an autodidact meets a dilettante…

‘Rise above yourself and grasp the world’ Archimedes – attribution

there’s no such thing as a fair election 1: the apportionment issue

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Canto: So we’ve been talking about how politics have been interacting with the Covid-19 pandemic, and came to the tentative conclusion that strong centralised governments, collaborationist and respected by their citizens, were faring better at managing the situation than right-wing quasi-dictatorial anti-government governments like Trump’s USA, Putin’s Russia and Bolsonaro’s Brazil…

Jacinta: And those three countries just happen to fill the top three places in Covid-19 cases, though to be fair, they have very large populations. Anyway, the Scandinavian countries we looked at all seemed to have coalition governments of some kind, and from our great distance we preferred to assume that they operated through some kind of more or less happy consensus – but maybe not.

Canto: So we’ve been reading David Deutsch’s book The beginning of infinity, and there’s an interesting chapter, ‘Choices’, which looks at voting systems and what we want from government…

Jacinta: Or perhaps what we need, or should expect. What is objectively best, something which Deutsch, being a progressivist optimist, believes we’re converging upon – what he calls, in the political sphere, ‘advancing from misconception to even better [i.e. less damaging] misconception’. Deutsch considers first the ‘apportionment problem’ in the USA, a problem that many electoral polities have, as they attempt to represent particular electoral regions, with their different populations, fairly within a federal electoral system. The USA, like Australia and many other countries, has a House of Representatives, to indicate the aim of representative government. There are 435 US House seats, and the Constitution requires that these seats be apportioned to the states according to their populations. For example if state x has 5% of the nation’s population, it should get 21.75 House members. This is of course impossible, so the obvious thing to do is round up to 22, right?

Canto: Obvious, maybe, but brimming with controversy, because this rounding up, or down, will affect states’ representation, often rather more than was ever suspected. Deutsch imagines a more simplified House with 10 seats, and 4 states. One state holds a little under 85% of the population, the other three have just over 5% each. Rounding will mean that the large state gets rounded down to 8 seats, the three smaller states get rounded up to 1. This means that you have to add an extra seat, but it also means that the smaller states are over-represented, population-wise, and the large state is under-represented. And if you don’t add an extra seat, and the rule is that all states must be represented, then the larger state is reduced to a grossly unrepresentative 7 seats. You could of course add two seats and allocate them to the large state, giving it 9 out of 12 seats, but that still under-represents that state’s population, while enlarging the House to a questionable degree.

Jacinta: In fact a quick calculation shows that, to provide that large state with 85% representation, while giving the other three states a seat, you’d have to add 10 more seats, but then you’d have to add more seats to make the other states more representative – unless I’m missing something, which I probably am. And so on, the point being that even with a simple model you can’t, just from a mathematical perspective, attain very precise representation.

Canto: You could, on that simple model, take a seat way from the least populated state, and give it to the most populated one, thereby keeping the state to ten seats, but having no representation at all seems grossly unfair, and in fact the US Constitution explicitly states that ‘Each State shall have at least one Representative’. The aim, of course is to have, as near as can be, the right measure of representativeness. Having no representation at all, even in one small region, contravenes the ‘no taxation without representation’ call-to-arms of the revolutionary American colonists and the founding fathers.

Jacinta: Yet all the argy-bargy that went on in the USA in the 19th century over apportionment rules and quotas – and it was often fierce – overlooked the fact that black peoples, native Americans, the poor, oh and of course women, were not entitled to be represented. As Deutsch points out, the founding fathers often bandied about the concept of the ‘will of the people’ in their work on the Constitution, but the only ‘people’ they were really talking about were the voters, a small fraction of the adult population in the early days of the nation.

Canto: Nevertheless the apportionment issue proved the bane of election after election, eminent mathematicians and the National Academy of Sciences were consulted, and various complicated solutions were mooted but none proved to everyone’s satisfaction as the system kept chopping and changing.

Jacinta: Of course this raises the question of whether majority rule is fair in any case, or whether fairness is the right criterion. We don’t decide our science or our judiciary by majority rule – and good science, at least, has nothing to do with fairness. Arguably the most significant weakness of democracy is the faith we place in it. In any case, as Deutsch reports:

… there is a mathematical discovery that has changed forever the nature of the apportionment debate: we now know that the quest for an apportionment rule that is both proportional and free from paradoxes can never succeed. Balinski and Young [presented a theorem which] proved this in 1975.

Deutsch calls this a ‘no-go theorem’, one of the first of which was proved by the Nobel Prize-winning economist Kenneth Arrow more than twenty years before. Arrow set out five basic axioms that a rule defining ‘the will of the people’ should satisfy:

Axiom 1: the rule should define a group’s preferences only in terms of the preferences of that group’s members.

Axiom 2: (the ‘no dictator’ axiom) the rule cannot designate the views of one particular person regardless of what the others want.

Axiom 3: if the members of the group are unanimous in their preference for something, then the rule must deem the group to have that preference.

These 3 axioms are expressions of the principle of representative government.

Axiom 4: If, under a given definition of ‘the preferences of the group’, the rule deems that the group has a particular preference, this remains the group’s preference if some members who previously disagreed with that preference now agree with it.

Axiom 5: If the group has some preference, and then some members change their minds about another matter, then the rule must continue to assign the original preference to the group.

These all seem like unproblematic axioms, but Arrow was able to prove that they were inconsistent, and this turns out to be problematic for social-choice theory in general, not just the apportionment issue. According to Deutsch at least, it reveals the mythical nature of ‘the will of the people’.

Canto: Did we really need to be told that? There is no ‘people’ in that sense. And I’m not talking about the Thatcherite claim that there’s no society, only individuals. I’m talking more literally, that there’s no such thing as an indivisible national entity, ‘the people’, which has made its preference known at an election.

Jacinta: Agreed, but that rhetoric is so ingrained it’s hard for people to let it go. I recall one of our prime ministerial aspirants, after losing the federal election, saying ‘graciously’ that he would bow to the ‘will of the people’ and, what’s more, ‘the people always get it right’. It was essentially meaningless, but no doubt it won him some plaudits.

Canto: In fact, voting doesn’t even reveal the will of a single person, let alone the ‘people’. A person might register a vote for person x mistakenly, or with indifference, or with great passion, or under duress etc. Multiply that by the number of voters, and you’ll learn nothing about the soi-disant will of the people.

Jacinta: Okay, we’ve talked about the problems of apportionment under the US multi-state system. Next time we’ll look at the different electoral systems, such as proportional-representation systems and plurality or ‘first past the post’ voting. Is any system more fair than another, and what exactly does ‘fair’ mean? Good government is what we want, but can this be described objectively, and can this be delivered by democracies?

Canto: Well, here’s a clue to that good government question, I think. I walk into my class and I’m faced with twenty students. If I’m asked ‘who’s the tallest person in the class?’ I can come up with an answer soon enough, even if I have to make a measurement. But if I’m asked ‘who’s the best person in the class (not the best student), I’m very likely to be lost for an answer, even if I’ve taught the class all year….

Jacinta: Interesting point, but we’re not talking about the best government. There might be a variety of good governments, and you might be able to point out a variety of students/persons in the class who’ve positively impressed you, for a variety of reasons. Good government is not one.

References

David Deutsch, The beginning of infinity, 2011

The Institutional Design of Congress

Written by stewart henderson

May 26, 2020 at 10:08 pm

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