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THE LAST WORD Make yours count
n Arrow’s 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 – A and C – 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. B could win if voters ranked candidates in order of preference and the candidate with the most “least preferred” votes was eliminated. But in a larger field, such methods tend to favour “donothing” 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, opinions of voters inevitably from 0 to 10. This method allows produce numerous small parties. voters to give some candidates There is a risk that these small the same score and to record parties may become entrenched “no opinion” for candidates in their positions and unable to they know nothing about. move forward. If they form Range voting is simple and coalitions, they blame others for easily understood, and it only any failures that result and do produces the failures of ranking not learn from their mistakes. systems if voters mistakenly think If re‑elected, they continue as it is better to vote tactically by before. The parties that result giving their preferred candidate top score and everyone else zero. “Voting systems that reflect In practice, giving an honest score many different opinions of voters inevitably produce to everyone gives the best result. numerous small parties” However, the method is unlikely to win favour because politicians get to choose the from such a system may be more method of voting, and most do idealistic than pragmatic because not want to know what score the they never have to accept full public would actually give them. responsibility for government. Brian Horton It is my opinion that the most West Launceston, democratic system is not one that Tasmania, Australia puts the “right” people into office, whoever they might be. It is a n When it comes to voting system that does not allow the systems, it is easy to assume that elected party or group to walk “fair and democratic” means away from blame when things “most directly reflects the views go wrong and, most importantly, of the voters”, but this is not that allows us to get rid of the always the case. people we voted into office the For example, a voting system last time, should we want. may produce a result where a Alan Urdaibay small, extremist party holds the Paignton, Devon, UK balance of power between two larger parties or coalitions. This n The short answer is that small party may then influence mathematics can prove that policy in a way that is neither there is no fair voting system. fair nor democratic. Also, when There are a number of theorems parties form a coalition, they to this effect, such as Arrow’s might reach compromises on impossibility theorem and the policies or form new policies that Gibbard-Satterthwaite theorem, the electorate did not vote for. among others. Voting systems that most These theorems are proved in closely reflect the many different essentially the same way. First, the
The writers of answers that are published in the magazine will receive a cheque for £25 (or US$ equivalent). Answers should be concise. We reserve the right to edit items for clarity and style. Please include a daytime telephone number and an email address if you have one. New Scientist retains total editorial control over the published content. Reed Business Information Ltd reserves all rights to reuse all question and answer material that has been
submitted by readers in any medium or in any format and at any time in the future. Send questions and answers to The Last Word, New Scientist, Lacon House, 84 Theobald’s Road, London WC1X 8NS, UK, by email to lastword@newscientist. com or visit www.newscientist.com/topic/ lastword (please include a postal address in order to receive payment for answers). Unanswered questions can also be found at this URL.
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?
idea of fairness is expressed as a number of axioms. For example, for Arrow’s theorem these are: if every voter prefers alternative X over alternative Y, then the group prefers X over Y; if every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs such as X and Z, Y and Z, or Z and W change); and there is no “dictator” (no single voter possesses the power to always determine the group’s 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 “fairness”. 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 week’s question Tense situation
Is it possible to increase the surface tension of water? My sci-fi writer friend claims it can be done, but is he just a bit wet? Stephen Partington
Will we ever speak dolphin? The latest book: packed full of wit, knowledge and extraordinary discovery Available from booksellers and at newscientist.com/dolphins
THE LAST WORD Make yours count
n Arrow’s 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 – A and C – 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. B could win if voters ranked candidates in order of preference and the candidate with the most “least preferred” votes was eliminated. But in a larger field, such methods tend to favour “donothing” 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, opinions of voters inevitably from 0 to 10. This method allows produce numerous small parties. voters to give some candidates There is a risk that these small the same score and to record parties may become entrenched “no opinion” for candidates in their positions and unable to they know nothing about. move forward. If they form Range voting is simple and coalitions, they blame others for easily understood, and it only any failures that result and do produces the failures of ranking not learn from their mistakes. systems if voters mistakenly think If re‑elected, they continue as it is better to vote tactically by before. The parties that result giving their preferred candidate top score and everyone else zero. “Voting systems that reflect In practice, giving an honest score many different opinions of voters inevitably produce to everyone gives the best result. numerous small parties” However, the method is unlikely to win favour because politicians get to choose the from such a system may be more method of voting, and most do idealistic than pragmatic because not want to know what score the they never have to accept full public would actually give them. responsibility for government. Brian Horton It is my opinion that the most West Launceston, democratic system is not one that Tasmania, Australia puts the “right” people into office, whoever they might be. It is a n When it comes to voting system that does not allow the systems, it is easy to assume that elected party or group to walk “fair and democratic” means away from blame when things “most directly reflects the views go wrong and, most importantly, of the voters”, but this is not that allows us to get rid of the always the case. people we voted into office the For example, a voting system last time, should we want. may produce a result where a Alan Urdaibay small, extremist party holds the Paignton, Devon, UK balance of power between two larger parties or coalitions. This n The short answer is that small party may then influence mathematics can prove that policy in a way that is neither there is no fair voting system. fair nor democratic. Also, when There are a number of theorems parties form a coalition, they to this effect, such as Arrow’s might reach compromises on impossibility theorem and the policies or form new policies that Gibbard-Satterthwaite theorem, the electorate did not vote for. among others. Voting systems that most These theorems are proved in closely reflect the many different essentially the same way. First, the
The writers of answers that are published in the magazine will receive a cheque for £25 (or US$ equivalent). Answers should be concise. We reserve the right to edit items for clarity and style. Please include a daytime telephone number and an email address if you have one. New Scientist retains total editorial control over the published content. Reed Business Information Ltd reserves all rights to reuse all question and answer material that has been
submitted by readers in any medium or in any format and at any time in the future. Send questions and answers to The Last Word, New Scientist, Lacon House, 84 Theobald’s Road, London WC1X 8NS, UK, by email to lastword@newscientist. com or visit www.newscientist.com/topic/ lastword (please include a postal address in order to receive payment for answers). Unanswered questions can also be found at this URL.
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?
idea of fairness is expressed as a number of axioms. For example, for Arrow’s theorem these are: if every voter prefers alternative X over alternative Y, then the group prefers X over Y; if every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs such as X and Z, Y and Z, or Z and W change); and there is no “dictator” (no single voter possesses the power to always determine the group’s 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 “fairness”. 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 week’s question Tense situation
Is it possible to increase the surface tension of water? My sci-fi writer friend claims it can be done, but is he just a bit wet? Stephen Partington
Will we ever speak dolphin? The latest book: packed full of wit, knowledge and extraordinary discovery Available from booksellers and at newscientist.com/dolphins