Max-Flow Refinement in Multi-Player Strategies: An Edmonds-Karp Approach to Graph-Based Iterated Prisoners Dilemma

In many real-world scenarios involving multiple participants, a fundamental tension exists between individual interests and collective well-being. While self-interested behavior may maximize short-term personal gains, mutual cooperation often produces superior outcomes for all parties. Such situations, termed social dilemmas, are highly relevant in addressing issues like resource allocation, environmental preservation, and collaborative decision-making. Traditional frameworks like Iterated Prisoner Dilemma (IPD) have been extensively used to study these dynamics. Hybrid approaches, including Reinforcement Learning (RL) and strategies like Tit- for-Tat (TFT), have demonstrated success in promoting cooperation. However, direct reciprocity strategies often falter when interactions involve intermediaries or circular dependencies. To address this limitation, we build upon the Graph-based simulation of Iterated version of Prisoner’s Dilemma (GIPD), a model that represents player interactions as a weighted directed graph. In this work, we incorporate the Edmonds-Karp algorithm, a computationally efficient implementation of the Max-Flow problem, to analyze and enhance cooperation pathways within this framework. By testing this extended model across diverse scenarios, we evaluate its effectiveness compared to traditional and graph- based strategies. Our findings reveal that integrating Edmonds- Karp into the GIPD significantly improves the ability to identify and sustain cooperative relationships, providing new insights into fostering collaboration in complex, multi-agent systems. .