Amending and enhancing electoral laws through mixed integer programming: the case of Italy

In this paper we discuss how a mathematical approach can be used to solve a serious drawback in the current Italian electoral system for the election of representatives at the Chamber of Deputies and suggest a methodology which is close to the one traditionally adopted in the Italian electoral history, but able to guarantee a more transparent, logical and fairer solution to the problem of transforming votes into seats. 1. The Italian electoral system and some of its drawbacks The Italian electoral law, like many others (for example, Mexico, Germany, Switzerland), wishes to achieve a double proportionality: on the one hand, Parliament seats should be assigned to parties, within each regional constituency, proportionally to the votes cast for the individual parties in the constituency; on the other hand, the amount of national seats obtained by any given party should be distributed among the different constituencies proportionally to the votes obtained by the party in the single constituencies. According to the Italian Constitution the size of the Chamber of Deputies is equal to 630 seats. The country is partitioned into 27 multi-member regional constituencies and the number of seats at stake in each regional constituency is proportional to the number of inhabitants, as provided by the latest population census. The only exception is the region of Valle dAosta which is a single-member district. Finally, 12 seats are assigned to a constituency of Italian citizens who are resident abroad. The current Italian system was introduced in 2005 and it allocates seats proportionally to the votes obtained by each party (and coalition of parties) at the national level and within multi-member regional constituencies. A majority prize is meant to ensure that the party or coalition with the greatest number of total votes wins a substantial majority of seats in the Chamber of Deputies (i.e., at least 340 seats), no matter how many votes the other parties receive. There is a single ballot and candidates are elected on the basis of a blocked list. Moreover, a complex scheme of thresholds is adopted to select which parties and coalitions are eligible to compete in the seat allocation. Despite these special features (majority prize and thresholds) the system was advertised as a proportional one. In fact, the principle of proportionality is adopted by many electoral systems as it is interpreted as a good approximation of the idea of “one-man-one-vote”: the percentage of votes that parties obtain in elections should be as close as possible to the percentage of seats they receive in the legislative assembly [4, 5]. Proportional representation is actually used by more nations than the plurality voting system. A general