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Multidimensional political apportionment.

Javier Cembrano1, José Correa2, Victor Verdugo3

  • 1Institute of Mathematics, Technische Universität Berlin, 10623 Berlin, Germany.

Proceedings of the National Academy of Sciences of the United States of America
|April 8, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces multidimensional proportional apportionment for fair seat allocation. While exact solutions aren't guaranteed and are computationally hard, approximate methods offer a practical approach for complex electoral systems.

Keywords:
apportionmentinteger programmingsocial choice

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Area of Science:

  • Political Science
  • Computational Social Science
  • Mathematics

Background:

  • Seat allocation in deliberative assemblies is a fundamental problem in political organization.
  • Divisor methods, like the Jefferson/D’Hondt method, capture the principle of proportionality.
  • The biproportional apportionment method extends divisor methods to two dimensions but is limited to this scope.

Purpose of the Study:

  • To initiate the study of multidimensional proportional apportionment, extending beyond two dimensions.
  • To formalize a notion of multidimensional proportionality.
  • To investigate the existence and computational complexity of multidimensional proportional apportionments.

Main Methods:

  • Formalizing a novel definition of multidimensional proportionality.
  • Analyzing an integer linear program to determine existence and complexity.
  • Applying discrepancy theory, inspired by the Beck–Fiala theorem, to find approximate solutions.

Main Results:

  • The existence of multidimensional proportional apportionments is not guaranteed.
  • Determining the existence of such apportionments is an NP-complete problem.
  • Approximate multidimensional proportional apportionments can be found with small deviations from marginals.

Conclusions:

  • Extending proportional seat allocation to multiple dimensions presents significant theoretical and computational challenges.
  • Approximate methods provide a viable path for implementing multidimensional proportional apportionment in real-world electoral systems.
  • The study's findings were validated using data from the 2021 Chilean Constitutional Convention election.