Here's a discussion at New Scientist on proportional representation vs. "first past the post": Can mathematics help?
On 5 May, the UK will hold a referendum to determine which voting system the country should use in future elections, with voters asked to decide whether they want to adopt the alternative vote (AV) or stick with the first-past-the-post (FPTP) system, which is currently used. Can mathematics tell us which system is the most fair?
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Highlighting a more radical solution is David Maclver, a software engineer with a background in mathematics. In 1963 economist Kenneth Arrow proved that no voting system can satisfy a few reasonable and democratic conditions – in other words, democracy is always unfair. MacIver points out that Arrow was only considering certain types of voting systems, and claims that there is an alternative, "perfect" system. Maclver's system is identical to FPTP in all but one respect. Voters in each constituency choose a single candidate, but then one voter is picked at random from each constituency and their choice determines which candidate gets elected. The random element means the system isn't covered by Arrow's theorem. It sounds horribly unfair but … – Jacob Aron, "Mathematicians weigh in on UK voting debate" (27 April 2011)
Sources have wondered whether a perfect system can exist without perfect voters, but those sources are not mathematicians.
Here is a completely biased source from a country* that does not use proportional representation, an ID sympathizer who is not a mathematician.
Denyse O'Leary is co-author of The Spiritual Brain.