty of the study is a comprehensive analysis of modern technologies, in particular generative artificial intelligence, their use in the development of computer games and trends in the formation of the space of such games. Conclusions. The main trends in the development of computer games have been identified. The factors that influence their market are identified and forecasts are made for the future development of this industry. The paper analyses the use of artificial intelligence, in particular generative artificial intelligence, in the development of computer games (to optimise game creation and reduce budgets) and the formation of the corresponding game space, generative artificial intelligence has opened a new era of development of the computer game space. The impact of artificial intelligence on the balance between integration and immersion in the real world is determined. The current state of virtual reality games with their achievements and problems is analysed, which is crucial for understanding their future development trajectory. The new trends and technologies considered point to the future of the computer games space.

Performance Analysis in Chinese Female Club Soccer Based on Offensive and Defensive Technical-Tactical Indicators

Abstract Introduction. This study aimed to analyze how technical and tactical performance indicators differ by different match outcomes in women’s soccer and apply a comprehensive approach to match analysis. Material and Methods. The investigation analyzed many technical and tactical indicators of female club players from 28 matches in the Chinese Women’s Super League during the 2020 and 2021 seasons. These included both offensive and defensive indicators, such as goal kicks, clearances, interceptions, and more. Standard scores and T-scores were considered in illustrating the typical performance status of the team. In evaluating the technical and tactical performance indicators across various match outcomes, we applied non-parametric Kruskal-Wallis tests. This approach was taken because the data were found appropriate for the non-normal distribution, allowing the comprehensive detection of the differences across game states. Results. Kruskal-Wallis test-based statistical analysis showed that there is a substantial number of differences in team performance across slumped and high-performance phases within the games. The passable offensive indicators, such as shot frequency and forward passes, dropped significantly during the slump (p < 0.05). Amongst the defensive actions, there were significant increases. By contrast, in high-performance phases, offensive actions-shots and forward passes were significantly more frequent (p < 0.01), indicating dynamics in the change of the team approach according to the state of play. Conclusions. Our analysis showed that more offensive actions relate to a win in a match, while defensive action usually leads to a lost match. This result will offer valuable information to coaches for training and game strategies.

Nash-equilibrium strategies of orbital Target-Attacker-Defender game with a non-maneuvering target

This study investigates the orbital Target-Attacker-Defender (TAD) game problem in the context of space missions. In this game, the Attacker and the Defender compete for a Target that is unable to maneuver due to its original mission constraints. This paper establishes three TAD game models based on the thrust output capabilities: unconstrained thrust output, thrust constrained by an upper bound, and fixed thrust magnitude. These models are then solved using differential game theory to obtain Nash equilibrium solutions for the game problems, and the correctness and effectiveness of the solution methods are verified through simulations. Furthermore, an analysis of the winning mechanisms of the game is conducted, identifying key factors that influence the game’s outcomes, including weight coefficients in payoffs, the maximum thrust acceleration limit, and the initial game state. Considering the unique characteristics of space missions, a specific focus is given to the analysis of the Defender’s initial states in the hovering formation and in-plane circling formation, revealing overall success patterns for defense strategies from these two formations. In summary, this study provides valuable insights into the control strategies and winning mechanisms of orbital TAD games, deepening our understanding of these games and offering practical guidance to improve success rates in real-world scenarios.

Balancing and Analyzing Player Interaction in the ESG+P Game with Machinations

This work explores the balancing of an educational game to teach sustainable development in organizations by focusing on player interaction and employing strategies. Game success is a challenge that relies on balancing the relationships among its elements. Balancing is a complex process performed over multiple iterations, starting at game conception and continuing throughout development and testing stages. This work extends our previous case study, which did not consider player interaction for the game balancing. We built two models that contains all game mechanics using the Machinations framework. The first model includes elements that randomly produce, distribute, and consume resources, while the second model analyzes player interaction and implements four player strategies. We simulated these models in batch plays, analyzed game states, and adjusted game economies. The random model simulation achieved a victory rate of 40%, while the interactive model simulation with player strategies increased victory rates to values between 66% and 81%. These results show that player interaction and decision-making can be more decisive than randomness in achieving victory. Machinations contributed to enhancing the game, proved its usefulness for simulating complex models, and deepened our understanding of game dynamics, including player actions, potential deadlocks, and feedback mechanisms. This work supports other authors’ findings by demonstrating that balancing the game as early as possible in the development process, considering player interaction, makes the design feasible; and provides evidence that computer simulations, such as Machinations, benefit the game balance and improve the game design without the need to build a prototype and conduct extensive playtests.

Toward Automated Knowledge Discovery in Case-Based Reasoning

Automated Case Elicitation (ACE) enables case-based reasoning (CBR) systems to automatically acquire knowledge through real-time exploration and interaction with environments. CBR is an explainable AI methodology, where decisions are based on previous encounters. ACE combined with CBR continues learning as it is being deployed, and produces specific cases that can be reviewed by humans, unlike pretrained large language models (LLMs) that learn by training offline on prior data. ACE and CBR may be useful methods to gather training data for use with generative AI, or to help them to adapt on the fly. This research explores ACE's potential by applying it to chess and conducting extensive experiments against Stockfish, the world's highest rated chess engine. An ACE agent was developed that combines random exploration with shallow alpha-beta search for novel game states. Results over 1000+ games showed the ACE player defeated Stockfish in nearly 10% of games—a notable achievement given Stockfish's extreme strength. Notably, the ACE agent required only 0.1 seconds per game compared an average of 8 minutes for Stockfish, while still gradually improving its win rate through accrued experience. Detailed analyses revealed how the relaxation of ACE's case matching criteria along with selective retention of useful cases enabled accumulation of strategic chess knowledge. The research provides valuable insights into ACE's proficiency for knowledge discovery in complex, adversarial domains. It lays groundwork for integrating ACE, an unsupervised CBR learner, with modern deep learning techniques like neural networks and large language models to combine the strengths of symbolic and subsymbolic AI. By demonstrating ACE's ability to extract strategic knowledge against world-class opponents, this work highlights its potential for impact across gaming, autonomous systems, and other complex problem-solving domains.

Computation Offloading Based on a Distributed Overlay Network Cache-Sharing Mechanism in Multi-Access Edge Computing

Multi-access edge computing (MEC) enhances service quality for users and reduces computational overhead by migrating workloads and application data to the network edge. However, current solutions for task offloading and cache replacement in edge scenarios are constrained by factors such as communication bandwidth, wireless network coverage, and limited storage capacity of edge devices, making it challenging to achieve high cache reuse and lower system energy consumption. To address these issues, a framework leveraging cooperative edge servers deployed in wireless access networks across different geographical regions is designed. Specifically, we propose the Distributed Edge Service Caching and Offloading (DESCO) network architecture and design a decentralized resource-sharing algorithm based on consistent hashing, named Cache Chord. Subsequently, based on DESCO and aiming to minimize overall user energy consumption while maintaining user latency constraints, we introduce the real-time computation offloading (RCO) problem and transform RCO into a multi-player static game, prove the existence of Nash equilibrium solutions, and solve it using a multi-dimensional particle swarm optimization algorithm. Finally, simulation results demonstrate that the proposed solution reduces the average energy consumption by over 27% in the DESCO network compared to existing algorithms.

A Study of Quantum Game for Low-Carbon Transportation with Government Subsidies and Penalties

Traditional classical game theory struggles to effectively address the inefficiencies in subsidizing and penalizing the R&D and production of low-carbon transportation vehicles. To avoid the shortcomings of classic game theory, this research integrates quantum game theory with Nash games to explore the characteristics of automakers’ behavior for low-carbon transportation with government subsidies and penalties. We first constructed a low-carbon transportation game model between the government and automakers. Then, the optimal profit strategies for both parties in a quantum entangled state were analyzed. Finally, the impact of quantum superposition states and the initial entangled state on the profit strategies of both parties was simulated and analyzed using Monte Carlo simulations. We find that under the joint effects of government subsidies and penalties, quantum game states and the initial quantum entangled state have a crucial influence on the game’s outcomes. They can encourage the realization of Nash equilibrium and perfect coordination in the quantum game, significantly increasing the profits for both parties. This in turn effectively stimulates automakers to research and produce low-carbon transportation solutions, promoting the rapid development of low-carbon transportation technology. In theory, this research can enrich the Quantum game for improvements in the Nash equilibrium solution for the government to subsidize and penalize the low-carbon transportation problem. Meanwhile, in practice, it can provide guidance and reference in optimal strategy selection conditions for government policymakers and automakers for low-carbon transportation.

New characterizations of the Shapley value using weak differential marginalities

The principle of differential marginality for cooperative games states that the payoff differential of two players does not change whenever their productivity differential, measured by the differentials of their marginal contributions to coalitions containing neither of them, does not change. The hypothesis of differential marginality is satisfied if and only if the two players are symmetric in the difference of the two games under consideration. In this paper we introduce two weakened variants of differential marginality in which symmetric players in the difference of the two games are replaced by mutually dependent players and necessary players, respectively. By combining these weakened versions of differential marginality with the standard axioms: efficiency and either the null player out property or the disjointly productive players property, we provide two new characterizations of the Shapley value for the setting of variable player sets.

Socio-economic determinants of pricing water for domestic use: A comparison of blocks and linear tariffs

This article explores a model in which an agency can adopt different tariff schemes (block tariff or linear tariff) for selling water to a domestic buyer (a household), considering socio-economic factors like household size and income level. Through the formulation and examination of a static game involving a buyer and an agency, where the latter considers also the buyer’s welfare in its decisions, the socio-economic determinants significantly affect the optimal pricing within each of the two suggested tariff schemes. This analysis leads to the identification of conditions under which the agency deems a block tariff more advantageous compared to a linear tariff, as well as circumstances where adopting a linear pricing strategy emerges as the optimal approach. By introducing a (discrete-time) dynamic extension of the model, in which the agency is assumed to operate as a boundedly rational agent, we investigate the role of the altruism that the agency decides to adopt toward the buyer, observing that it ends up affecting the price dynamics (defined by a gradient-like mechanism). In addition, we apply the model to the real case of the Apulia region in Italy, comparing the unit price currently applied in the region with the results of applying the model in the Apulian territory.

Nash equilibrium selection using a hybrid two-player static game with trade-off ranking method

The paper aims to suggest the ranking of an optimal solution when there exists more than one Nash equilibrium in the game theory solution concept. Many studies tend to merge the game theory with the multi criteria decision-making (MCDM) method to cater the real-situation problems. In the paper, a novel hybrid non-cooperative static game in game theory is combines with the trade-off ranking (TOR) method in MCDM. The proposed hybrid method is used to rank multiple Nash equilibria concerning some criteria. The methodology for both static game and TOR method are explained in the paper. The game theory model used is a two-player non-constant-sum static game. The proposed methodology is tested using international cooperation in Iran. The result suggests the ranking of the combined strategies using the proposed method.

Quantal Response Equilibrium and Rationalizability: Inside the Black Box

This paper aims to connect epistemic and behavioral game theory by examining the epistemic foundations of quantal response equilibrium (QRE) in static games. We focus on how much information agents possess about the probability distributions of idiosyncratic payoff shocks, in addition to the standard assumptions of rationality and common belief in rationality. When these distributions are transparent, we obtain a solution concept called Δp-rationalizability, which includes action distributions derived from QRE; we also give a condition under which this relationship holds true in reverse. When agents only have common belief in the monotonicity of these distributions (for example, extreme value distributions), we obtain another solution concept called ΔM-rationalizability, which includes action distributions derived from rank-dependent choice equilibrium, a parameter-free variant of QRE. Our solution concepts also provide insights for interpreting experimental and empirical data.

Decision Making in Monopoly Using a Hybrid Deep Reinforcement Learning Approach

Abstract: This research introduces a novel approach to decision-making in the classic board game Monopoly by leveraging a hybrid deep reinforcement learning framework. Traditional methods for decision-making in Monopoly often rely on heuristic strategies or rule-based systems, which may lack adaptability and fail to capture the dynamic and complex nature of the game. In contrast, this study proposes a hybrid model that combines the strengths of deep learning and reinforcement learning to enable an agent to autonomously learn effective decision-making strategies through interactions with the Monopoly environment. The hybrid deep reinforcement learning model is designed to process and interpret the intricate game state representations, learning optimal decision policies over time. The incorporation of deep neural networks allows the model to capture complex patterns and dependencies in the data, while reinforcement learning enables the agent to iteratively improve its decision-making abilities through trial and error. The research focuses on developing a suitable reward structure and feature representation that enhances the model's ability to navigate the diverse scenarios encountered in Monopoly, ultimately aiming to achieve superior decision-making performance compared to traditional approaches.

Research of artificial intelligence in imperfect information card games

Artificial intelligence (AI) in games has advanced significantly, notably in perfect information games such as Go and Chess. Imperfect information games, in which participants do not have complete information about the game state, create more difficulties. They incorporate both public and private observations, where strategies must be improved to achieve a Nash equilibrium. This study investigates artificial intelligence and reinforcement learning approaches, in which agents learn to maximize future rewards through interactions with their surroundings. The paper then focuses on card game research platforms such as RLCard and OpenAI Gym. It gives a comprehensive summary of research in No Limit Texas Hold'em, a difficult two-player poker game with a large decision space. DeepStack and Libratus are successful systems that have attained expert-level and superhuman play, respectively. Pluribus, a superhuman artificial intelligence for six-player poker, and DouZero, a pure reinforcement learning technique for the multiplayer card game, DouDiZhu, are both investigated. Overall, this paper provides background information on reinforcement learning and imperfect information games, analyzes commonly used research platforms, evaluates the effectiveness of AI algorithms in various card games, and offers future research areas and directions.

Retracted: On Stackelberg Leader with Min-Max Followers to Solve Fuzzy Continuous Static Games

Retracted: On Stackelberg Leader with Min-Max Followers to Solve Fuzzy Continuous Static Games

Strategies to Mitigate Cutthroat Competition in E-Commerce: Insights from Game Theory

As e-commerce industries advanced irresistibly in recent years, some platforms chose to reduce prices again and again to seize the market, which lead to vicious competition and price war. So, this article tried to find out the best method for different companies to deal with price war. Based on several articles about how to avoid price war for enterprises, this article analyzes price war from the game theory perspective by establishing static game models to analyze the feasibility of each method. The result show that the main reason of this phenomenon is people are trapped in a prisoner's dilemma, and price war is an economic necessity behavior. As a result, government is suggested to establish market superior department and companies are suggested to produce differential products and cooperate with each other to deal with price war. For different kind of enterprises, oligopolies can produce products suit for different people and collaborate with influential companies in different areas while supply chain companies should reach a cooperation treaty with upstream and downstream enterprises achieve win-win cooperation.

Collusion, inattentive consumers and shrouded prices

We explore the incentives of firms to partition and shroud prices in a model where consumers may ignore a price component. In the static game shrouding can only arise under restrictive conditions if the market is sufficiently concentrated and the shrouded price component is sufficiently high. In the dynamic game we find strong pro-collusive effects of shrouding in that firms can more easily collude on shrouded prices than in the static game and collusion is also facilitated compared to standard Bertrand markets. Such collusive strategies are moderated if consumer learning is important.

Repeated foreign body ingestion as a dynamic game: A case report

In this case report, we correct the analysis of mitigating intentional foreign body ingestion offered by Sarah J. Diamond and Amnon Sonnenberg. Patients who notoriously swallow foreign objects generate significant economic costs to hospitals. In previous publications, it has been argued that hospitals might reduce the need for endoscopy by offering such patients a paid position to discourage foreign body ingestions. However, the game-theoretical analysis offered in the literature was based on a static game which did not justify the relevant equilibrium. To obtain the actual goal of keeping the patient away from foreign body ingestion, we consider a repeated game with an infinite horizon. We show that there exists a combination of strategies applied by the hospital and the patient that leads to a steady state in which the patient will be discouraged from foreign body ingestion.

Toward Project Portfolio Selection Applying Game Theory: A Colombian Case Study

The project portfolio selection process aims to maximize synergy among chosen projects while delivering significant value to the parent organization. Due to the complexity of this decision, several approaches to project portfolio selection have been developed. In this paper, we adopt a cooperative and static game with complete information and implement the Shapley Value to determine the optimal mix of projects in a case study of a Colombian project portfolio. The application of the Shapley Value demonstrates its equitable nature, as it fairly distributes benefits considering the number of coalitions and the agreements made by each project within these coalitions.

On the Structure of Feedback Potential Difference Games

A static potential game is a non-cooperative game for which there exists a fictitious function, also referred to as a potential function, whose optimizers provide a Nash equilibrium of the associated non-cooperative game. In this paper, we study non-zero-sum finite-horizon difference games with feedback information structure which admit a potential game structure. We provide conditions for the existence of an optimal control problem such that the optimal solution of this problem provides a feedback Nash equilibrium of the corresponding non-cooperative game. We specialize the obtained results to a linear-quadratic setting so as to obtain these verifiable existence conditions in terms of the problem data. Finally, we illustrate our results using a network flow control problem.