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Square Root Voting System, Optimal Threshold and π

In: Voting Power and Procedures

Author

Listed:
  • Karol Życzkowski

    (Jagiellonian University
    Polish Academy of Sciences)

  • Wojciech Słomczyński

    (Jagiellonian University)

Abstract

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favor of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic “union” of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: $$q \simeq 1/2 + 1/\sqrt{\pi M}$$ . The prefactor $$1/\sqrt{\pi }$$ appears here as a result of averaging over the ensemble of “unions” with random populations.

Suggested Citation

  • Karol Życzkowski & Wojciech Słomczyński, 2014. "Square Root Voting System, Optimal Threshold and π," Studies in Choice and Welfare, in: Rudolf Fara & Dennis Leech & Maurice Salles (ed.), Voting Power and Procedures, edition 127, pages 127-146, Springer.
    • Handle: RePEc:spr:stcchp:978-3-319-05158-1_8
    DOI: 10.1007/978-3-319-05158-1_8
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    Cited by:

    1. Laurent Bartholdi & Wade Hann‐Caruthers & Maya Josyula & Omer Tamuz & Leeat Yariv, 2021. "Equitable Voting Rules," Econometrica, Econometric Society, vol. 89(2), pages 563-589, March.
    2. Kirsch, Werner & Toth, Gabor, 2022. "Collective bias models in two-tier voting systems and the democracy deficit," Mathematical Social Sciences, Elsevier, vol. 119(C), pages 118-137.

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