Abstract
It studies the problem of immigration proof partition for communities (countries) in a multidimensional space. This is an existence problem of Tiebout type equilibrium, where migration stability suggests that every inhabitant has no incentives to change current jurisdiction. In particular, an inhabitant at every frontier point has equal costs for all available jurisdictions. It is required that the inter-country border is represented by a continuous curve.
The paper presents the solution for the case of the costs described as the sum of the two values: the ratio of total costs on the total weight of the population plus transportation costs to the center presented as a barycenter of the state. In the literature, this setting is considered as a case of especial theoretical interest and difficulty. The existence of equilibrium division is stated via an approximation reducing the problem to the earlier studied case, in which centers of the states never can coincide: to do this an earlier proved a generalization of conic Krasnosel’skii fixed point theorem is applied.
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Notes
- 1.
As far as I know, the last version of the paper is more general and admits flexible centers, but still not general enough.
- 2.
The general idea of this equilibrium is that individuals can “vote with their feet” by leaving situations they do not like or going to situations they believe to be more beneficial, i.e. the inhabitants of countries or municipalities are able to move and to choose the place of residence which is more suitable for them. And perhaps it is not necessary to physically move—for example, it is so in the formation of football fans clubs—sometimes it is enough to register and then “consume” the benefits and disadvantages of this membership.
- 3.
Here \(\delta _{-i}=(\delta _j)_{j\in N\setminus \{i\}}\).
- 4.
This means that \(\mu (B)>0\iff \int _B dxdy>0\) for every measurable \(B\subseteq \mathcal{A}\).
- 5.
- 6.
Recall that we assumed \(\mu (\mathcal{A}\))=1.
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Marakulin, V. (2019). Spatial Equilibrium in a Multidimensional Space: An Immigration-Consistent Division into Countries Centered at Barycenter. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_46
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