Jagiellonian compromise

Electoral system for two-tier voting bodies

The Jagiellonian compromise is an electoral system for two-tier voting bodies originally proposed in 2004[1] for the Council of the European Union as a way of achieving "one person, one vote" within the union. The compromise was analysed by various authors[2] and received attention in the popular press.[3] The system is based on the square root law of Penrose, which implies that a priori voting power defined by the Penrose–Banzhaf index of a member of a voting body is inversely proportional to the square root of its size. Hence the number of votes obtained by a representative of a state  j {\displaystyle j} with population  N j {\displaystyle N_{j}} is proportional to  N j {\displaystyle {\sqrt {N_{j}}}} . Jagiellonian Compromise is based on a single criterion only. Decision of the Council of the union of  M {\displaystyle M} member states is taken if the sum of the weights of states voting in favour of a given proposal exceeds the qualified majority quota  q {\displaystyle q} equal to

q = 1 2 ⋅ ( 1 + ∑ i = 1 M N i ∑ i = 1 M N i ) {\displaystyle q={\frac {1}{2}}\cdot \left(1+{\frac {\sqrt {\sum _{i=1}^{M}N_{i}}}{\sum _{i=1}^{M}{\sqrt {N_{i}}}}}\right)}

For a generic distribution of population among  M {\displaystyle M} states of the union, the optimal threshold  q ∗ {\displaystyle q_{*}} decreases with  M {\displaystyle M} as  q ∗ ≈ 1 / 2 + 1 / π M {\displaystyle q_{*}\approx 1/2+1/{\sqrt {\pi M}}} .[4]

## See also

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* Mathematics portal
* Political science portal

* List of countries and dependencies by population
* Penrose method
* Jagiellonian University

## References

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### Footnotes

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1. ^ Słomczyński & Życzkowski 2004.
2. ^ Kirsch 2010; Pukelsheim 2010; Ratzer 2006.
3. ^ Highfield, Roger (14 July 2004). "The EU Constitution Is 'Unfair', According to Game Theorists". The Telegraph. London. Retrieved 30 August 2017.
4. ^ Życzkowski & Słomczyński 2012.

### Bibliography

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* Kirsch, Werner (2010). "The Distribution of Power in the Council of Ministers of the European Union". In Cichocki, Marek A.; Życzkowski, Karol (eds.). Institutional Design and Voting Power in the European Union. Farnham, England: Ashgate Publishing. pp. 93ff. ISBN 978-1-4094-9995-4.
* Pukelsheim, Friedrich (2010). "Putting Citizens First: Representation and Power in the European Union". In Cichocki, Marek A.; Życzkowski, Karol (eds.). Institutional Design and Voting Power in the European Union. Farnham, England: Ashgate Publishing. pp. 235–254. ISBN 978-1-4094-9995-4.
* Ratzer, Edward (2006). "On the 'Jagiellonian Compromise': Voting in the European Union" (PDF). Cambridge, England: University of Cambridge. Retrieved 30 August 2017. `{{cite journal}}`: Cite journal requires `|journal=` (help)
* Słomczyński, Wojciech; Życzkowski, Karol (2004). "Voting in the European Union: The Square Root System of Penrose and a Critical Point". arXiv:cond-mat/0405396.
* Życzkowski, Karol; Słomczyński, Wojciech (2012). "Square Root Voting System, Optimal Threshold and $$ \uppi $$ π". Power, Voting, and Voting Power: 30 Years After. pp. 573–592. arXiv:1104.5213. doi:10.1007/978-3-642-35929-3_30. ISBN 978-3-642-35928-6. S2CID 118756505.

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