  A preference domain is called a non-dictatorial domain if it allows the
design of unanimous social choice functions (henceforth, rules) that are
non-dictatorial and strategy-proof. We study a class of preference domains
called unidimensional domains and establish that the unique seconds property
(introduced by Aswal, Chatterji, and Sen (2003)) characterizes all
non-dictatorial domains. The principal contribution is the subsequent
exhaustive classification of all non-dictatorial, unidimensional domains and
canonical strategy-proof rules on these domains, based on a simple property of
two-voter rules called invariance. The preference domains that constitute the
classification are semi-single-peaked domains (introduced by Chatterji, Sanver,
and Sen (2013)) and semi-hybrid domains (introduced here) which are two
appropriate weakenings of single-peaked domains and are shown to allow
strategy-proof rules to depend on non-peak information of voters' preferences;
the canonical rules for these domains are the projection rule and the hybrid
rule respectively. As a refinement of the classification, single-peaked domains
and hybrid domains emerge as the only unidimensional domains that force
strategy-proof rules to be determined completely by voters' preference peaks.
