  Fiber-fed multi-object spectroscopic surveys, with their ability to collect
an unprecedented number of redshifts, currently dominate large-scale structure
studies. However, physical constraints limit these surveys from successfully
collecting redshifts from galaxies too close to each other on the focal plane.
This ultimately leads to significant systematic effects on galaxy clustering
measurements. Using simulated mock catalogs, we demonstrate that fiber
collisions have a significant impact on the power spectrum, $P(k)$, monopole
and quadrupole that exceeds sample variance at scales smaller than
$k\sim0.1~h/Mpc$.
  We present two methods to account for fiber collisions in the power spectrum.
The first, statistically reconstructs the clustering of fiber collided galaxy
pairs by modeling the distribution of the line-of-sight displacements between
them. It also properly accounts for fiber collisions in the shot-noise
correction term of the $P(k)$ estimator. Using this method, we recover the true
$P(k)$ monopole of the mock catalogs with residuals of $<0.5\%$ at
$k=0.3~h/Mpc$ and $<4\%$ at $k=0.83~h/Mpc$ -- a significant improvement over
existing correction methods. The quadrupole, however, does not improve
significantly.
  The second method models the effect of fiber collisions on the power spectrum
as a convolution with a configuration space top-hat function that depends on
the physical scale of fiber collisions. It directly computes theoretical
predictions of the fiber-collided $P(k)$ multipoles and reduces the influence
of smaller scales to a set of nuisance parameters. Using this method, we
reliably model the effect of fiber collisions on the monopole and quadrupole
down to the scale limits of theoretical predictions. The methods we present in
this paper will allow us to robustly analyze galaxy power spectrum multipole
measurements to much smaller scales than previously possible.
