It is amazing how quickly the titans of the middle of the century have passed. Paul Samuelson and his mathematization, Ronald Coase and his connection of law to economics, Gary Becker and his incorporation of choice into the full sphere of human behavior, John Nash and his formalization of strategic interaction, Milton Friedman and his defense of the market in the precarious post-war period, Robert Fogel and his cliometric revolution: the remaining titan was Kenneth Arrow, the only living economist who could have won a *second* Nobel Prize without a whit of complaint from the gallery. These figures ruled as economics grew from a minor branch of moral philosophy into the most influential, most prominent, and most advanced of the social sciences. It is hard to imagine our field will ever again have such a collection of scholars rise in one generation, and with the tragic news that Ken has now passed away as well, we have, with great sadness and great rapidity, lost the full set.

Though he was 95 years old, Arrow was still hard at work; his paper with Kamran Bilir and Alan Sorensen was making its way around the conference circuit just last year. And beyond incredible productivity, Arrow had a legendary openness with young scholars. A few years ago, a colleague and I were debating a minor point in the history of economic thought, one that Arrow had played some role in; with the debate deadlocked, it was suggested that I simply email the protagonist to learn the truth. No reply came; perhaps no surprise, given how busy he was and how unknown I was. Imagine my surprise when, two months letter, a large manila envelope showed up in my mailbox at Northwestern, with a four page letter Ken had written inside! Going beyond a simple answer, he patiently walked me through his perspective on the entire history of mathematical economics, the relative centrality of folks like Wicksteed and Edgeworth to the broader economic community, the work he did under Hotelling and the Cowles Commission, and the nature of formal logic versus price theory. Mind you, this was his response to a complete stranger.

This kindness extended beyond budding economists: Arrow was a notorious generator of petitions on all kinds of social causes, and remained so late in life, signing the Economists Against Trump that many of us supported last year. You will be hardpressed to find an open letter or amicus curiae, on any issue from copyright term extension to the use of nuclear weapons, which Arrow was unaware of. The Duke Library holds the papers of both Arrow and Paul Samuelson – famously they became brothers-in-law – and the frequency with which their correspondence involves this petition or that, with Arrow in general the instigator and Samuelson the deflector, is unmistakable. I recall a great series of letters where Arrow queried Samuelson as to who had most deserved the Nobel but had died too early to receive it. Arrow at one point proposed Joan Robinson, which sent Samuelson into convulsions. “But she was a communist! And besides, her theory of imperfect competition was subpar.” You get the feeling in these letters of Arrow making gentle comments and rejoinders while Samuelson exercises his fists in the way he often did when battling everyone from Friedman to the Marxists at Cambridge to (worst of all, for Samuelson) those who were ignorant of their history of economic thought. Their conversation goes way back: you can find in one of the Samuelson boxes his recommendation that the University of Michigan bring in this bright young fellow named Arrow, a missed chance the poor Wolverines must still regret!

Arrow is so influential, in some many areas of economics, that it is simply impossible to discuss his contributions in a single post. For this reason, I will break the post into four parts, with one posted each day this week. We’ll look at Arrow’s work in choice theory today, his work on general equilibrium tomorrow, his work on innovation on Thursday, and some selected topics where he made seminal contributions (the economics of the environment, the principal-agent problem, and the economics of health care, in particular) on Friday. I do not lightly say that Arrow was the greatest living economist, and in my reckoning second only to Samuelson for the title of greatest economist of all time. Arrow wrote the foundational paper of general equilibrium analysis, the foundational paper of social choice and voting, the foundational paper justifying government intervention in innovation, *and* the foundational paper in the economics of health care. His legacy is the greatest legacy possible for the mathematical approach pushed by the Cowles Commission, the Econometric Society, Irving Fisher, and the mathematician-cum-economist Harold Hotelling. And so it is there that we must begin.

Arrow was born in New York City, a CCNY graduate like many children of the Great Depression, who went on to study mathematics in graduate school at Columbia. Economics in the United States in the 1930s was not a particularly mathematical science. The formalism of von Neumann, the late-life theoretical conversion of Schumpeter, Samuelson’s *Foundations*, and the soft nests at Cowles and the Econometric Society were in their infancy.

The usual story is that Arrow’s work on social choice came out of his visit to RAND in 1948. But this misstates the intellectual history: Arrow’s actual encouragement comes from his engagement with a new form of mathematics, the expansions of formal logic beginning with people like Peirce and Boole. While a high school student, Arrow read Bertrand Russell’s text on mathematical logic, and was enthused with the way that set theory permitted logic to go well beyond the syllogisms of the Greeks. What a powerful tool for the generation of knowledge! His Senior year at CCNY, Arrow took the advanced course on relational logic taught by Alfred Tarski, where the eminent philosopher took pains to reintroduce the ideas of Charles Sanders Peirce, the greatest yet most neglected American philosopher. The idea of relations are familiar to economists: give some links between a set (i.e, xRy and yRz) and some properties to the relation (i.e., it is well-ordered), and you can then perform logical operations on the relation to derive further properties. Every trained economist sees an example of this when first learning about choice and utility, but of course things like “greater than” and “less than” are relations as well. In 1940, one would have had to be extraordinarily lucky to encounter this theory: Tarski’s own books were not even translated.

But what great training this would be! For Arrow joined a graudate program in mathematical statistics at Columbia, where one of the courses was taught by Hotelling from the economics department. Hotelling was an ordinalist, rare in those days, and taught his students demand theory from a rigorous basis in ordinal preferences. But what are these? Simply relations with certain properties! Combined with a statistician’s innate ability to write proofs using inequalities, Arrow greatly impressed Hotelling, and switched to a PhD in economics with inspiration in the then-new subfield on mathematical economics that Hotelling, Samuelson, and Hicks were helping to expand.

After his wartime service doing operation research related to weather and flight planning, and a two year detour into capital theory with little to show for it, Arrow took a visiting position at the Cowles Commission, a center of research in mathematical economics then at the University of Chicago. In 1948, Arrow spent the summer at RAND, still yet to complete his dissertation, or even to strike on a worthwhile idea. RAND in Santa Monica was the world center for applied game theory: philosophers, economists, and mathematicians prowled the halls working through the technical basics of zero-sum games, but also the application of strategic decision theory to problems of serious global importance. Arrow had been thinking about voting a bit, and had written a draft of a paper, similar to that of Duncan Black’s 1948 JPE, essentially suggesting that majority voting “works” when preferences are single-peaked; that is, if everyone can rank options from “left to right”, and simply differ on which point is their “peak” of preference, then majority voting reflects individual preferences in a formal sense. At RAND, the philosopher Olaf Helmer pointed out that a similar concern mattered in international relations: how are we to say that the Soviet Union or the United States have preferences? They are collections of individuals, not individuals themselves.

Right, Arrow agreed. But economists had thought about collective welfare, from Pareto to Bergson-Samuelson. The Bergson-Samuelson idea is simple. Let all individuals in society have preferences over states of the world. If we all prefer state A to state B, then the *Pareto criterion* suggests society should as well. Of course, tradeoffs are inevitable, so what are we to do? We could assume cardinal utility (e.g., “how much money are willing to be paid to accept A if you prefer B to A and society goes toward A?”) as in the *Kaldor-Hicks criterion* (though the technically minded will know that Kaldor-Hicks does not define an order on states of the world, so isn’t really great for social choice). But let’s assume all people have is their own ordinal utility, their own rank-order of states, an order that is naturally hard to compare across people. Let’s assume for some pairs we have Pareto dominance: we all prefer A to C, and Q to L, and Z to X, but for other pairs there is no such dominance. A great theorem due to the Polish mathematician Szpilrain, and I believe popularized among economists by Blackwell, says that if you have a quasiorder R that is transitive, then there exists an order R’ which completes it. In simple terms, if you can rank some pairs, and the pairs you do rank do not have any intransitivity, then you can generate a complete rankings of all pairs which respects the original incomplete ordering. Since individuals have transitive preferences, Pareto ranks are transitive, and hence we know there exist social welfare functions which “extend” Pareto. The implications of this are subtle: for instance, as I discuss in the link earlier in this paragraph, it implies that pure monetary egalitarianism can *never* be socially optimal even if the only requirement is to respect Pareto dominance.

So aren’t we done? We know what it means, via Bergson-Samuelson, for the Soviet Union to “prefer” X to Y. But alas, Arrow was clever and attacked the problem from a separate view. His view was to, rather than taking preference orderings of individuals as given and constructing a social ordering, to instead ask whether there is any mechanism for constructing a social ordering from *arbitrary* individual preferences that satisfies certain criteria. For instance, you may want to rule out a rule that says “whatever Kevin prefers most is what society prefers, no matter what other preferences are” (non-dictatorship). You may want to require Pareto dominance to be respected so that if everyone likes A more than B, A must be chosen (Pareto criterion). You may want to ensure that “irrelevant options” do not matter, so that if giving an option to choose “orange” in addition to “apple” and “pear” does not affect any individual’s ranking of apples and pears, then the orange option also oughtn’t affect society’s rankings of apples and pears (IIA). Arrow famously proved that if we do not restrict what types of preferences individuals may have over social outcomes, there is *no* system that can rank outcomes socially and still satisfy those three criteria. It has been known that majority voting suffers a problem of this sort since Condorcet in the 18th century, but the general impossibility was an incredible breakthrough, and a straightforward one once Arrow was equipped with the ideas of relational logic.

It was with this result, in the 1951 book-length version of the idea, that social choice as a field distinct from welfare economics really took off. It is a startling result in two ways. First, in pure political theory, it rather simply killed off two centuries of blather about what the “best” voting system was: majority rule, Borda counts, rank-order voting, or whatever you like, every system *must* violate one of the Arrow axioms. And indeed, subsequent work has shown that the axioms can be relaxed and still generate impossibility. In the end, we do need to make social choices, so what should we go with? If you’re Amartya Sen, drop the Pareto condition. Others have quibbled with IIA. The point is that there is no right answer. The second startling implication is that welfare economics may be on pretty rough footing. Kaldor-Hicks conditions, which in practice motivate all sorts of regulatory decisions in our society, both rely on the assumption of cardinal or interpersonally-comparable utility, *and* do not generate an order over social options. Any Bergson-Samuelson social welfare function, a really broad class, must violate some pretty natural conditions on how they treat “equivalent” people (see, e.g., Kemp and Ng 1976). One questions whether we are back in the pre-Samuelson state where, beyond Pareto dominance, we can’t say much with any rigor about whether something is “good” or “bad” for society without dictatorially imposing our ethical standard, individual preferences be damned. Arrow’s theorem is a remarkable achievement for a man as young as he was when he conceived it, one of those rare philosophical ideas that will enter the canon alongside the categorical imperative or Hume on induction, a rare idea that will without question be read and considered decades and centuries hence.

Some notes to wrap things up:

1) Most call the result “Arrow’s Impossibility Theorem”. After all, he did prove the impossibility of a certain form of social choice. But Tjalling Koopmans actually convinced Arrow to call the theorem a “Possibility Theorem” out of pure optimism. Proof that the author rarely gets to pick the eventual name!

2) The confusion between Arrow’s theorem and the existence of social welfare functions in Samuelson has a long and interesting history: see this recent paper by Herrada Igersheim. Essentially, as I’ve tried to make clear in this post, Arrow’s result does *not* prove that Bergson-Samuelson social welfare functions do not exist, but rather implicitly imposes conditions on the indifference curves which underlie the B-S function. Much more detail in the linked paper.

3) So what is society to do in practice given Arrow? How are we to decide? There is much to recommend in Posner and Weyl’s quadratic voting when preferences can be assumed to have some sort of interpersonally comparable cardinal structure, yet are unknown. When interpersonal comparisons are impossible and we do not know people’s preferences, the famous Gibbard-Satterthwaite Theorem says that we have no voting system that can avoid getting people to sometimes vote strategically. We might then ask, ok, fine, what voting or social choice system works “the best” (e.g., satisfies some desiderata) over the broadest possible sets of individual preferences? Partha Dasgupta and Eric Maskin recently proved that, in fact, good old fashioned majority voting works best! But the true answer as to the “best” voting system depends on the distribution of underlying preferences you expect to see – it is a far less simple question than it appears.

4) The conditions I gave above for Arrow’s Theorem are actually different from the 5 conditions in the original 1950 paper. The reason is that Arrow’s original proof is actually incorrect, as shown by Julian Blau in a 1957 Econometrica. The basic insight of the proof is of course salvageable.

5) Among the more beautiful simplifications of Arrow’s proof is Phil Reny’s “side by side” proof of Arrow and Gibbard-Satterthwaite, where he shows just how related the underlying logic of the two concepts is.

We turn to general equilibrium theory tomorrow. And if it seems excessive to need four days to cover the work on one man – even in part! – that is only because I understate the breadth of his contributions. Like Samuelson’s obscure knowledge of Finnish ministers which I recounted earlier this year, Arrow’s breadth of knowledge was also notorious. There is a story Eric Maskin has claimed to be true, where some of Arrow’s junior colleagues wanted to finally stump the seemingly all-knowing Arrow. They all studied the mating habits of whales for days, and then, when Arrow was coming down the hall, faked a vigorous discussion on the topic. Arrow stopped and turned, remaining silent at first. The colleagues had found a topic he didn’t fully know! Finally, Arrow interrupted: “But I thought Turner’s theory was discredited by Spenser, who showed that the supposed homing mechanism couldn’t possibly work”! And even this intellectual feat hardly matches Arrow’s well-known habit of sleeping through the first half of seminars, waking up to make the most salient point of the whole lecture, then falling back asleep again (as averred by, among others, my colleague Joshua Gans, a former student of Ken’s).