Which Brilliant Ideas Are Far Less Widely Known Than They Ought To Be?

It's a great question which a friend and I were discussing over a group dinner Friday night. I can offer plenty of suggestions to whet your appetite; there are numerous ideas that would enrich people's understanding of everyday societal matters that are not currently captured with enough depth to do so. Examples would be things like comparative advantage, that which is seen and unseen, deadweight costs, Coase's theorem, Chesterton's fence and the concept of the invisible hand. Then there are absolutely life-changing ideas such as Aumann's Agreement Theorem and Harsanyi's Amnesia Principle that bear repeating regularly to get them as widely disseminated as possible. All of those ideas make an appearance in this blog from time to time.

However, the idea I'll pick as the idea of choice in response to this blog post's title will be Arrow's Theorem, as a tribute to Kenneth Arrow who died this week. He was, up until a few days ago, one of the greatest living economists. Arrow's Theorem is not the easiest to describe simplistically because it is mathematically dense, and is quite counterintuitive too (I had a go once before using voting as the illustrative template). In fact, past reading of economics has given me the impression that several economist social commentators have described Arrow's Theorem as the hardest complex idea to simplify for a layperson. I'll have a try with a version I typed up yesterday in the hope that you find it both understandable and intriguing.

To do so, let's discuss my favourite dessert - cheesecake. What I'm going to illustrate is that no democratic system, however fair it appears in terms of participants and number of choices, can satisfy all the axioms associated with the preferences. So for example, if there are three, then: if two are satisfied the third must be violated, whether those two are 1 and 2, 2 and 3, or 1 and 3.

Let's illustrate this. Every day, Julie, Frank and Timothy split a cheesecake with one flavour - Chocolate, Strawberries or Lemon. Their preference orderings change from day to day - some days Julie is in the mood for Strawberries, other days the very thought of Strawberries makes her wish she was eating another flavoured cheesecake. Every evening at dinner time they have to call in their first, second and third choice cheesecake orders (let's say the cheesecake patisserie insists that you specify your second and third choices in case they run out of something.) So Julie, Frank and Timothy need a method for translating their individual preferences to a cheesecake order. Now as it happens, on the 22nd on February, their preferences ran as follows:

 

	Julie	Frank	Timothy
First Choice	Chocolate	Strawberries	Lemon
Second Choice	Strawberries	Lemon	Chocolate
Third Choice	Lemon	Chocolate	Strawberries

I’m not going to divulge what system these three were using to determine their order, but I will tell you that on the 22nd they phoned the patisserie and expressed Chocolate as their first choice. On the 23rd, Julie ranked Chocolate over Lemon again (and let's suppose we don’t know anything about Frank’s or Timothy’s rankings). On the 22nd, only Julie ranked Chocolate over Lemon, whereas on the 23rd, Julie plus possibly one or both of the others ranked Chocolate over Lemon. So if the cheesecake order ranked Chocolate above Lemon on the 22nd, then it should certainly have ranked Chocolate over Lemon on the 23rd, no?

In principle, yes (disallowing for a change of mind for variety - not something that's as likely to happen in political voting). Any other result would have seemed unreasonable to Julie, Frank and Timothy, so when they designed their system, they designed it with the following feature. If we list Chocolate over Lemon on our order on one day, and if none of the people who prefer Chocolate over Lemon change their minds about that the next day, then we should list Chocolate over Lemon the next day as well.

Because this was implicit in their system (and because they’d listed Chocolate over Lemon on the 22nd when only Julie had that preference), they always listed Chocolate over Lemon on any day when Julie preferred Chocolate to Lemon. That is to say, Julie was the principal figure on the Chocolate/Lemon consideration - the 'authoritarian', if you like.

Now on the 24th, Julie was in a Chocolate/Lemon/Strawberries frame of mind. Suppose again we don't know much about Frank’s or Timothy’s moods except that they both favoured Lemon over Strawberries. Since everyone preferred Lemon to Strawberries, Lemon was, of course, listed higher than Strawberries in the cheesecake hierarchy. Once again, any other result would have appeared unreasonable to Julie, Frank and Timothy, so they’d built their system along the following lines:

Whenever we unanimously prefer option X to option Y, option X should rank higher than option Y on our order.

First, Julie preferred Chocolate to Lemon, so of course Chocolate ranked higher than Lemon. Second, everyone preferred Lemon to Strawberries, so Lemon ranked higher than Strawberries. Logic says that Chocolate must have ranked higher than Strawberries. Logic also says that when Julie prefers Chocolate/Lemon/Strawberries in that order, and everyone else prefers Lemon to Strawberries, Chocolate must be more preferable than Strawberries.

They also agreed the following:

Our preferences regarding Lemon should not affect the relative positions of Chocolate and Strawberries in the ranking. Therefore, the above should hold if we drop all the Lemon-related assumptions. That is to say, on any day when Julie prefers Chocolate to Strawberries, Chocolate must rank higher than Strawberries.

The next day, the 25th, Julie’s preferences ran Strawberries/Chocolate/Lemon, while the other two both preferred Strawberries to Chocolate. Since they all preferred Strawberries to Chocolate, Strawberries came out higher than Chocolate on the cheesecake order. Since Julie was a Chocolate/Lemon authoritarian, Chocolate came out higher than Lemon. Logic tells us also that Strawberries came out higher than Lemon. And the same would be true on any day when Julie preferred Strawberries/Chocolate/Lemon and everyone else preferred Strawberries to Chocolate.

But the ranking of Strawberries vs. Lemon was designed to be unaffected by how anyone cared about Chocolate, so the Chocolate-related information cannot be relevant. This tells us that on any day when Julie prefers Strawberries to Lemon, Strawberries rank higher than Lemon. She’s not just a Chocolate/Lemon authoritarian and a Chocolate/Strawberries authoritarian; she’s a Strawberries/Lemon authoritarian too.

What we’ve uncovered is that any Chocolate/Lemon authoritarian is also a Chocolate/Strawberries authoritarian and a Strawberries/Lemon authoritarian. Interchanging the flavours, we could have just as easily discovered that any Chocolate/Strawberries authoritarian (e.g. Julie) is also a Chocolate/Lemon and a Lemon/Strawberries authoritarian - and so on as every pair of flavours appear. In other words, despite the table of preferences above, Julie is an absolute cheesecake authoritarian as all of her preferences are entirely reflected in the cheesecake order on any given day.

Now here's the upshot of it all. We started by declaring that Chocolate came out on top on the 22nd. But if Strawberries had come out on top, then using Arrow's model I could have given evidence to show that Frank is an absolute authoritarian, and if Lemon had come out on top, then using Arrow's model I could have given evidence to show that Timothy is an absolute authoritarian. Regardless of what happened on the 22nd, someone must be an absolute authoritarian - counterintuitive as that may have first appeared.

Extending that further, if there had been more people than just Julie, Frank and Timothy, and more flavours than Chocolate, Strawberries and Lemon, then although the argument becomes more mathematically intractable, it is not at heart any further from the nub of the wisdom just covered - that is, if you have a scenario that translates a series of individual preference choices into a single group preference of choice, that scenario will throw up an authoritarian.

Crucially here, we're seeing the distinction between what are very coherent individual preferences (called transitivity), and how at a group level it becomes incoherent (intransitivity). Transitivity is formally expressed as: if A > B, and B > C, then A > C - so in other words, with cheesecake choices, if Julie likes Chocolate more than Strawberries and Strawberries more than Lemon, she should like Chocolate more than Lemon.

Groups, however, don't necessarily have a transitive preference order - they have intransitive 'cycling' of preferences, which means Chocolate beats Strawberries, which beats Lemon, which beats Chocolate, which beats Strawberries, which beats Lemon, which beats Chocolate - meaning for group scenarios you might just as well put the flavours of cheesecake in a hat.