Fair Qualified Majorities in Weighted Voting Bodies

In parliaments elected by proportional systems the seats are allocated to the political parties roughly proportionally to the shares of votes for the party lists obtained in elections. Assuming that members of the parliament representing the same party are voting together, it has sense to require that distribution of the influence of the parties in parliamentary decision making is proportional to the dis-tribution of seats. There exist measures (so called voting power indices) reflecting an ability of each party to influence outcome of voting. Power indices are functions of distribution of seats and voting quota (where voting quota means a minimal number of votes required to pass a proposal). By a fair voting rule we call such a quota that leads to proportionality of influence to relative representation. Usually simple majority is not a fair voting rule. That is the reason why so called qualified or constitu-tional majority is being used in voting about important issues requiring higher level of consensus. Qualified majority is usually fixed (60% or 66.67%) independently on the structure of political repre-sentation. In the paper we use game-theoretical model of voting to find a quota that defines the fair voting rule as a function of the structure of political representation. Such a quota we call a fair major-ity. Fair majorities can differ for different structures of the parliament.Concept of a fair qualified majority is illustrated on the data for the Lower House of the Czech Parliament elected in 2010. It can be applied also in corporate governance (decision making rules in share-holding companies).