So, I’m a formal language theorist and computer security researcher. One of the most practical results of my research has been the observation that if two implementations of the same protocol differ slightly, such that they are actually implementing grammars for mostly-mutually-intelligible “dialects” of the same formal language, this can lead to all sorts of interesting shenanigans. (We broke SSL this way; the paper is in Financial Crypto 2010.)

One naive but (apparently) simple way around this problem is, of course, the (software) monoculture: if everyone runs Windows on both client and server, then of course there won’t ever be any mistranslation-based vulnerabilities … until, of course, Microsoft end-of-lifes Windows XP and hundreds of businesses are still stuck on it, not to mention a good chunk of China, and of course IE 6 and IE 8 speak the exact same dialects of HTTP and HTML, right? And those are totally the same dialect that every version of IIS speaks, right? And so does every other webserver written in Java or Ruby or whatever and … yeah, you see where this ends up.

So, too, I think, with a monoculture of moral traditions, but I think Randall Munroe’s take on aesthetic traditions suffices.

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In that case, you’re miscalibrated about (1) the ability of the philosophical community to identify their own greatest works and researchers, and about (2) the degree to which your “respect” for people correlates with their philosophical ability.

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Used on inhomogeneous cakes and particularly if the other party’s preferences are unknown your cake dividing algorithm does have the problems you claim it avoids. The audience can see this without much math. For example, I might prefer vanilla and you might prefer chocolate except I think you also prefer vanilla. The cake is half chocolate and half vanilla. So I divide it giving both lots equal parts of chocolate and vanilla and then it doesn’t matter which lot you chose, and we both would have done better by giving me all the vanilla and you all the chocolate. In math-speak that means the division is not Pareto-optimal. That particular problem wouldn’t occur if we both new each other’s preferences, but then suppose the cake has only one indivisible good part and is otherwise gross. Then the chooser will get the good part and the divider will get only junk. In math-speak that’s dictatorship.

I don’t see what your coin example is about, but some interpretation of it probably can be made into an example of Arrow’s theorem. That definitely doesn’t mean the theorem is only about coins.

In that interview – and, free strategy advice, you could really have clicked through to the actual interview rather than making it obvious you’re getting your entire subject knowledge from a Wikipedia article on a political subject – Arrow is talking about practical real-life voting systems and what kind of trade-offs to make from a least bad perspective. In that context it’s fairly clear there is more information on the problem approval voting choses to solve, but no pretension that it is isomorphic to the original problem of preference or utility aggregation. I’ve actually read Arrow saying ordinality doesn’t help when he actually was writing about preference aggregation in some paper, but I don’t have it at hand right now so I guess you are free to disbelieve me.

With that I’ll leave it to the audience to make their minds up about which if any of us two knows what he’s talking about.

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As to the rest, you simply have no idea what Arrow’s theorem says. In particular, it does say that it impossible to divide a pile of coins, based purely on the preferences of which agents would prefer to have or not have which coins.

PS – Arrow himself says that the theorem does not apply to cardinal systems.

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Your cake and money examples both assume that both parties have identical preference schedules over possible resource baskets. That is plausible if there is only one resource (cake and money respectively in your examples) and Arrow’s theorem does indeed not ban aggregation of preferences basically identical in that way. But then utilitarianism working in all cases except where people have different preferences doesn’t sound that hot either.

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The way to understand the conflict between Arrow’s theorem and utilitarian aggregation is to think about the case of two parties. Everyone knows how to split a cake and everyone knows that there’s no way to adjudicate between two voters electing one of them dictator.

Or a formal model: the question of how to split a pot of money (infinitely divisible, or just two coins), with square root utility. The aggregate utility function prefers more equitable divisions to less, with equal division the maximum. It is not a total order, being indifferent between (a,1-a) and (1-a,a), but we can break that tie by letting the first player choose between them.

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