On the cold, foggy northwestern shore of Lake Superior lies the small city of Thunder Bay, Ontario. Not long after the fiery birth of planet Earth, magma poured through Mesoproterozoic rifts into the Earth’s crust and left behind sheets of hard diabase rock, which icy Canadian streams eroded over the eons into flat-topped mesas that today cast long shadows onto the vast body of water nearby. But just a few hours after the much less dramatic birth of time on January 1, 1970, at midnight Eastern Standard Time, something a bit more puzzling happened in this spot. Two towns merged into one.
Port Arthur and Fort William were two nearby towns, both growing fast and both about to grow into each other. The local governments decided that the two towns needed to merge. They had one question: what to name the new town? They also had an answer: ask the residents! The new city at the head of the lake would be named by popular vote. Suggestions were taken, ballots were drawn up with the leading suggestions, and a referendum was scheduled. The day came and citizens flocked to the polls to choose their favorite of the final three options. But when all the votes finally came in, the tally was … interesting:
Measuring happiness is hard. Philosophers don’t even have a consistent definition for it. But I think we can all agree that the residents of the new city would’ve been happier with either of the last two options than the first (and The Rev. Bayes would probably agree). Spoiler candidates constantly hassle first-past-the-post voting systems. But is there a better alternative?
All’s Fair in Love, War, and Politics
Serious discussion about electoral systems is in vogue these days. You may have heard that, in 2018, Maine switched to ranked-choice voting for its state elections. Unless you’ve been living under a rock (or outside of the United States) since 2000, you certainly haven’t missed the constant controversy over the Electoral College’s winner-take-all style system that decides U.S. presidential elections.¹ But before we pillory any particular method, let’s first prove that a fair system exists. We will explore some foundational results of social choice theory, a branch of mathematics dedicated to formally studying such problems.
When the field of social choice theory blossomed in the middle of the 20th century, mathematician Kenneth Arrow theorized three² conditions that any mathematically fair voting system should satisfy. Make sure you agree with each one before moving on — they’re all quite reasonable.
A voting system respects unanimity if when every single person votes for one candidate, that candidate wins. First-past-the-post systems obviously satisfy unanimity. For any ranked voting system, we phrase unanimity as follows: if every person ranks candidate A higher than candidate B, then the outcome will rank candidate A higher than candidate B. Note that candidate A doesn’t necessarily win, but only beats candidate B. For all we care, a third, dark horse candidate C could beat both A and B. And, if one single person ranks B above A, we don’t have any constraints on the outcome anymore. A could beat B or B could beat A. It doesn’t concern unanimity anymore. Unanimity is sometimes referred to as the weak Pareto condition or Pareto efficiency.
A voting system respects nondictatorship if there is no person P who can always determine the outcome of the election. For example, a voting system that discards every ranking except person P’s and then uses P’s ballot to determine the outcome violates nondictatorship.
3. Independence of Irrelevant Alternatives
A voting system respects independence of irrelevant alternatives if a candidate dropping out doesn’t affect the results. Let’s walk through an example before we state it more formally. Suppose Senators Henry Clay, Daniel Webster, and John C. Calhoun all decide to run for the same Senate seat. Clay barely wins, Webster follows in second, and Calhoun ends up third. The narrowly decided election falls within the margin for a mandatory recount. But before the second tally, Senator Webster decides that he would rather allocate his time to further developing his apportionment algorithm, and withdraws. Fortunately, everyone specified a second choice, so the count can continue without a second whole election. Then the second time, Calhoun beats Clay! Apparently, many of Webster’s voters specified Calhoun as their second choice. This outcome confuses the voters — if the people preferred Calhoun to Clay in a direct match-up, obviously Clay should never beat Calhoun! A voting system satisfying independence of irrelevant alternatives must preserve the relative ranking of candidates, regardless of any other candidate’s addition into or removal from the election.
Now we will tell the story of an election with a voting system satisfying all of these conditions. It will have three parts.
Part I: The Pivotal Voter
Our story will begin with three³ main characters: Alice, Bob, and Carol. Alice, Bob, and Carol have each campaigned for weeks to earn a seat on the local board of supervisors. Now it’s time for the county to vote. The outcome isn’t yet determined, so we’ll prepare by brute force — checking every possible way that people could vote and making sure the voting system doesn’t break.
We’ll start by assuming that everyone hates Bob. Perhaps Bob tweeted in favor of locating a nuclear waste dump site in the county six or seven years ago, and Carol’s campaign dug it up and plastered it all over television. Then (from now on using A, B, and C to represent Alice, Bob, and Carol respectively) the ballots of all n citizens would look like this:
Since the system in use satisfies unanimity, the outcome will rank both A and C higher than B. We will call this Configuration 1.
Next let’s assume that everyone loves Bob. Maybe, in an alternate universe, Bob tweeted that he would give everyone in the county $1,000. Then the ballots might look more like this:
Again, because of unanimity, the outcome will rank B higher than both A and C. We will call this Configuration n. Now that we have two of the configurations, we can define all the other ones. Let Configuration i be the configuration where i voters hate B and n – i voters love B.
Clearly there’s some minimum k for which Configuration k has B beating A. It might not be when B gets to a majority, since we don’t know exactly how we’re counting votes. But at one end B loses to A and at the other B wins to A, so somewhere in between, even if it’s at k = n – 1 or even k = n, B must beat A at some point. We’ll refer to citizen k, the citizen whose ballot causes B to win, as the pivotal voter for B over A. We’ll name her Karen.
Part II: A Dictator
Observe that it doesn’t matter how the voters in any configuration rank A and C relative to each other. Because of independence of irrelevant alternatives, we can take C out of the running and Karen still determines B over A. Karen, however, has decided to become an active participant in the election, and she’s no longer satisfied with such an incomplete influence.
Karen splits the configurations into two sections: all the ballots before her and all the ballots after her. She notices that in the configuration where the first section is
and the second section is
and she votes [A, B, C] then A beats B (by independence of irrelevant alternatives, the new ballots rank A and B the same way as Configuration k − 1) and, by unanimity, B must beat C (since B ranks above C on every ballot).
Karen also considers voting [B, A, C]. Removing C from the ballots, the configuration matches Configuration k. That means the order of A and B switches (because of independence of irrelevant alternatives): B will beat A. By independence of irrelevant alternatives again, the relative ranking of A and C remains unaltered. Then B beats C.
A serious problem now arises. What if every other voter switched the order of B and C on their ballots? Because of independence of irrelevant alternatives, the relative rankings of B and A and A and C both would be preserved, so B would still beat C. But only Karen prefers B to C! Karen has become a dictator for B over C.
Part III: The Dictator
A, B, and C were arbitrary placeholders, so Karen has counterparts: a dictator for A over C, a dictator for C over B, and dictators for the three other permutations of two candidates from three. But as fate dooms every triumvirate (or sextumvirate, in this case) to collapse into autocracy, dictating B over C no longer quells Karen’s ambition — not when others possess equivalent power.
But then Karen notices something. She may not have any competition after all. The pivotal voter for B over C (the dictator for B over A) must appear in line before or at the same place as Karen, or else Karen wouldn’t be able to truly choose B over C. By the same logic, the dictator for C over B must appear after or in the same place as the pivotal voter for B over C. But nothing about the story we’ve told so far depends on B and C specifically. That means we could switch B and C on everyone’s ballots and everything established so far would remain (with B and C switched). Once again, Karen’s observation leads to a startling realization. Since the dictator for C over B must appear both before or at the same place as Karen and after or at the same place as Karen, Karen must be the dictator for C over B. Karen, then, chooses which of B or C beats the other in every case. Karen is a dictator for B and C. But the dictator for B over A was also sandwiched in between Karen and Karen. That means Karen must also be the dictator for B over A. Repeating the argument (with different letters), only Karen decides which of B and A beats the other. Therefore Karen, from her humble beginnings as citizen k, used deductive reasoning to determine that she completely dictates the outcome of the election.
On Aristotelian Democracy
The story told here did not tie itself to any one specific voting system. Any voting system that weakly⁴ ranks candidates cannot satisfy unanimity, nondictatorship, and independence of irrelevant alternatives at the same time. Mathematics refers to this bleak conclusion as Arrow’s theorem.⁵
In practice, voting systems usually break independence of irrelevant alternatives. As the city of Thunder Bay learned at its foundation, first-past-the-post certainly does. But do not lose all hope — though no perfect system exists, some systems break only rarely, and we do have ways to measure which voting systems work better than others.
Like the ancient plateaus overlooking the shores of Thunder Bay, many truths transcend human scales. Mathematics provides deep insight into the structure of the universe, helping to reveal these majestic laws. But Arrow’s theorem, while embedded in that beautiful structure, really says something that is on a human scale. It’s so familiar a truth to us that it may even bring us comfort to hear. Arrow’s theorem tells us we will never make everyone happy. People will always disagree, someone will always lose, and someone will always win. Human society isn’t perfect and can never be. But that’s okay: in utopia, no one ever invents. Problems drive innovation; in their absence, we stagnate. Stability breeds stability, but who knows what wonders chaos can conceive.
- Controversy over the Electoral College doesn’t fall entirely within the scope of this discussion. It’s not even close to a mathematically fair voting system, but it also isn’t supposed to be.
- Depending on which assumptions you include in your definition of a social choice function, you may need up to five conditions.
- A story faithful to the original mathematics would consider n main characters. But the same tale can be told more simply and more concretely with only three.
- In a weak ranking, a voter can rank two (or more) candidates as equal. In a strong ranking, voters must always make a choice between two candidates. Any strong ranking is a weak ranking, but not vice versa. The story told here, then, applies to any weak or strong ranking.
- Named, of course, after the same Kenneth Arrow from before.
This article was adapted from a lecture I gave to the Virginia Tech MAA math club on October 18, 2018. The informal proof of Arrow’s theorem is based heavily on the proof given on the Wikipedia page, but with significant influences and background gathered from other sources. The lecture also contained information about alternative voting systems and how to effectively compare voting systems, but that has been abbreviated here for the sake of this already lengthy article.