Abstract:

The crucial problem in the theory of integer proportional divisions is how to find the
allocation that is both a sequence of integers and also the best approximation of the sequence of quotas of such division, i.e. a not necessarily integer sequence determined by the exact proportions. It turns out that very intuitive and natural solutions in this area lead to quite unexpected properties called paradoxes. The best-known and widely researched in the literature are the population paradox, the Alabama paradox and the new state paradox. It was proved that there is no repeatable algorithmic way to solve this problem and free from all defects. One of the streams of theoretical analysis of degressively proportional divisions relates to the methods applied in proportional allocation. As a result, one can consider the problem of existence of similar paradoxes in this case. The paper demonstrates that there exist the circumstances when any degressively proportional integer approximation leads to the occurrence of all basic paradoxes.