# Probabilistic Approach to the Rounding Problem with Applications to Fair Representation

## Abstract

Failure to add to 100% occurs frequently for sums of percentages in reported sets of tables. It occurs so frequently, that if many sums of percentages add to exactly 100% in a reported set of tables, one begins to suspect the reporter of forcing the situation. Extending the pioneer works of Mosteller, Youtz and Zahn (1967) and of Diaconis and Freedman (1979) who assess the probability that a table of conventionally (MYZ) rounded proportions adds to 1, Balinski and Rachev (1992) introduced some rules of rounding that can improve the conventional rule. Investigating and developing further the so-called K-stationary divisor rules of rounding we compute, for several of these rules, the limiting probability that the rounded percentages add to 100%. We build up a bridge between the problem of rounding and the problem of apportionment. We apply the theory of apportionment in allocating representation among geographical regions in Greece and among states in U.S.A. We investigate and comment on the methods of apportionment currently being used in the two countries, as well as on other possible options including some K-stationary methods.

## Keywords

Electoral System Fair Share Vector Problem Fair Representation Hamilton Method## Preview

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