ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 299–307.
Pleiades Publishing, Inc., 2016.
Original Russian Text
M.L. Blank, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 108–116.
The Problem of Fair Division
for a Hybrid Resource
M. L. Blank
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
National Research University—Higher School of Economics, Moscow, Russia
Received November 25, 2015; in ﬁnal form, February 16, 2016
Abstract—We propose an elementary solution to the apartment rent division problem. This
problem belongs to the class of problems of “fair division,” but diﬀers from its standard setting
by “heterogeneity,” i.e., the presence of both a conventional continuous component and a dis-
crete one, a ﬁxed set of rooms. A combinatorial-topological approach to solving this problem in
a ﬁnite number of steps (each of which requires a survey of all participants), actively used in the
literature, allows to obtain an approximate decision only. We propose a fundamentally diﬀerent
setting, based on a priori estimates of each room by the participants and allowing, in principle,
to consider various optimization tasks as well. Our approach is particularly relevant in the case
of a large number of participants. We also note that the proposed approach allows to ﬁnd a
solution in a number of cases where the combinatorial-topological approach does not work.
One of the most fundamental problems of mathematical economics is the task of optimizing
the division of resources in the presence of a given set of constraints according to some (possibly
conﬂicting) criteria. Moreover, often it is not clear a priori whether there exists at least some
(though not optimal) division under the given constraints. The simplest example is the situation of
constructing basic solutions in the classical simplex method. It is easy to understand that the set
of feasible solutions in the simplex method forms a polyhedron, whose vertices, in principle, can be
calculated explicitly. However, in the case of a large number of constraints, this task is extremely
time-consuming, and it turns out that it is easier to solve it by constructing a dynamical system
whose trajectories converge to a feasible solution or diverge if the solution does not exist.
Despite these diﬃculties, the problems in the above example are mostly technical in nature.
A fundamentally diﬀerent situation arises in an also classic problem (we do not know the ﬁrst
publication on this topic, but the mathematical formulation appeared already in 1948 , and its
discussion can be found in books on entertaining mathematics for children in the 1960s) of the “fair
division of the cake,” which in the simplest case can be formulated as follows. Assume that we want
to divide a cake among n participants in such a way that each of them thought that he had not
been cheated. It is important that the participants have their own (maybe mutually conﬂicting)
criteria for comparing diﬀerent pieces of the cake. For example, one participant may better like
a piece with a cherry, and another notices that the cake is burnt on the left, etc. The question
The research was carried out at the Institute for Information Transmission Problems of the Russian
Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.