Single-Peakedness and Condorcet Domains
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, b to c and c to a at the same time. Thus, majority preference exhibits a so called Condorcet cycle: a > b > c > a.
The absence of (Condorcet) majority cycles can only be guaranteed under suitable domain restrictions. For example, majority voting (over pairs of alternatives) results in a consistent ranking of alternatives if preferences are ‘single-peaked’: voters prefer alternatives that are ‘closer’ to their unique peak. In this case, majority voting corresponds to choosing the median voter peak. Single-peaked preferences play a dominant role in models of political economy. (This may be due to both their simplicity and intuitive appeal as well as the fact that they constitute the only minimally rich and connected domains that contains two completely reversed strict preference orders.
Other domain restrictions that prevent majority preferences from being cyclic include single-crossing preferences or preferences with ‘value-restrictions'. More generally, all of the just mentioned are instances of so called ‘Condorcet Domains’. Puppe and Slinko (2015) prove that all (closed) Condorcet domains satisfy an intermediateness condition with respect to an appropriate median graph. This fact allows us to study the properties of Condorcet domains in more detail.
The following figures depict graphical representations of such domains: