Internal slackening scoring methods

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 scoringmethods, denoted by λα, 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 λα and λ1 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. © The Author(s) 2011.