  In fair division of indivisible goods, using sequences of sincere choices (or
picking sequences) is a natural way to allocate the objects. The idea is the
following: at each stage, a designated agent picks one object among those that
remain. This paper, restricted to the case where the agents have numerical
additive preferences over objects, revisits to some extent the seminal paper by
Brams and King [9] which was specific to ordinal and linear order preferences
over items. We point out similarities and differences with this latter context.
In particular, we show that any Pareto-optimal allocation (under additive
preferences) is sequenceable, but that the converse is not true anymore. This
asymmetry leads naturally to the definition of a "scale of efficiency" having
three steps: Pareto-optimality, sequenceability without Pareto-optimality, and
non-sequenceability. Finally, we investigate the links between these efficiency
properties and the "scale of fairness" we have described in an earlier work
[7]: we first show that an allocation can be envy-free and non-sequenceable,
but that every competitive equilibrium with equal incomes is sequenceable. Then
we experimentally explore the links between the scales of efficiency and
fairness.
