A GENERALIZED (alpha, beta) NORMALIZATION FRAMEWORK FOR SUPER-ADDITIVE GAMES WITH APPLICATIONS TO THE RANK-SHAPLEY VALUE

Resumen

Normalization in super-additive cooperative games is a process that transforms a game into an equivalent representation while preserving its core strategic and structural properties. This paper introduces a generalized normalization framework that unifies and extends existing methodologies by defining a flexible family of normalization schemes. The proposed approach involves specifying and estimating the boundary (transformation) parameters to ensure that all coalition values are mapped within predefined bounds. This transformation enables the adjustment of coalition values through either compression or expansion, depending on the original distribution and the desired normalization limits. The framework is designed to be broadly applicable across all super-additive games and maintains essential game-theoretic properties, including transferable utility invariance, player symmetry, and the inessential nature of the original game. These preserved characteristics are fundamental to ensuring that the normalized game retains the strategic equivalence of the original formulation. By offering a unified structure, the framework facilitates consistent analysis and comparison across diverse cooperative scenarios. A practical application is presented to demonstrate the effectiveness of the normalization procedure, illustrating how coalition values can be systematically adjusted to fit within specified constraints. This example underscores the utility of the framework in real-world cooperative settings, particularly in contexts involving resource allocation and strategic decision-making. The generalized normalization scheme proposed in this study provides valuable insights and tools for researchers and practitioners seeking to model, analyze, and optimize cooperative interactions in a mathematically rigorous and operationally meaningful way.