  Suppose that red and blue points form independent homogeneous Poisson
processes of equal intensity in $R^d$. For a positive (respectively, negative)
parameter $\gamma$ we consider red-blue matchings that locally minimize
(respectively, maximize) the sum of $\gamma$th powers of the edge lengths,
subject to locally minimizing the number of unmatched points. The parameter can
be viewed as a measure of fairness. The limit $\gamma\to-\infty$ is equivalent
to Gale-Shapley stable matching. We also consider limits as $\gamma$ approaches
$0$, $1-$, $1+$ and $\infty$. We focus on dimension $d=1$. We prove that almost
surely no such matching has unmatched points. (This question is open for higher
$d$). For each $\gamma<1$ we establish that there is almost surely a unique
such matching, and that it can be expressed as a finitary factor of the points.
Moreover, its typical edge length has finite $r$th moment if and only if
$r<1/2$. In contrast, for $\gamma=1$ there are uncountably many matchings,
while for $\gamma>1$ there are countably many, but it is impossible to choose
one in a translation-invariant way. We obtain existence results in higher
dimensions (covering many but not all cases). We address analogous questions
for one-colour matchings also.
