Abstract
This paper considers a Schelling model in an arbitrary fixed network where there are no vacant houses. Agents have preferences either for segregation or for mixed neighborhoods. Utility is non-transferable. Two agents exchange houses when the trade is mutually beneficial. We find that an allocation is stable when for two agents of opposite-color each black (white) agent has a higher proportion of neighbors who are black (white). This result holds irrespective of agents’ preferences. When all members of both groups prefer mixed neighborhoods, an allocation is also stable provided that if an agent belongs to the minority (majority), then any neighbor of opposite-color is in a smaller minority (larger majority).
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Notes
Segregation also exists between followers of different religions, between men and women in an office canteen, between tourists and locals at a city square, between staff and students in a seminar room, between different nationalities at a conference dinner, between workers with different skills in different firms, or between different species occupying their own territory (Pancs and Vriend 2007; Schelling 1971).
In recent times, racial-ethnic segregation has declined in the United States, but income (or class) segregation has grown (Massey et al. 2009).
In the literature on residential segregation, the definition of neighborhood is different. Given that houses are located in a checkerboard network or torus, a von Neumann neighborhood implies that each agent only considers the four adjacent agents as neighbors whilst a Moore neighborhood includes the eight surrounding agents as neighbors.
For any finite set S, |S| is the number of elements belonging to S.
Pancs and Vriend (2007) consider the symmetric single-peaked utility function where \(p_{m}=p_{0}=0.5\) and \(u_{l}(\cdot )\) is an increasing linear function on the interval [0, 0.5].
The naming follows the notion of p-cohesive groups in Morris (2000) where it is used to study contagion through a myopic best-response rule. A subset is p-cohesive if each element has at least a proportion p of neighbors within the subset.
In the literature on residential segregation, two kinds of neighborhoods are assumed: continuous neighborhoods and bounded neighborhoods as illustrated by the left and middle panels of Fig. 2, respectively.
Part (II) of Theorem 1 is independent of the assumption that the preference for mixed neighborhoods is represented by a symmetric single-peaked or symmetric single-plateaued utility function.
In the left and right panels, \(h(\mathbf {A}_{1})\) and \(h(\mathbf {A}_{2})\) form a counterpart partition of \(\mathbf {H}\) while in the middle panel, \(h(\mathbf {A}_{1})\) and \(h(\mathbf {A}_{2})\) do not form a counterpart partition of \(\mathbf {H}\).
We would like to thank two anonymous referees so much for pointing out these possibilities and encouraging this discussion.
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We thank the editor and two anonymous referees for suggesting ways to improve the substance and exposition of this paper. Part of this work was undertaken while Zhiwei Cui was visiting the Department of Applied Mathematics at National Dong Hwa University, whose hospitality is greatly acknowledged. This work was financially supported by the National Science Foundation of China (No. 61202425) and the Fundamental Research Funds for the Central Universities (YWF-13-D2-JC-11, YWF-14-JGXY-016).
Proofs
Proofs
Proof of Proposition 1
According to the specific values of \(\displaystyle \min _{i\in \mathbf {H}^{'}} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}\), the proof is divided into two cases.
Case I \(\displaystyle \min _{i\in \mathbf {H}^{'}} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}=1\).
In this case, \(|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|=\eta _{i}(g)\) holds for any \(i\in \mathbf {H}^{'}\). That is, for any \(i\in \mathbf {H}^{'}\), \(\mathbf {N}_{i}(g)\subset \mathbf {H}^{'}\). Therefore, \(g_{ij}=0\) for any \(i\in \mathbf {H}^{'}\) and \(j\in \mathbf {H}{\setminus }\mathbf {H}^{'}\). Following from the assumption that \(g_{ij}=g_{ji}\) for any \(i,j\in \mathbf {H}\), we have that \(\mathbf {N}_{j}(g)\subset \mathbf {H}{\setminus }\mathbf {H}^{'}\) for any \(j\in \mathbf {H}{\setminus }\mathbf {H}^{'}\). As a result,
Case II \(\displaystyle \min _{i\in \mathbf {H}^{'}} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}<1\).
In this case, \(\mathbf {N}_{\mathbf {H}^{'}}(g)\ne \emptyset \) and \(\mathbf {N}_{\mathbf {H}{\setminus } \mathbf {H}^{'}}(g)\ne \emptyset \). First of all, we introduce a lemma to facilitate the proof.
Lemma 3
Consider a house distribution network g. For any proper, nonempty subset of houses \(\mathbf {H}^{''}\subset \mathbf {H}\), if \(\mathbf {N}_{\mathbf {H}^{''}}(g)\ne \emptyset \),
The proof of this lemma is trivial. We omit it here.
We then verify that the subset \(\mathbf {H}{\setminus }\mathbf {H}^{'}\) is internally cohesive as follows.
where the “\(\le \)” inequality follows from the internal cohesion of \(\mathbf {H}^{'}\) and the third and the last “\(=\)” inequalities are applications of Lemma 3. \(\square \)
Proof of Proposition 3
Assume that there exists a house \(i_{0}\in \mathbf {H}^{'}\) such that \(\mathbf {N}_{i_{0}}(g)\cap \mathbf {H}^{'}=\emptyset \). Therefore,
On the other hand, according to the nonexistence of isolated houses, \(\mathbf {N}_{i_{0}}(g) \ne \emptyset \). Combining this with \(\mathbf {N}_{i_{0}}(g)\cap \mathbf {H}^{'}=\emptyset \), both \(j\in \mathbf {H}{\setminus } \mathbf {H}^{'}\) and \(j\in \mathbf {N}_{\mathbf {H}^{'}}(g)\) hold for any \(j\in \mathbf {N}_{i_{0}}(g)\). It follows that
Inequalities (5) and (6) yield a contradiction to the internal cohesion of \(\mathbf {H}^{'}\). \(\square \)
Proof of Lemma 1
Without losing generality, assume that each utility function \(u_{l}(\cdot )\), \(l=1,2\), is symmetric single-plateaued. That is, \(p_{m}<0.5\). Let \(h^{'}\) denote the allocation resulting from h by agents a and b exchanging houses. Depending on the specific positions of houses h(a) and h(b), the proof is divided into two cases.
Case I \(h(a)\in h(\mathbf {A}_{1}){\setminus } \mathbf {N}_{h(\mathbf {A}_{2})}(g)\) or \(h(b)\in h(\mathbf {A}_{2}){\setminus } \mathbf {N}_{h(\mathbf {A}_{1})}(g)\).
It is sufficient to consider the situation \(h(a)\in h(\mathbf {A}_{1}){\setminus } \mathbf {N}_{h(\mathbf {A}_{2})}(g)\). It follows that
If \(h(b)\in h(\mathbf {A}_{2}){\setminus } \mathbf {N}_{h(\mathbf {A}_{1})}(g)\), owing to the specification of \(u_{1}\) and \(u_{2}\), \(\pi _{a}(h)=\pi _{a}(h^{'})=0\) and \(\pi _{b}(h)=\pi _{b}(h^{'})=0\). If \(h(b)\in N_{h(\mathbf {A}_{1})}(g)\), \(\pi _{b}(h)>0\) and \(\pi _{b}(h^{'})=0\) hold. According to Definition 1, the exchange is not beneficial.
Case II \(h(a)\in \mathbf {N}_{h(\mathbf {A}_{2})}(g)\), \(h(b)\in \mathbf {N}_{h(\mathbf {A}_{1})}(g)\) and \(g_{h(a)h(b)}=0\).
In this case, it follows that for agent a and agent b,
If \(\displaystyle \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}\in [p_{m},1-p_{m}]\) or \(\displaystyle \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(b)}(g)}\in [p_{m},1-p_{m}]\), the exchange between agents a and b is not beneficial. It is sufficient to consider the situation \(\displaystyle \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}\in [p_{m},1-p_{m}]\). \(\pi _{a}(h^{'})\ge \pi _{a}(h)\) implies that \(\displaystyle \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(b)}(g)}\in [p_{m},1-p_{m}]\). It follows that
Therefore, both agents a and b obtain the highest possible level of utility and have no incentive to exchange their houses.
Now we turn to the situation \(\displaystyle \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}\notin [p_{m},1-p_{m}]\) and \(\displaystyle \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(b)}(g)}\notin [p_{m},1-p_{m}]\). If \(h^{'}\) results from h by agents a and b beneficially exchanging houses,
and
where there exists at least one strict inequality. By substituting Eqs. (7) and (8), the above condition can be rewritten as
and
where there exists at least one strict inequality. When verifying the above conditions, a contradiction yields. \(\square \)
Proof of Lemma 2
Consider agent a. According to the specification of \(u_{1}(\cdot )\),
Without loss of generality, assume that \(\displaystyle \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)} >\frac{1}{2}\). In this case, \(\pi _{a}(h)\le \pi _{a}(h^{'})\) implies that
Simplifying the above expression, \(\pi _{a}(h)\le \pi _{a}(h^{'})\) only if
Hence, Inequality (3) is verified.
The above reasoning also applies to agent b; that is, Inequality (4) holds. Following from Definition 1, the conclusion can be verified. \(\square \)
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Cui, Z., Hwang, YA. House exchange and residential segregation in networks. Int J Game Theory 46, 125–147 (2017). https://doi.org/10.1007/s00182-015-0526-2
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DOI: https://doi.org/10.1007/s00182-015-0526-2