Let this set be called V, and let it be almost decisive for x against y.
It is shown that given conditions C1, C2, and C3, there must be a lineage which is almost decisive over some ordered pair of alternatives.
Suppose that lineage J is almost decisive for some x against some y, i.e., there exists x, y in X such that A x, y).
A lineage J is picked out to denote A x, y) to mean that J is almost decisive for x against y, and denote D x, y) to mean that J is decisive for x against y.
If there is some individual J who is almost decisive for any ordered pair of alternatives, then a LTM satisfying conditions C1, C2, and C3 implies that J must be dominant.
But only the individual in V 1 ranks x over z, the rest rank z over x, so that a certain lineage has turned out to be almost decisive.
Since V is almost decisive for x against y, and since every linage in V ranks x higher than y, and every lineage not in V does the opposite, we must have xPy.
DEFINITION 2. A set of lineages V is almost decisive for x against y if xPy whenever xP iy for every i in V, and yP ix for every i not in V. DEFINITION 3. A set of lineages V is decisive for x against y if xPy when xP iy for every i in V.
"The economy is almost always decisive in general elections," says YouGov president Peter Kellner.
The appeal of armed resistance, being able to fight and not merely to talk, was almost certainly decisive.
