## 1 Introduction  
Federated Learning (FL) has emerged as a transformative paradigm for collaborative machine learning (ML), enabling a network of distributed clients such as smartphones, edge devices, and IoT nodes to train a shared global model without sharing their raw data [1]. This decentralized approach offers substantial privacy advantages but introduces new challenges in managing system resources and client diversity. A core challenge in FL is the efficient selection of participating clients at each communication round, as this decision directly affects model convergence, battery consumption, and privacy leakage [2]. In practical deployments, FL systems often consist of highly heterogeneous devices: resource-rich smartphones, latency-sensitive edge devices, and battery-constrained IoT sensors. These devices vary not only in computational power and battery capacity but also in privacy sensitivity and data quality. As a result, client selection becomes a multi-objective decision-making problem, where a trade-off must be struck between model accuracy, battery sustainability, and privacy preservation. The multi-objective client selection problem in FL requires not only rigorous theoretical modeling but also careful consideration of real-world system constraints. These include managing energy consumption on battery-powered devices, preserving user privacy, and adapting to the heterogeneity of modern IoT and edge computing environments. By combining game-theoretic concepts with these practical challenges, our approach delivers a scalable and adaptive solution that bridges theoretical foundations with deployment realities. This integration ensures that FL can be effectively implemented in diverse, resource-constrained settings, moving beyond purely conceptual frameworks toward impactful, operational systems. Moreover, our strategy accounts for realistic client diversity and operational variability, fostering solutions that are both theoretically sound and practically viable. This comprehensive perspective enables the design of mechanisms that meet multiple objectives simultaneously (accuracy, sustainability, and privacy) reflecting the complex demands of modern distributed learning applications. Prior works typically optimize for a single objective, for example selecting clients solely based on data quality [3] or battery availability [4,5], and therefore fail to adapt to the complex, multi-objective nature of real-world FL scenarios. To address this limitation, we propose a centralized reward-based payoff strategy (RBPS) with cooperative intent for client selection. RBPS formulates the selection process as a cooperative game, where clients collectively align with a system-wide objective by optimizing a shared reward function. Using the principles of Nash Equilibrium (NE), RBPS balances three critical metrics: model accuracy, battery level, and privacy sensitivity. Each client evaluates a reward function that integrates these factors, with adaptive weights assigned according to its device type (smartphone, edge, or IoT). Unlike prior approaches that treat clients as passive participants or optimize one dimension at a time, RBPS is implemented as a centralized strategy with cooperative intent, dynamically adapting to client contexts. In each round, clients with favorable reward scores are selected to participate and receive the updated model, while non-selected clients remain idle to preserve battery life and privacy budgets. Our RBPS strategy is designed to reflect real-world deployment constraints. We simulate a heterogeneous pool of devices with varying levels of battery, computation, and privacy settings, and evaluate the strategy using four benchmark datasets of increasing complexity: MNIST, EMNIST, Fashion-MNIST, and CIFAR-10. Clients are drawn from realistic distributions, including smartphones (high accuracy, moderate privacy, and high battery consumption), edge devices (balanced capabilities), and IoT sensors (low battery consumption and high privacy risk).