Lexicographic and Transfer Orders on a Set of Allocations

Abstract :

When allocating a given number of indivisible and identical goods among contenders with given claims, one specifies the boundary conditions which are to be satisfied by such allocation. These conditions are not always sufficiently precise so as to obtain an unambiguous apportionment. If there are many allocations satisfying the conditions, one has to add more of them to the list. Most frequently a specified method or algorithm of division is accepted, however this generates an insoluble dilemma of why a particular method should be selected out of many other existing ones. Another option discussed in this paper is a compromise developed by contenders participating in the distribution and based on the assumption that no individual contender or no coalition of them can be favoured. This compromise can be achieved by subsequent reductions of the set of feasible allocations by eliminating favouring allocations. A favouring allocation satisfies the list of contenders in such a way that the first contender from the list is allocated as many goods as possible, the remaining goods are distributed by allocating as many goods as possible to the second contender, and so on. Identifying the allocations of this type requires considering all possible arrangements of contenders and establishing a relation of order depending on the sequence of contenders.?In this paper the two types of order relations and their properties are analyzed as well as how establishing a given type of relation influences the reduction of the set of feasible allocations. The first type of relation are lexicographic orders which compare the values of coordinates of allocation vectors in a specified sequence. The second type of relation are transfer orders where one considers a chance to transfer goods in a specified sequence from a selected group of contenders to remaining contenders.