We consider two-tier voting system and try to determine optimal weights for a fair representation in such systems. A prominent example of such a voting system is the Council of Ministers of the European Union. Under the assumption of independence of the voters, the square root law gives a fair distribution of power (based on the Penrose-Banzhaf power index) and a fair distribution of weights (based on the concept of the majority deficit), both given in the book by Felsenthal and Machover. In this paper, special emphasis is given to the case of correlated voters. The cooperative behaviour of the voters is modeled by suitable adoptions of spin systems known from statistical physics. Under certain assumptions we are able to compute the optimal weights as well as the average deviation of the council's vote from the public vote which we call the democracy deficit.

The Penrose-Banzhaf index and the Shapley-Shubik index are the best-known and the most used tools to measure political power of voters in simple voting games. Most methods to calculate these power indices are based on counting winning coalitions, in particular those coalitions a voter is decisive for. We present a new combinatorial formula how to calculate both indices solely using the set of minimal winning coalitions.

In this article the voting rules in the Council of the European Union are investigated. It is known that both the current system, according to the Treaty of Nice, and the voting system proposed in the Lisbon Treaty strongly deviate from Penrose's square root law, which under certain assumptions can be shown to be the ideal power distribution. Since it seems easier to make corrections to the current systems than to agree upon completely different new voting rules, one may hope that adjustments of the quota in the Lisbon Treaty might give rise to a system that is close to the square root law. In this article, this question is investigated. The computations it contains show that a mere change of quota in the treaties of Nice and Lisbon cannot bring the system substantially closer to the ideal distribution of power as given by the square root law.

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