  Using the first three years of data from the Dark Energy Survey, we use
ratios of small-scale galaxy-galaxy lensing measurements around the same lens
sample to constrain source redshift uncertainties, intrinsic alignments and
other nuisance parameters of our model. Instead of using a simple geometric
approach for the ratios, we use the full modeling of the galaxy-galaxy lensing
measurements, including the corresponding integration over the power spectrum
and the contributions from intrinsic alignments and lens magnification. We
perform extensive testing of the small-scale shear ratio (SR) modeling by
studying the impact of different effects such as the inclusion of baryonic
physics, non-linear biasing, halo occupation distribution descriptions and lens
magnification, among others, and using realistic $N$-body simulations. We
validate the robustness of our constraints in the data by using two independent
lens samples, and by deriving constraints using the corresponding large-scale
ratios for which the modeling is simpler. The DES Y3 results demonstrate how
the ratios provide significant improvements in constraining power for several
nuisance parameters in our model, especially on source redshift calibration and
intrinsic alignments (IA). For source redshifts, SR improves the constraints
from the prior by up to 38\% in some redshift bins. Such improvements, and
especially the constraints it provides on IA, translate to tighter cosmological
constraints when SR is combined with cosmic shear and other 2pt functions. In
particular, for the DES Y3 data, SR improves $S_8$ constraints from cosmic
shear by up to 31\%, and for the full combination of probes (3$\times$2pt) by
up to 10\%. The shear ratios presented in this work are used as an additional
likelihood for cosmic shear, 2$\times$2pt and the full 3$\times$2pt in the
fiducial DES Y3 cosmological analysis.

Cosmology and Nongalactic Astrophysics