Inverse problems in voting theory

In this paper we deal with certain inverse problems arising in the use of bi-proportional apportionment systems. These are highly sensitive integer problems and their solution allows to determine approximately the number of regional party seats in different electoral regions on the base of representative nation-wide predictions for the total number of party seats. Such information may be of great value in the political planning prior to the elections. If there are m electoral regions and n parties then the problem solved is the restoration of mn integers (the numbers of seats for regional party seats) using the information for n expected shares of votes cast at nation-wide level.