Basic concepts of mechanics and elections.

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It may be possible to give an electoral interpretation of the essential arithmetic or algebra of classical and relativistic mechanics. That is to show how an electoral system may work in terms of the so-called Galilean transformations and the Lorentz transformations, respectively.

The point of these transformations is that they translate the different measurements differently placed and timed observers make of the same event, so that they agree precisely what theyve seen. This should be true no matter what choices of observation points are made.

In this way, a general theory of physical observation was also becoming a general method of observational choices. Physics was having to become consistent with electoral method. Meanwhile, some political reformers were developing voting method that would recognise the general choice of the voters.

It follows that the electoral methods of physics and politics must be consistent with each other, if they are to hope to offer the general choice of physical observers or political voters.
It should be possible to use the Galilean and Lorentz transformations as models for political electoral systems.

Basic concepts of mechanics and elections.

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In the centuries following Galileo and Newton, it became fashionable to model new sciences on classical mechanics. The natural sciences did, with considerable success. Sir James Jeans was one of the earlier doubters of this approach's continued value, when he remarked: The universe is becoming to look less like a great engine than a great idea.

The very idea of a social science was often founded on quite literal applications of the language of mechanics to social systems. The collection of statistics about society revealed, in many cases, almost mechanical regularities in the incidences of human behavior.
The influence was not all one way: statistical mechanics was actually a borrowing of physics from sociology.

In politics, there is even the term 'the Machine', whereby each party organisation attempts to manufacture election results in their favor. Parties became determined to triumph, not only over each other, but over the freedom of the voters to discard them.
Here, the concern is to compare the mathematical form of classical and relativistic mechanics with electoral system: that is to show the inter-dependence of knowledge and freedom.

Classical mechanics has three basic concepts: space, time and mass. These were independent until special relativity formulated space-time and joined the conservation laws of mass and energy into a conservation of mass-energy.

Looking at elections, the most important players are voters, representatives and candidates. A mathematical analogy follows, between these two triads. Mass is a scalar measurement. This means mass is just a number, representing its magnitude. Space is a vector, which means it has a number representing a magnitude of distance, and it has a direction, like a compass.

In classical physics, time has no direction. Classical physics treats time like the reversible motion picture of a vase falling and smashing. Common sense knows which direction 'time's arrow' took. But popularisers of physics tell us that the theoretical view-point has been that, however unlikely, the vase could have behaved like the reversed film.

Language disposes us to think in terms of a mass of votes, as the voters were termed 'the masses'. This can be justified by the fact that choice is directed to the candidates and not the voters: votes are a scalar; choice of candidates is a vector quantity.

For a while, political scientists have devised 'candidate-spaces'. One used Euclid's geometry to show how candidates could best get themselves elected by how well positioned they were on various strategic issues, that were key influences on the voters. This example only used single-choice voting ( with a cross ).

On my first page about Penrose dodecahedrons, I showed how the dodecahedron can geometrically represent the 120 permutations of preference for five candidates. Any order of preference, for the five candidates, is signified by an orientation in space.

The quantum rules, governing the behavior of two separately observed Penrose dodecahedrons, are supposed to represent a conservative correlation, between two separated parts of a system of quantum physical events, seen before they can be linked by a light-speed signal.
Indeed, an experimental version, of the imagined dodecahedrons, could not be used as a communication device, so the special relativity principle still holds, for light-speed as a maximum limit, such as, at which a message can be sent.

Special relativity, itself, involves separate observers, who may be differently oriented in space, with respect to the event they are both measuring. The Lorentz transformations transform their respective space and time co-ordinates into each other.
That is the basis of the analogy between a spatial vector and a candidate vector.

The probabilistic nature of time's arrow may also characterise the concept of representation. Consider the probabilistic distribution of the votes for a range of candidates. Say the candidates range from left wing thru the centre to the right wing. Most voters' opinions may be in the middle, in which case there will be something like a binomial distribution of votes for candidates.
Or public opinion might be skewed to one or other political wing. In this case, there may be something like a Poisson distribution of the votes for candidates.

One could say there is a probability of candidates being elected as representatives, according to some such probability distribution, favoring some candidates over others. We might estimate such probabilities by placing statistical confidence limits on a distribution of votes for candidates. This would give betting odds, if you like, on the likelihoods of each candidate becoming a representative.

In this respect, a political career may career, into representation, like time's arrow.

When voters are given ranked choices, the number of permutations of preference for candidates soon becomes large. And when there are many voters, counting, to find out who are the most prefered candidates, is enormously complex. It is as complex as the distribution of molecules in a gas, described by statistical mechanics.

Elections compare to thermo-dynamics, in that they are both about a run-down from organised to disorganised states, whether of choice or of energy. A reasonably competant or efficient electoral system may be able to discern the most popular candidates. But as the last seats remain to be taken by the marginal candidates, limitations in the electoral procedure may cause 'noise' that drowns out a clear decision whether one candidate is really more electable than another.

Redistributed preference votes may result somewhere down the line in an element of perversity. But to dismiss them, with a flourish of fearsome jargon, as 'non-monotonic', as if this were a final judgement on their unworthiness, is to make oneself as ridiculous as a perpetual-motion machine-maker of electoral systems.
It is less mentioned that x-voting systems of quota counting party lists are non-monotonic.

Representation, like time, seems a probabilistic and problematic concept of somewhat indecisive direction.
This possibility, of making representation stand in for time, suggests its properties are not so much unique as a matter of mathematical form. They are both something like a 'probability vector.'

Distributions of the vote and count.

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In discussing voting distributions, the nature of the vote used can affect the distribution. The inefficiency of the single preference or spot vote, as a statement of preference, is liable to cause tactical voting, among middling voters, for a second choice to the left or the right of their true political position. For fear of wasting the vote on middling candidates, the distribution of votes becomes polarised between right and left wing parties.

The crudity of the spot vote, as a measure of choice, causes the 'noise' that prevents some first preference choices from being heard in the election.
Those who ignore the waste of the spot vote, for the non-monotonicity of the preference vote, gulp at a fly and swallow a camel. The spot vote's one-stage count hasnt even got the potential to detect a marginal irrationality that may crop up at some stage in the count of a preference vote. The spot vote, unlike the preference vote, wasnt even designed for a rational count but only a simple majority count.

This polarisation by the spot vote may exaggerate or aggravate or petrify social polarisation. Proportional representation, if by transferable voting, would allow voters to transcend the polarities, if they wished. And most probably this would happen, to some extent.

For explanation of a monotonic element to voting systems, see Robert A Newland. Comparative Electoral Systems published by the Electoral Reform Society.

Richard Lung
11 october 2003.

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