An odd kind of cycle

The UK Parliament seems a mad place.

I used to teach micro theory - or at least when Seamus was on sabbatical. He understood the math better than I did, and would likely have corrected one or two things here.

Completeness was pretty easy. Nobody actually keeps the full set of preference relations in their heads. Most of the time, preference is only discovered in choice anyway. But it's conceptually easy.

Reflexivity - that's just like dotting the i in completing the requirements. Honestly, I've never thought hard about the reflexivity inequality. That an apple is at least as preferred as an apple seemed more like dotting the i in completing the axioms - not something of any particular consequence. Varian's text had the assumption as trivial. For all x in X, x is at least as preferred as x. Yawn.

It would be madness to imagine otherwise. How could something be less preferred than itself?

For my not-all-that-mathematical brain, identity meant this had to reduce to an equality rather than an inequality: if x is x, then flipping the x to the other side of the inequality meant the only way it could hold was if it were actually an equality rather than an inequality - but it never mattered. How could an apple not be as preferred as that very same apple? How could anyone possibly be other than indifferent between an option and itself? And if they aren't, how can we say that they're really identical? The failure of it would have to mean that x and x aren't really the same thing. We wouldn't look at somebody willing to pay zero for a bottle of water when standing next to a tap, but willing to pay $5 for it at a music festival, and conclude that reflexivity had failed; we'd say that the goods were different because of differences in time and place.

In any case, we never spent any time on it. It didn't really matter.

When we'd get to the transitivity assumption, things would get more fun. I'd tend to illustrate by example.

If someone had intransitive preferences, you could quickly turn that person into a money pump. If they really prefer an apple to a banana, and the banana to a carrot, and the carrot to an apple, you can structure a series of trades where they keep giving you cash plus their banana for your apple, then cash plus their apple for your carrot, then cash plus their carrot for your banana, and so on. Sylvester McMonkey McBean for the win.

It's a mad sort of preference configuration.

But it comes up in social choice theory where it's far easier to generate intransitive social preferences out of underlying transitive individual preferences, so long as enough people have non-single-peaked preferences if it's a unidimensional choice, or if there's no median in all directions in multidimensional realms. And we'd illustrate that in the public choice classes with the beautiful McKelvey result. Individual rational preferences leading to what seems collective madness and the ability for a strong agenda-setter to arbitrarily choose outcomes.

And that all then brings us to Brexit. I have no clue what's there going on.

Here's the BBC's tally on yesterday's series of votes.

How MPs voted

	For	Against	Defeated by
Confirmatory referendum	268	295	27
Customs union	264	272	8
Labour's Brexit plan	237	307	70
Common Market 2.0	188	283	95
Revoking Article 50 to avoid no deal	184	293	109
No-deal exit on 12 April	160	400	240
Malthouse Plan B	139	422	283
EFTA and EEA membership	65	377	312

Recall that in these up-down votes, they're all implicitly running against the status quo. If they're all defeated, the status quo obtains. In the absence of other action by Parliament, no-deal exit on 12 April obtains.

None of the options enjoys majority support. But does that make for an Arrow/Black/Condorcet cycle? Not really. One option did beat all the other options. That option is the status quo. Every one of these options lost to the status quo. That makes the status quo - Brexit with no deal on 12 April - the Condorcet Winner.

It's a Condorcet Winner powerful enough to defeat itself as option 400 to 160 though.

Reflexivity says that an option should be at least as preferred as itself. No-deal exit on 12 April (the status quo) is preferred over No-deal exit on 12 April (the alternative voted on option) by a margin of 240 votes. That makes it at least as preferred as itself. But not if we flip which side of the inequality the status quo and the identical voted-on option are on.

We haven't had the sequence of pairwise votes of each option against the others that might reveal the cycle. But if completeness and transitivity kinda imply reflexivity, perhaps the apparent failure of reflexivity here suggests the failure of transitivity.

I expect the real answer in this particular case is that the voted-on option was not the same as the status quo: it was instead being compared to the other options put in a sequence of up-down votes against the status quo. So Members were registering their dissatisfaction with the status quo as against the other available options, and their desire to maintain the option value of keeping other options open.

It would also help make sense of the utter inability of the agenda-setter to use the McKelvey result to actually get anywhere. No option can ever beat the status quo because the status quo is supported both by those who want a no-deal exit and by those who like the option value of trying again for their more preferred option later.

What a mess.