What is Delta R (ΔR) and why is it important?

Delta R is the looseness in the system.
It is formally defined in ICC AC316 1.4.7 and is called Average Travel and Seating Increment.

1.4.7 Device Average Travel and Seating Increment ("ΔR") reads: ΔR is defined as the average of the movement required to cause incremental motion from a seated position and the opposite movement required to reseat the device after the actuation (or ratcheting). The movement causing actuation or ratcheting is defined as Δ1 and the seating movement for either continuously varying or incremental or ratcheting travel devices is defined as Δ2. Therefore ΔR= (Δ1+ Δ2)/2 for incremental or ratcheting travel devices, and ΔR= Δ2 for continuously varying travel devices. Testing or measuring for Δ1and Δ2 is defined in Section 4.3 of this criteria.
Download a PDF of AC316 from ICES site Released 07/01/2010 (590kb)

Delta R is the "lost motion" when the device reverses direction from advancing (as the building shrinks) to resisting a load (as during a seismic or wind event). Think "Backlash".

Delta R is important because during a wind or seismic event the whole structure undergoes repeated load reversals. Load reversals open and close any gaps or potential gaps causing dynamic impact forces each time the gap slams closed. Impact forces are notoriously difficult to calculate but are generally many times the static loads.

The AutoTight Catalog p5 has more information about how to calculate elongation including Delta R.
Catalog Page on Elongation

This paper describes the issue of Delta R in more depth:
"Tie-Down Systems: Critical Code Changes"
Download a PDF of this paper