Random choice models for a GM derivative of a Thermo-dynamic Relativity.

This page is the second of a continuation from page, Statistical basis of Differentiation and Geometric Mean Differentiation of an Interval Acceleration. (The first continuation was about an Hour-glass universe etc.) As on earlier pages (which I can now consign to My Archive section) I shall be comparing relativity with electoral method.

Or, more exactly, as the above title says, I compare a thermo-dynamic relativity with election science. This is because a geometric mean derivative, (hopefully) of general from special relativity, also allows defining time probabilisticly, as thermo-dynamics does. This is in conformity with our common sense notion of time, as more or less (probably) in line with disintegration.

The transferable voting system may consist of a constituency system of multi-member constituencies that form a random distribution of constituencies. The most constituencies, for any given number of seats per constituency, is at the average number of seats per constituency.

The sample size, N, is most simply set at the maximum seats per constituency, which is the maximum of J, the range of possible seats per constituency from zero to one, to two, etc. This is not always so, in work with the normal distribution, but it is easier to work with its discrete number version, the binomial distribution.

This random distribution is reached by an expansion of the binomial theorem. This is a two-term factor - the bi in binomial - raised to a certain power. In the head and tails example, on the page: Statistical Differentiation etc (relegated to My Archive) the power is four. Two to the power of four derives the sixteen possibilities mentioned: 2^4 = 16.

The two terms in the binomial factor are the number of urban or rural districts in any of, say, sixteen constituencies. Urban simply stands for population dense, or populously prefered, which will yield about a seat's worth of representation, while rural stands for population scarce, or populously unprefered, which wont. The same logic is at work, for urban and rural, as for coin heads and tails.

Physics has a duality of kinematics and dynamics. In effect, the latter mimics the former except that the mass variable stands in for the time variable. Election science has a similar duality. A binomial distribution of constituencies is a function of the random number of seats per multi-member constituencies. Compared to constituencies as a function of seats per constituency, voters could be a function of number of candidates prefered per voter, with a single transferable vote (STV).

The dual nature of an electoral analogy with physics (of kinematics and dynamics) can be maintained by considering transferable seats from one multi-member constituency to another, to compensate proportionly for population shifts, when people vote with their feet. Foot-voting and voting are thus the "kinematics" and "dynamics" of elections.

The Single Transferable Vote (STV) Proportionly Represents voters in multi-member constituencies. In a five-member constituency, the candidates are elected on achieving a quota of one-sixth of the votes. Thus, the PR would be (at least) five-sixths of the voters. Some candidates may win more than their quota, q = v/(s + 1), which is the total votes, v, in the constituency, divided by the number of seats, s, plus one.

Using small numbers, say there are 120 voters. For 5 seats, the quota is 20. Say some candidate gets (a total vote, w, of) 60 votes, as first preferences. Then she gets to keep 20 votes, while the other 40 votes are transferable to next preferences, on ballot papers that rank order of choice for candidates.

The so-called keep-value, k, here, is 20/60 =1/3. The winning candidate gets to keep the 60 first preferences at one-third of their value, because that is just enough to gain her election. The other forty votes-worth are not wasted but transferable to next preferences at a transfer value, f, of 1 - 20/60 = 2/3.

It is possible to make electoral analogies with both kinematic and dynamic relativistic versions of the equation of the normal distribution. As shown on the page, Statistical basis of Differentiation etc, by inventing a new kind of (geometric mean) derivative, an original equation, T = I/V, where I is a constant, the Special Relativity Interval, can derive:

F = (dT/dV)^dV = e^-V = e^-(vt - ct)/2ct(1-t).

This is a kinematic version of SR expresssed as a normal distribution. Its electoral equivalent might be: C = e^-(s - [s])/2[s]{1-[s]/D}.

The square brackets, round s, for seats per constituency, [s] represent the average seats per constituency.
The neutral form of the normal distribution expressed in general statistical terms is given here as:

F = e^-(J - NP)/2NP(1-P).

(Note, I have not included coefficient, like the inverse of the square root of two pi, which standardises the area of the bell-curve to unity or totality of possibilities of a given occurence.)

The equivalent terms, of the three equations, can be put in table 1.

Table 1: Compared statistics of SR kinematics and STV.
Statistical term SR kinematics term Matching STV term
P, probability of the chosen one of two kinds of possibility. t, time [s]/D, Ratio of average seats (or populous districts) per constituency to number of populous and unpopulous districts: probability the constituency districts have seats.
Q = 1-P, complementary probability. 1-t 1- [s]/D, Probability the constituency districts dont have seats.
N, sample size: total of the two kinds of possibilities, randomly differing in the number of each. c, light speed. D, Districts populously prefered or unpopulously unprefered.
J, number of times one of two possibilities occurs, for any given trial. y = vt, distance equals velocity times time, in observers' local reference frames. s, Seats per constituency.
F, frequency of trials randomly occuring for every possible J. F, frequency as randomly varying acceleration event with observers' reference frames. C, Frequency of constituencies with any given number of seats.
NP, the average J. ct, an average distance. [s], Average number of seats per constituency.

The above SR kinematics comparison, with STV, is strictly followed by the SR dynamics comparison with STV, where:

V = e^-(x - [x])/2[x]{1-[x]/K}.

Now to set-out the dynamics comparision in Table 2.

Table 2: Compared statistics of SR dynamics and STV.
Statistical term SR dynamics term Matching STV term
P, probability. m, mass. [x]/K, probability the voter prefers the candidates.
Q, complementary probability. 1 - m 1-[x]/K, probability the voter does not prefer the candidates.
N, sample size. c, light speed. K, number of candidates, popularly prefered or unprefered.
J, possiblities occurence. p, momentum. x, number of preferences, per voter, for candidates
F, frequency. (Another frequency interpretation of observers' reference frames, in similar dimensions but with mass instead of time.) V, frequency of voters with a given number of preferences, x.
NP, average of J. mc = E/c, energy over light speed. [x], average number of preferences per voter.

In table 3, the binomial distribution is used to illustrate either the random distribution of constituencies for any given number of seats per constituency, or, random distribution of voters for any given number of candidate preferences. For instance, in table 3, there is a constituency system of up to 8 seats per constituency. (Notice, that on a random basis, one case of zero seats is allowed for: a constituency too sparsely populated to justify any representation.)

The same distribution, as a distribution of voters, could apply to the case of one constituency with an average number of seats, which is: PN = (1/2)8 = 4 seats. Assuming the average constituency has 8 candidates, the probability of their election is one half. The voters will have from zero to eight preferences.

Table 3: A binomial distribution of either constituencies with varying numbers of seats or voters with varying numbers of preferences.
Number of seats (populously prefered districts) per constituency or number of preferences (popularly prefered candidates) per voter. 0 1 2 3 4 5 6 7 8
Number of constituencies with a given number of seats or number of voters with a given number of preferences. 1 8 28 56 70 56 28 8 1
Seats times constituencies or preferences times voters. 0 8 56 168 280 280 168 56 8

Richard Lung.
25 november 2010.

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