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Signpost sequence

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Generalized rounding rule

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

Formal definition

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

round ⁡ ( x ) = { ⌊ x ⌋ x < s ( ⌊ x ⌋ ) ⌊ x ⌋ + 1 x > s ( ⌊ x ⌋ ) {\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.

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.


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