House exchange and residential segregation in networks

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).

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,

$$\begin{aligned} \min _{j\in \mathbf {H}{\setminus }\mathbf {H}^{'}} \frac{|\mathbf {N}_{j}(g)\cap (\mathbf {H}{\setminus }\mathbf {H}^{'})|}{\eta _{j}(g)}=1>0=\max _{i\in \mathbf {N}_{\mathbf {H}{\setminus }\mathbf {H}^{'}}(g)}\frac{|\mathbf {N}_{i}(g)\cap (\mathbf {H}{\setminus }\mathbf {H}^{'})|}{\eta _{i}(g)}. \end{aligned}$$

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 $\mathbf {N}_{\mathbf {H}^{''}}(g)\ne \emptyset$,

$$\begin{aligned} \min _{i\in \mathbf {N}_{\mathbf {H}{\setminus }\mathbf {H}^{''}}(g)} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{''}|}{\eta _{i}(g)}=\min _{i\in \mathbf {H}^{''}} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{''}|}{\eta _{i}(g)}. \end{aligned}$$

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.

$$\begin{aligned}&\max _{i\in \mathbf {N}_{\mathbf {H}{\setminus }\mathbf {H}^{'}}(g)}\frac{|\mathbf {N}_{i}(g)\cap (\mathbf {H}{\setminus }\mathbf {H}^{'})|}{\eta _{i}(g)}\\&\quad =\max _{i\in \mathbf {N}_{\mathbf {H}{\setminus }\mathbf {H}^{'}}(g)}\Big (1-\frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}\Big )=1-\min _{i\in \mathbf {N}_{\mathbf {H}{\setminus }\mathbf {H}^{'}}(g)} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}\\&\quad = 1-\min _{i\in \mathbf {H}^{'}}\frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}\\&\quad \le 1-\max _{j\in \mathbf {N}_{\mathbf {H}^{'}}(g)} \frac{|\mathbf {N}_{j}(g)\cap \mathbf {H}^{'}|}{\eta _{j}(g)} =\min _{j\in \mathbf {N}_{\mathbf {H}^{'}}(g)}\Big (1- \frac{|\mathbf {N}_{j}(g)\cap \mathbf {H}^{'}|}{\eta _{j}(g)}\Big )\\&\quad = \min _{j\in \mathbf {N}_{\mathbf {H}{\setminus }(\mathbf {H}{\setminus }\mathbf {H}^{'})}(g)} \frac{|\mathbf {N}_{j}(g)\cap (\mathbf {H}{\setminus }\mathbf {H}^{'})|}{\eta _{j}(g)} = \min _{j\in \mathbf {H}{\setminus }\mathbf {H}^{'}} \frac{|\mathbf {N}_{j}(g)\cap (\mathbf {H}{\setminus }\mathbf {H}^{'})|}{\eta _{j}(g)} \end{aligned}$$

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,

$$\begin{aligned} 0\le \min _{i\in \mathbf {H}^{'}} \frac{|\mathbf {N}_{i}(g)\cap \mathbf {H}^{'}|}{\eta _{i}(g)}\le \frac{|\mathbf {N}_{i_{0}}(g)\cap \mathbf {H}^{'}|}{\eta _{i_{0}}(g)}=0. \end{aligned}$$

(5)

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

$$\begin{aligned} \max _{j\in \mathbf {N}_{\mathbf {H}^{'}}(g)} \frac{|\mathbf {N}_{j}(g)\cap \mathbf {H}^{'}|}{\eta _{j}(g)}\ge \max _{j\in \mathbf {N}_{i_{0}}(g)}\frac{1}{\eta _{j}(g)}\ge \frac{1}{|\mathbf {H}|-1}>0. \end{aligned}$$

(6)

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

$$\begin{aligned} \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}=1 \text { and then } \frac{|\mathbf {N}_{h^{'}(b)}(g)\cap h^{'}(\mathbf {A}_{2})|}{\eta _{h^{'}(b)}(g)}=0. \end{aligned}$$

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,

$$\begin{aligned} \frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)}= & {} \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(b)}(g)}, \end{aligned}$$

(7)

$$\begin{aligned} \frac{|\mathbf {N}_{h^{'}(b)}(g)\cap h^{'}(\mathbf {A}_{2})|}{\eta _{h^{'}(b)}(g)}= & {} \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(a)}(g)}. \end{aligned}$$

(8)

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

$$\begin{aligned} \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(b)}(g)} =1-\frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(b)}(g)}\in [p_{m},1-p_{m}]. \end{aligned}$$

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,

$$\begin{aligned} \left| \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}-\frac{1}{2}\right| \ge \left| \frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)}-\frac{1}{2}\right| \end{aligned}$$

and

$$\begin{aligned} \left| \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(b)}(g)}-\frac{1}{2}\right| \ge \left| \frac{|\mathbf {N}_{h^{'}(b)}(g)\cap h^{'}(\mathbf {A}_{2})|}{\eta _{h^{'}(b)}(g)}-\frac{1}{2}\right| \end{aligned}$$

where there exists at least one strict inequality. By substituting Eqs. (7) and (8), the above condition can be rewritten as

$$\begin{aligned} \left| \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}-\frac{1}{2}\right| \ge \left| \left( 1-\frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(b)}(g)}\right) -\frac{1}{2}\right| \end{aligned}$$

and

$$\begin{aligned} \left| \frac{|\mathbf {N}_{h(b)}(g)\cap h(\mathbf {A}_{2})|}{\eta _{h(b)}(g)}-\frac{1}{2}\right| \ge \left| \left( 1-\frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}\right) -\frac{1}{2}\right| \end{aligned}$$

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 $u_{1}(\cdot )$,

$$\begin{aligned} \pi _{a}(h)\le \pi _{a}(h^{'})\Longleftrightarrow \left| \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}-\frac{1}{2}\right| \ge \left| \frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)}-\frac{1}{2}\right| . \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{c c} \displaystyle \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}\!\ge \! \frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)} &{} \text {if } \displaystyle \frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)}\!\ge \!\frac{1}{2};\\ \displaystyle \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}-\frac{1}{2}\ge \frac{1}{2}-\frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)} &{} \text {otherwise}.\\ \end{array} \right. \end{aligned}$$

Simplifying the above expression, $\pi _{a}(h)\le \pi _{a}(h^{'})$ only if

$$\begin{aligned} \frac{|\mathbf {N}_{h(a)}(g)\cap h(\mathbf {A}_{1})|}{\eta _{h(a)}(g)}+ \frac{|\mathbf {N}_{h^{'}(a)}(g)\cap h^{'}(\mathbf {A}_{1})|}{\eta _{h^{'}(a)}(g)}\ge 1. \end{aligned}$$

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. $\square$