Why? Well, had the statesman been more familiar with math – or rather, theoretical economics – then he might have known that democracy, as barely acceptable as it is, is actually impossible. Or at least, that’s what one famous result from economist Kenneth Arrow seems to suggest – and it’s convincing enough that even the Nobel committee, in the 1972 award ceremony that bestowed him with the Prize in Economics, described the theorem as “rather discouraging”.

But is Arrow’s theorem really all that pessimistic? Arrow himself didn’t think so – and neither do today’s experts. So, what does the result actually say? And, perhaps more importantly… what does it mean?

What is Arrow’s Impossibility Theorem?

Imagine you’ve been tasked with creating a country from scratch. You know you want this new nation to be a democracy – but ironically, you and your fellow founders are at odds about what exact form of rule-by-the-people to choose.

What you can agree on, though, is a set of more general criteria for your system. They’re common sense, really: firstly, you want it to have “unrestricted domain” – a fancy way of saying you want it to actually produce a result every time; second, it can’t be a dictatorship, so no one single vote can override any other; thirdly, you want it to be possible for any one candidate to beat any other candidate, so long as they get the right combination of votes – no dud candidates allowed.

On top of that, there are some things you’re going to assume about your population. Again, they’re nothing wacky: first, their preferences are going to be transitive – so if they prefer candidate A to candidate B, and candidate B to candidate C, then they must also prefer candidate A to candidate C; finally, they’re going to decide “independent of irrelevant alternatives” – so, if candidate D suddenly enters the fray, it shouldn’t result in people deciding they now prefer B over A instead.

Basically, you’re assuming your populace is rational and your voting system is sensible. It doesn’t seem like a lot to ask – but here’s the thing: according to Arrow, it’s totally impossible.

“Arrow’s theorem […] can be interpreted as saying that there is no perfect voting rule,” explains Edith Elkind, Ginni Rometty Professor of Computer Science at Northwestern University and an expert in the area of computational social choice. 

“In more detail, Arrow’s theorem considers the setting where there is a set of at least three candidates, each voter reports a ranking of these candidates (from best to worst, no ties allowed), and the goal is to come up with an overall ranking,” Elkind tells IFLScience. “The statement of the theorem […] is that no voting rule satisfies all of these conditions.”

Let’s consider an example. Suppose we have three candidates and three voters – as simple a setup as we can get – and we’re working within a ranked voting system. Voter one prefers candidate A to candidate B, and candidate B to candidate C. Voter two prefers candidate B to candidate C, and candidate C to candidate A. Finally, voter three prefers candidate C to candidate A, and candidate A to candidate B. 

Already, you can see we’re at an impasse. There’s no way to satisfy the demands of the group: overall, they prefer A to B, B to C, and C to A. 

Not only is it usually the case that you can’t please everyone, it’s sometimes the case that you can’t even please most people.

Ben Abrams

And sure, you may reply if you live in, say, the US or UK, that’s why we don’t use ranked voting systems. But here’s the kicker: Arrow’s theorem applies to all ranking-based systems, from First-Past-The-Post – which often breaks the independence of irrelevant alternatives criteria – to instant-runoff voting, to single transferable vote systems, and so on.

“What Arrow and similar scholars show is that ranked-choice voting systems can’t be made flawlessly,” explains Ben Abrams, a lecturer in sociology at University College London whose research focuses on democracy, populism, and revolution. “There will always be a chance of an imperfect outcome in close elections with divided voting blocs.”

“Not only is it usually the case that you can’t please everyone,” he tells IFLScience, “it’s sometimes the case that you can’t even please most people.”

So, is democracy impossible?

Traditionally, it’s at this point that readers start to feel a little hopeless. Democracy is doomed; it’s official, math has proven it; might as well set up a dictatorship and get it over with. But is that really the case?

“Not at all!” Abrams tells IFLScience. “Arrow was very supportive of democracy, and so too are all the people who use his work.”

There are in fact a few ways to get around the “problem” of Arrow’s result. First, you can notice that the theorem only applies to ranking-based systems – and while that may be the kind we’re most familiar with, that’s not the only way to vote.

 “There are other systems, like rating-based systems (a type actually used at the UN), which don’t fall foul of Arrow’s paradox,” Abrams points out, “even if they might have other issues.”

Increasing the number of assumptions we make can help, too. For example, “if all candidates can be placed on a left-to-right axis and all votes are consistent with this axis […] then there is also a rule that circumvents Arrow’s impossibility result,” Elkind tells IFLScience.

Arrow’s rule, taken literally, applies to a rather specific scenario, which does not quite match what happens in politics […] we want winners, not collective rankings.

Edith Elkind

In other words, if you can assume that, say, a voter whose top preference is Bernie Sanders would also rank Kamala Harris over Donald Trump, then you can avoid the kinds of stand-off Arrow’s theorem would otherwise produce. It is, like the original set of conditions that created the paradox, not an outlandish assumption to make – and while it’s “only formally guaranteed to work if no voter ever submits a ballot that is inconsistent with the axis,” Elkind explains, “it still works nicely if almost all voters’ rankings are consistent with the axis.”

Which neatly brings us to probably the most important factor limiting the impact of Arrow’s Theorem: real life. 

“Arrow’s rule, taken literally, applies to a rather specific scenario, which does not quite match what happens in politics,” Elkind tells IFLScience. In the real world, “we want winners, not collective rankings,” she points out. 

Like all theoretical results, the theorem is set in a perfect world filled with sensible actors – voters who never rank their choices tactically, or vote based on vibes or charisma. “Arrow is also very concerned about ‘dictators’ – individuals whose votes can decide whole elections,” says Abrams, “but in reality you don’t get them very often, and even when you do, they only emerge after the fact, so they don’t know their own influence (and may never, in the case of secret ballots).”

Arrow’s point

All of which leads to the ultimate question: what’s the point of all this?

“The reason why Arrow’s theorem is important is that it told researchers and practitioners alike that the quest for a perfect opinion aggregation method is hopeless,” Elkind tells IFLScience. “Before it was discovered, people kept finding faults with existing voting rules and then tried to tweak them to avoid these specific faults; this resulted in more sophisticated voting rules, which in turn, were later found to exhibit undesirable behavior in some cases, and were replaced with yet more complex rules.”

“While Arrow’s theorem did not quite stop this process (there are voting rules that were proposed in the 21st century), it showed that we cannot hope to converge to a rule that always produces unobjectionable collective decisions.”

In other words, Arrow’s Impossibility Theorem can be thought of as a kind of political Incompleteness Theorem – a result that says, hey, sometimes, there just is no perfect solution. And while that might sound dispiriting, it’s actually a very important and impactful result: with the ideal officially out of reach, scholars could concentrate on developing whole new theories of social choice. It was so profound, in fact, that it has since been characterized as the “Big Bang” of modern social choice theory.

Plus, Arrow’s theorem applies widely outside of politics. It “can apply to all sorts of situations where groups are deciding on things,” Abrams says, “so you see it come up in plenty of different contexts, from welfare economics to philosophy.”

So, when you and your pals are out and trying to decide what restaurant to eat at? That’s Arrow’s Theorem. If you’re trying to create laws or moral choices based on utility, you’ll likely come up against Arrow’s Theorem again. “It is relevant whenever we need to aggregate rankings,” Elkind tells IFLScience, whether that’s in politics, or “in various competitions (e.g., Eurovision), producing university rankings based on experts’ opinions, etc.”

Ultimately, then, is Arrow’s Impossibility Theorem really as “discouraging” as the Nobel committee declared? Not really – in fact, far from being the end of democracy, it kickstarted a whole new field of voting research. 

Perhaps a more optimistic view was the one taken by Arrow himself. The conditions on ranked voting may be “mutually contradictory”, he admitted in his Nobel Memorial Lecture the same year he received his Prize in Economics, and “the philosophical and distributive implications […] are still not clear.”

Nevertheless, he said, giving up is not the answer. “I hope that others will take this paradox as a challenge,” he concluded, “rather than as a discouraging barrier.”