Estimating the Efficiency of Voting in Big Size Committees

In a simple voting committee with a finite number of members, in which each member has a voting weight, the voting rule is defined by the quota (a minimal number of voting weights is required to approve a proposal), and the efficiency of voting in the committee is defined as the ratio of the number of winning coalitions (subsets of the set of members with total voting weights no less than the quota) to the number of all possible coalitions. A straightforward way of calculating the efficiency is based on the full enumeration of all coalitions and testing whether or not they are winning. The enumeration of all coalitions is NP-complete problem (the time required to find the solution grows exponentially with the size of the committee) and is unusable for big size committees. In this paper we are developing three algorithms (two exact and one heuristic) to compute the efficiency for committees with high number of voters within a reasonable timeframe. Algorithms are applied for evaluating the voting effi ciency in the Lower House of the Czech Parliament, in the European Parliament and in the Council of Ministers of the EU.