Given some resource and a set of agents (players), one of the goals of welfare economics is to divide the resource among the agents in an envy-free manner. Envy-freeness, as a criterion of fair division, is the principle where every player feels that their share is at least as good as the share of any other agent, and thus no player feels envy.
In a mathematical simplification a group ofr players want to divide among themselves a commodity, commonly referred to as the “cake”. The simplest model of the cake is the interval $[0, 1]$, which should be cut into $r$ pieces by $r-1$ cuts.
The classical approach to envy-free division and equilibrium problems arising in mathematical economics typically relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner’s lemma or some extension of these results involving mapping degree.
We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology, and the so called configuration space/test map scheme https://www.msri.org/workshops/378/schedules/2492, which was originally developed for applications in discrete and computational geometry and topology(Tverberg type problems, necklace splitting problem in the sense of N.Alon and D. West, etc.).
We illustrate the method by proving several relatives (extensions) of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake. We also show how this technique allows us to improve the splitting necklace theorem of N. Alon by adding natural constraints on the distribution of pieces of the necklace.
References
[1] D. Jojić, G. Panina, R.T. Živaljević. Splitting necklaces, with constraints. SIAM J. Discr. Math.
[2] G. Panina, R.T. Živaljević. Envy-free division via configuration spaces, arXiv:2102.06886v3 [math.CO].
[3] D. Jojić, G. Panina, R.T. Živaljević. A Tverberg type theorem for collectively unavoidable complexes, Israel J. Math. (2021).
[4] R.T. Živaljević. Topological methods in discrete geometry, Chapter 21 inHandbook of Discrete and Computational Geometry(third edition), CRC Press LLC, 2017.