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

Formal definition

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Mathematically, a signpost sequence is a localized sequence, meaning the  th signpost lies in the  th interval with integer endpoints:   for all  . This allows us to define a general rounding function using the floor function:

 

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

Applications

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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

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  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.