Abstract
Several real-life complex systems, like human societies or economic networks, are formed by interacting units characterized by patterns of relationships that may generate a group-based social hierarchy. In this paper, we address the problem of how to rank the individuals with respect to their ability to “influence” the relative strength of groups in a society. We also analyse the effect of basic properties in the computation of a social ranking within specific classes of (ordinal) coalitional situations. We show that the pairwise combination of these natural properties yields either to impossibility (i.e., no social ranking exists), or to flattening (i.e., all the individuals are equally ranked), or to dictatorship (i.e., the social ranking is imposed by the relative comparison of coalitions of a given size). Then, we turn our attention to an algorithmic approach aimed at evaluating the frequency of “essential” individuals, which is a notion related to the (ordinal) marginal contribution of individuals over all possible groups.
We are grateful to Hossein Khani for pointing out a mistake in the proof of Proposition 2. We also thank four anonymous referees for their valuable suggestions and comments on a former version of this paper. This work benefited from the support of the projects AMANDE ANR-13-BS02-0004 and CoCoRICo-CoDec ANR-14-CE24-0007 of the French National Research Agency (ANR).
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Notes
- 1.
Roughly speaking, two pairs of single elements i, j and p, q are said to be symmetric if, for coalitions S with the same cardinality, the number of times that \(S \cup \{i\}\) is stronger than \(S \cup \{j\}\) equals the number of times that \(S \cup \{p\}\) is stronger than \(S \cup \{q\}\), and the number of times that \(S \cup \{j\}\) is stronger than \(S \cup \{i\}\) equals the number of times that \(S \cup \{q\}\) is stronger than \(S \cup \{p\}\) (for more details, see Definition 3).
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Moretti, S., Öztürk, M. (2017). Some Axiomatic and Algorithmic Perspectives on the Social Ranking Problem. In: Rothe, J. (eds) Algorithmic Decision Theory. ADT 2017. Lecture Notes in Computer Science(), vol 10576. Springer, Cham. https://doi.org/10.1007/978-3-319-67504-6_12
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