A VOTING GAME (N,W) is a set N of VOTERS with a family W of its subsets such that: (1) N∈W; (2) If C⊂C'⊂N and C∈W, then C' ∈W; (3) If C∈W, then N-C is not in W. The set P(N) of all subsets of N consists of 3 sets: W, L={C: N-C∈W}, and B=P(N)-W-L, with elements termed WINNING, LOSING, and BLOCKING COALITIONS. A winning/blocking coalition C is called MINIMAL if no proper subset of C is in W/B. In practice, voters are assigned positive WEIGHTS p_i and W is defined - given a QUOTA q>½p(N) - as the set of such Cs that p(C)≥q; p(C) stands for Σp_i over all i in C.

Measuring the degree to which group decisions depend on each voter is the key topic of the THEORY OF VOTING GAMES. The theory assumes that VOTING POWER must not be confused with a voter's relative weight p_i/p(N), but should be construed as dependent on the number of winning coalitions in which i's presence is necessary to stay winning. Next, one proceeds to define indices of which the Banzhaf and Shapley-Shubik are favored by most ACADEMIC analysts.

Voting power became a hot issue in mathematical political science when the decision rules for the enlarged EU were set up by the Nice Treaty (2001). Since then various voting games have been proposed for the E. Council and analyzed by many experts (see http://www.cyf-kr.edu.pl/~ussozans/voting.htm). In June 2004, some 50 scientists advocated (in a letter to the governments) a game with weights computed as square roots of the EU states populations. This meant to reject both the crude demographic weights, retained in the Constitution Treaty, and "political" weights reflecting a negotiated division of power. The scholars argued that the system based on Penrose's theorems more faithfully renders democratic principles and yields a flatter power distribution than the Constitution game.

The aim of this paper is not to convert politicians to scientific methods of constructing voting systems, but to propose a scientific reconstruction of their priorities and to explain such outcomes as the consent of France, UK and Italy to the game which - IF voting power is measure by means of classical indices - gives Germany a big power advantage over them. The author claims that what the negotiators wanted to maximize for their states was, in fact, blocking power described as follows: the BLOCKING POWER of an actor decreases with the number of other actors needed to form with him a minimal blocking coalition and increases with the number of voters from among whom he may choose partners for SMALL SIZE minimal blocking coalitions. The WINNING POWER has a similar meaning. The paper brings a mathematical elaboration of these two relatively independent facets of voting power and presents an analysis of the EU Constitution game in terms of certain new indices.

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