The title may seem a bit counter-intuitive, but I can't get my head around this:
Almost Decisiveness: If, when all individuals in a (finite or infinite) group G prefer an alternative x to another alternative y, and other individuals (individuals in N\G) prefer y to x, the society prefers x to y, then G is almost decisive for x against y.
Decisiveness: If, when all individuals in a group G prefer x to y, the society prefers x to y regardless of the preferences of other individuals, then G is decisive for x against y.
So, in the definition of ADC, that sounds decisive to me, as they've basically decided what the group prefers, they aren't almost deciding. Maybe that's just poor naming. However it confuses me even more when the definition of DC says the group prefers x to y if G prefers x to y, regardless of the other individuals' opinions, because surely if everyone is against G in ADC, yet what G prefers the group prefers, then the group prefers what G prefers regardless of the preferences of the other individuals?
I'm not making myself very clear, I'm trying to say that in the definition of ADC, that seems like the worst case scenario for G when everyone else is against them, yet they still prevail. Surely then in ADC the society prefers what G prefers regardless of other individuals' preferences?
I'm hoping someone understands the point I'm trying to make here, your help would be appreciated!
You are assuming that extra support for x cannot turn x into a losing alternative. That is, you assume that all voting methods satisfy "monotonicity". However, there are methods, such as "plurality with elimination", where increased support for x can actually make x lose. See http://math.hawaii.edu/~marriott/teaching/summer2013/math100/violations.pdf for an example. So, "almost decisiveness" means decisive, except for perverse decision rules that violate monotonicity