Probabilistic Approach to the Rounding Problem with Applications to Fair Representation

  • Bessy Athanasopoulos


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.


Electoral System Fair Share Vector Problem Fair Representation Hamilton Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Balinski, M.L. and Young, H.P., 1982, Fair Representation: Meeting the Ideal of One Man, One Vote, Yale University, New Haven.Google Scholar
  2. Balinski, M.A., Demange, G., 1989, Algorithms for Proportional Matrices in Reals and Integers, Math Programming, North Holland, 45:193–210.MathSciNetMATHCrossRefGoogle Scholar
  3. Balinski, M.L. and Rachev, S.T., 1992, Rounding proportions: rules of rounding, Technical Report No. 384, Laboratoire d’Econometrie, École Polytechnique.Google Scholar
  4. Billingsley, P., 1986, Probability and Measure, 2nd edition, Wiley.Google Scholar
  5. Birkhoff, G., 1976, Monotone apportionment Schemes, Proceedings of the National Academy of Sciences, U.S.A., 684–686.Google Scholar
  6. Diaconis, P. and Preedman, D., 1979, On rounding percentages, Journal of the American Statistical Association, 74:359–364.MathSciNetMATHGoogle Scholar
  7. Feller, W., 1970, An Introduction to Probability Theory and Its Applications, John Wiley & Sons, New York, Vol. II, 2nd ed., 504–515.Google Scholar
  8. Hoffman, Mark S., 1992, The World Almanac and Book of Facts, 588:74–75.Google Scholar
  9. Legislative decrees concerning the election of the Greek deputies, 1928-1990, Journals of the governments of the Greek Republic, Athens.Google Scholar
  10. Maejima M., Rachev, S.T., 1987, An ideal metric and the rate of convergence to a self-similar process, Annals of Probability, 15:702–727.MathSciNetCrossRefGoogle Scholar
  11. Mosteller F., Youtz, C. and Zahn, D., 1967, The distribution of sums of rounded percentages, Demography, 4:850–858.CrossRefGoogle Scholar
  12. Nikolakopoulos, I., 1989, Introduction in the Theory and Practice of Electoral Systems, Sakkoula A., Athens.Google Scholar
  13. Pyke, R., 1965, Spacings, The Journal of the Royal Statistical Society, Series B, 27, No. 3:395–449.MathSciNetMATHGoogle Scholar
  14. Rachev, S.T., 1991, Probability Metrics and the Stability of Stochastic Models, Wiley, New York.MATHGoogle Scholar
  15. Turing, A.M., 1948, Rounding-off errors in matrix processes, Quart. J. Mech., 1:287–308.MathSciNetMATHCrossRefGoogle Scholar
  16. Wilkinson, J.H., 1963, Rounding errors in algebraic processes, Prentice-Hall, Englewood Cliffs, NJ.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Bessy Athanasopoulos
    • 1
  1. 1.Department of Statistics and Applied ProbabilityUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations