## An R Package for Social Ranking

Teaser

This M.Sc. thesis was written under the joint supervision of Stefano Moretti, the Paris-Dauphine University, and Jochen Staudacher, Kempten University of Applied Sciences, in April 2022. The concept of social ranking solutions is introduced, giving way to the implementation of a software package aimed at better illustrating and solving these sets of problems. Several social ranking solutions from the literature are introduced and illustrated via simple examples as well as implemented in R language, providing a complete software package aimed at solving this kind of problems.

During this collaboration the project was presented at a conference at L’Université Paris-Dauphine on March 30, 2022 (https://sites.google.com/view/anr-themis/home-page/events?pli=1).

Short Description

In many areas of life we aim to rank elements. For example, we may be interested in individual performances based on their contributions to groups, be it voters in a political party, players in a football club or stockholders.

Cooperative game theory is widely known to tackle these sets of problems. These require each group (also coalition) to have a value assigned to them. More often than not however, the strengths of these coalitions are hardly quantifiable. Instead of conjuring up highly volatile evaluation methods, regarding the ordinal information may be of more use.

This concept has been introduced by Stefano Moretti in 2015[1]. Given a set of elements N and its power set $2^N$, can we construct a function that, from a given power relation over $2^N$, can produce a power relation over the set of N.

$f:{\mathcal{T}(2^N)} \rightarrow {\mathcal{T}(N)}$

The name “social ranking solution” is termed, and in the years following this publication several solutions have been proposed, such as the Ordinal Banzhaf Relation[2], Copeland-like and Kramer-Simpson-like method[3], Lexicographical-Excellence and Dual-Lexicographical-Excellence solution[4].

It is clear that this field of research would profit from a general purpose library, not only to help in the researching process, but also to reach a wider audience and to make it more accessible to newcomers. As such, this thesis aims to implement a flexible package in the versatile and popular R programming language.

Results

By the end of the thesis on April 19, 2022, the socialranking package was successfully implemented and published on the Comprehensive R Archive Network (https://CRAN.R-project.org/package=socialranking). The author continues his work on social rankings in a PhD program at the Paris-Dauphine University.

Author

Felix Fritz
Submitted on April 19, 2022
Email: felix.w.fritz@gmail.com

References

[1] Stefano Moretti. An axiomatic approach to social ranking under coalitional power relations. Homo Oeconomicus, 32(2):183–208, 2015.

[2] Hossein Khani, Stefano Moretti, and Meltem Öztürk. An Ordinal Banzhaf Index for Social Ranking. In 28th International Joint Conference on Artificial Intelligence (IJCAI 2019), pages 378–384, 2019.

[3] Encarnación Algaba, Stefano Moretti, Eric Rémila, and Philippe Solal. Lexicographic solutions for coalitional rankings. Social Choice and Welfare, pages 1–33, 2021.

[4] Giulia Bernardi, Roberto Lucchetti, and Stefano Moretti. Ranking objects from a preference relation over their subsets. Social Choice and Welfare, 52(4):589–606, nov 2018. doi: 10.1007/s00355-018-1161-1.