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There is a serious technical flaw in the newly approved Italian electoral law. The flaw lies in the method used to allocate the Chamber of Deputies seats to parties (or coalitions) within multi-member regional constituencies. The procedure stated in the law could produce contradictory results: it could end up assigning a party more (or less) seats than it is entitled to receive on the basis of the same law. At least two types of paradoxes may occur. Although they have been utterly overlooked in the debate over electoral reform, they can be critical in practice when trying to determine the actual seat allocation. The failure of the current Italian electoral law was inherited from the previous one but the consequences are worse. Moreover, a correction mechanism introduced into the law at the last-minute does not prevent it from producing contradictory results. The paradoxes that undermine the Italian electoral law are pointed out and a solution is proposed. A broad conclusion is that a more extensive use of mathematics in the design and evaluation of electoral systems would help identify flaws and deliver more transparent, logical and fairer electoral laws.

Three apportionment problems are addressed that are of current interest in Germany and Switzerland: the assignment of committee seats in a way that preserves the parliamentary majority-minority relation, the introduction of minimum restrictions in a two-ballot system to accomodate the direct seats won by the constituency ballots, and biproportional apportionment methods for systems with multiple districts so as to achieve proportionality between party votes as well as between district populations.

The divisor method with standard rounding (Sainte-Laguë/Schepers) is amended by a gentle majority clause, in order to map the government majority in parliament into a seat majority in committees.

In this paper we show results for some unipersonal and pluripersonal elections in Ecuador and in Spain, observing that the used methods can be replaced by better ones. We present a method for individual elections based on one-on-one comparisons, and preferential voting. This method verifies CONDORCET and PARETO, and it is furthermore better than the Two Round method. For multicandidate elections we give proportional and monotone methods based on preferential vote. We also analyse the electoral system of Ecuador and propose alternatives for it.

In this work we present several methods for distributing the seats of the European Parliament amongst the States of the European Union, in accordance with the restrictions established in article I-20 of the projected European Constitution. The proposed methods can be applied to the current composition of the EU, but also if there is a change in the number of states or their populations. They are based on of every country so that they verify the constitutional restrictions, and so that their rounding to an integer number will constitute an allotment of the seats of the Parliament.

The complexities of voting theory are captured by Arrow’s Impossibility Theorem and McKelvey’s chaos result in spatial voting. A careful analysis of Arrow’s theorem, however, proves that not all of the supplied information is used by the decision rule. As such, not only does this seminal result admit a benign interpretation, but there are several ways to sidestep Arrow’s negative conclusion. McKelvey’s result is described in terms of more general voting rules. Then a new solution concept, called the ‘finesse point’, is introduced. This centrally located point generalizes the core and minimizes what it takes to respond to any proposal by another person.

The Faroese electoral system uses a method of proportional representation for distributing the seats in the Faroese Parliament (The Løgting). The electoral formulae attempt to give each political party a number of seats that is close to its vote share. In addition, each district should receive a number of delegates that is proportional to the number of voters in the district. We show that the current electoral formula has significant weaknesses, and propose 7 alternative electoral formulae which consider various subsets of constraints — such as lower bounds on the number of seats in districts and electoral thresholds. Numerical simulations with the current and proposed electoral formulae on the elections from 1978 to 2004, and on randomly generated election results, are presented. The results show that some of the proposed alternatives clearly are superior to the current electoral formula with respect to well-known quality measures.