  Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm
version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists
a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y
of \Pi such that translating \Pi by -Y yields a configuration whose law is that
of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of
\Pi, and prove that there is no shift coupling in which \Pi is a function of
\Pi^*. The key ingredient is a deterministic translation-invariant rule to
allocate sets of equal volume (forming a partition of R^d) to the points of
\Pi. The construction is based on the Gale-Shapley stable marriage algorithm
[Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random
element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A
shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show
that there exists an extra head scheme which is a function of \Gamma if and
only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law
of \Gamma is product measure and d\geq3, we prove that there exists an extra
head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of
Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].
