Three Averages Binomial STV.

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Traditional STV.

There is no question that it is possible to conduct a manual count of the transferable vote by acceptable scientific standards. It can be done by transfering a random sample of the total transferable vote of any elected candidate with a surplus above the quota. The so-called law of large numbers observes the increasing probability with size that the random sample is representative. The randomising is done, in a rough and ready way, by shaking up the ballot papers in a ballot box, as in the Republic of Ireland.

I believe the automated counts, in Cambridge Massachusetts, works on some random number algorithm.

The margin of error in the random sample transfer depends on the statistical rigor with which it is carried out. Hence it is controlable error.

The random sample transfer is not reliable for STV elections with small populations. J B Gregory did the arithmetic of how all voters for a winning candidate get a proportionate share in how they transfer the surplus remainder, of their one vote each, to next prefered candidates.

In a large election, it turns out that giving the appropriate weight to successive preferences, at every stage of the count, is too laborious manually. Returning officers had to resort to various shifts and short-cuts that probably would not affect the result. These became formalised in sets of rules, gradually amended over the decades.

Meek STV and Binomial STV.

We have to thank their labors for the fact that Brian Meek recognised that automation made possible more systematic use of the Gregory method. The surplus value is the fraction of a vote, that is not needed, in all votes, for a candidate elected with a surplus. The remainder of each one vote is called the keep value, because it is the fraction of every vote, that helps elect a candidate, that that elected candidate keeps.

Hence, keep value plus surplus value equals one vote.

In the traditional STV count, the keep value never changes, because once a candidate is elected, no more further preferences to him are counted. The returning officer passes them over for any further preference for election to remaining seats.

But Meek STV keeps track of further preferences to already elected candidates. This recognises the increased surplus of votes to that candidate, and correspondingly decreases that candidates keep value. Thus the smallness of the keep value measures the popularity of the candidate. And this keep value diminishes as more surplus votes accrue to the candidate, in the course of the Meek method count.

I guess Meek STV is the one beacon of light in the twentieth century Dark Ages of Democracy. (Other beacons tending to be "all smoke and mirrors".) The 18th century French Enlightenment got off to a creditable start on election method. The 19th century made considerable progress (more than the 20th). If Meek had just programmed the traditional hand count, skipping surplus transfers for candidates already with a quota, it would not have been all that remarkable.

Instead, Meek used the key concept of the "keep value." A keep value of one means that a candidate is just elected to a quota, with no surplus votes over the quota. The larger the surplus of votes that an elected candidate accumulates, in the Meek count, the smaller the keep value, as a ratio of the quota to that candidates total transferable vote.

Binomial STV (BTV).

Meek STV keep-valuing is the key that opens the doors of Binomial STV. (For short, call it BTV.)

BTV extends the use of the keep value for candidates in deficit of a vote, as well as a surplus. As keep value plus transfer value equals one, and deficit keep values are greater than one, this implies a negative transfer value.

The first door BTV opens is that every candidate is ranked from smallest to largest keep value each possesses.

The second door BTV opens is that the all-candidates keep value makes possible a rational exclusion count, as well as a rational election count.

Count the preferences in reverse and thus find the exclusion keep values. Invert them and average them with the election keep values, using the geometric mean. This is first order Binomial STV. The bi-, in binomial refers to the two terms of an election count of preferences and an exclusion count of unpreferences.

The third door BTV opens is that, following (non-commutative) expansion of the binomial theorem, second, third, fourth etc order BTV elections are possible. In general, statistical analysis of the vote may be consistently and indefinitely refined.

And it does not have to stop at the binomial theorem!
This may be of more immediate practical importance for the science of data mining or data retrieval, than elections.

However, BTV should minimise the problem of "premature exclusion" of candidates, to which all existing official election methods (including the best of them, Meek STV) are prone, especially in single vacancy elections.

The fourth door that BTV opens is over-all level of satisfaction with the candidates: all preferences, expressed or absented, count equally. Partly exhausted ballots or Totally exhausted ballots (None of the Above) help towards a quota for an unfilled seat.

Indeed, all preferences, expressed or not, have to be counted equally, in order not to give undue weight to last expressed preferences, in the reverse preference count. For, the last preferences, normally are not expressed. However, BTV may encourage a fuller expression of later preferences, helping to count against most disliked candidates.

Also, this means that BTV differs from Meek STV, which reduces the quota as ballots become exhausted or express no further preference.

With BTV the (Droop) quota stays constant.

Besides Binomial STV, I also invented the Harmonic Mean quota. This is a harmonic mean or average of the Hare quota and the Droop quota. These are respectively maximum and minimum quotas for a proportional count of voters prefered candidates in a multi-member constituency. Thus they are out-lying quotas to the Harmonic Mean quota, which by taking their average, is more representative and less likely to give a stray result, one way or the other.

This is what taking averages is generally about: calculating the representative result, most typical of the main body of results, and farthest away from stray results at the most error-prone margins of a range.

This was also the purpose, in Binomial STV, of averaging each candidates election count keep value and exclusion count keep value, by the geometric mean, to get a more representative final keep value for each candidate.

The purpose of this up-date page on my invention of Binomial STV is to introduce a third average, the arithmetic mean, for the same purpose of averaging out out-lying results, one way or the other, in a range of calculations.

My modified procedure applies to second or higher order Binomial STV. First-order BTV is just a straightforward election count and a straightforward exclusion count. But second or more order BTV involves more counts, in which a candidates vote is re-distributed.

Up till now, I decided which candidates votes redistributed, simply on a plurality basis: such as the candidate with the most preferences in an election count or the candidate with the most un-preferences in an exclusion count.

Now I realise that a more representative result might depend on each candidate, in turn, having their preferences in an election count (or unpreferences in an exclusion count) redistributed. This sub-routine of redistribution counts would modify each candidates keep value, in varying ways. The most representative of these redistribution keep values would be found by taking their arithmetic mean.

Three foremost averages can be used in Binomial STV, to minimise error from their distinct ranges of data, which they represent. This is Three Averages Binomial STV (BTV3).

Note: Before putting-up this page, I discovered a fourth average, my own innovation, which I called the factorial mean. And applied it to averaging orders of Binomial STV counts: Factorial Mean and four averages Binomial STV (BTV4).

Richard Lung.