Mathematically speaking, which kind of voting system produces the most democratic and fair result in a general election? There are several to choose from: first past the post, alternative vote, single transferable vote and many others, but surely mathematics can decree which is the fairest system?
鈥� Arrow鈥檚 impossibility theorem shows that any voting system in which candidates are ranked in order will not provide a fair result in all cases.
Take an example where there are two candidates with extreme views 鈥� A and C 鈥� each preferred by 35 per cent of the population, but hated by everyone else. Candidate B is moderate but only scores 30 per cent when all three candidates are standing. However, B will win (with 65 per cent) in a two-horse race against either A or C. Yet if all three candidates stand for election in almost all voting systems in common use, B will be eliminated, leaving either A or C to win, even though B could beat either A or C in a two-way vote.
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B could win if voters ranked candidates in order of preference and the candidate with the most 鈥渓east preferred鈥� votes was eliminated. But in a larger field, such methods tend to favour 鈥渄o-nothing鈥� candidates who are not disliked by anyone but would not gain first place against any other candidate either.
An alternative is range voting, a system in which every candidate is given a score: for example, from 0 to 10. This method allows voters to give some candidates the same score and to record 鈥渘o opinion鈥� for candidates they know nothing about.
Range voting is simple and easily understood, and it only produces the failures of ranking systems if voters mistakenly think it is better to vote tactically by giving their preferred candidate top score and everyone else zero. In practice, giving an honest score to everyone gives the best result.
However, the method is unlikely to win favour because politicians get to choose the method of voting, and most do not want to know what score the public would actually give them.
Brian Horton, West Launceston, Tasmania, Australia
鈥� When it comes to voting systems, it is easy to assume that 鈥渇air and democratic鈥� means 鈥渕ost directly reflects the views of the voters鈥�, but this is not always the case.
For example, a voting system may produce a result where a small, extremist party holds the balance of power between two larger parties or coalitions. This small party may then influence policy in a way that is neither fair nor democratic. Also, when parties form a coalition, they might reach compromises on policies or form new policies that the electorate did not vote for.
Voting systems that most closely reflect the many different opinions of voters inevitably produce numerous small parties. There is a risk that these small parties may become entrenched in their positions and unable to move forward. If they form coalitions, they blame others for any failures that result and do not learn from their mistakes. If re-elected, they continue as before. The parties that result from such a system may be more idealistic than pragmatic because they never have to accept full responsibility for government.
鈥淰oting systems that reflect many different opinions of voters inevitably produce numerous small parties鈥�
It is my opinion that the most democratic system is not one that puts the 鈥渞ight鈥� people into office, whoever they might be. It is a system that does not allow the elected party or group to walk away from blame when things go wrong and, most importantly, that allows us to get rid of the people we voted into office the last time, should we want.
Alan Urdaibay, Paignton, Devon, UK
鈥� The short answer is that mathematics can prove that there is no fair voting system. There are a number of theorems to this effect, such as Arrow鈥檚 impossibility theorem and the Gibbard-Satterthwaite theorem, among others.
These theorems are proved in essentially the same way. First, the idea of fairness is expressed as a number of axioms. For example, for Arrow鈥檚 theorem these are: if every voter prefers alternative X over alternative Y, then the group prefers X over Y; if every voter鈥檚 preference between X and Y remains unchanged, then the group鈥檚 preference between X and Y will also remain unchanged (even if voters鈥� preferences between other pairs such as X and Z, Y and Z, or Z and W change); and there is no 鈥渄ictator鈥� (no single voter possesses the power to always determine the group鈥檚 preference).
This set of axioms is then mathematically proved to be inconsistent: that is, no voting system can satisfy them all.
This means that any voting system has to abandon at least one criterion of 鈥渇airness鈥�. Which criterion to abandon is a matter for social and political debate. Mathematics can prove that this debate is necessary but cannot, of course, dictate how it should be resolved. It all depends on what you mean by fairness.
John Dobson, Hexham, Northumberland, UK
This article appeared in print under the headline 鈥淢ake yours count鈥�