  Cooperative localization with map matching has been shown to reduce Global
Navigation Satellite System (GNSS) localization error from several meters to
sub-meter level by fusing the GNSS measurements of four vehicles in our
previous work. While further error reduction is expected to be achievable by
increasing the number of vehicles, the quantitative relationship between the
estimation error and the number of connected vehicles has neither been
systematically investigated nor analytically proved. In this work, a
theoretical study is presented that analytically proves the correlation between
the localization error and the number of connected vehicles in two cases of
practical interest. More specifically, it is shown that, under the assumption
of small non-common error, the expected square error of the GNSS common error
correction is inversely proportional to the number of vehicles, if the road
directions obey a uniform distribution, or inversely proportional to logarithm
of the number of vehicles, if the road directions obey a Bernoulli
distribution. Numerical simulations are conducted to justify these analytic
results. Moreover, the simulation results show that the aforementioned error
decrement rates hold even when the assumption of small non-common error is
violated.
