Links:

- Introduction.
- Complex systems.
- Zipf's law, self-similarity and fractals.
- Borda's method.
- 'Landslide' majorities.

Murray Gell-Mann's popular work ( published by Little, Brown and Co in 1994 ) tells a little about his personal life, mostly his youth -- tho there are a few genial anecdotes about colleagues. This is just as well, because as C S Lewis said: He'd never read an autobiography yet in which the early years werent by far the best. Lewis appears to have discovered a law of nature, or human nature. Gell-Mann is a student of both.

Gell-Mann's chapter, 'Quarks and all that' echoes '1066 and all that', as if we would laugh-off his most famous discoveries as ancient history. In fact, he gives a typical account you might read in other popular physics books.

He is much better on current research to demystify quantum theory. The paradox of Schrodinger's cat is laid to rest ( tho attempts may be made to revive it ).

Nevertheless, the focus has changed, from : what are the basic parts of the world? to : how does everything fit together so it works properly? Gell-Mann helped found the Santa Fe Institute concerned with the general properties of complex systems and their emergent features that make successive chemical, biotic and social systems irreducible wholes.

*The Quark and the Jaguar* sets out to define complexity.
Complexity is in the observer and the observed. Observations are most
complex when they are not so apparently random, that no rules can be
abstracted by the observer, and when the observations are not so
regular, that they can be summed-up in a simple rule.

Consequently, the skill of observers is most tried, as themselves 'complex adaptive systems', when they have to distinguish most carefully the essential patterns in the data - the sensory 'signals', if you like, from the random noise. ( That is, if you consider one's perception of the real world rather like receiving a radio signal, so one has to fine tune out the interference to its message. )

The 'noise' could be superstitions caused by one's conditioning to chance associations between events that have no rational connection. However, Malinowski's anthropology and Jung's psychology have impressed on us that apparently silly customs may have a ritual value for the integration of society and of the personality.

No doubt much of the paranormal is credulous. But I dont agree with Gell-Mann's throwaway dismissal of 'psychic detectives'. The police are scientific investigators ( in a democracy ). If they find such people useful at times, that is surely being practical rather than dogmatic about things we dont understand.

Anyway, Gell-Mann's treatment of complexity may be illustrated by voting methods. Candidates first past the post in marginal constituencies depend largely on chance factors to win. 'There is no greater gamble than a British general election,' admitted one devotee of the simple majority system. An opposite fault applies in the safe seats, where results are too determined. An agent boasted he could put up a pint pot of beer in this constituency and still get it elected.

The random effects of marginal constituencies and the pre-determined effects of safe seats are both examples of low 'effective complexity'.

The voters are caught between two extremes and have difficulty adapting to the system either way. If you are in a safe seat, you know your vote is unlikely to make any difference. That's why party politicians tend to favor single member systems. A safe seat is a local monopoly for some party, whose candidate does not have to earn an elective proportion of the vote, in competition with candidates of his own party, as well as of other parties.

In a marginal constituency, you may have to vote tactically for the best chance to make your vote count. The information value of the X-vote is too low to register more than a single preference, unlike a ranked choice.

A combination of an elective proportion and a ranked choice ( which exists as a voting system called the single transferable vote ) therefore increases the effective complexity of a voting system in two ways. The ranked choice of a preference vote reduces all the 'noise' from split votes that interferes with and frustrates the popular will. A proportional count prevents votes being wasted in predictable pile-ups that make safe seats.

In short, the voters have the best chance of adapting the political system with transferable voting.

That's why the Establishment least wants that system, to disestablish its opposition to the world's changing needs. Michels called this evident state of affairs 'the iron law of oligarchy'. But government is supposed to be the cybernetic principle of the rulers responding to the ( especially voting ) information feedback of the ruled. The least effective government as cybernetic system has minimal feedback methods of voting.

Typically, these are partisan systems that only tell the rulers what they want to know from the ruled, namely that they follow their party lines. Indeed, the voters can do no other, as the likes of party list systems pre-define the terms of the popular vote.

I read somewhere that at a conference, Stephen Hawking had just
quoted off the top of his head an equation about a mile long, when
Murray Gell-Mann promptly stood up to point out a missed term. Yet *The
Quark And The Jaguar* takes an interest in the simplest of arithmetic
laws.

They may apply thru-out the sciences. Zipf's law is one of many 'scaling laws' or 'power laws' about which '...we see what is going on but do not yet understand it.' For example, you can rank 1st, 2nd, 3rd etc the cities of a country by their population size, which turns out to be inversely proportional to that rank.

If the first city has about 10 million people, the second city turns out to have about half that number or around 5 million. The third largest city will have one-third the population of the biggest, or some three and one-third million people. And so on, down to, say, the hundredth city at about 100,000 citizens.

Similar relations hold for ranking countries by their volume of business in exports, or for ranking firms by their volume of business in sales.

Modified versions of Zipf's law may produce a formula that is a better fit of the data, but the point is that there is an underlying regularity. Gell-Mann says this is reminiscent of self-similarity found in nature. Trees from their largest branches to their smallest twigs, or rivers down to their smallest tributaries, have a characteristic shape at every scale. The same is true to some extent of clouds and mountains and many natural features.

Such features do not have regular dimensions, one, two or three. But they were found to have fractional dimensions. A screwed-up ball of paper is not a proper ball of three dimensions but is more than two dimensions. It may typically measure over 2.7 dimensions. Likewise, the squiggly lines, say, of rivers on maps, have a characteristic fractional dimension of slightly more than one dimension.

Hence, the term 'fractals', which relate to 'chaos' theory. In
*Does God Play Dice?* Ian Stewart says: 'The same complexity of structure
that lets fractals model the irregular geometry of the natural world is
what leads to random behaviour in deterministic dynamics.'

Knowing the fractals of natural phenomena enables them to be
modelled realistically as in computer 'landscapes'.

You could also simulate a society and an economy, with the help of
scaling laws like Zipf's law.

The Santa Fe Institute includes political science in its array of systems studies. But it is possible Gell-Mann's colleagues havent heard of Borda's Method of counting votes for a single vacancy. This is actually an electoral version of Zipf's law.

Voters can order their choice of candidates, 1st, 2nd, 3rd, 4th,
etc. These preferences are given due weight in the count, as a measure of
their order of importance. If there were five candidates, your first
preference would get five points; your second would get four points,
and so on to your last preference getting one point.

Laplace gave an involved proof of Borda's Method.

In *Elections and Electors*, JFS Ross pointed out that the more
candidates standing, the less important the first preference, using
Borda's method of weighting the count with an arithmetic series.

Ross suggested the preferences be weighted by a geometric series.
The first preference would count as one vote, the second as half a vote,
the third preference as one-quarter of a vote, the fourth as one-eighth
of a vote...

A happy medium, between weighting by the arithmetic series and by the geometric series, would be to weight preferences with the harmonic series. Choice 1 counts as one vote; choice 2 counts as 1/2 a vote; choice 3 counts as 1/3 of a vote; choice 4 counts as 1/4 of a vote...

This modified version of Borda's Method was once favored by Sir Robin Day. And it is Zipf's law for an election, whereby the count is inversely proportional to the vote.

You could imagine Zipf's law applied to cities as an 'electoral' system of how people vote with their feet. The largest city attracts twice as many as the second largest, three times as many as the third largest, etc. Borda's political justice turns out to be a case of art unconsciously imitating nature.

Borda's method was designed to overcome an objection to the Second Ballot, which does not weight preferences to account for their order of importance. If three candidates contest one seat and none wins over half the votes, the candidate with least votes has to stand down. A second ballot decides between the two remaining candidates.

But Condorcet pointed out that the eliminated candidate ( say, a centre candidate ) might have won more votes from either a right or a left wing candidate than they would have won from each other. ( By the way, this isnt necessarily the case. Extremes may have more in common than moderates. )

Borda's method, in turn, is open to the objection that the lesser weights given to lesser preferences, count to some extent against a voter's first preference. That candidate has a better chance of winning if the voter refrains from adding further choices.

This problem is overcome by the transfer of votes, surplus to a
quota or proportion of votes needed to elect the most prefered candidate,
according to the voters' succeeding preferences for candidates, elected
in multi-member constituencies.

The size of the most prefered candidate's surplus vote determines how
much weight to assign to the next preferences of the most popular
candidate's voters.

Borda's method has to assume what value voters assign to their preferences. But with ( the so-called Senatorial rules of ) transferable voting, this is a real value based on the size of surplus votes, which does not count against more prefered candidates already elected.

Benoit Mandelbrot generalised Zipf's law by adding a constant,
c, to its inverse proportion. That is, 1/1, 1/2, 1/3,... becomes
1/(1+c), 1/(2+c), 1/(3+c),... Let that constant equal one, and you have
the Droop quota, which gives the elective proportion of votes to become
a representative in one, two, or three member constituencies etc.

The Droop quota is used with transferable voting. Candidates, winning more than their quota, have their surplus votes
transfered to their voters' next preferences.

Zipf's law describes natural structures. Whereas Borda's method is a similar structure, consciously imposed by the rules of an electoral system.

Murray Gell-Mann cites the work of Per Bak and associates on how structures arise naturally without imposed constraints.

Cone-like heaps of sand had more grains of sand piled on them. As their steepened slopes became more unstable, a critical value was passed for avalanches, which left the slope back at the critical value. This cycle was called self-organised criticality.

The single transferable vote is analgous to such 'self-organised systems'. The surplus votes transfered to next prefered candidates are akin to the avalanche, a political 'landslide majority', caused by the piling of extra sand on a mound or cone, above the value for a stable heap. This critical value compares to the quota, or proportion of votes needed to elect the most prefered candidate ( and in turn the next prefered candidates ).

This voting system is not self-organizing, however, except in the attenuated sense of automating the count in a computer program.

The actual way that grains of sand tumble together is extremely complicated, just as is the way that thousands of voters' preferences combine. But each scenario clearly follows a typical structural development. The contrast is that Per Bak and his colleagues evolved formulas from a phenomenum. Whereas the pioneers of electoral science, from Borda and Condorcet, Andrae and Hare, Clark, Droop and Gregory, onwards evolved a phenomenum from formulas.

The former is natural science, the latter is 'moral science'.

The introduction to Gell-Mann's *The Quark And The Jaguar* and the
last chapter on a sustainable world is an admirable survey that perhaps
speaks for many as to the kind of world they would like to work for.

Richard Lung.

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