The Mathematics of Democracy

15 January 2021

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 

Comments (4)

Sharonartist's picture


Sunday, January 24, 2021 -- 1:08 PM

I participated extensively in

I participated extensively in local politics (Berkeley, California) between 2000 and 2010, and based on my experience, I wrote a 3-part series for the local paper on the problems with the district election system. District elections have many bad consequences, some of which are discussed in the commentaries below. Later, recently, when California cities are under court pressure to find a way to provide more minority representation, I discussed the issue with someone who was trying to devise an election system that would provide "perfect" representation, as if that would somehow fix our policy-making problems. Personally, I doubt that the half of the public that doesn't vote at all is too concerned about receiving mathematically perfect representation. Instead, they don't participate because they perceive something wrong with the policy-making result, no matter WHO is elected. In fact, the incentives for good behavior once officials are elected to office are just as important as, or even more important than, HOW they get elected in the first place. With all their focus on IRV, proportional voting, gerrymandering, etc. people forget about the second half of the equation. The problem with district elections is that they incentivize bad behavior.

The three commentaries are here:

Sharon Hudson
Oakland, CA

MJA's picture


Friday, January 29, 2021 -- 10:50 AM

I think the truest and

I think the truest and strongest form of Democracy is self-control. Any other forms of government or control be it elected or dictated only weakens the masses of their own power, a true loss of Oneself. We are taught that it is patriotic to elect another to control us, when the real question is: who needs to be controlled, the governed or the governors? +

Tim Smith's picture

Tim Smith

Wednesday, February 3, 2021 -- 1:46 PM

This is a well done blog post

This is a well done blog post.

I am both relieved and alarmed by the recent national, local and even state elections here in Portland Oregon. This makes this show timely to boot. Now in the lull of unfortunate gridlock in the case of our nation and group think in the case of my city is a good time to think about the philosophy of democracy. Now is the time to think about what is best, not based on outcomes, but on representation of what we saw in the recent elections. That so much of my recent philosophical thought has been based on alarm harkens back to the fact that all philosophy is personal on a certain level. Maybe this show and post will lighten that load... probably not.

Moon presents the issues; One person one vote as founded in the 60's, Packing/Cracking, Arrow's Impossibility theorem , many ways to any one thing, efficiency gap and data transparency. Even random lottery based systems and naming conventions are touched on. Certainly ranked voting is the key reform that would compliment our current system. Even a simple two position ranking would greatly reduce the Nader effect that robbed Gore in his presidential run.

Ranked voting can also be manipulated. Michigan and Nebraska shared a college football championship when one coach manipulated his ranked voting to give Nebraska the coaches award. This sparked reform that we endure today.

Not all votes are equally valued. Not all votes should be equally valued. A large portion of US citizens would fail the test given recent naturalized citizens. Competency is undervalued as a criteria for voting. This aside, tests for competency have been used to disenfranchise voters however unfairly.

The Federalist Papers reflection on districting pre dates the party system in the US. Party politics changes everything with respect to those arguments which assumed a geographical and social reality that really never existed in the first place.

Hmm... what is best? Reversing Citizens United would seem first and foremost on my list. Regardless of competency we should not allow commercial entities suffrage on any scale. Money drives politics. Wealth inequity drives injustice.

Topology would seem to be the best approach if the spaces of interest could be defined in all the correct dimensions and to the proportions of opinion. Then the edges of this space could be minimized to allow the best reflection of democracy. It is likely that ultimately we are all best organized into water conservation districts than any math driven social or political network.

I like Moon's understanding. The Republicans won seats in the House and lost seats in the Senate and the Presidential election. Some might say this is democracy working. It is the government itself that is failing. It would be good to quantify the debates in our society so we all could understand each others thoughts better, if not our own. Data transparency with a nod to data privacy would go along way to getting our government and lives working again.

The revival of Conundrums in remote and interactive space was interesting. I wouldn't have been able to represent as well as George did to Ray and Josh. I thought that more a continuation of the discussion of Democracy as economic boycotts have their own reality. Early on, Conundrums were taken by one of the philosopher hosts, explicated and advice dispensed. This format was much more inclusive, back and forth and directed to the folk. Let's get that spirit back into our Democracy.

This was a good show. I didn't know about Arrow prior to reading Ray's blog or listening to this show.

Posted in the recent show notes as well as here...

Harold G. Neuman's picture

Harold G. Neuman

Saturday, April 24, 2021 -- 5:30 PM

Have made comments on 1.

Have made comments on 1. Democracy; 2. Meritocracy. 3. One person, one vote, 4. Several more connections that need not be repeated. Remarks I last made, today, on reason were signal. Because those illustrated the connectivity of things you regularly revisit, every few years Your formula, though logical, does not bring answers to the questions raised. Your contributors, though academic enough, do not solve questions raised.. I cannot write books on philosophy. It would be a waste of time because I have no academic credentials in the field. As an autodidact, I an a pariah. To repeat several known notions regarding philosophy and related concepts: philosophy is the attempt to see how things hang together
Sellars. Beliefs are ....personal affairs, which are adventures, etc... Dewey. And, according to Davidson, there are propositional attitudes: belief, opinion, desire, and expectation---others. Here's the rub, seems to me. If you want to pursue philosophy, you need to sometimes answer questions. I try to do that. Sans degree. Avec experience.