<p>We propose the Generalized Egalitarian Banzhaf value (GEBV), a noval solution concept for cooperative games with transferable utility, which integrates egalitarian principles with the classical notion of marginal contribution. Motivated by the need to balance equity and strategic power in coalition formation, the GEBV distributes the total value of the grand coalition using a weighted mechanism based on coalition sizes. This value generalizes the Egalitarian Banzhaf value by allowing a flexible, non-decreasing sequence of parameters enabling a smooth transition from purely egalitarian allocations in small coalitions to marginalist allocations in larger ones. We provide a rigorous axiomatic characterization of the GEBV, showing that it uniquely satisfies Dummy player in a non-negative environment property, Equal treatment, Linearity, and Super additivity. Further, we demonstrate its applicability through examples and compare it with existing values such as the classical Banzhaf value and the Egalitarian Shapley value. The GEBV is particularly suited for voting systems, corporate governance, and networked decision-making, where both fairness and individual influence must be respected. This work contributes to the broader literature by offering a unified and adaptable framework that bridges egalitarian and marginalist principles in cooperative game theory.</p>