Felix Böhm

What's wrong with plurality voting?

November 2021 Home

Imagine you're twelve again and it's the summer holidays. Outside, the sun is boiling the asphalt. The fan is whirring. You're bored. Suddenly, the doorbell snaps you out of your lethargy. Your friends — a big group of fifteen — desperately need some action as well. One proposes to go roller-skating, another to play some board games.

How would you come to a decision?

Probably by letting everybody raise their hand for their preferred activity and choosing the one with the most votes, right? This is what's called plurality voting. It's easy and intuitive, so why should there be anything wrong with it?

Let's visualize our example. Say that there are only two preferences someone can have about spending their free time:

Your fifteen friends and you are represented as dots on this spectrum of preferences. With plurality voting, everyone votes for the option that is closest to them.

Nine of you prefer to go roller-skating and seven to play board games, and you go dig your skates out of the garage. Seems fair.

Spoiling the vote

Now one of your friends just loves board games a little too much and knows a little bit about voting theory. They cunningly propose that apart from roller-skating and playing board games, you could also go shoot some hoops. You repeat the vote. Same group, same preferences.

And this time the board games win, not the roller-skating. Wait a second.

If roller-skating had won again, or the new basketball option, that's reasonable. But adding the option of playing basketball made the previously losing board game option win against the roller-skating one. That doesn't seem fair.

On our preference spectrum playing basketball is similar to roller-skating. As a result, it "steals" votes away from the roller-skating option, spoiling its election. This phenomenon is fittingly called the spoiler effect and is one of the biggest flaws of plurality voting.

The basketball voters now face a dilemma: Do they continue to vote honestly for playing basketball, according to their true preferences, essentially throwing away their votes? Or do they vote for their second favorite option (roller-skating) to avoid letting their least preferred option (board games) win?

Such strategic voting — caused by the spoiler effect — makes plurality voting systems converge to two options over time. In general, the mechanics of plurality voting leave voters feeling misrepresented, and discourages new candidates from entering the race.

Instant-runoff voting

Now, maybe we can modify plurality voting to avoid the spoiler effect. What if voters submitted not only their favorite choice but a ranking of all choices? Such voting systems are called ranked voting systems.

The most popular ranked voting system is instant-runoff voting. It works by adding up all the first preferences and successively eliminating the option with the least first preferences until only the winner is left. When an option is eliminated, all options below it move up in the rankings.

Let's see how instant-runoff voting works with our last example. Each voter ranks the options by distance. A voter's first preference is thus always the activity closest to them.

In three steps first playing basketball, then playing board games is eliminated.

Note that in the second step, the basketball voters now vote for their second preference, which in this case is roller-skating for all three of these voters.

Roller-skating is back! Did we beat the spoiler effect for good?

Unfortunately not.

Instant-runoff voting still suffers from the spoiler effect, although to a lesser extent than plurality voting. If we include the option to play video games in our original vote — something that requires neither much thinking nor much sweating — first roller-skating, then playing video games itself is eliminated, and the board games win again:

Other ranked voting systems different from instant-runoff voting cannot avoid the spoiler effect either. An economist named Kenneth Arrow proved that there simply exists no (reasonable) ranked voting system that does not suffer from the spoiler effect.

Approval voting

That leads us to our last voting system, called approval voting. Approval voting is like plurality voting, but instead of raising your hand exactly once, you can raise it as often as you want, "approving" of options you like.

This way, you can express that you'd be fine with roller-skating or playing basketball. Or with all three options. Or none.

In our example, approval is best represented as you and your friends voting for (approving of) every option that is closer than some specific distance to you. To keep the graphic clear, it only shows example distances for four of the voters.

The largest circle only contains the board games option, so that voter only approves of playing board games. One circle contains roller-skating and basketball, so this voter approves of both. The other two circles contain no options. These voters approve of nothing at all.

Since approval voting is not a ranked voting system, it's exempt from Arrow's impossibility theorem. And lo and behold, approval voting does indeed not suffer from the spoiler effect! This is because adding options does not change the number of votes that existing options receive.

Additionally, there's no incentive to not vote for one's first preference, which prevents the convergence to few options over time.

In 2010, the Voting Power in Practice workshop organized a vote among the participating experts to elect "the best voting procedure". Out of 18 different voting methods, approval voting ranked first, instant-runoff voting ranked second, and plurality voting dead last.

I think that's crazy, experts are unanimous that plurality voting is terrible, and nonetheless, it's used in political elections all over the world and is the system that most would intuitively choose to decide whether to go roller-skating or play board games.

Sadly, approval voting isn't flawless either.

In certain cases, it can incentivize voters to only vote for their first preference, thus practically degenerating into plurality voting. However, philosopher Allan Gibbard and economist Mark Satterthwaite proved that there exists only a single voting method that does not create strategic incentives for some voters in some situations to submit untruthful preferences. This magical voting system is called dictatorship.

In the end, even though a perfect voting system does not exist, approval voting comes pretty close. So, next time you're out and about with your fifteen friends, just raise your hands as often as you like — whenever you approve of an option — and no board-game-loving friend will be able to spoil the vote in their favor.


We only covered single-winner methods in this introduction, meaning voting methods that elect a single winner out of the available options. Another type of voting methods are multi-winner methods, which are useful for dividing up finite, but divisible, resources, like seats in parliament.

If you're interested to learn more about voting theory, you could check out this great detailed introduction and the concept of voter satisfaction efficiency. These two posts also cover score voting, where instead of ranks or approvals, the voters assign scores (e.g. from 1 to 10) to the available options.

Thanks to Moritz Makowski, Simon Böhm, and Julia Godart for reading drafts of this.


Footnotes

  1. This is called Duverger's law.
  2. If you add a new option, every voter who has that option in their circle will approve of it. But they will also continue to approve of all options that were in their circle before.
  3. Suspiciously enough, they used approval voting to vote on this ranking ...
  4. Meaning that a single voter decides the result alone, and all other votes don't matter. This is obviously not a very reasonable voting system, but funnily enough, it also satisfies many other positive properties. Long live the dictatorship, I guess?