signpost sequence

In mathematics and apportionment theory, a signpost sequence is a sequence of real numbers, called signposts, used in defining generalized rounding rules. A signpost sequence defines a set of signposts that mark the boundaries between neighboring whole numbers: a real number less than the signpost is rounded down, while numbers greater than the signpost are rounded up.{{Citation |last=Pukelsheim |first=Friedrich |title=From Reals to Integers: Rounding Functions, Rounding Rules |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=71–93 |url=https://doi.org/10.1007/978-3-319-64707-4_4 |access-date=2021-09-01 |place= |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_4 |isbn=978-3-319-64707-4|url-access=subscription }}

Signposts allow for a more general concept of rounding than the usual one. For example, the signposts of the rounding rule "always round down" (truncation) are given by the signpost sequence $s_0 = 1, s_1 = 2, s_2 = 3 \dots$

Formal definition

Mathematically, a signpost sequence is a localized sequence, meaning the $n$ th signpost lies in the $n$ th interval with integer endpoints: $s_n \in (n, n+1]$ for all $n$ . This allows us to define a general rounding function using the floor function:

$\operatorname{round}(x) = \begin{cases}
\lfloor x \rfloor & x < s(\lfloor x \rfloor) \\
\lfloor x \rfloor + 1 & x > s(\lfloor x \rfloor)
\end{cases}$

Where exact equality can be handled with any tie-breaking rule, most often by rounding to the nearest even.

Applications

In the context of apportionment theory, signpost sequences are used in defining highest averages methods, a set of algorithms designed to achieve equal representation between different groups.{{cite book |last1=Balinski |first1=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}

References

Category:Sequences and series

Category:Apportionment methods