Challenges for the statistical gravitational-wave method to measure the
  Hubble constant
  Gravitational waves (GW) can be employed as standard sirens that will soon
measure the Hubble constant with sufficient precision to weigh in on the $\sim
5\sigma$ Hubble tension. Most GW sources will have no identified
electromagnetic counterpart, leading to uncertainty in the redshift of the
source, and in turn a degeneracy between host galaxy distance, redshift, and
$H_0$. In the case where no electromagnetic counterparts are identified, it has
been proposed that a statistical canvassing of candidate GW hosts, found in a
large galaxy survey for example, can be used to accurately constrain the Hubble
constant. We study and simulate this "galaxy voting" method to compute $H_0$.
We find that the Hubble constant posterior is in general biased relative to the
true value even when making optimistic assumptions about the statistical
properties of the sample. This bias is caused by the fundamental degeneracy
between redshift and $H_0$, and is effectively irreducible without accurate
information about the redshift distribution from which the GW sources come.
