The Mathematics of Democracy

Shouldn’t everybody have an equal vote? Isn’t majority rule just an excuse to keep minorities down? Is a truly fair democracy even possible? And how do we decide what counts as fair in the first place? This week on Philosophy Talk, we’ll explore answers to these questions!

Shouldn’t everybody have an equal vote? Isn’t majority rule just an excuse to keep minorities down? Is a truly fair democracy even possible? This week on Philosophy Talk, we’ll explore answers to these questions!

Rousseau said that the law (when it’s legitimate) expresses the general will; it’s what people in general want. But how can we determine the general will under conditions of widespread disagreement? We should probably take a vote. But that doesn’t settle the matter; we’ll still have to decide what to vote on, and how to tally up all the votes fairly.

This turns out to be trickier than it looks. In his 1785 “Essay on the Application of Analysis to the Probability of Majority Decisions,” the Marquis de Condorcet describes a paradoxical phenomenon: even if each individual voter has consistent preferences, deciding the group’s preferences by majority rule can have inconsistent results, so that the group prefers candidate A over candidate B, prefers B over a third candidate C, and prefers C over A. (For a detailed explanation of the paradox, see the Stanford Encyclopedia of Philosophy entry on voting methods.) 

A famous 1951 proof by Kenneth Arrow challenges not just majority rule, but any possible voting system. Arrow considered ranked-choice voting systems, where each voter provides a ranking of all the candidates on a ballot, and these individual rankings are put together to create group ranking. He claimed that a good voting system should satisfy all five of the following requirements (which are labeled with their names):

• Universal Domain: No matter how individual voters rank the candidates, as long as each of them is consistent, the system outputs a group ranking.

• Ordering: The group ranking should be consistent, and should not allow for ties.

• Pareto Principle: If every voter ranks A above B, the group ranks A above B as well.

• Independence of Irrelevant Alternatives: Whether the group ranks A above B depends only on how the individual voters rank A and B; it doesn’t depend on their preferences about other candidates.

• Non-dictatorship: The voting system shouldn’t pick one individual whose preferences dictate the group’s ranking, regardless of how others vote.

(For a more formal statement of Arrow’s conditions, see the Stanford Encyclopedia of Philosophy entry on social choice theory.) Unfortunately, as Arrow showed, no possible voting system fulfills all five of the requirements.  

It’s not so easy to avoid the problem by tinkering. You could try changing the setup to let voters pick their favorite candidate, or changing ordering to allow ties, but a growing collection of voting impossibility theorems, which show that the problem is robust under a range of assumptions.

Does all this math mean that democracy is impossible? Only if your standards for democracy are very demanding. Universal Domain is a very stringent requirement; it says that the voting method has to yield a result, even in weird situations that seldom or never arise. And Independence of Irrelevant Alternatives starts to look less plausible when you recognize the logical relationships between preferences about different pairs of candidates. Preferring A to B and B to C means you can’t rationally prefer C to A, so your preferences concerning B are at least sometimes relevant to your preference between A and C.

Impossibility theorems are somewhat abstract and theoretical, but mathematics also helps to illuminate more immediate problems for democracy, such as gerrymandering—the practice of dividing up election districts in a way that favors one party over others. By “packing” opposing voters into districts where they form the overwhelming majority, and “cracking” the remaining opponents into separate districts where they can’t achieve even a simple majority, gerrymandering gives the party an unfair advantage.

But how can you tell when a district plan is gerrymandered? You can’t always discover the intentions of the people who drew the plan, but you can ask questions about its effects. You might be suspicious of districts that are irregularly shaped… but there are sometimes good reasons for this irregularity (like a district that’s bordered by a river or a coastline), and someone might cheat by using districts that skew the results of elections, but look ordinary enough to fly under the radar.

Our guest this week, mathematician Moon Duchin, has considered other options. One option is to check for gerrymandering by measuring partisan symmetry—roughly, whether different parties are treated alike in similar scenarios. For instance, if Republicans can be elected to 65% of the seats in a state with 60% of the vote, then partisan symmetry requires that Democrats be able to win 65% of the seats in scenarios where they get 60% of the vote. Another way to measure gerrymandering is the efficiency gap, which compares the number of “wasted” votes across parties. A vote is wasted if it’s cast for a losing candidate, or if it’s for a winning candidate who was already above the margin required to win. And you might think gerrymandering involves mismatched efficiency gaps between parties.

A third option, pioneered and defended by Moon, asks: how many ways of re-drawing the districts would actually affect the results of the election? If the actual districting plan has extremely unusual or surprising results, that’s a red flag for gerrymandering. The third approach is mathematically difficult, because there are so many possible ways to draw election districts that it’s usually impossible to describe and reason about them all, but today’s computer scientists and mathematicians have ways of drawing a random sample of them.

Mathematics raises theoretical challenges to democracy, but it also gives us practical solutions. While I don’t think mathematical and philosophical thought are enough to heal our divided democracy, I’m excited about their ability to help. And I’m excited to learn more from talking to Moon on our upcoming show!

Image by Matthias Wewering from Pixabay