Non-Proportional Allocation of Indivisible Goods

Fair division of goods has been an issue since people began forming communities. This applies not only to physical goods but also to participation in the profits, costs or work. Continuous social and economic development creates further needs to seek new solutions. Diverse forms of shared properties involve using various methods as well. Different solutions have to be applied, among others, while the division of indivisible goods among agents with different claims for the goods is required. In many cases the main emphasis is put on the optimal representation of the structure of claims made by agents participating in division. The exact reflection of this structure is generally impossible, because most often the proportional division is not integer. However, in some instances the structure of claims made by participants in division does not allow to apply the principle of proportionality. In such cases some modified solutions are searched for. One of such modifications is degressive proportionality, which allows unequal representation of the structure of claims. However, this idea turned out to be understood differently, even in its main recent area of application - the apportionment of seats in the European Parliament. Currently the problem of degressively proportional division of indivisible goods is being studied in the framework of fair division theory, with many questions still open. The paper presents different interpretations of the idea of degressive proportionality together with its practical source, i.e. the allocation of mandates among Member States of the European Union in the European Parliament. Each of the interpretations is formalized and briefly analyzed.