  Any successful model of dark matter must explain the diversity of observed
Milky Way (MW) satellite density profiles, from very dense ultrafaints to
large, low density satellites such as Crater~II that appear to be larger their
anticipated host dark matter haloes. We consider cold dark matter (CDM), warm
dark matter (WDM, 3.3keV thermal relic power spectrum), and a self-interacting
dark matter model (SIDM) that induces gravothermal collapse in low mass
subhaloes. Predictions for these density profiles are complicated by the
limitations of simulation resolution in the stripping of subhaloes by the MW
system, therefore we make predictions for satellite properties in these three
models using $N$-body simulations combined with a semi-analytic halo stripping
algorithm. We find that most CDM and WDM subhaloes of mass $>10^{8}$$M_{\odot}$
are large enough after stripping to fit most satellites; however, the required
amount of stripping often requires a stronger tidal field than is available on
the subhalo's orbit. The lower concentrations of WDM subhaloes enable this
model to explain the required satellite masses with less stripping than is
necessary for CDM, and is thus consistent with orbits of larger pericentres.
SIDM cores offer the best fits to massive, low density satellites at the
expense of predicting many $>10^{9}$$M_{\odot}$ subhaloes to host low density
satellites with no observed analogue. We conclude that an SIDM model must have
a very high velocity-dependent cross-section in order to match all satellites,
and that WDM offers a marginally better fit than CDM to the MW satellite mass
function.
