Application of Classical and Fuzzy Clustering to the Fair Division of Homogeneous Divisible Resources

Résumé

This thesis applies clustering, both classical and fuzzy, to fair division for homogeneous divisible resources, in situations where individuals have not contributed to the creation of the resource. On this basis, we designed an original fair division approach, APCR, which integrates classification into the redistribution process in order to reduce inequalities and improve the conditions of the poorest individuals. Within the APCR framework, we created and developed several allocation algorithms (PRRC/D, PRRC/F, PRRG/D, PRRG/F, PCO), which we applied and compared with existing rules P and PA. The JM index that we introduced proves in this thesis to be an essential tool for distributive justice. We also developed the MI algorithm (Index Method) as a social choice method that maximizes the sum of the values of the poorest individuals. The study shows that the PRRC/D allocation algorithm is the least unequal and therefore the most favorable to poor individuals according to the JM index. The index also reveals that our algorithms in a classical environment (PRRC/D, PRRG/D) are more favorable to the less well-off than those in a fuzzy environment (PRRC/F, PRRG/F). Clustering highlights that PRRC/D, PRRC/F, and PCO are closer to each other, while PRRG/D and PRRG/F are closer to traditional rules. Moreover, MI as a social choice method confirms the collective preference for PRRC/D. Finally, to handle larger datasets, we also developed a package and functions in R. One of these functions made it possible to automatically compute shares according to the PCO rule; the results obtained were identical to those determined manually, confirming the robustness, speed, and reliability of the computer implementation. These results highlight the originality of our contributions : the APCR, its allocation algorithms, the JM index, the MI method, and the development of software tools in R, which constitute scientific and practical advances in favor of a fairer, faster, and more automated redistribution of resources.

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