[EM] Truncation error in STV

[James Gilmour]

> At 17:25 -0400 4.8.2003, Dgamble997 at aol.com wrote:
> >The point he was trying to make is that ERS97 gives the impression 
> >that vote totals are accurate to 0.01 votes and that in fact they 
> >are not.

The important word here is "impression".  Brian Wichmann was applying the conventional mathematical
interpretation of accuracy to any value reported as "0.01".  It is clear from reading all the rules
that the calculations are not done to that accuracy, but that is the impression that could be
conveyed.

> >Take the following example:
> >
> >A surplus of 20 votes is to be transferred. This surplus arises from 
> >a candidate achieving a quota after having 45 votes transferred to 
> >him/her.

We also need to know the number of ballot papers that conveyed those 45 votes.  Without that
specific info, I guess these are 45 first preference votes and therefore there are 45 papers each
with a value of 1.00.

> >Each vote is transferred at a value of 0.44 to 2 decimal 

I think "vote" here should be "ballot paper".  The value conveyed on each ballot paper can change
through successive transfers, but the value of one vote remains constant.

> >Each vote [ballot paper] is transferred at a value of 0.44 to 2 decimal 
> >places to the next candidate. 19.8 votes are transferred not 20, 0.2 
> >votes are lost due to rounding to 2 dp. In order to obtain accurate 
> >totals to 2 dp individual votes must be taken to 3,4 or more dp.( 
> >0.4444 X 45 =  19.998).
> 
> Thanks for the reply. How do you get "about (number of digits in 
> total votes)+1" from this? 

In David's example the surplus was 20 votes carried on 45 papers each with a value of 1.00.  As his
calculation shows, the value of the votes "lost" by the truncated calculation is 0.20.  But suppose
this had been a major public election and the surplus had been 2000 votes carried on 4500 papers
each with a value of 1.00.  Then the "loss" due to the truncated arithmetic would be 20 votes.  So
to obtain any given accuracy,  the number of decimal places in the calculation must vary with the
magnitude of the numbers involved.  Hence Brian Wichmann's suggestion of (number of digits in total
votes + 1).

> What are the complexities?

Cannot comment specifically as Brian never said what he had in mind, but I am sure you can see the
complications that might arise, for example, from varying the number of decimal places in the
calculation according to the number of digits in the total votes, and in the calculations of
fractional values of papers that had been transferred several times, reducing in value at each
transfer.

> Why can't we 
> round off 19.998 to 20.00? Is it because the rounding error might 
> help a candidate to reach the quota?

You did not "round OFF", you rounded UP".  The problem with conventional rounding is that you may at
some stage find you have more votes than you started with.  That is not valid and so is not
acceptable.  There are rounding rules that might avoid this (always round to an even value), but
that could be construed as conferring an advantage on the candidate whose vote was thereby rounded
up compared with the candidate whose vote was rounded down.

James