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Elementary Solution to the Fair Division Problem

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Abstract

A new and relatively elementary approach is proposed for solving the problem of fair division of a continuous resource (measurable space, pie, etc.) between several participants, the selection criteria of which are described by charges (signed measures). The setting of the problem with charges is considered for the first time. The problem comes down to analyzing properties of trajectories of a specially constructed dynamical system acting in the space of finite measurable partitions. Exponentially fast convergence to a limit solution is proved for both the case of true measures and the case of charges.

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Notes

  1. The resource is represented by a unit segment, and partition elements are intervals.

  2. As a reviewer pointed out to us, a particular example for 3 participants was considered in [13].

  3. Here and in what follows, \(\bigsqcup\) denotes the union of disjoint (nonintersecting) sets.

  4. A measure \(\mu\) on \((M, \Sigma)\) is called probabilistic if \(\mu(M)=1\).

  5. Sets \(\{A_i\}\) are called disjunctive if they are pairwise disjoint.

  6. As a reviewer pointed out to us, a similar construction of preferences was described in [15], where some examples were analyzed for 3 and 4 participants.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia

    M. L. Blank & M. O. Polyakov

  2. Higher School of Economics—National Research University, Moscow, Russia

    M. L. Blank & M. O. Polyakov

  3. Higher School of Modern Mathematics, Moscow Institute of Physics and Technology, Moscow, Russia

    M. L. Blank

Authors
  1. M. L. Blank
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  2. M. O. Polyakov
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Blank, M.L., Polyakov, M.O. Elementary Solution to the Fair Division Problem. Probl Inf Transm 60, 53–70 (2024). https://doi.org/10.1134/S003294602401006X

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