[EM] Olli - Your Truncation error in STV:

[Olli Salmi]

At 10:01 -0400 9.8.2003, Donald Davison wrote:
>Olli - Your Truncation error in STV:
>
>Dear Olli Salmi, you wrote: "Could somebody please explain this to me?  Why
>is "about (number of digits in total votes)+1"  implied? What are the
>complexities elsewhere?"
>
>Donald here:  The fractional transfer of votes can draw a fine line between
>the candidates that should be elected and the candidates that should not be
>elected, but the lost of votes due to the dropping of decimals can move
>that line and elect a different candidate somewhere in the calculations.
>
>If we are to have the vote sums of each candidate have an accuracy to 0.001
>of a vote, then the transfer value will have to contain more decimal places
>because the effect of decimals is going to be increased the more the
>tranfer value is multiplied, but I do not agree with this Wichmann that the
>number should be based on total votes.  Most votes in a STV election are
>never transferred.  My thinking is that it should be based on only the
>number of papers of the candidate with the surplus.

Thanks for your reply. I thought Wichmann meant you'd have to 
estimate the maximum number of tranfers but I didn't get very far 
along this line of thought. It's much simpler if it's just a question 
of significant digits. There are standard conventions for that.

In the election of the Australian senate the transfer value is taken 
to eight decimals but only whole votes are transferred and the quota 
is an integer.
http://www.aec.gov.au/_content/what/voting/count_senate.htm
In the example on this site the value of the transfer is 1513870 x 
0.65350505 = 989 321, which means it's rounded down from 
989321.69004. The ERS rules would transfer 1513870 x 0.65=984015.5 
votes. It's the same with Northern Ireland rules, but they only 
transfer whole votes. That's more than 5000 votes wasted. The value 
of the votes is accurate only to two significant digits.

The number of ballot papers and and the number of seats are exact 
figures, so the quota could theoretically be calculated to any number 
of decimal places.

>An extreme example would be if only one fractional part of a vote was to be
>transferred.  In that case we would only need three decimal places in the
>transfer value in order to maintain three decimal places in the accuracy of
>the candidate's vote sum.
>
>Three decimal places in the candidates' vote sums would reduce any error
>down to less than one thousandth of a vote.  In order to maintain this
>accuracy, the transfer value will need more places because it is multiplied
>times the number of papers, hence any error will also be multiplied.  The
>transfer value will need three plus the number of digits of the papers.
>Example: If there are 25,000 papers held by the candidate with the surplus
>votes then the transfer value needs three plus five decimal places in order
>to keep the error in the candidates' sums down to one thousandth of a vote.
>Any less decimals will not do the job correctly.

You might want to keep an extra digit in intermediate results.

>The complexities mentioned by Wichmann may be in the hand counting of an
>election, but there are hand calculators that can handle more decimal
>places, so that should not be a problem.

He's concerned about the form of the rules and I don't think he's 
thinking about hand counting.

>Complexities may merely mean that
>people just don't like to deal with decimals.  The trouble with decimals
>for some people is that these people often get the point in the wrong
>place.  They like to take an easy solution and not have anything to do with
>decimal places, get rid of them, and that is just what some jurisdictions
>do, they drop any fractional part of any transfer so that all candidate
>sums are kept in whole numbers - so much for accuracy.

STV has a long history, that's one of the reasons. In some places 
they still transfer actual papers.

Olli Salmi