### Abstract

We deal with the ranking problem of the nodes in a directed graph. The bilateral relationships specified by a directed graph may reflect the outcomes of a sport competition, the mutual reference structure between websites, or a group preference structure over alternatives. We introduce a class of scoring methods for directed graphs, indexed by a single nonnegative parameter α. This parameter reflects the internal slackening of a node within an underlying iterative process. The class of so-called internal slackening scoring methods, denoted by λ

^{α}, consists of the limits of these processes. It is seen that λ^{0}extends the invariant scoring method, while λ^{∞}extends the fair bets scoring method. Method λ^{1}corresponds with the existing λ-scoring method of Borm et al. (Ann Oper Res 109(1):61–75, 2002) and can be seen as a compromise between λ^{0}and λ^{∞}. In particular, an explicit proportionality relation between λ^{α}and λ^{1}is derived. Moreover, the internal slackening scoring methods are applied to the setting of social choice situations where they give rise to a class of social choice correspondences that refine both the Top cycle correspondence and the Uncovered set correspondence.Original language | English |
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Pages (from-to) | 445-462 |

Journal | Theory and Decision |

Volume | 72 |

Issue number | 4 |

Publication status | Published - 2012 |

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## Cite this

Slikker, M., Borm, P. E. M., & van den Brink, R. (2012). Internal slackening scoring methods.

*Theory and Decision*,*72*(4), 445-462. http://www.springerlink.com/content/y637147880857427/