A stability index for local effectivity functions

The structure of unstable power mechanisms

“…The present study extends the theory and the results of Abdou (2010). In that study, the scope was limited to coalitional power distributions.…”

“…2.3), one has to check whether the definition of Abdou (2010) is consistent with the one given here. Indeed, one can verify that any cycle in the sense of Abdou (2010) gives rise to a cycle of the same order in the sense of definition 3.2, and conversely. It follows that the stability index defined here extends the stability index defined in that paper.…”

“…Remark 3.5 In the context of local effectivity functions, a stability index has been defined in Abdou (2010). This index is based on the notion of cycles introduced in definition 3.1 of that paper.…”

“…An instability type would determine the general features along which instability is likely to emerge and, consequently, the lines along which the society is likely to split. This idea applied to local effectivity functions has led to the definition of a stability index (Abdou 2010). We extend that definition to interaction forms.…”

“…The stability index is related closely to the Nakamura number for simple games (Nakamura 1979), the difference being that the Nakamura number is defined on the winning coalition structure only, whereas the stability index depends on the whole interaction form. In Abdou (2010), it is shown that, in case of instability, the index of a maximal local effectivity function is either 2 or 3. Here, even for maximal interaction forms, the index can take any value between 2 and the cardinality of the alternative set.…”