Anthony Gottlieb digs beneath the peculiar mathematics of voting in a review of Numbers Rule at The New Yorker

This excerpt taken from a larger article available at The New  Yorker:

Whenever the time came to elect a new doge of Venice, an official went to pray in St. Mark’s Basilica, grabbed the first boy he could find in the piazza, and took him back to the ducal palace. The boy’s job was to draw lots to choose an electoral college from the members of Venice’s grand families, which was the first step in a performance that has been called tortuous, ridiculous, and profound. Here is how it went, more or less unchanged, for five hundred years, from 1268 until the end of the Venetian Republic.

Thirty electors were chosen by lot, and then a second lottery reduced them to nine, who nominated forty candidates in all, each of whom had to be approved by at least seven electors in order to pass to the next stage. The forty were pruned by lot to twelve, who nominated a total of twenty-five, who needed at least nine nominations each. The twenty-five were culled to nine, who picked an electoral college of forty-five, each with at least seven nominations. The forty-five became eleven, who chose a final college of forty-one. Each member proposed one candidate, all of whom were discussed and, if necessary, examined in person, whereupon each elector cast a vote for every candidate of whom he approved. The candidate with the most approvals was the winner, provided he had been endorsed by at least twenty-five of the forty-one.

Don’t worry if you blinked: bewildering complexity was part of the point. The election aimed to reassure Venetians that their new ruler could not have been eased into place by backroom deals. Venetians had been coming up with inventive ways to make political decisions for a couple of hundred years before they concocted this rigmarole. Earlier elections in Venice, and in other Italian communes, required a winner to be endorsed by two-thirds, or sometimes three-quarters, of the voters. The hallmark of the Venetian approach has come to be known as “approval voting,” in which electors do not need to pick a favorite but may vote for several candidates they like.

In 1179, two years after a stay in Venice, Pope Alexander III reformed papal elections, perhaps because he liked some of what he saw there. Among other things, he abolished a tradition of requiring unanimity among the cardinals, and settled for a two-thirds majority instead. You would expect a two-thirds consensus to be easier to reach than unanimity, but papal conclaves in the thirteenth century seemed to go on forever. On six occasions, it took several months to choose a Pope. In 1241, by some accounts, the head of the civil administration in Rome threatened to exhume the corpse of the defunct Pope and parade it through the city in full regalia if the cardinals didn’t settle on a new one. Eventually, the cardinals got the hang of it. After some tinkering over the years, the two-thirds rule was reconfirmed by the present Pope, in 2007.

“What is done by two-thirds of the Sacred College, that is surely of the Holy Ghost,” Pius II said of his own election, in 1458. He did not explain why divine approval kicks in only at the two-thirds mark. Since then, mathematicians, economists, and political theorists have made their own attempts to elucidate the math of voting, and figure out better electoral systems. The story of these efforts is told in “Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present” (Princeton; $26.95), by George Szpiro, a journalist and mathematician.