New voting methods and fair elections : The New Yorker - 0 views
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history of voting math comes mainly in two chunks: the period of the French Revolution, when some members of France’s Academy of Sciences tried to deduce a rational way of conducting elections, and the nineteen-fifties onward, when economists and game theorists set out to show that this was impossible
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The first mathematical account of vote-splitting was given by Jean-Charles de Borda, a French mathematician and a naval hero of the American Revolutionary War. Borda concocted examples in which one knows the order in which each voter would rank the candidates in an election, and then showed how easily the will of the majority could be frustrated in an ordinary vote. Borda’s main suggestion was to require voters to rank candidates, rather than just choose one favorite, so that a winner could be calculated by counting points awarded according to the rankings. The key idea was to find a way of taking lower preferences, as well as first preferences, into account.Unfortunately, this method may fail to elect the majority’s favorite—it could, in theory, elect someone who was nobody’s favorite. It is also easy to manipulate by strategic voting.
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If the candidate who is your second preference is a strong challenger to your first preference, you may be able to help your favorite by putting the challenger last. Borda’s response was to say that his system was intended only for honest men.
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After the Academy dropped Borda’s method, it plumped for a simple suggestion by the astronomer and mathematician Pierre-Simon Laplace, who was an important contributor to the theory of probability. Laplace’s rule insisted on an over-all majority: at least half the votes plus one. If no candidate achieved this, nobody was elected to the Academy.
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Another early advocate of proportional representation was John Stuart Mill, who, in 1861, wrote about the critical distinction between “government of the whole people by the whole people, equally represented,” which was the ideal, and “government of the whole people by a mere majority of the people exclusively represented,” which is what winner-takes-all elections produce. (The minority that Mill was most concerned to protect was the “superior intellects and characters,” who he feared would be swamped as more citizens got the vote.)
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The key to proportional representation is to enlarge constituencies so that more than one winner is elected in each, and then try to align the share of seats won by a party with the share of votes it receives. These days, a few small countries, including Israel and the Netherlands, treat their entire populations as single constituencies, and thereby get almost perfectly proportional representation. Some places require a party to cross a certain threshold of votes before it gets any seats, in order to filter out extremists.
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The main criticisms of proportional representation are that it can lead to unstable coalition governments, because more parties are successful in elections, and that it can weaken the local ties between electors and their representatives. Conveniently for its critics, and for its defenders, there are so many flavors of proportional representation around the globe that you can usually find an example of whatever point you want to make. Still, more than three-quarters of the world’s rich countries seem to manage with such schemes.
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The alternative voting method that will be put to a referendum in Britain is not proportional representation: it would elect a single winner in each constituency, and thus steer clear of what foreigners put up with. Known in the United States as instant-runoff voting, the method was developed around 1870 by William Ware
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In instant-runoff elections, voters rank all or some of the candidates in order of preference, and votes may be transferred between candidates. The idea is that your vote may count even if your favorite loses. If any candidate gets more than half of all the first-preference votes, he or she wins, and the game is over. But, if there is no majority winner, the candidate with the fewest first-preference votes is eliminated. Then the second-preference votes of his or her supporters are distributed to the other candidates. If there is still nobody with more than half the votes, another candidate is eliminated, and the process is repeated until either someone has a majority or there are only two candidates left, in which case the one with the most votes wins. Third, fourth, and lower preferences will be redistributed if a voter’s higher preferences have already been transferred to candidates who were eliminated earlier.
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At first glance, this is an appealing approach: it is guaranteed to produce a clear winner, and more voters will have a say in the election’s outcome. Look more closely, though, and you start to see how peculiar the logic behind it is. Although more people’s votes contribute to the result, they do so in strange ways. Some people’s second, third, or even lower preferences count for as much as other people’s first preferences. If you back the loser of the first tally, then in the subsequent tallies your second (and maybe lower) preferences will be added to that candidate’s first preferences. The winner’s pile of votes may well be a jumble of first, second, and third preferences.
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Such transferrable-vote elections can behave in topsy-turvy ways: they are what mathematicians call “non-monotonic,” which means that something can go up when it should go down, or vice versa. Whether a candidate who gets through the first round of counting will ultimately be elected may depend on which of his rivals he has to face in subsequent rounds, and some votes for a weaker challenger may do a candidate more good than a vote for that candidate himself. In short, a candidate may lose if certain voters back him, and would have won if they hadn’t. Supporters of instant-runoff voting say that the problem is much too rare to worry about in real elections, but recent work by Robert Norman, a mathematician at Dartmouth, suggests otherwise. By Norman’s calculations, it would happen in one in five close contests among three candidates who each have between twenty-five and forty per cent of first-preference votes. With larger numbers of candidates, it would happen even more often. It’s rarely possible to tell whether past instant-runoff elections have gone topsy-turvy in this way, because full ballot data aren’t usually published. But, in Burlington’s 2006 and 2009 mayoral elections, the data were published, and the 2009 election did go topsy-turvy.
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Kenneth Arrow, an economist at Stanford, examined a set of requirements that you’d think any reasonable voting system could satisfy, and proved that nothing can meet them all when there are more than two candidates. So designing elections is always a matter of choosing a lesser evil. When the Royal Swedish Academy of Sciences awarded Arrow a Nobel Prize, in 1972, it called his result “a rather discouraging one, as regards the dream of a perfect democracy.” Szpiro goes so far as to write that “the democratic world would never be the same again,
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There is something of a loophole in Arrow’s demonstration. His proof applies only when voters rank candidates; it would not apply if, instead, they rated candidates by giving them grades. First-past-the-post voting is, in effect, a crude ranking method in which voters put one candidate in first place and everyone else last. Similarly, in the standard forms of proportional representation voters rank one party or group of candidates first, and all other parties and candidates last. With rating methods, on the other hand, voters would give all or some candidates a score, to say how much they like them. They would not have to say which is their favorite—though they could in effect do so, by giving only him or her their highest score—and they would not have to decide on an order of preference for the other candidates.
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One such method is widely used on the Internet—to rate restaurants, movies, books, or other people’s comments or reviews, for example. You give numbers of stars or points to mark how much you like something. To convert this into an election method, count each candidate’s stars or points, and the winner is the one with the highest average score (or the highest total score, if voters are allowed to leave some candidates unrated). This is known as range voting, and it goes back to an idea considered by Laplace at the start of the nineteenth century. It also resembles ancient forms of acclamation in Sparta. The more you like something, the louder you bash your shield with your spear, and the biggest noise wins. A recent variant, developed by two mathematicians in Paris, Michel Balinski and Rida Laraki, uses familiar language rather than numbers for its rating scale. Voters are asked to grade each candidate as, for example, “Excellent,” “Very Good,” “Good,” “Insufficient,” or “Bad.” Judging politicians thus becomes like judging wines, except that you can drive afterward.
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Range and approval voting deal neatly with the problem of vote-splitting: if a voter likes Nader best, and would rather have Gore than Bush, he or she can approve Nader and Gore but not Bush. Above all, their advocates say, both schemes give voters more options, and would elect the candidate with the most over-all support, rather than the one preferred by the largest minority. Both can be modified to deliver forms of proportional representation.
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Whether such ideas can work depends on how people use them. If enough people are carelessly generous with their approval votes, for example, there could be some nasty surprises. In an unlikely set of circumstances, the candidate who is the favorite of more than half the voters could lose. Parties in an approval election might spend less time attacking their opponents, in order to pick up positive ratings from rivals’ supporters, and critics worry that it would favor bland politicians who don’t stand for anything much. Defenders insist that such a strategy would backfire in subsequent elections, if not before, and the case of Ronald Reagan suggests that broad appeal and strong views aren’t mutually exclusive.
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Why are the effects of an unfamiliar electoral system so hard to puzzle out in advance? One reason is that political parties will change their campaign strategies, and voters the way they vote, to adapt to the new rules, and such variables put us in the realm of behavior and culture. Meanwhile, the technical debate about electoral systems generally takes place in a vacuum from which voters’ capriciousness and local circumstances have been pumped out. Although almost any alternative voting scheme now on offer is likely to be better than first past the post, it’s unrealistic to think that one voting method would work equally well for, say, the legislature of a young African republic, the Presidency of an island in Oceania, the school board of a New England town, and the assembly of a country still scarred by civil war. If winner takes all is a poor electoral system, one size fits all is a poor way to pick its replacements.
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Mathematics can suggest what approaches are worth trying, but it can’t reveal what will suit a particular place, and best deliver what we want from a democratic voting system: to create a government that feels legitimate to people—to reconcile people to being governed, and give them reason to feel that, win or lose (especially lose), the game is fair.