# Fixing Elections while Mocking Math

This is a picture of Democratic Institutions Minister Maryam Monsef of the current Government of Canada from a CBC Story. Her job is empowering Canadians to have their say about our democracy. What she is doing in this picture is mocking a simple formula for measuring the fairness of an election. This formula appeared in a report on electoral reform in Canada. This is not her job. This is the opposite of her job. When an expert panel is convened to advise the citizens and government of their options to make elections more nearly fair, mocking the options (other than the one that let you win the last election) is malfeasance, fraud, and betrayal of duty. This is triply so in an official in charge of enhancing fairness.

No-one should mock an attempt to measure fairness. Mocking it by exploiting fear of math is especially vile.

The minister is exploiting fear of math to dismiss a report commissioned by the government. A possible outcome made more likely by this is retention of political power by her party. Given all the talk recently about fixing elections, a sober consideration of the options would far more palatable than deriding those options, a fact that Ms. Monsef seems to have belatedly recognized. Math is the Right of all Free People which places Ms. Monsef’s act squarely in opposition to freedom and democracy. In this post Occupy Math will go through the details of the Gallagher index — the formula Ms. Monsef is mocking — as well as a simple formula for telling if the election in your state or county has been rigged with a gerrymander

The Gallagher index

In pure math notation, the formula is a little intimidating, but let’s take this puppy apart and understand it.

1. This formula contains a sum over all political parties.
2. For each party, $V_i$ is the percentage of votes the party got.
3. For each party, $S_i$ is the percentage of seats the party got.
4. Take the difference between the percent of seats and votes for each party.
5. Square the differences — this gets rid of minus signs and make bigger differences more noticeable.
6. Add up the squared differences, divide by two, and take the square root.

Since we squared our differences, taking the square root puts the units back into “percent of” units. The number that comes out of the formula is zero if the election was completely fair (which is impossible because of rounding errors). Numbers near zero indicate a relatively fair election, by one measure of fairness. The index measures disproportionality of an election — the difference between the proportion of the vote a party got and the seats they obtained in the parliament or congress.

This link leads to a whole book by Michael Gallagher on electoral math and a spreadsheet for calculating the Gallagher index. This is the Gallagher who thought up the index.

Similar to the loathsomeness of mocking math is secretly using it to rig an election.

The Republican party probably managed to rig the 2016 election in a big way. This is a harsh claim that requires strong support. The mechanism was gerrymandering — creating electoral districts that make one party waste their votes. What is a wasted vote? There are two sorts of wasted vote — votes cast by someone whose candidate did not win and votes cast for a winning candidate in excess of those needed to win. In a democracy there must be wasted votes. If the number of wasted votes is very large, it is likely that someone is up to no good. A wonderful example of an election rigged with gerrymandering appears in the way Republicans used their control of the state government of Wisconsin to create a “hyperpartisan gerrymander”. This attempt was so odious that the courts ordered Wisconsin to try again. How was the court convinced a gerrymander had taken place?

Two sorts of unfair districts are created to generate an advantage for one party.

1. A packed district places as many voters from one party into a district as possible. This means that party will win the district — but by a huge margin causing a whole lot of wasted votes beyond those needed to win.
2. Cracked districts divide the votes from one party among many districts so that they are not a majority in any of those districts. This causes wasted votes where the party’s candidate does not win.

The fact there are two ways to cook the election with bogus voting districts makes gerrymandering a type of optimization problem with many different solutions to the problem of fixing an election. A consequence of this is that gerrymanders can be “cooked to order” to fit current laws.

Clearly, we need a simple measure that detects gerrymanders!

The Wisconsin Republicans held secret meetings, designed a tight gerrymander, and then destroyed the evidence. What simple formula, based on publicly available information, can be used to detect this sort of anti-democracy chicanery? The answer is simple: divide the wasted votes of one party by the wasted votes of the other. If this number is not near one, either above or below, there was a de facto gerrymander.

The ratio of wasted votes is simple and easy to compute, it is based on public information, and it is fairly hard to argue with. It detects not only intentional gerrymanders but accidental ones, which are possible. It also removes the need to prove intent to steal the election. Having an objective, non-partisan measure of electoral fairness is even more valuable when it avoids the need to cast blame or actually provide legal proof of nefarious intent. The ability to provide strong, quantitative, non-partisan evidence that an election was unfair was what convinced the court. A measure like this can be embedded in law in a manner that makes it clear objections to the measurement are anti-democracy.

This brings us back to Ms. Monsef.

One would think it difficult to find similarities between the Canadian Liberal Party and the American Republican Party, but Ms. Monsef pulls it off. Both parties have declared opposition to objective, quantitative evidence by mocking it. Occupy Math has little hope that the Republicans will embrace evidence, but this behavior is not entirely typical of the Liberal party. Occupy Math urges the Government of Canada to stop publicly mocking mathematics!

Most people think that democracy is a simple system, and pure democracy is simple, but it can also be tyrannical. A big complicated government like the Democratic Republic of the United States or a parliamentary government like Canada or the United Kingdom creates many points at which citizens must participate with clear eyes to keep their government officials and politicians from getting up to no good. One of Occupy Math’s themes is that math will set you free but one could also say math gives you clear eyes. In a future post we will look at more math relevant to fair and democratic government including the terrifying Arrow’s theorem. Got a good topic in this area? Please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics