page forty

The Ballot

Thirty-nine rooms have weighed, dated, bred and forecast, and in none of them has anybody ever had to agree. Today's number founds a learned society, lets its President die and has every member rank the candidates. The papers are honest. The trouble is that there is more than one honest way to count them.

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    The papers, sorted into piles that read the same, first choice at the top

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    Society
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    The arrow keys walk to other days. Every day is a different society, a different field of candidates and a different set of papers.

    What is real here and what is not

    The five ways of counting and the theorems are real. The society, its members and its candidates are the day's number.

    1. 01

      Borda and Condorcet

      Jean-Charles de Borda put his points system to the French Academy of Sciences in 1770, because he could see that a plurality can elect the one candidate most members like least. The Marquis de Condorcet answered in 1785 that the right winner is whoever beats every rival head to head, and in the same essay found that sometimes nobody does: a majority can prefer A to B, B to C and C to A. Ramon Llull had written down both ideas in the thirteenth century, and they were lost.

    2. 02

      Hare and the Oxford don

      Thomas Hare published his transferable vote in 1857 for electing many members at once; its one-seat form, the instant runoff, was put forward by W. R. Ware at MIT in 1871. In the 1870s Charles Dodgson, who wrote Alice as Lewis Carroll, printed pamphlets on how Christ Church ought to count its own elections, after watching a well-liked candidate lose to a scheme. A Victorian society with no rule for counting is exactly where these arguments were had.

    3. 03

      Arrow, Gibbard, Satterthwaite

      Kenneth Arrow proved in 1951 that with three or more candidates no method of turning ranked papers into a ranking of the whole society can meet a short list of reasonable conditions all at once, except a dictatorship. Allan Gibbard (1973) and Mark Satterthwaite (1975) proved the sharper thing the caucus shows you: every such method that is not a dictatorship and can elect anybody can sometimes be beaten by a voter who lies.

    4. 04

      How the day is drawn

      Candidates and members are placed on a hidden map of opinion and members rank candidates by distance, with some noise and a few who rank at random. The page keeps drawing electorates until at least three of the five counts disagree and some faction could gain by lying under one of them. The house always has an odd number, so no head-to-head is tied, and other ties go by lot. The caucus is scored against a search of every paper your faction could hand in.

    Forty rooms, and one number doing all the work. Today it calls a meeting, hands out the papers and leaves you to find out that the count, not the vote, decides who wins.