Can Everyone Get What They Want? Arrow’s Impossibility Theorem

Imagine three friends trying to decide what to do after school. Alice wants to go to the movies. Bob wants to play basketball. Charlie wants to get ice cream. They decide to vote on it, pair by pair: movies versus basketball, basketball versus ice cream, ice cream versus movies.

Here’s what happens. When they vote on movies versus basketball, Alice and Charlie both prefer movies—so movies wins. When they vote on basketball versus ice cream, Alice and Bob prefer basketball—so basketball wins. You’d think that means movies should beat ice cream too, right? But when they vote on that, Bob and Charlie both prefer ice cream. Ice cream wins.

This is weird. The group seems to prefer movies to basketball, and basketball to ice cream, but then ice cream to movies. It’s a cycle. No one’s individual preferences are confused—Alice knows she’d rather watch a movie than play basketball—but when you try to add up what everyone wants, you get something that makes no sense at all.

This puzzle was discovered in the 1700s by a French mathematician named the Marquis de Condorcet (say “con-dor-SAY”). It’s called the paradox of voting. And for a long time, people thought it was just a quirk of one particular voting method—maybe if they tried a different way of counting votes, the problem would go away.

Then, in the 1950s, an economist named Kenneth Arrow proved something that shocked people. He showed that the problem isn’t just with one voting method. It’s deeper than that. There is no fair way to take a bunch of people’s individual preferences and turn them into a single group preference—at least, not if you want that group preference to behave sensibly.

This is Arrow’s Impossibility Theorem. Let’s understand what it says, why it matters, and whether there’s any way around it.

What Arrow Was Trying to Do

Arrow wanted to find a procedure—any procedure—that could take a group of people’s rankings of some options and produce a single “social ranking” of those options. The procedure could be a voting method, or something else entirely. The question was: could any such procedure meet five simple requirements that seem obviously fair?

Here are the five requirements. See if they strike you as reasonable.

First: Handle any possible preferences. The procedure has to work no matter what weird preferences people have. People could rank the options in any order at all, and the procedure still has to produce a result. (This is called “Unrestricted Domain.”)

Second: Always produce a ranking. The result can’t be a cycle like the one above. It has to be a real ranking, from best to worst, with no loops. (This is called “Social Ordering.”)

Third: Respect unanimous agreement. If absolutely everyone prefers one option to another, the group ranking has to agree. (This is called “Weak Pareto,” after an economist named Vilfredo Pareto.)

Fourth: No dictators. There can’t be one person whose preferences always overrule everyone else’s. If one person says “I want this,” and everyone else disagrees, the group can still sometimes go with everyone else. (This is called “Non-Dictatorship.”)

Fifth: Ignore irrelevant alternatives. When deciding whether the group prefers movies to basketball, you should only need to know how people feel about movies and basketball. You shouldn’t need to know how they feel about ice cream. (This is called “Independence of Irrelevant Alternatives.”)

These all sound pretty reasonable, right? You’d think there must be some voting method that does all of these things.

Arrow proved there isn’t. Not if there are more than two options to choose among. Every possible procedure violates at least one of these requirements.

Why This Is So Upsetting

Before Arrow’s theorem, a lot of people believed in something called “the will of the people.” The idea was that if you just listened carefully enough to what everyone wanted, you could figure out what society as a whole wanted. Democracy wasn’t just about counting votes—it was about discovering a real thing, the common good, that existed out there somewhere.

Arrow’s theorem suggests this might be an illusion. There may be no such thing as “the will of the people”—not in the way people imagined. You can have the preferences of individual people, but there’s no guaranteed way to combine them into a single coherent group preference.

Some political scientists have taken this to mean that democracy itself is incoherent. If there’s no such thing as what the people really want, then what are we even doing when we vote? Others say Arrow’s requirements are too strict, and we shouldn’t give up on democracy just because perfect fairness is impossible.

The Trickiest Requirement

The requirement that causes the most trouble is the last one: Independence of Irrelevant Alternatives. It sounds innocent, but it’s actually very powerful.

Here’s why it matters. Think about a real election. Say you’re trying to decide between two candidates for class president. You ask everyone: “Who do you prefer?” But maybe some people would choose a third candidate if they could, and that changes how they feel about the first two. Independence says: too bad. Your feelings about the third candidate shouldn’t affect how we compare the first two.

This seems weirdly strict. In real life, whether you prefer Alice or Bob for class president might depend a lot on whether Charlie is also running. Maybe you’d pick Alice over Bob if Charlie is in the race (because Alice and Charlie agree on some issue you care about), but you’d pick Bob over Alice if Charlie drops out. Independence says the procedure shouldn’t care about that—it should compare Alice and Bob the same way regardless of Charlie.

Why did Arrow include this requirement? Partly because he wanted procedures that couldn’t be manipulated. If how we compare two options depends on people’s opinions about a third option, then someone might try to game the system by pretending to feel differently about the third option. Independence makes that harder.

But here’s the thing: lots of real voting methods violate this requirement. And sometimes that’s actually a good thing.

One Way Out: Restricting What People Can Prefer

Here’s one way to escape Arrow’s theorem: don’t let people have just any preferences. If you can guarantee that people’s preferences will have a certain shape, then you can avoid the problem.

For example, imagine you’re trying to decide how hot to make the porridge. Papa bear likes it as hot as possible. Mama bear likes it as cold as possible. Baby bear likes it warm best, then hot, then cold. These preferences are “single-peaked”—each person has a favorite temperature, and the further you get from their favorite, the less they like it, on either side.

When preferences are single-peaked like this, simple majority voting actually works! It always produces a coherent ranking. And the winner turns out to be the option that’s the favorite of the “median” voter—the person whose favorite is right in the middle.

This is comforting, because a lot of real decisions have this structure. When people are choosing between political candidates on a left-right spectrum, or deciding how much money to spend on a school trip, their preferences are often single-peaked. So in practice, majority voting can work fine even though Arrow’s theorem says it shouldn’t always.

But single-peakedness is a restriction. You can’t guarantee it will hold in every situation. Sometimes people’s preferences just don’t line up that nicely.

Another Way Out: Let People Grade Instead of Rank

Here’s another escape route, and it’s one that Arrow himself came to appreciate late in his life.

Arrow’s framework assumed people would rank the options—first, second, third, and so on. But what if instead of ranking, people could grade the options? What if you could say: movies are “excellent,” basketball is “good,” and ice cream is “okay”?

This seems like a small change, but it’s huge. When you grade things, you give more information than just a ranking. You say not just that you prefer movies to basketball, but how much better you think movies are. And this extra information, even though it’s still just grades (not precise numbers), opens up new possibilities.

For instance, you could use “median grading.” Take each option and find the middle grade—the one right in the middle of all the grades people gave it. Then rank the options by their median grades. This procedure turns out to satisfy all of Arrow’s requirements, once you rephrase them to work with grades instead of rankings.

Well, almost. It satisfies something very close to Arrow’s requirements. Whether it satisfies the spirit of them is something philosophers still argue about. The person who developed this approach, Rida Laraki, has argued that grading is actually more democratic than ranking, because it lets people express how they really feel rather than forcing them to choose between awkward comparisons.

But there’s a catch. For this to work, everyone has to use the same “grade language.” If one person’s “good” means something very different from another person’s “good,” the system breaks down. And in real life, people do have different standards.

What This All Means

Arrow’s theorem is one of those rare discoveries that changed how people think about democracy. Before Arrow, it was common to talk about “the will of the people” as if it were a real thing that existed, waiting to be discovered. After Arrow, philosophers and political scientists realized this was more like a myth.

But the lesson isn’t that democracy is hopeless. It’s that we need to be honest about what voting can and can’t do. Voting doesn’t discover some pre-existing “will of the people.” Instead, it’s a tool for making decisions together, under constraints, knowing that no tool will be perfect.

Different voting methods have different strengths and weaknesses. Some are more resistant to manipulation. Some handle certain kinds of disagreement better than others. Some produce clearer winners. None is perfect. The job of figuring out which one to use in which situation is part of what makes democracy hard—and part of what makes it worth doing.

Philosophers still argue about whether Arrow’s requirements are really as reasonable as they seem. Maybe we shouldn’t expect group decisions to be as orderly as individual decisions. Maybe a little bit of cycling is okay, as long as we have ways of breaking the tie. Maybe we should give up on one of the requirements rather than trying to satisfy them all.

What’s clear is that Arrow discovered something real and important. There is a tension between respecting individual preferences and having a coherent group preference. And no clever voting method can make that tension disappear.

Appendices

Key Terms

Term	What it does in this debate
Social ordering	A ranking of all options from best to worst, without cycles or loops
Preference profile	A complete list of everyone’s rankings of the options
Unrestricted domain	The requirement that a voting method must work for any possible set of preferences
Weak Pareto	The requirement that if everyone prefers one thing to another, the group must also prefer it
Non-Dictatorship	The requirement that no single person’s preferences always overrule everyone else’s
Independence of Irrelevant Alternatives	The requirement that how we compare two options depends only on people’s feelings about those two, not about others
Single-peaked preferences	Preferences where each person has a favorite, and likes things less as they get farther from it
Median grading	A method where each option gets the grade that’s right in the middle of all the grades people gave it

Key People

Kenneth Arrow (1921–2017) — An economist who proved that no perfect voting system exists. He won the Nobel Prize for his work.
Marquis de Condorcet (1743–1794) — A French mathematician who discovered the voting paradox that started this whole line of thinking. He died during the French Revolution.
Duncan Black (1908–1991) — A Scottish economist who showed that majority voting works when preferences are single-peaked.
Rida Laraki (born 1965) — A computer scientist who developed the idea of using grades instead of rankings to escape Arrow’s theorem.

Things to Think About

If there’s no perfect voting system, how should we decide which voting system to use? What trade-offs are worth making?
The “Independence of Irrelevant Alternatives” requirement says we shouldn’t need information about third options to compare two options. But in real life, sometimes the third option matters. Can you think of a situation where knowing about a third option changes how you’d compare two others? Should that be allowed in voting?
Arrow’s theorem says that if you have any voting method that meets all five requirements, you’ll end up with a dictator. But the “dictator” in Arrow’s sense isn’t necessarily someone powerful—it could be someone who just happens to always agree with the outcome. Does this mean the non-dictatorship requirement is too strict? Or does it reveal something troubling about democracy?
Grades seem to solve the problem, but only if everyone uses the same standard. What might go wrong if different people have different ideas about what counts as “excellent” or “good”?

Where This Shows Up

Student council elections — The debate over how to count votes (plurality, ranked choice, approval voting) is directly about which trade-offs to make, given Arrow’s theorem.
Movie ratings on websites — When you see a movie’s average star rating, you’re seeing one answer to the aggregation problem. Different sites use different methods (median, average, weighted averages), and they often produce different results.
Artificial intelligence — When an AI system has to combine different people’s preferences to make a recommendation, it faces exactly the problem Arrow studied.
Sports rankings — How should we decide which team is number one? Different ranking systems (like the BCS in college football) try to combine many different opinions and statistics, and they all have flaws that Arrow’s theorem helps explain.