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 ${\displaystyle s_{0}=1,s_{1}=2,s_{2}=3\dots }$

Formal definition

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

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

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.^[2]

References

1. Pukelsheim, Friedrich (2017), "From Reals to Integers: Rounding Functions, Rounding Rules", 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, retrieved 2021-09-01
2. 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.