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[1]. 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 [math]\displaystyle{ s_0 = 1, s_1 = 2, s_2 = 3 \dots }[/math]
Formal definition
Mathematically, a signpost sequence is a localized sequence, meaning the [math]\displaystyle{ n }[/math]th signpost lies in the [math]\displaystyle{ n }[/math]th interval with integer endpoints: [math]\displaystyle{ s_n \in (n, n+1] }[/math] for all [math]\displaystyle{ n }[/math]. This allows us to define a general rounding function using the floor function:
[math]\displaystyle{ \operatorname{round}(x) = \begin{cases} \lfloor x \rfloor & x \lt s(\lfloor x \rfloor) \\ \lfloor x \rfloor + 1 & x \gt s(\lfloor x \rfloor) \end{cases} }[/math]
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.[2]
References
- ↑ Pukelsheim, Friedrich (2017), "From Reals to Integers: Rounding Functions, Rounding Rules" (in en), Proportional Representation: Apportionment Methods and Their Applications (Springer International Publishing): pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN 978-3-319-64707-4, https://doi.org/10.1007/978-3-319-64707-4_4, retrieved 2021-09-01
- ↑ Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9. https://archive.org/details/fairrepresentati00bali.
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