5 Paradoxes About How Voting Works

5 The Winner May Be Someone the Majority Oppose

Let’s start with something you’ve surely heard about but that we still haven’t solved.

When we declare whoever gets the most votes to be the winner, that’s not the one-and-only system of elections but something called first-preference plurality. The winner received a plurality of votes (more votes than anyone else) but perhaps not a majority of votes. It’s even possible that the majority of voters specifically opposed this candidate. 

You’ll see this in the form of spoiler candidates — people who never had a shot at winning but draw just enough votes to ensure an otherwise viable candidate loses. As you might know, Al Gore lost the 2000 election because he lost Florida by 550 votes, and Ralph Nader got 97,000 votes in Florida that year. But did you know there were actually eight different third-party candidates in that election who all managed to get more votes in Florida than the difference between Bush’s and Gore’s final tallies?

An upset like that also might not come from a fringe candidate. If a new candidate enters the race, who’s an exact clone of the current leading candidate, the two of them will precisely split the vote. First place will go to a third candidate, even though a majority of people would have preferred either of the clones. We call this the cloning paradox, and one of the goals of any voting system is that the invasion of a clone shouldn’t upset the race.

Political parties and primaries exist to keep clones from battling in the ultimate election. Another solution, which nullifies any kind of spoiler candidate, is runoff elections. After the first round of voting, we eliminate whoever came in last place and let people vote again. This might sound complicated, but when a presidential race takes years and costs $16 billion, a second or third round of voting isn’t that huge of an extra step. 

Runoffs also needn’t take place in different stages. We can have an instant runoff by instructing voters to rank candidates in order of preference, instead of choosing just one. Then we can use those rankings to determine how they’d vote if we did hold a series of rounds, each with one fewer choice. 

When we describe it like that, an instant runoff, also known as ranked-choice voting, appears to have no shortcomings at all. 

Well, about that...

4 Runoffs Can Give Someone the Win After People Like Them Less

Now, let’s leave the realm of actual problems and look at the theoretical, in the form of a scenario modeled after one in this YouTube video. 

Imagine an election with three candidates. We’ll call one “the villain,” one “the moderate” and one “the nut.” In this hypothetical, the nut isn’t some total fringe candidate but is narrowly trailing the other two. Under first-preference plurality, the villain wins. With a runoff, the nut gets eliminated, and their supporters otherwise favor the moderate, so the moderate wins in the end. 

But now, let’s imagine instead that the villain does something really dumb right before Election Day. Say, he refers to mailmen as “people,” and this enrages one segment of his base, who were counting on all mailmen being executed if their guy won. These voters don’t switch to the moderate. They switch to the nut. The moderate comes in third on Election Day and is eliminated.

The moderate voters came from across the spectrum, so with their candidate gone, half of them move to the nut, and half move to the villain. The villain ends up winning the election, even though he wouldn’t have had he never scared away that section of his base who oppose postage.

Could something like this actually happen? This is something philosophers worry about each night, philosophers who also come up with such other scenarios as...

3 Condorcet’s Paradox

The idea behind these runoffs is that while we may not find one candidate who’d get a majority when they go against everyone at once, we can find one who’d get a majority if they went head-to-head with everyone individually. But what if we can’t? What if we fall into a election where the majority prefer the nut to the moderate, a majority prefer the villain to the nut and a majority prefer the moderate to the villain? Who should win?

“We don’t have to worry about that,” you might say, “because the situation you described is completely impossible.” It is not impossible, and we can calculate the probability of it occurring this way:

Explaining that math is beyond the scope of this article, so let’s just tell you that we know about this paradox thanks to the Marquis de Condorcet, a mathematician who lived in Revolution-era France. His research into proposed systems of voting should have come in handy when his country threw out the monarchy, as Condorcet was on the side of the revolutionaries. Then it turned out that he wasn’t the exact variety of revolutionary that was most favored, so he ended up going into hiding to avoid being killed. He ended up dying by poison, either by his own hand or through assassination. 

We first knew about the paradox from Condorcet, but he wasn’t the first person to come up with it. In 2001, two centuries after Condorcet, we discovered a lost manuscript from the 13th century called Ars Magna. The writer was Ramon Llull, a Spanish missionary, and he came up with the same voting system Condorcet later would and foresaw the same possible flaws. 

The manuscript doubtless contains many other secrets, including how to turn lead into gold, so we need to parse it all while we still can. 

2 A State Can Lose a Representative If We Raise the Total Number of Reps

In 1793, the United States had a population of under four million and a House of Representatives with 105 members. The population rose steadily for more than a century, and we kept expanding the House accordingly, till around 1930, when we decided we were going to cap the House at 435 no matter what. 

New representatives always had a chance of making some states unhappy. If the country’s population rose enough for the House to gain exactly one new member, one new state would gain a representative, and every other state would find itself with proportionately less representation. But then, in 1880, the clerks at the census office discovered something weirder. If the total number of reps rose from 299 to 300, Alabama would find itself with fewer representatives. No, it wouldn’t just find itself with fewer than whichever state gained a seat. Alabama would have fewer reps when the House seats 300 than Alabama would have when the House seated 299. 

At the time, Alabama had 2.56 percent of the country’s population. With a 299-seat House, that meant it got a quota of 7.65 reps, since 7.65 is 2.56 percent of 299. Each human being is, under God, indivisible, so that figure got rounded to eight reps. With a 300-seat House, they’d now have a quota of 7.68 reps, and you might assume that would round to eight as well. But no: Here, it got rounded down to seven.

If we rounded to the nearest whole number the quota each state receives under a 299-seat House, those rounded figures actually wouldn’t add up to 299. They add up to 301, since so many of those quotas happened to have decimals more than 0.5 on the end. Plus, rounding would give Nevada zero reps, but we have to give it at least one, so we end up with a total of 302.

So, we wouldn't do that. To get a 299-seat House, we instead start by assigning every state a quota and then round every quota down (except for Nevada, which must round up to get its one rep). Add up all those rounded quotas, and we get 278. To bring the total to 299, we now add one rep to each of 21 states, starting with whichever one lost the most by rounding and then working our way down the list. Kentucky happened to lose the most, then Indiana. Alabama happened to be state number 21, so it was one of those states that must receive another rep after rounding done, bringing its total to eight seats.

Now, let’s try that with a 300-seat House. We again assign each state a quota and round each down to the nearest whole number (again, except for Nevada, which rounds up to one). This time, when we add all those rounded quotas, we get 281. That means 19 states must get an additional rep. The order of states is different this time, with Wisconsin coming on top, followed by Michigan. Alabama now appears 20th in the list, so it doesn’t get another rep. It’s stuck with seven seats.

This hit the census people as crazy in 1880, and we now refer to this possibility (which has had the chance to pop up a bunch more times in history) as the Alabama Paradox. But when they dug through the logs, they saw their predecessors had made the same observation about a different state 10 years earlier. 

In 1870, if the House had 270 members, Rhode Island would have two reps, but if the House had 280 members, Rhode Island would have just one. It would lose half its reps, which is a bigger deal than what would happen to Alabama, but everyone immediately forgot this because Rhode Island is so small. 

1 The Paradox of Voting

One final paradox related to voting is so basic, it’s known simply as “the paradox of voting.” It’s about how, for any individual person, the benefits from voting are so small, it can’t ever be worth the trouble. You might have felt this when voting for president anywhere but a swing state, since voting for the party your state will choose, the party your state won’t choose or a third party no one will choose will all have no effect on who wins. But this is really a broader issue, which political scientists have argued over for centuries. It was argued by our old friend, the Marquis de Condorcet...

...as well as by Lewis Carroll, when he wasn’t writing about Alice’s Adventures in Wonderland. The issue is that in any election, especially large elections, the chance that you’ll cast the deciding vote is close to zero, so you incur some cost in voting but no proven benefit. Of course, if many people decline to vote, the results really will change, and if everyone does, the whole system collapses, so we’d recommend against skipping voting based on that calculus. 

We’ve spent a couple centuries puzzling over what to do about the voting paradox. We can tell people their vote really does make a difference. We can suggest that they receive benefits beyond affecting the results, including the satisfaction of having fulfilled one’s duty. Or we can argue that not voting comes with its own unseen cost: regret. When all that sways no one, we turn to the other route of striving to make voting as easy as possible. If people are struggling with cost-benefit analysis over voting, we should reduce the cost. 

But there is also a far more effective solution that we should implement. We must serve fresh apple pie. Every single person who votes should receive a slice of pie. Suddenly, the benefit of voting becomes obvious, and that benefit is pie.

If your own polling station fails to offer pie incentives, we hereby authorize you after voting to make your own pie, and then to eat it all. Go ahead. You deserve it. 

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