An impossible theorem made a young David Dreyer Lassen change direction

I began studying economics in 1992, straight after upper secondary school, hoping to learn about exchange rates and macroeconomics. This was when the European monetary cooperation that would later lead to the euro was taking shape. But after my second-year exams, my interests took a completely different turn.

It was spring 1994, in Anneks A and B — the lecture halls on Studiestræde in central Copenhagen that will be familiar to many.

Microeconomics was taught using the so-called green notes. Anyone who studied economics back then will remember the green notes. These were teaching notes written by the macroeconomics and microeconomics lecturers, and the microeconomics notes were — to be perfectly honest — heavy going.

They were typewritten, the paper was rather coarse and, on a good day, there were no more than one or two sentences in plain Danish that actually explained why the questions mattered. On top of that, it was fiendishly difficult.

One of the topics in the spring of 1994 was welfare economics: how we, as a society, assess whether and how different allocations of resources and different policy measures affect both individual and overall welfare.

Yes, it’s complicated. But decisions still have to be made. And what is overall, or aggregate, welfare? Does it exist, and can it be measured?

Best wishes, Kenneth Arrow

It was here, in my second year, that I was introduced to Arrow’s impossibility theorem.

A theorem is mathematical jargon for a statement that has been proved and is particularly profound or important.

Many decisions are made by and affect groups — families, boards, organisations, Denmark as a whole. So when can we say that a collective decision, A or B, is best for the group?

The answer is: generally, we cannot. Best wishes, Kenneth Arrow. He analysed the problem in his PhD thesis in the late 1940s and was awarded the Nobel Prize partly for this work.

So what does the theorem mean? Consider a group that has to choose between at least three options.

Arrow’s theorem states — roughly speaking — that no voting system can turn individual group members’ rankings of the three options into a single ranking for the group while also meeting four reasonable requirements: it must be able to accommodate every conceivable combination of preferences; if everyone prefers A to B, the group must do so too; the group’s choice between A and B must not depend on what people think of a third option, C; and no single person may dictate the outcome.

Arrow’s work laid the foundations of modern social choice theory and turned welfare economics upside down

Each requirement can be met individually, but they cannot all be met at once.

I was hooked

For me, the experience back then, after my second-year exams, had two dimensions: first, there was the substance of the theorem itself — no, economic analysis alone cannot actually tell us what Denmark should do — and second, and more importantly, the broader insight that rigour, logic and mathematical tools could be used to study democratic processes and how economic policy is made.

Arrow’s work laid the foundations of modern social choice theory and turned welfare economics upside down.

I never went on to work specifically on social choice myself, but it sparked my interest in what is known as political economics: rather than studying the consequences of a given economic policy, political economics focuses on why one policy is chosen over another.

I was hooked on the subject. I specialised in it during my master’s degree, wrote my thesis on it and was awarded a PhD scholarship at a basic research centre where I had worked as a student assistant for several years.

As part of my PhD, I spent a year at Harvard University, which was — and remains — particularly strong in this field.

Two Nobel laureates and an email

Although I was not working directly on social choice myself, I could not resist taking a course on it when I had the opportunity at Harvard.

The course was taught by Amartya Sen, another Nobel laureate, and in many ways it was an extraordinary experience: alongside around 30 students from various disciplines, there were many professors in attendance, as well as a couple of visiting professors (and their spouses!) of a similar calibre.

It combined some of the most challenging material at one of the world’s best universities with something best described as a public lecture series — albeit a highly selective one. All delivered by a man of the world who drew freely on examples in many languages and from across cultures.

Several lectures focused specifically on the impossibility theorem and Sen’s own subsequent work: the idea that the impossibility is partly of our own making.

Arrow allows only information about what people prefer — not how strongly they prefer it. Allow for the latter, and for the fact that some people have more at stake than others, and the impossibility becomes less absolute.

Debate was heated, and during the course Sen had to consult Arrow by email to clarify an interpretation. So there we were, watching two Nobel laureates correspond about what exactly one of them had proved half a century earlier.

Since my PhD, I have worked across economics and political science. Arrow’s impossibility theorem never became my own field of research, but encountering it on Studiestræde in 1994 changed the question that interested me: not just what consequences economic policy has, but why we choose one policy over another.

This article was translated with the assistance of artificial intelligence and subsequently reviewed by a member of the editorial staff.