The Government of India, through an order released on April 7, 2025, approved the merger of Rajasthan Marudhara Gramin Bank (RMGB) and Baroda Rajasthan Kshetriya Gramin Bank (BRKGB), effective from May 1, 2025, forming the new Rajasthan Gramin Bank under the “One State, One RRB” initiative. Sponsored by the State Bank of India, this strategic consolidation integrated over 1,593 branches across 41 districts, combining a business volume of nearly 1 trillion rupees. This study employs game theory to examine the strategic behavior surrounding the merger, emphasizing regulatory clarity, mutual trust, and the equilibrium outcomes among stakeholders. Drawing parallels from the successful 2017 SBI merger—which led to significant profitability and digital transformation—the Rajasthan Gramin Bank merger aims to replicate these benefits by unifying financial and technological resources. A key objective is to bridge the rural–urban divide and create an inclusive digital banking ecosystem using platforms like SBI’s YONO. Early developments point to strong digital integration and adaptive human resource strategies modeled after SBI’s post-merger reforms, enabling responsive, feedback-driven decision-making. Game-theoretic simulations suggest that cooperative approaches yield the Pareto-optimal outcomes, while risks such as asymmetric asset quality and weak signaling may trigger strategic defection or inefficiencies. This research underscores the vital role of strategic consolidation and game-theoretic frameworks in shaping resilient, inclusive, and efficient rural banking systems, offering valuable insights for policymakers and financial institutions aiming to enhance stability and economic outreach.

This paper analyzes the economic and environmental implications of mandatory recycling policies within a dynamic duopoly framework. Using a two-period Cournot model with identical firms producing a homogeneous good, we explore the impact of an exogenously imposed recycling rate on firms' production decisions, profits, consumer surplus, welfare, and environmental damage. In the first period, firms rely entirely on fresh inputs. In the second, they are required to recycle a portion of their initial production, though they may also continue using fresh inputs. Three types of equilibrium emerge, depending on the costs' parameters and the recycling rate. In equilibrium, firms always produce positive quantities in the first period. However, in the second period, either they both produce using a mix of fresh and recycled materials, both rely solely on recycled materials, or one firm uses both input types while the other uses only recycled materials. Notably, the analysis uncovers counterintuitive effects: stricter recycling regulations do not always reduce the use of fresh inputs or environmental harm and can even decrease economic welfare. These findings underscore the importance of carefully calibrating recycling policies, as overly stringent mandates can have unintended negative consequences. The model offers a novel theoretical lens on circular economy interventions, contributing to the broader debate on the design of sustainable production policies.

Introduction：It is well known that the pharmaceutical industry is crucial to human health as well as social well-being. This paper uses a differential game approach to investigate coordination strategies in China’s pharmaceutical supply chain under a two-invoice mechanism.

Methods：Using a differential game approach, this paper modeled a three-tier system to analyze dynamic interactions between R&D effort, logistics effort, market demand, and quality evolution. A state equation described drug quality evolution, influenced by R&D effort, logistics service effort, and natural decay rate. A market demand function (positively related to quality, negatively to retail price) and long-term profit functions (with quadratic convex costs for R&D and logistics) were established. Three scenarios were examined: centralized optimization, decentralized Stackelberg game, and a novel combination contract incorporating government subsidies and cost-sharing. For each scenario, optimal strategies, drug quality trajectories, and profits were derived using backward induction and solving differential equations. Finally, numerical experiments were conducted to verify the models by comparing key indicators.

Results：Drug quality converges to a steady state independent of initial levels. Outcomes vary significantly: centralized decision-making yields the highest quality and profit, followed by the combination contract, with decentralized decision-making the lowest.

Conclusions：This paper integrates pharmaceutical supply chain performance and drug quality with time by using differential games. The convergence of drug quality to a steady state indicates that long-term quality depends on decision behaviors, not initial conditions. The combination contract effectively coordinates decentralized decision-making, improving quality, efforts, and profits, making it more feasible.

The aim of this work is to apply the dynamic programming framework developed by S. Mirica in 2004 in order to solve the classical Dolichobrachistochrone differential game, which was originally formulated by R. Isaacs in 1965, in a rigorous manner. A key objective of this study is to identify the optimal feedback strategies for the first time, which is an original contribution that enhances the theoretical landscape of differential games. These strategies offer significant benefits, including the ability to dynamically optimise the system performance, allocate resources more effectively and achieve targets efficiently. Their inherent simplicity also allows for easier implementation and analysis, contributing to reduced computational complexity.

Our method is based on a refined version of Cauchy's method of characteristics, adapted to stratified Hamilton–Jacobi equations. This technique enables the characterisation of a broad class of the optimal trajectories and helps to define the domain of existence of the value function. To demonstrate the effectiveness of the proposed feedback strategies, we apply the well-known Verification Theorem for locally Lipschitz value functions as a sufficient condition for optimality.

Introduction:

Complex adaptive systems (CASs) consist of interacting adaptive agents whose collective behavior leads to emergent properties not found in individual components and not caused by external forces. Examples include the brain, immune system, and economy. Understanding such systems requires a holistic approach rather than traditional reductionism. Due to their memory and nonlocal behavior, fractional differential equations are well-suited for modeling CASs. However, studies on fractional-order CASs remain limited. In this paper, we investigate the existence, uniqueness, equilibrium points, and uniform stability of fractional-order differential equations in games with non-uniform interaction rates.

Method and results: Let exist and be bounded on D. Condition (2) implies that the functions fi satisfy the Lipschitz condition where and .

Theorem 1: (Existence and uniqueness) Let the assumptions (1)-(2) be satisfied. Then, the initial value problem has a unique solution .

Theorem 2: (Games with non-uniform interaction rates) Let . Then, the initial value problem has a unique solution .

Theorem 3: (Asymmetric games) The initial value problem with the initial data has a unique solution .

Conclusion: The results confirm solution reliability for diverse fractional game models with complex interaction structures.

In today’s polarized societies, individuals’ opinions evolve not only through rational deliberation and peer influence, but also as a result of psychological, social, and institutional costs that inhibit belief change. We introduce a novel continuum-trait partial differential equation (PDE) model to describe how opinions evolve across two ideological axes (e.g., economic and social views), capturing both spatial and temporal dynamics in a continuous opinion space.

Our model couples diffusion, drift due to cognitive/social movement costs, and nonlocal peer competition, yielding the evolution equation. We rigorously formulate this system with Neumann boundary conditions and prove well-posedness in the appropriate Sobolev spaces. We show that the system admits a Lyapunov (free-energy) functional, and its minimizers—representing evolutionarily stable opinion states—display spontaneous segregation into polarized clusters.

Our theoretical results demonstrate that the combination of rugged cost landscapes and peer crowding effects leads to persistent opinion fragmentation. This work bridges mathematical biology and social dynamics and offers new tools for analyzing the stability and evolution of ideologically divided societies using continuum PDE methods.

Introduction:
The integration of Cooperative Game Theory (CGT) into engineering education has increasingly emerged as a transformative approach to assessing collaborative problem-solving (CPS) competencies. Unlike traditional individual-based assessments, CGT fosters a cooperative learning environment by aligning individual incentives with collective goals, thereby promoting communication, critical thinking, and teamwork among students.

Methods:
This study explores the application of CGT-based evaluation models—such as CoGame and the Cobb–Douglas framework—within undergraduate engineering courses. A range of gamified tools, including intergroup competitions, commercial board games (e.g., Pandemic), and simulation-based activities, were implemented to measure CPS behaviors. Quantitative models were employed to track student engagement, individual contributions, and team dynamics, while reflective tools such as peer assessments and learning journals were used to validate learning outcomes.

Results:
The findings indicate that CGT-based assessments significantly improve academic performance and student engagement. Participants demonstrated enhanced CPS skills, particularly in task planning, communication, and collaborative execution. Behavioral indicators derived from gameplay data enabled real-time monitoring of group interaction and revealed the value of balancing individual and group evaluation. Notably, students exposed to gamified learning environments reported higher motivation and lower performance anxiety compared to those in traditional settings.

Conclusions:
Cooperative Game Theory provides a robust and flexible framework for evaluating collaborative problem-solving in engineering education. Its practical application not only enhances technical and soft skills but also supports fairer, more interactive assessment methods. While challenges remain in implementation and design complexity, CGT offers a promising pathway for cultivating essential 21st-century engineering competencies.

This paper presents a game theoretic model of primitive economic exchange, designed to capture strategic human behavior in pre-monetary societies. The model draws on insights from evolutionary theory, anthropology, and psychology to represent fundamental behavioral dimensions such as ethicality, selfishness, and forgiveness. It provides a framework to analyze how cooperation and reciprocity, considered to be core components of economic exchange, may have evolved and stabilized in small populations.

Understanding human behavior in such environments requires a synthesis of both the sciences and the social sciences. While disciplines like history, anthropology, and psychology offer detailed empirical accounts of human behavior, fields such as evolutionary biology and ecology provide insight into the origins of and adaptive logic behind such behaviors. This model integrates these perspectives to investigate how individuals might behave under conditions where reputation management, fairness, and social exchange are necessary for survival.

The game is structured around three core behavioral axes: Ethical vs. Unethical, Selfish vs. Unselfish, and Agreeable vs. Non-Agreeable. Payoff analysis suggests that a strategy profile characterized by ethicality, selfishness, and Non-Agreeableness emerges as evolutionarily stable. This outcome highlights the importance of maintaining a reciprocal balance, meeting one’s own needs, and selectively discriminating against exploiters in environments where repeat interactions are likely. In contrast, unethical or unfair strategies are only advantageous in isolated, one-shot exchanges, consistent with the literature on evolutionary games, human behavioral ecology, and anthropology.

This interdisciplinary approach offers three key contributions:

1. It is synthesized from underutilized but complementary theories across multiple disciplines.
2. It models human-specific behaviors with a theoretical grounding in established fields.
3. It produces results consistent with those of earlier game models, despite a more complex starting foundation.

Overall, the paper tries to demonstrate how evolutionarily consistent strategies for primitive exchange can be better understood through a unified model of decision-making grounded in both the natural and social sciences.

Abstract

This paper explores the intersection of game theory, evolutionary learning, and auction-based market design in digital environments. With the rapid expansion of e-commerce, NFTs, and decentralized finance platforms, auction systems have evolved from traditional formats into dynamic, gamified models that impact participant behavior and market efficiency. We analyze how learning algorithms embedded in auction games can simulate evolutionary behavior, leading to optimized bidding strategies and fairer market outcomes.

Drawing on experimental game simulations and real-world data from online auctions, we assess the effectiveness of various gamified features such as reputation systems, adaptive pricing, and incentive-driven participation in shaping user engagement and learning curves. The study also examines the implications of market design choices on long-term user behavior, particularly in environments where asymmetric information and strategic adaptation are prevalent. By blending behavioral economics with artificial intelligence and auction theory, the paper offers new insights into how digital auction platforms can be engineered to promote transparency, trust, and sustainability. These findings contribute to the design of next-generation online markets where learning and evolution are not by-products but central to user experience and economic efficiency.

Research suggests that large-scale human cooperation is driven by shared narratives that encode common beliefs and values. This study explored whether such narratives can similarly nudge large language model (LLM) agents toward collaboration through a (networked) finitely repeated public goods game after priming them with different stories.

Our method employed LLM agents playing repeated networked public goods games characterized by collective optimality, individual incentives, and iterative adaptation properties. We implemented two complementary variants: single-pool experiments used one shared pool to test the scaling effects across group sizes and robustness to defection, while multi-pool experiments introduced strategic complexity through overlapping pools. We manipulated the behavioral homogeneity through narrative priming, where agents received a story-based behavioral context via system prompts. The story corpus comprised eight cooperation-themed narratives emphasizing teamwork and collective benefits, plus four control conditions. Depending on the experimental condition, the agents either received identical stories (homogeneous condition) or randomly sampled distinct stories (heterogeneous condition).

The results demonstrate that story-based priming affects collaboration. Common stories improved collaboration and benefited all the participants, while priming using different stories reversed this effect, favoring self-interested agents. In homogeneous groups, using cooperation-themed stories achieved near-perfect collaboration scores, significantly outperforming the baseline controls. In heterogeneous groups, self-interested agents achieved the highest cumulative payoffs while cooperation-primed agents obtained the lowest returns. These patterns persisted across network sizes and structures in both single-pool and multi-pool architectures.

The findings reveal that narrative coherence among agents influences the viability of cooperation, with implications for multi-agent coordination and AI alignment in networked environments.

The code is available at https://github.com/storyagents25/story-agents.