The Condorcet paradox shows that pairwise majority voting may produce cyclic outcomes on an unrestricted preference domain. For instance consider three voters with their preferences:
Voter 1: a > b > c Voter 2: b > c > a Voter 3: c > a > b
A majority of voters prefers a to b, a majority prefers b to c, and a majority prefers c to a. This leads to the Condorcet cycle
a > b > c > a.
The absence of majority cycles can only be guaranteed under suitable domain restrictions, such that single-peaked preferences, preferences with ‘value-restrictions' or single-crossing preferences – these are so called Condorcet domains. Puppe and Slinko (2015) prove that all (closed) Condorcet domains satisfy a so called intermediateness condition with respect to an appropriate median graph. This property allows us to investigate the Condorcet domains in more detail.
The following figures depict graphical representations of those domains: