Elections do not work at all; you need to blame math

Voters make bulletins at a polling station located in San Francisco Columbia, June 5, 2018

San Francisco Mayor Ed Lee died in December 2017. His replacement was held in early June. Almost a week later, it was still unknown who won [ original article of June 6, 2018 / approx. trans. ]. Partly due to the fact that voices were still coming. Voting by mail can simply be sent on voting day, and six days later, the city authorities reported that they had 87 more votes left to process. But this is not the only obstacle. Another problem is mathematics.

You see, the election of the mayor of San Francisco is not just another story “the winner is the one with the most votes.” This race is another example of cultural innovation that California sometimes unexpectedly releases to unprepared America, as it did with smartphones and fashionable toasts. Surprise! We do not even vote like everyone else.

The scheme of work is called rating voting . Voters rank three choices by degree of preference. In the process of counting, the candidate who scored the smallest number of first places is discarded, then the votes cast for him are transferred to the person who was in second place with voters who prefer this candidate, and then the process repeats. Sounds crazy? Actually it is ingenious. And crazy.

The bulletins list eight candidates for the post of mayor, including the unlikely ones, such as a lawyer who has already participated in elections three times, an adherent of holistic medicineand republican. San Francisco residents focused on three options: London Breed, Jane Kim and Mark Leno, local elected people with interconnected life paths that could happen only in the midst of aggressive municipal politics in a region where an insane amount of money from technology companies turns which, for the most part, the government can not impose a paw, due to the fact that corporations agree on tax breaks, and property owners underpay taxes, understating the cost of housing). So far, Breed has the highest number of votes for the first place - 10% more than Leno, who is in second place. But the redistributed votes cast for Kim, who was in third place, gave Leno such a slight advantage that he was not visible if you looked at him from the edge.

Why complicate simple, direct choices? The fact is that elections are not easy. The theory of social choice produces many different ways a group can make a decision, and the “majority of votes” —the one who gets the most votes wins — only one of them. It works great if there are only two options in your newsletter. But add more options and you will have problems.

When reformist candidate Jess Ventura defeated Republican Norma Coleman and Democrat Skip Humphrey in an election in Minnesota in 1998, policy experts saw that voters were disgusted with how the system worked. Ventura scored 37%, Coleman 35%, Humphrey - 28. But as the mathematics from Emory University Victoria Powers wrote in 2015, the exit polls said that almost everyone who voted for Coleman would have given Humphrey second place, Coleman was the second option for almost everyone who voted for Humphrey. “Voters preferred Coleman to both other candidates, and he also lost the election,” Powers wrote.

Here you have the most votes. The same problem with the "anti-majority", in which everyone chooses whom he hates the most, and the person with the least number of votes wins. Both are potentially capable of violating the Condorcet jury theorem . In 1785, the philosopher and mathematician Marie Jean-Antoine Nicolas de Carite, the Marquis de Condorcet , in particular, said that the candidate must win the election, ahead of all other candidates one by one. Sequential pair-wise voting in which you remove the losers gives us a clear Condorcet winner. But this winner will be different for different sequence of pairwise comparisons .

So yes, the majority of votes is bad. “This is very restrictive for voters,” said Daniel Ulman, a mathematician at George Washington University, co-author of The Mathematics of Politics. “If you allow voters to choose two of their best candidates, or sort all ten in order, or indicate who they like or dislike, or come up with some other voting options, then everything becomes more interesting.”

Indeed. Other systems allow voters more choices, but they also generate what mathematicians call paradoxes. Here is an example: the choice of ranking lacks “monotony”. This means that sometimes people have to vote against a candidate supported by them in order to increase their chances of winning. “This is unpleasant, because when you enter the voting booth, you are not sure whether you should disclose your true desire,” says Ulman.

Indeed, in some election campaigns, people are told which two candidates should be voted for, regardless of the order - in fact, vote against the third corner of the triangle. On the other hand, imagine how different the history of America could have been if the 2000 elections (Al Gore had practically equaled George W. Bush, and Ralph Nader and Patrick Buchanan also participated) had a choice through ranking.

Ranking and sequential pairwise comparison are not even the strangest features. You can assign each candidate a rating, some points to the most preferable, fewer points to the second, even less to the third, and so on. And the one who scores more points will win. This is the Borda method"A funny problem: in the same elections, with the same number of votes, the majority of votes, the anti-majority votes and the Bord method can give completely different winners. And the Bord method contradicts the Condorcet method. That's the same.

" Many years ago a meeting of experts on voting systems was held, and they voted on the most preferred method of voting. It turned out that the majority method did not receive a single vote, ”says Ulman.“ One of the most beloved ones was an approving vote when you for each candidate, make the choice “yes” or “no”, and the winner is the one who gets more choices “yes.”

Yes, I asked how they voted. “They actually used an approving vote,” said Ulman.

Many professional communities do the same, including mathematicians. You may think that only the most innocent and least controversial candidate can win this way, but the winners are actually revealed - and these will be the winners of Condorcet - with broad support. (Engineers do not like it very much; the Institute of Electrical and Electronics Engineers abandoned this practice). You can take care even more and cross the different options, or add a ranking to the approval of yes / no. One of the drawbacks may be that voters must have their own opinion about all the candidates on the ballot. “If someone tells you that you need to fill out a newsletter and rank all 20 options,” a lot of people will most likely be able to make the first and second choice by preference, well, maybe the third one - and then they will say “yes, I’m talking about the rest and did not hear. "

But the mystery of the identity of the next mayor of San Francisco was not even the main drama of the day. Instead of dividing candidates by party, in California, they all end up on the same ballot, and two people who got more votes than the others go to additional elections in November. And still, if they are from the same party.

California has long sought to make democracy surgery to correct both cosmetic and vital deficiencies. In the Gilded Age in California, politics was so corrupt that progressive reformers established an initiative that allowed all people who collected enough signatures to put the law to a vote. The two-person primaries, which are also used in Washington and Nebraska, also serve as tools to combatjerrimending . Like many Californian ideals, the voting system is both slightly insane and wonderful.

And she is doomed. In the 1950s, economist Kenneth Arrow set out to choose one, the best method of voting, one election, to rule all. As a result, he proved that the ideal method does not exist. Arrow's paradox , for which he received the Nobel Prize in 1971, says that beyond the choice by a majority of two candidates, there is no method for accurately determining the choice of majority.

Here it is, democracy. We cannot make the state ideal - we can only bring it closer to the ideal.