In 1880, trade unionist George Howell published a pamphlet titled “One Man, One Vote”. Since then, versions of the slogan “one person, one vote” have been used in a variety of settings to express a democratic ideal: elections should provide every citizen with an equal say in governance.  But in America, the reality still falls short of the ideal.

One Person, Zero Votes?

One problem is that many Americans do not vote.  According to the Pew Research Center, only 56% of voting-age Americans participated in the 2016 presidential election, as compared to 80% in Denmark, 83% in Sweden, and 89% in Belgium.  (These numbers measure the voter turnout in the most recent national election in each country as a percentage of voting-age population.)

Countries with higher turnout often achieve higher rates of voter registration by automatically registering eligible citizens; according to the statistics from Pew, only 64% of voting-age Americans were registered to vote at the time of the 2016 election, compared with voter registration rates of 94% in Denmark, 96% in Sweden, and 98% in Belgium.

This isn’t just the fault of the voting public.  Many states have policies that actively discourage voting, sometimes deliberately adopted as a way to disenfranchise poor and black voters.  Since section 4b of the Voting Rights Act was struck down in the 2013 Supreme Court case Shelby vs. Holder, states with a history of racist discrimination are no longer required (at least in practice) to get approval from the federal government before changing their election laws.  Common ways of lowering voter turnout include restrictive ID requirements, making it harder to vote, limiting the hours and days at which people can vote, and preventing people with criminal convictions from voting.

Does Your Vote Count?

Even when people manage to vote, their votes don’t count equally.  There are, in fact, philosophical questions about what equality would even amount to.

Of course, there’s one sense in which your vote is unlikely count at all; if your preferred candidate or ballot initiative wins by more than one vote, than the outcome would have been the same even if you’d stayed home.  Even supposing your preferred candidate did win by one vote, who’s to say that it was your vote that was responsible, rather than one of the other votes for the same candidate?  But somehow, our ballots add up to make a difference, even if the responsibility for an election outcome can’t be pinned on any individual.

How Much Does Your Vote Count?

Voting districts present a more immediate problem for equality and fairness.  Representatives in the US are elected to particular districts by a majority vote.  But if you take the same bunch of voters, and find different ways of dividing them up into districts, you can change the outcome of an election without changing how any individual votes.  

Dividing up districts to advantage a particular political party is called gerrymandering (after a map designed by Governor Elbridge Gerry, where which one of the districts was so oddly shaped that the cartoonist from the Boston Gazette decided it looked like a salamander).  Corrupt representatives use the techniques of “packing” (sticking all the opponent’s constituents into one district so that the extra are wasted) and “cracking” (spreading the opponent’s constituents out over many districts, so that they can’t win the majority in any).

In her remarks to the American Academy of Arts and Sciences, mathematician Moon Duchin discusses three possible ways to measure gerrymandering.  

The measure of partisan symmetry uses an equation called the seats-votes curve, which represents how many seats each party would have won if it had won x percent of the vote (for every x between 0 and 100).  Judging by partisan symmetry, gerrymandering occurs when the two parties would have won different numbers of seats with the same proportion of the vote—for instance, if Democrats could win 30 out of 90 seats with 25 percent of the vote, while Republicans would only win 15 seats with 25 percent of the vote.  (This way of explaining things assumes a two-party system, but that’s appropriate for the examples that the method of partisan symmetry aims to cover.)

A problem with the measure partisan symmetry is that it relies on speculation about what would have happened if people have voted differently.  A newer measure which avoids this problem is the efficiency gap.  The efficiency gap begins with the concept of a wasted vote—one that makes no difference to the outcome of an election.  Every vote for a losing candidate is a wasted vote, but so is every vote for a winning candidate above the bare minimum needed to win.  There is an efficiency gap (and therefore gerrymandering, according to the measure) if one party wastes significantly more of its votes than the other.  

The efficiency gap measure is clear, well motivated, and easy to use but it suffers from a few oddities.  In a scenario where 65% of the people vote for a party that wins 65% of the seats, the efficiency gap measure counts this outcome as unfair to the winning party (which could have leveraged its votes to win even more seats).  As Duchin points out, no single number can capture all the complexity of gerrymandering.

A third measure of gerrymandering involves sampling.  The idea is to consider all possible ways of drawing district boundaries, and assign a probability to each based on its similarity to the other possibilities.  An outcome then counts as gerrymandered if it was highly unlikely for the votes cast to lead to that outcome. The sampling method offers up some counter-intuitive results, and is still under-explored, but it suggests another promising way to understand gerrymandering.

Whither Democracy?

Disenfranchisement and gerrymandering are significant threats to democracy.  They are not, of course, the only threats. Achieving the ideal of “one person, one vote” is not enough to ensure freedom: citizens also need enough information to make reasonable decisions, enough freedom and education to think for themselves, and good options to choose among.  But achieving more equal representation through voting would nonetheless represent an important kind of progress.