What do the people really want? The Condorcet paradox and the referendum

What do the people really want? The Condorcet paradox and the referendum:

I have not encountered the Condorcet paradox before, but it is truly fascinating. Even if you are not a math geek! 

From wikipedia:

The Condorcet paradox (also known as voting paradox or the paradox of voting) in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century,[1][2][3] in which collective preferences can be cyclic, even if the preferences of individual voters are not cyclic.

Okay, lets stop here before the heavy maths starts and break this down. First thing we need to understand is that the political spectrum is not 1-Dimensional (left-right), there are many unrelated subjects that people might agree or disagree upon. 

Below one of the most common 2D graphs uses today to arranges parties on a political spectrum. (And yes, that is also an oversimplification, but it is good enough to explain the problem at hand, so I will stick with this from now on.)

It is easy to see how for example two people might agree on economic freedom, but disagree on personal one.

So looking at the very first example of the Condorcet Paradox wiki page: Where voter interests (1,2,3) do not align with what the candidates (A,B,C) stand for.

So whichever candidate gets elected here, it will only represent the majority interest on one out of three topics. So the majority is misrepresented.

(It is important to note that this is not the same as the Spoiler Effect, which is one of the majour problems with First Past The Post, but a more general problem with voting systems.)

If you overlay this to a referendum and try to compress multiple issues down to a simple yes/no answer, it gets even worse. Even a well educated population might vote an option that is not representing the wishes of the majority.

So each individual votes on what is most important to them, but the accumulative result is not representing the wishes of the collective.

Lets just say simple yes/no questions are not good solutions to complex questions asked to a populous.

There is also the Nakamura numbers, which go much further and try to highlight when a given choice for a specific type of question, is likely to result in the wrong outcome:

In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationality of preference aggregation rules (collective decision rules), such as voting rules. It is an indicator of the extent to which an aggregation rule can yield well-defined choices.

… but that is for another post.