# Jagiellonian Compromise

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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.

## Reaction and explanation

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 ${\displaystyle j}$ with population ${\displaystyle N_{j}}$ is proportional to ${\displaystyle {\sqrt {N_{j}}}}$. Jagiellonian Compromise is based on a single criterion only. Decision of the Council of the Union of ${\displaystyle M}$ Member States is taken if the sum of the weights of States voting in favour of a given proposal exceeds the qualified majority quota ${\displaystyle q}$ equal to

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

For a generic distribution of population among ${\displaystyle M}$ States of the Union, the optimal threshold ${\displaystyle q_{*}}$ decreases with ${\displaystyle M}$ as ${\displaystyle q_{*}\approx 1/2+1/{\sqrt {\pi M}}}$.[4]

## References

### Notes

1. ^
2. ^
3. ^ Highfield, Roger (14 July 2004). "The EU Constitution is 'Unfair', According to Game Theorists". The Telegraph. London. Retrieved 30 August 2017.
4. ^

### Bibliography

Kirsch, Werner (2010). "The Distribution of Power in the Council of Ministers of the European Union". In Cichocki, Marek A.; Życzkowski, Karol. 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. 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.
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 π". arXiv:1104.5213.