PROPORTIONALITY IN TWO DIMENSIONS Applied to the representation of Cantons and Constituent Peoples in the House of Peoples of the Parliament of the Federation of Bosnia and Herzegovina

The entities are characterized by two attributes. Therefore, they can be arranged in a two-dimensional matrix. Let I and J be the number of rows and columns in the matrix. Hence there are a total of I  J entities competing for the H units of the good. The variables i and j vary over the sets M = {1, 2, ... I} and N = {1, 2, ... J}, respectively. There are given positive real numbers pi,j. They represent the entities' claims on a share of the H units of the good. As far as possible the distribution should be proportional to the numbers pi,j. Row and column sums in the pi,j-matrix, and the sum of all entries in the matrix, are denoted as follows: pi,N = jN pi,j pM,j = iM pi,j pM,N = iM,jN pi,j Similar notation is used for other indexed variables. It is convenient to normalize the numbers pi,j by defining qi,j = pi,j  H / pM,N for all i and j Then qM,N = H. Let ai,j denote the amount of the good allocated to entity (i, j). These must be nonnegative numbers satisfying aM,N = H. The ai,j-matrix is called an allocation. For now, there is no requirement that ai,j be an integer. Perfect proportionality can be achieved by letting ai,j = qi,j. If there are no further constraints, this is the obvious solution to the problem. Further restrictions are imposed. There are given positive numbers ri for i  M and sj