However, instead of proportionality, the authors posed "representation of the median voter's preferences" as their criterion, supposing it to be the goal of plurality systems. By their description of the majoritarian vision of politics, the point of elections is to allow 'citizens' to make a clear decision about who governs 'them'. Working on this assumption, they tried to test the Downsian theory of plurality systems, which says that the two main parties will converge toward the preferences of the median voter.
But this supposed expression of preferences isn't the point of elections at all. For a start, any study in this field should acknowledge that, there is no rational, self-consistent way to express the preferences of a diverse group – as Arrow's theorem showed. Therefore there can be no ideal in this matter, because every possible form of representation will contain some paradox or other. But Arrow's theorem isn't the only trouble with this sort of approach. There is no foolproof way to choose a ruler – not only in democracies, but in every kind of system – because all rulers, people, are fallible. The reader may object: "Yes, there is no perfect way, but surely there are better and worse ways to choose a ruler."
No. Accept for the sake of argument that any ruler, or method of choosing one, is prone to error. Therefore, if it is taken to be the 'right' method, and is established dogmatically and without any redeeming institutional features, its inherent errors – whatever they happen to be – become entrenched. For this reason, Popper suggested in the Open Society and its Enemies that the question we ask about politics ought not be "Who should rule?" but rather "How can we limit the damage they do?" In other words, it doesn't matter how the ruler is chosen per se – there is nothing *inherently* better about rule of the many versus rule of the few – but rather that the system contains mechanisms for its own improvement. Such improvement is greatly helped by a procedure for carrying out changes of power without violence. That is the real virtue of democracy.
Moving on from this basic objection, the next question is, how exactly do we measure the "median voter's preferences"? The authors go with the left-right scale, using it to compare policy-positions of citizens with policy-positions of the parties that are supposed to represent them. They make an ostensibly reasonable defence of this tool: It is true that 'left' or 'right' are good summaries the political views of most people in democracies. Nevertheless, I don't think that makes it viable for this study. It merely compares how right- or left-wing the policy positions of the voters are with how right- or left-wing are those of the representatives, taking no notice of the relative importance of different policies to the voters (which could easily cause a left-leaning person to vote for a more right-wing party, for example).
An even worse difficulty, not acknowledged as such by the author, is that the polarisation of left and right varies wildly between countries. The same units are used to measure them, yet the difference between, say, 1 and 2 on the left-right scale might be considerably greater for France than for the United States. This is not merely a lack of 'rigour'; the same units are being used to represent wildly different values (which themselves may not be measurable in the first place). It effectively makes the study meaningless.