  A Peskun ordering between two samplers, implying a dominance of one over the
other, is known among the Markov chain Monte Carlo community for being a
remarkably strong result. It is however also known for being a result that is
notably difficult to establish. Indeed, one has to prove that the probability
to reach a state $\mathbf{y}$ from a state $\mathbf{x}$, using a sampler, is
greater than or equal to the probability using the other sampler, and this must
hold for all pairs $(\mathbf{x}, \mathbf{y})$ such that $\mathbf{x} \neq
\mathbf{y}$. We provide in this paper a weaker version that does not require an
inequality between the probabilities for all these states: essentially, the
dominance holds asymptotically, as a varying parameter grows without bound, as
long as the states for which the probabilities are greater than or equal to
belong to a mass-concentrating set. The weak ordering turns out to be useful to
compare lifted samplers for partially-ordered discrete state-spaces with their
Metropolis--Hastings counterparts. An analysis in great generality yields a
qualitative conclusion: they asymptotically perform better in certain
situations (and we are able to identify them), but not necessarily in others
(and the reasons why are made clear). A quantitative study in a specific
context of graphical-model simulation is also conducted.
