Internal slackening scoring methods

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 a. 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 ¿^sup a^, consists of the limits of these processes. It is seen that ¿^sup 0^ extends the invariant scoring method, while ¿^sup 8^ extends the fair bets scoring method. Method ¿^sup 1^ corresponds with the existing ¿-scoring method of Borm et al. (Ann Oper Res 109(1):61-75, 2002) and can be seen as a compromise between ¿^sup 0^ and ¿^sup 8^. In particular, an explicit proportionality relation between ¿^sup a^ and ¿^sup 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.