01 · From two to N
The same ratio, scaled up
Following Harsanyi's (1963) route from two-person to N-person bargaining, the chapter extends the power ratio directly: when a leader L is the sole individual exerting influence on the rest of a group, her bargaining power is her total inducing force over the sum of everyone's forces — the sum of her pushes on each of the N−1 others, against the sum of their resistances:
αL = ΣfL→k / (ΣfL→k + Σrk)
Power stays a number between 0 and 1, individually traceable per follower — which is exactly what makes the next two results visible.
02 · Horizontal
Power thins with group size
Conformity within a group falls as the group grows (Gerard, Wilhelmy & Conolley, 1968).
VB-ANBS offers a mechanism: for a leader to hold the same level of bargaining power over a larger group, her inducing force must spread evenly across its members — so the push each follower actually feels shrinks proportionally. In a coordination game where she prefers option 1 and her followers prefer option 2, that per-follower thinning makes it less likely the whole group lands on her option.
A public announcement to fifty people simply carries less force per ear than a word to five.
03 · Vertical
Power thins with distance down the hierarchy
Groups aren't only flat. In a "tall" hierarchy, L influences her follower j, who in turn influences his follower k (French Jr, 1956) — and we even represent power mentally as position on a vertical axis (Giessner & Schubert, 2007). Comparing L's power over a direct follower with her power over a follower one level removed, the chapter shows the individual at the higher level possesses less bargaining power over followers as the power distance between them increases.
Put it succinctly, the announcement fades not just across the room, but down the stairs.
Set aside, deliberately: coalitions — followers combining forces against a leader — are bracketed by the chapter as a separate issue. The panel's decay numbers are illustrative; the two negative associations are the chapter's.