[EM] Quota averages

[Richard Lung]

Quota averages

Representative Government, by John Stuart Mill, describes what we now 
call the Hare quota, as an average. Add the district electorates and 
divide by the number of districts. That is the arithmetic mean voters 
per seats.

The Droop quota allows an election, to a single vacancy, with only half 
the votes to the candidate. Whereas, with the Hare quota, only the total 
vote is elective. The latter is undemocratic, because it requires total 
deference to one candidate. The Droop quota is also undemocratic, in 
that it allows the election of a candidate, with a negligibly greater 
vote than another.

To avoid the undemocratic extremes of maximum and minimum proportional 
representation, I introduced average proportional representation. The 
average of the Hare and Droop quotas, that is to say, a quota 
representative of their range, is their harmonic mean, which I therefore 
called the Harmonic Mean quota, V/(S + 1/2).

The working is: invert the Hare and Droopquotas; add, and divide by two; 
then re-invert. Or:

[S/V + (S+1)/V]/2 = (2S+1)/2V. Hence: 2V/(2S+1).

The Monty Hall (multiple doors) problem isa successive choice and 
exclusion procedure that probability sums to Droopquota proportional 
representation, the ratio: S/(S+1).

The Harmonic Mean quotaseries is alternate terms to the Droop quota. 
Therefore, the Harmonic Mean quota follows Monty Hall procedure. As must 
an alternate quota foralternate terms of the Droop quota.

The alternate HM quota PR is: (2S-1)/2S.It is an average of the Hare 
quota PR ratio and alternate Droop quota ratio: S/S and(S-1)/S. The 
latter multiplied by the inverse Droopquota ratio gives a squared 
geometric mean quota ratio: (S-1)(S+1)/S^2 = (S^2 –1)/S^2.

{The GM ratio is: [(S^2 -1)^1/2)]/S. Multiplyby the Hare quota, V/S, for 
the geometric mean quota: V[(S^2 -1)^1/2)]/S^2.}

The effect of multiplying alternate and inverse Droop quota ratios is to 
give analmost complete or unitary proportional representation, while 
retaining the essential Droop quota ratio, at a much higher level. The 
Monty Hall procedure, per number of door choices, produces Droop quota 
proportional representation.

Suppose the Monty Hall problem starts with three doors, then the above 
GM formula squares the number of doors to 9, and the PR is in the ratio 
8/9.

The contestant is asked to make a choice, from nine doors, with a prize 
behind one of them, and a probability of success of one ninth.The 
compere opens one of the other doors, lacking the prize, and offers the 
contestant another choice. The probability of success is now one eighth 
times a smaller statistical “universe” eight ninths the size of the 
original: 1/8 x 8/9 = 1/9. Thischoice may be repeated eight times, 
giving an eight ninths probability of success to finding the prize 
behind one of the nine doors.

If the prize has not been found behind one of the chosen eight doors, it 
is certain to be behind the ninth door. But equally, the compere is 
certain to remove the one remaining door, leaving zero probability of 
success. This is comparable to the situation, coming to the residual 
unrepresented quota, caused by the Droop quota denominator, in terms of 
seats plus one.

A quota is an average, and can be different kinds of average, the 
arithmeticmean, harmonic mean and geometricmean. The geometric mean is a 
namefor a power arithmetic mean. There is also a power harmonic mean, in 
which oneof the binomial theorem terms is a fraction. This was the 
fourth average, in FABSTV: Four Averages Binomial Single Transferable 
Vote. There, I used the power harmonic means, of individual candidates 
keep values, toaverage higher orders of count, in accord with the 
binomial theorem.

  Regards,

Richaed Lung.


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