Signpost sequence explained

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.

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 ...

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.^[1]

Notes and References

1. Book: Balinski . Michel L. . Fair Representation: Meeting the Ideal of One Man, One Vote . Young . H. Peyton . Yale University Press . 1982 . 0-300-02724-9 . New Haven . registration.