Abstract
The existence of immigration proof partition for communities (countries) in a multidimensional space is studied. This is a Tiebout type equilibrium its existence previously was stated only in one-dimensional setting. The migration stability means that the inhabitants of a frontier have no incentives to change jurisdiction (an inhabitant at every frontier point has equal costs for all possible adjoining jurisdictions). It means that inter-country boundary is represented by a continuous curve (surface).
Provided that the population density is measurable two approaches are suggested: the first one applies an one-dimensional approximation, for which a fixed point (via Kakutani theorem) can be found after that passing to limits gives the result; the second one employs a new generalization of Krasnosel’skii fixed point theorem for polytopes. This approach develops [8] and extends the result to an arbitrary number of countries, arbitrary dimension, possibly continuous dependence on additional parameters and so on.
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Notes
- 1.
This being combined means that \(\mu (A)>0\iff \int _Adxdy>0\) for every measurable \(A{\subseteq }\square ABCD\).
- 2.
We use standard notations \(z^+=\sup \{z,0\}\) and \(z^-=\sup \{(-z),0\}\) for any real z.
- 3.
Here as above \(\mu (\cdot )\) is absolutely continuous measure on \(\mathcal{A}\), specifying the resettlement of the population.
- 4.
This is Lemma 1 from [7], where its comprehensive proof is also presented.
- 5.
Here \(a\wedge b=\min \{a,b\}\).
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Marakulin, V.M. (2016). On the Existence of Immigration Proof Partition into Countries in Multidimensional Space. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds) Discrete Optimization and Operations Research. DOOR 2016. Lecture Notes in Computer Science(), vol 9869. Springer, Cham. https://doi.org/10.1007/978-3-319-44914-2_39
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