Cooperative decision-making for the provision of a locally undesirable facility

Abstract

We consider the decentralized provision of a global public good with local externalities in a spatially explicit model. Communities decide on the location of a facility that benefits everyone but exhibits costs to the host and its neighbors. They share the costs through transfers. We examine cooperative games associated with this so-called Not In My Back-Yard problem. We derive and discuss conditions for core solutions to exist. These conditions are driven by the temptation to exclude groups of neighbors at any potential location. We illustrate the results in different spatial settings. These results clarify how property rights can affect cooperation and shed further light on a limitation of the Coase theorem.

Appendices

Appendix A: Proof of Proposition 1

Let ${\varvec{x}}$ be an allocation which meets the conditions stated in Proposition 1, that is, the efficiency condition (1), individual rationality conditions (2), and the following lower bounds for every $S\in \mathcal {Y}=\{N\backslash S|S\in \overset{\circ }{\mathcal {N}}\}\cup \{N\backslash \{i\}|i\in N\}$,

$$\begin{aligned} \sum _{i\in S}x_{i}\ge v(S) \end{aligned}$$

(5)

We first show that it satisfies the core lower bounds (5) for any arbitrary coalition. Let $T\subseteq N$.

• If T is a non-building coalition, we have $v(T)=0$. $\forall i\in T$, ${\varvec{x}}$ meets the individual rationality constraint $x_{i}\ge 0$. The sum of these constaints yields Condition (5) for T.

• If T is a building coalition, we have $v(T)=b(T)-c(T)$. Let us consider $j^{*}\in argmin_{j\in T}\sum _{i\in T}c_{ij}$ an optimal site in T and $S^{*}=\overset{\circ }{\mathcal {N}}(j^{*})\backslash T$, the set of strict neighbors of $j^{*}$ that are not in T and $\bar{T}=N\backslash S^{*}$. Since $\bar{T}=N\backslash S^{*}\in \mathcal {Y}$,

  $$\begin{aligned} \sum _{i\in \bar{T}}x_{i}\ge b(\bar{T})-c(\bar{T}) \end{aligned}$$

  Besides, $c(\bar{T})\le c(T)$ so:

  $$\begin{aligned} \sum _{i\in \bar{T}}x_{i}\ge b(\bar{T})-c(T) \end{aligned}$$

  (6)

  As for every $i\in N$, $N\backslash \{i\}\in \mathcal {Y}$, we have $\sum _{j\in N\backslash \{i\}}x_{j}\ge v(N\backslash \{i\})$. This inequality can be rewritten, using the efficiency condition (1), as $x_{i}\le v(N)-v(N\backslash \{i\})$. We have $\forall i\in N\backslash \mathcal {H},v(N)-v(N\backslash \{i\})\le b_{i}$ and Assumption 2 additionally implies $\forall h\in \mathcal {H},v(N)-v(N\backslash \{h\})\le b_{h}$. Thus, $\forall i\in N,-x_{i}\ge -b_{i}$. From the summation of the latter inequalities for all agents in $\bar{T}\backslash T$ to inequality (6), we obtain $\sum _{i\in T}x_{i}\ge b(T)-c(T)=v(T).$ Hence condition (5) holds for T.

We have shown that the core lower bounds can be restricted to coalitions in $\mathcal {Y}$. From Assumption 1, coalitions in $\{N\backslash S|S\in \mathcal {\overset{\circ }{\mathcal {N}}}\}$ are all building coalitions so the constraints associated with them are: $\sum _{i\in N\backslash S}x_{i}\ge b(N\backslash S)-c(N\backslash S)$. Combining them with the efficiency constraints yields conditions (4) in Proposition 1.

Appendix B: Proof of Proposition 2

From Proposition 1, the core can be defined as:

$$\begin{aligned}&\left\{ {\varvec{x}}\in {\mathbb {R}}_{+}^{n}\left| \sum _{N}\right. x_{i}\right. = v(N) \text{ and } \forall S\in \mathcal {\bar{N}},\sum _{S}x_{i}\le b(S)-(c(N)-c(N\backslash S))\nonumber \\&\qquad \left. \text{ and } \forall i\in N,x_{i}\ge 0\right\} \end{aligned}$$

(7)

A necessary and sufficient condition for the non-emptiness of this set involves the linear program (LP1):

$$\begin{aligned} \max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \bar{\mathcal {N}},\sum _{i\in S}x_{i}\le b(S)-(c(N){-}c(N\backslash S)),\forall i\in N,x_{i}\ge 0\right\} \!\ge \! v(N) \end{aligned}$$

(8)

To show this equivalence, first note that if the set involved in condition (7) is non-empty. Therefore any element of it constitutes a feasible solution in the linear program defined in condition (8). A fortiori, its optimal solution also satisfies the condition. Therefore (7)$\Rightarrow$(8). For the converse, consider a solution ${\varvec{x^{*}}}$ to the linear program and assume it satisfies condition (8). Consider the allocation ${\varvec{x^{\epsilon }}}$ defined by $x_{i}^{\epsilon }=\max (0,x_{i}^{*}-\epsilon )$, for all $i\in N$ and some $\epsilon >0$. For any $\epsilon$, the resulting allocation still pertains to the feasible set of the linear program and, by a continuity argument, we can always find $\epsilon$ such that $\sum _{N}x_{i}^{\epsilon }=v(N)$. This allocation pertains to the set defined in condition (7), which, therefore, is non-empty.

Assumption 3 implies that the saving induced by the withdrawal of a community will never exceed its benefit so for any $S\subseteq T\subseteq N,$ we have $v(S)\le v(T)\le v(N)$. In particular, for every $S\in \mathcal {\bar{N}}$, $v(N\backslash S)\le v(N)$ which implies $b(S)-(c(N)-c(N\backslash S))\ge 0$. Hence, this linear program is feasible when Assumption 3 is met (take, for all $i\in N,x_{i}=0$). Besides, it is bounded (by $\sum _{N}b_{i}$ for instance) so it admits a finite value.

We now show that the individual rationality constraints $x_{i}\ge 0$ are non-binding in (LP1). We start to show that, for any optimal solution, no community, $i_0 \in N$, pays more than $\max _{j\in N\backslash \{i_{0}\}}c_{ji_{0}}$, the highest external cost it can bear. Let ${\varvec{x^{*}}}$ be an optimal solution to (LP1) and $i_{0}\in N$. Assume that:

$$\begin{aligned} x_{i_{0}}^{*}<b_{i_{0}}+\min _{T\in \mathcal {\bar{N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

(9)

We can then increase $x_{i_{0}}^{*}$ by some $\epsilon >0$ such that:

$$\begin{aligned} x_{i_{0}}^{*}+\epsilon <b_{i_{0}}+\min _{T\in \mathcal {\bar{N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

(10)

Such an increase improves the objective. We shall show that it also leads to a feasible solution. Let $S\in \bar{\mathcal {N}}$ such that $i_{0}\in S$. Because $S\in \bar{\mathcal {N}}$, $S\backslash \{i_{0}\}$ also pertains to $\bar{\mathcal {N}}$ (except for the case $S=\{i_{0}\}$, in which the result is direct). By feasibility of ${\varvec{x^{*}}}$, we have:

$$\begin{aligned} \sum _{i\in S\backslash \{i_{0}\}}x_{i}^{*}\le b(S\backslash \{i_{0}\})-(c(N)-c((N\backslash S)\cup \{i_{0}\})) \end{aligned}$$

(11)

Summing inequalities (10) and (11), we get:

$$\begin{aligned}&\sum _{i\in S}x_{i}^{*}+\epsilon <b(S)-(c(N)-c((N\backslash S)\cup \{i_{0}\}))\nonumber \\&\quad +\min _{T\in \mathcal {\bar{N}}:i_{0}\in T}\{c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\} \end{aligned}$$

Since $\min _{T\in \mathcal {\bar{N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \le c(N\backslash S)-c((N\backslash S)\cup \{i_{0}\})$, we have:

$$\begin{aligned} \sum _{i\in S}x_{i}^{*}+\epsilon <b(S)-(c(N)-c(N\backslash S)) \end{aligned}$$

All the constraints involving $x_{i_{0}}$ are met. This contradicts the optimality of ${\varvec{x^{*}}}$. Hence, inequality (9) cannot hold by contradiction. We have:

$$\begin{aligned} x_{i_{0}}^{*}\ge b_{i_{0}}+\min _{T\in \mathcal {\bar{N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

(12)

Besides, for any $T\in \mathcal {\bar{N}}$ such that $i_{0}\in T$,

$$\begin{aligned} c((N\backslash T)\cup \{i_{0}\})-c(N\backslash T)=\min _{j\in (N\backslash T)\cup \{i_{0}\}}\sum _{k\in (N\backslash T)\cup \{i_{0}\}}c_{jk}-\min _{j\in N\backslash T}\sum _{k\in N\backslash T}c_{jk} \end{aligned}$$

Let us denote by $j^{*}$ an optimal host in $N\backslash T$. Since $\min _{j\in (N\backslash T)\cup \{i_{0}\}}\sum _{k\in (N\backslash T)\cup \{i_{0}\}}c_{jk}\le \sum _{k\in (N\backslash T)\cup \{i_{0}\}}c_{j^{*}k}$ , by definition of the minimum, we have:

$$\begin{aligned} c((N\backslash T)\cup \{i_{0}\})-c(N\backslash T)\le \sum _{k\in (N\backslash T)\cup \{i_{0}\}}c_{j^{*}k}-\sum _{k\in N\backslash T}c_{j^{*}k}=c_{j^{*}i_{0}} \end{aligned}$$

Hence:

$$\begin{aligned} c((N\backslash T)\cup \{i_{0}\})-c(N\backslash T)\ge -\max _{j\in N\backslash \{i_{0}\}}c_{ji_{0}} \end{aligned}$$

(13)

From conditions (12) and (13), we get $x_{i_{0}}^{*}\ge b_{i_{0}}-\max _{j\in N\backslash \{i_{0}\}}c_{ji_{0}}$ and, from Assumption 3, $x_{i_{0}}^{*}\ge 0$. Thus, individual rationality constraints can be discarded from (LP1) without altering the value of the objective. This leads us to consider the linear program (LP2):

$$\begin{aligned} \max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \mathcal {\bar{N}},\sum _{i\in S}x_{i}\le b(S)-(c(N)-c(N\backslash S))\right\} \end{aligned}$$

Again, this linear program is bounded and feasible. Therefore, it admits a finite value and so its dual $(LP2^{*})$:

$$\begin{aligned} \min _{{\varvec{x}}}\left\{ \left. \sum _{S\in \mathcal {\bar{N}}}\chi _{S}(b(S)-(c(N)-c(N\backslash S)))\right| \forall i\in N,\sum _{S\in \mathcal {\bar{N}}:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} \end{aligned}$$

Which can be further simplified to:

$$\begin{aligned} \min _{{\varvec{x}}}\left\{ \left. b(N)-\sum _{S\in \mathcal {\bar{N}}}\chi _{S}(c(N)-c(N\backslash S))\right| \forall i\in N,\sum _{S\in \mathcal {\bar{N}}:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} \end{aligned}$$

A necessary and sufficient condition for non-emptiness of the core is that the value of $(LP2^{*})$ is lower than $v(N)=b(N)-c(N)$. This leads to the following condition:

$$\begin{aligned} \max _{{\varvec{\chi }}}\left\{ \left. \sum _{S\in \mathcal {\bar{N}}}\chi _{S}\left( 1-\frac{c(N\backslash S)}{c(N)}\right) \right| \forall i\in N,\sum _{S\in \mathcal {\bar{N}}:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} \le 1 \end{aligned}$$

Appendix C: Discarding Assumption 2 in Proposition 2

If $|\mathcal {H}|>1$, the proof of Proposition 2 holds. Here we assume $|\mathcal {H}|=1$ and show that a similar result to Proposition 2 can still be obtained. The difference lies in the fact that the host can get more than $b_{h}$ in core allocations, which prevents an immediate focus on neighborhoods. However, we show that, as soon as an additional assumption is met, requiring that $x_{h}\le b_{h}$ does not alter the value of the linear program. This allows a focus on neighborhoods. The proof proceeds as the proof of Propositions 1 and 2: we first discard redundant constraints and simplify non-binding constraints in a linear program related to the emptiness of the core.

Let us denote by h the unique optimal host in N and let ${\varvec{x}}$ be an allocation which meets the efficiency condition (1), individual rationality constraints (2) and the following core lower bounds for every $S\in \mathcal {Y}'=\mathcal {E}\cup \mathcal {E}_{h}\cup \{N\backslash \{i\}|i\in N\}$,

$$\begin{aligned} \sum _{i\in S}x_{i}\ge v(S) \end{aligned}$$

(14)

Where $\mathcal {E}=\{N\backslash S|S\in \overset{\circ }{\mathcal {N}} \text{ and } c(N\backslash S)\le c(N)\}$ and $\mathcal {E}_{h}=\{S\subset N|h\not \in S\}$.

We first show that it satisfies the core lower bounds (14) for any arbitrary coalition. Let $T\subseteq N$.

• If T is a non-building coalition, we have $v(T)=0$. $\forall i\in T$, ${\varvec{x}}$ meets the individual rationality constraint $x_{i}\ge 0$. The sum of these constraints yields Condition (14) for T.

• If T is a building coalition, we have $v(T)=b(T)-c(T)$. Let us consider $j^{*}\in argmin_{j\in T}\sum _{i\in T}c_{ij}$ an optimal site in T and $S^{*}=\overset{\circ }{\mathcal {N}}(j^{*})\backslash T$, the set of strict neighbors of $j^{*}$ that are not in T. We define $\bar{T}=N\backslash S^{*}$. If $c(N\backslash S^{*})\le c(N)$, then $\bar{T}\in \mathcal {E}\subset \mathcal {Y}'$. If $c(N\backslash S^{*})>c(N)$, then it must be that h is not in $N\backslash S^{*}$ hence $\bar{T}\in \mathcal {E}_{h}\subset \mathcal {Y}$’. Therefore:

  $$\begin{aligned} \sum _{i\in \bar{T}}x_{i}\ge b(\bar{T})-c(\bar{T}) \end{aligned}$$

  Besides, $c(\bar{T})=\min _{j\in \bar{T}}\sum _{k\in \bar{T}}c_{jk}\le \sum _{k\in \bar{T}}c_{j^{*}k}=\sum _{k\in T}c_{j^{*}k}=c(T)$, where the third equality comes from the fact that communities in $\bar{T}\backslash T$ do not belong to the neighborhood of $j^{*}$ by construction. Hence:

  $$\begin{aligned} \sum _{i\in \bar{T}}x_{i}\ge b(\bar{T})-c(T) \end{aligned}$$

  (15)

  The rationality of coalitions $N\backslash \{i\}$ yields $\forall i\in N\backslash \{h\},-x_{i}\ge -b_{i}$. From the summation of the latter inequalities for all agents in $\bar{T}\backslash T$ to inequality (15), we obtain $\sum _{i\in T}x_{i}\ge b(T)-c(T)=v(T).$ Hence condition (14) holds for T.

We have shown that the core lower bounds can be restricted to coalitions in $\mathcal {Y}$’. Combining them with the efficiency constraints and defining $\bar{\mathcal {E}}=\{T|T\in \overset{\circ }{\mathcal {N}} \text{ and } c(N\backslash T)\le c(N)\}$ and $\bar{\mathcal {E}}_{h}=\{T|h\in T\}$, the respective complementary of $\mathcal {E}$ and $\mathcal {E}_{h}$, the core is non-empty if and only if:

$$\begin{aligned}&\max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \bar{\mathcal {E}}\cup \bar{\mathcal {E}}_{h}\cup \{i|i\in N\},\sum _{i\in S}x_{i}\le b(S)-(c(N)\right. \nonumber \\&\left. \qquad -c(N\backslash S)),\forall i\in N,x_{i}\ge 0\right\} \ge v(N) \end{aligned}$$

We now eliminate constraints in $\bar{\mathcal {E}}_{h}$. Let us denote by (LP3) the former linear program. Let us consider ${\varvec{x^{*}}}$ as an optimal solution to (LP3) and let us assume $x_{h}^{*}>b_{h}$ so that we can write $x_{h}^{*}=b_{h}+\epsilon$, $\epsilon >0$. At this stage, an additional assumption is required:

Assumption 6

$\exists S\in \overset{\circ }{\mathcal {N}}$ such that $h\not \in S$ and $c(N\backslash S)\le c(N)$

This assumption implies that it is always possible to exclude some agents different from h and save on the cost of the project. We will show there always exists another optimal solution, ${\varvec{x'}}$, such that $x'_{h}\le b_{h}$. From Assumption 6, there exists $S\in \overset{\circ }{\mathcal {N}}$ such that $h\not \in S$ and $c(N\backslash S)\le c(N)$. Let us consider $S\cup \{h\}\in \bar{\mathcal {E}_{h}}$; we have, by feasibility of ${\varvec{x^{*}}}$ in (LP3):

$$\begin{aligned} \sum _{i\in S}x_{i}^{*}+x_{h}^{*}\le \sum _{i\in S}b_{i}+b_{h} \end{aligned}$$

Hence,

$$\begin{aligned} \sum _{i\in S}x_{i}^{*}\le \sum _{i\in S}b_{i}-\epsilon \end{aligned}$$

Besides the rationality of coalitions $N\backslash \{i\}$ requires $\forall i\in S$, $x_{i}^{*}\le b_{i}$. Hence, there exists $(\epsilon _{i})_{i\in S}\in \mathbb {R}_{+}^{|S|}$ such that, $\sum _{i\in S}\epsilon _{i}=\epsilon$ and, for all $i\in S$, $x_{i}^{*}\le b_{i}-\epsilon _{i}$. Let us define ${\varvec{x'}}$ as follows:

$$\begin{aligned} x'_{h}= & {} x_{h}^{*}-\epsilon =b_{h}\\ x'_{j}= & {} x_{j}^{*}+\epsilon _{i}\quad \hbox {for all} \quad j\in S\\ x'_{i}= & {} x_{i}^{*}\quad \hbox {for all}\quad i\not \in S\cup \{h\} \end{aligned}$$

By construction this solution yields the same objective. We want to show it is feasible as well. Let $T_{S}$ be such that $T_{S}\cap S\ne \emptyset$ and $T_{S}\in \bar{\mathcal {E}}\cup \bar{\mathcal {E}}_{h}\cup \{i|i\in N\}$, an arbitrary coalition of $\bar{\mathcal {E}}\cup \bar{\mathcal {E}}_{h}\cup \{i|i\in N\}$ containing elements of S. Three cases arise:

• If $T_{S}\in \{i|i\in N\}$, then the associated constraint $x_{i}\le b_{i}$ is met by construction.

• If $T_{S}\in \bar{\mathcal {E}}_{h}$, we have, where the first inequality comes from the fact that $\sum _{i\in T_{S}\cap S}\epsilon _{i}-\epsilon \le 0$ and the second is the feasibility of ${\varvec{x^{*}}}$ in (LP4):

  $$\begin{aligned} \sum _{i\in T_{S}}x'_{i}\le \sum _{i\in T_{S}}x_{i}^{*}\le b(T_{S})-(c(N)-c(N\backslash T_{S})) \end{aligned}$$

• If $T_{S}\in \bar{\mathcal {E}}$, $T_{S}\cup \{h\}\in \bar{\mathcal {E}}_{h}$ and by feasibility of ${\varvec{x^{*}}}$ in (LP4):

  $$\begin{aligned} \sum _{i\in T_{S}}x_{i}^{*}+x_{h}^{*}\le b(T_{S})+b_{h}-(c(N)-c(N\backslash (T_{S}\cup \{h\}))) \end{aligned}$$

Simplifying $b_{h}$ and because $\sum _{i\in T_{S}\cap S}\epsilon _{i}\le \epsilon$,

$$\begin{aligned} \sum _{i\in T_{S}}x_{i}'\le \sum _{i\in T_{S}}x_{i}^{*}+\epsilon \le b(T_{S})-(c(N)-c(N\backslash (T_{S}\cup \{h\}))) \end{aligned}$$

Because $c(N\backslash T_{S})\le c(N)$, the optimal location in $N\backslash T_{S}$ cannot be h. Hence, the withdrawal of h can only lead to a decrease in cost, so that $c(N\backslash (T_{S}\cup \{h\}))\le c(N\backslash T_{S})$.

Finally, we have, for any constraint $T_{S}$ involving elements of S:

$$\begin{aligned} \sum _{i\in T_{S}}x'_{i}\le b(T_{S})-(c(N)-c(N\backslash T_{S})) \end{aligned}$$

This establishes that ${\varvec{x'}}$ is feasible. Hence it is an optimal solution as well. Finally, we can require that $x_{h}\le b_{h}$ without altering the value of the linear program. This defines the linear program (LP4):

$$\begin{aligned} \max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \bar{\mathcal {E}}\cup \bar{\mathcal {E}}_{h},\sum _{i\in S}x_{i}\le b(S)-(c(N)-c(N\backslash S)),\forall i\in N,x_{i}\le b_{i} \text{ and } x_{i}\ge 0\right\} \end{aligned}$$

It is straightforward to show that, following the introduction of the additional constraint $x_{h}\le b_{h}$, all constraints in $\mathcal {\bar{E}}_{h}$ are redundant in (LP4). Hence (LP4) can be rewritten:

$$\begin{aligned} \max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \bar{\mathcal {E}},\sum _{i\in S}x_{i}\le b(S)-(c(N)-c(N\backslash S)),x_{h}\le b_{h},\forall i\in N,x_{i}\ge 0\right\} \end{aligned}$$

And, adding some redundant constraints to simplify the notations:

$$\begin{aligned} I({\varvec{C}})=\max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \bar{\mathcal {N}},\sum _{i\in S}x_{i}\le b(S){-}(c(N){-}c(N\backslash S)),\forall i\in N,x_{i}\le b_{i} \text{ and } x_{i}\ge 0\right\} \end{aligned}$$

We eventually get an expression similar to the one introduced in Proposition 2: Assumption 2 can be replaced by Assumption 6 provided we impose the additional condition $x_{h}\le b_{h}$ in the former linear program. Hence, an expression of $I({\varvec{C}})$ can be obtained by defining the function $c'$ such that $c'(N\backslash \{h\})=min\{c(N\backslash \{h\}), c(N)\}$ and, for all $S\subset N$ different from $N\backslash \{h\}$, $c'(S)=c(S)$. Then:

$$\begin{aligned} I({\varvec{C}})=\max _{{\varvec{\chi }}}\left\{ \left. \sum _{S\in \bar{\mathcal {N}}}\chi _{S}(1-\frac{c'(N\backslash S)}{c'(N)})\right| \forall i\in N,\quad \sum _{S\in \mathcal {\bar{N}}:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} \end{aligned}$$

Appendix D: Proof of Corollary 1

Consider the NIMBY problem $\sigma =(N,{\varvec{b}},{\varvec{C}})$ and let $t\in \mathbb {R}_+$. Define $\sigma '=(N,{\varvec{b}},{\varvec{C'}})$, where ${\varvec{C'}}={\varvec{C}}+t{\varvec{I_n}}$ and ${\varvec{I_n}}$ denotes the identity matrix. For any building coalition in $\sigma '$, we have $c'(S)=c(S)+t$ where c(S) and $c'(S)$ denote the cost of the project for coalition S in $\sigma$ and $\sigma '$ respectively.

The linear programs defining $I({\varvec{C'}})$ writes:

$$\begin{aligned} I({\varvec{C'}})=\max _{{\varvec{\chi }}}\left\{ \left. \sum _{S\in \bar{\mathcal {N}}}\chi _{S}\left( 1-\frac{c(N\backslash S)+t}{c(N)+t}\right) \right| \forall i\in N,\quad \sum _{S\in \bar{\mathcal {N}}:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} \end{aligned}$$

Defining $\tau = \frac{t}{c(N)}$ and substituting t, we get:

$$\begin{aligned} I({\varvec{C'}})=\frac{1}{1+\tau }\max _{{\varvec{\chi }}}\left\{ \left. \sum _{S\in \bar{\mathcal {N}}}\chi _{S}\left( 1-\frac{c(N\backslash S)}{c(N)}\right) \right| \forall i\in N, \quad \sum _{S\in \bar{\mathcal {N}}:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} \end{aligned}$$

Therefore

$$\begin{aligned} I({\varvec{C'}})=\frac{I({\varvec{C}})}{1+\tau } \end{aligned}$$

Appendix E: Proof of Corollary 2

Let $\sigma =(N,{\varvec{b}},{\varvec{C}})$ and $\sigma '=(N,{\varvec{b}},{\varvec{C'}})$ be two NIMBY problems meeting Assumptions 1 to 4. Define c and $c'$ the cost function in the problem $\sigma$ and $\sigma '$ respectively, and assume

• 1. $c(N)=c'(N)$;

• 2. ${\varvec{C}}\ge {\varvec{C'}}$.

Let (LP) and $(LP')$ be the linear programs defining respectively $I({\varvec{C}})$ and $I({\varvec{C'}})$ and let ${\varvec{\chi }}$ be an optimal solution to (LP). In case, some additional coalitions appears in the set $\bar{\mathcal {N}}$ in $\sigma$, extend ${\varvec{\chi }}$ by assigning a weight of 0 to them. This defines a feasible solution ${\varvec{\bar{\chi }}}$ in $(LP')$. Besides, because ${\varvec{C}}\ge {\varvec{C'}}$ and $c(N)=c'(N)$, we have $\forall S\in \bar{\mathcal {N}},1-\frac{c(N\backslash S)}{c(N)}\le 1-\frac{c'(N\backslash S)}{c'(N)}$, so the objective of $(LP')$ at ${\varvec{\bar{\chi }}}$ is not lower than $I({\varvec{C}})$. Therefore, the value of $(LP')$, $I({\varvec{C'}})$, cannot be lower than $I({\varvec{C}})$.

Appendix F: Proof of Corollary 3

In the linear case, Assumption 2 holds, and we explicitly compute the value of $I({\varvec{C}})$. In this section we will use the notion of balanced collections. A collection $\mathcal {B}$ of subsets of N is said to be balanced if and only if there exist strictly positive weights ${\varvec{\chi ^{\mathcal {B}}}}=(\chi _{S}^{\mathcal {B}})_{S\in \mathcal {B}}$ such that, for any $i\in N,$ $\sum _{S\in \mathcal {B}:i\in S}\chi _{S}^{\mathcal {B}}=1$. Denoting by $\mathbb {B}(\mathcal {\bar{N}})$ the set of balanced collections over N composed of elements of $\bar{\mathcal {N}}$ only, we can write:

$$\begin{aligned} I({\varvec{C}})=\frac{1}{c(N)}\max _{\mathcal {B}\in \mathbb {B}(\bar{\mathcal {N}})}\left\{ \sum _{S\in \mathcal {B}}\chi _{S}^{\mathcal {B}}(c(N)-c(N\backslash S))\right\} \end{aligned}$$

We compute the costs saved by excluding a set of neighboring communities from the grand coalition $c(N)-c(N/S)$ for every $S\in \bar{\mathcal {N}}$. In the linear case, S is of size 1 or 2.

• Case $|S|=1$. Some cost is saved by excluding a single community only if the community excluded is neighbor of one of the optimal hosts: 1 or n. The external cost $\delta c$ is then saved: $c(N)-c(N \backslash S)=\delta c$ for $S \in \{\{2\},\{n-2\}\}$.

• Case $|S|=2$. Let S be a coalition of two communities $S=\{j,j+2\}$ neighbor of a community not located at the extreme of the line $j+1 \in \{1,\ldots ,n-2\}$. The cost saved by excluding S is $\delta c$ because the optimal host becomes $j+1$ with $c(N \backslash S)=c$ while it is 1 or n in the grand coalition with $c(N)=c+\delta c$. For all other coalitions of size 2 neighbor of the same community, no cost is saved: $c(N)-c(N \backslash S)=0$.

Therefore, for any $S\in \bar{\mathcal {N}}$, we have the corresponding values:

$$\begin{aligned} c(N)-c(N\backslash S)=\left\{ \begin{array}{l} \delta c\quad \hbox { if }S\in \{{2},{n-1}\}\\ \delta c\quad \hbox { if }S\in \{\{j,j+2\}|j\in \{1,\ldots ,n-2\}\}\\ 0\quad \hbox { otherwise} \end{array}\right. \end{aligned}$$

$\mathcal {\bar{N}}$ is a set of coalitions of no more than two players. Hence, for any balanced collection $\mathcal {B}$ of elements of $\mathcal {\bar{N}}$, there exists a partition of N into pairwise disjoint sets $N_{1},\ldots ,N_{l},l=0 \ldots L$ where each $N_{l}$ with $l>0$ is a coalition of at least three communities such that $\mathcal {B}$ consists of full cycles on each $N_{l}$ and a partition of $N_{0}$ (Balinski 1970, as stated in Le Breton and Weber 1995:316). Because no cycle can be formed out of elements of $\mathcal {\bar{N}}$ in the linear case, all balanced collections over $\mathcal {\bar{N}}$ are partitions. In summary, we are interested in finding partitions $\mathcal {P}$ of N, composed with elements of $\mathcal {\bar{N}}$ which maximize $\sum _{S\in \mathcal {P}}(c(N)-c(N\backslash S))$. We now explain how to find such optimal partitions.

First, for any partition involving coalitions in which 2 or $n-1$ belongs to a two-agent coalition, we weakly improve on the objective by splitting such coalitions into singletons. Hence, we can restrict our attention to coalitions in which such communities appear as singletons. The construction of an optimal partition then consists in maximizing the number of coalitions of the form $\{\{j,j+2\}|j\in \{1,\ldots ,n-2\}\}$. In the case $n\in \{4,5,6,7\}$, such optimal partitions are trivial as soon as communities 2 and $n-1$ appear as singletons. Figure 6 presents optimal partitions and the corresponding value of $\sum _{S\in \mathcal {P}}(c(N)-c(N\backslash S))$.

Fig. 6

Initial patterns. The reasoning adopted for finding the optimal partitions consists in considering all possible cases. We detail the case n = 7. First, we know that there is always an optimal partition containing $\{2\}$ and $\{6\}$ as singletons. The value associated with each is $\delta c$. The value associated with any other single individual is 0 whereas the value associated with any pair of $\mathcal {\bar{N}}$ is $\delta c$. An optimal partition thus contains as many pairs of $\mathcal {\bar{N}}$ as possible. This is achieved with the partition $\mathcal {P}=\{\{1,3\},\{2\},\{4\},\{6\},\{5,7\}\}$

For any $n>7$, we know that n can be decomposed as $n=4k+i$, $k\in \mathbb {N} \text{ and } i\in {0,1,2,3}$. According to this decomposition, an optimal partition can be found by combining the initial patterns above and the iterative pattern presented in Fig. 7 which maximizes the value that can be obtained by adding 4 communities to the initial pattern.

Fig. 7

Iterative pattern

We eventually find the following optimal partitions:

• If $n=4k,k\in \mathbb {N}$, $\mathcal {P}=\{\{1\},\{2\},\{n-1\},\{n\}\}\cup _{j=1}^{k-1}\{\{4j-1,4j+1\},\{4j,4j+2\}\}$

• If $n=4k+1,k\in \mathbb {N}$, $\mathcal {P}=\{\{1,3\},\{2\},\{n-1\},\{n\}\}\cup _{j=1}^{k-1}\{\{4j,4j+2\},\{4j+1,4j+3\}\}$

• If $n=4k+2,k\in \mathbb {N}$, $\mathcal {P}=\{\{1,3\},\{2\},\{n-1\},\{n-2,n\}\}\cup _{j=1}^{k-1}\{\{4j,4j+2\},\{4j+1,4j+3\}\}$

• If $n=4k+3,k\in \mathbb {N}$, $\mathcal {P}=\{\{1,3\},\{2\},\{4\},\{n-1\},\{n-2,n\}\}\cup _{j=1}^{k-1}\{\{4j+1,4j+3\},\{4j+2,4j+4\}\}$

And the associated values are:

$$\begin{aligned} I({\varvec{C}})=\left\{ \begin{array}{l} \frac{n}{2}\frac{\delta }{1+\delta }\quad \text{ if } n=4k,k\in \mathbb {N}\\ \frac{n+1}{2}\frac{\delta }{1+\delta }\quad \text{ if } n=4k+1,k\in \mathbb {N}\\ \frac{n+2}{2}\frac{\delta }{1+\delta }\quad \text{ if } n=4k+2,k\in \mathbb {N}\\ \frac{n+1}{2}\frac{\delta }{1+\delta }\quad \text{ if } n=4k+3,k\in \mathbb {N} \end{array}\right. \end{aligned}$$

The condition on $\delta$ expressed in Corollary 3 directly follows from the comparison of $I({\varvec{C}})$ with 1.

Appendix G: Proof of Proposition 4

Let R be an exogenous expectation formation rule and $v^{R}$ its associated characteristic function. We want to show that under Assumptions 1, 2, 4 and 5, the R-core is non-empty if and only if $I({\varvec{C}})\ge 1$. We extend the proof of Propositions 1 and 2.

First, we eliminate redundant constraints in the system defining the core. We distinguish between building and non-building coalitions. $NB=\{T\subset N|b(T)<c(T)\}$ is the set of non-building coalitions. Replicating the proof of Proposition 1, the constraints for building coalitions can be restricted to $\{N\backslash S|S\in \mathcal {\bar{N}}\}$. However, the constraints for non-building coalitions cannot be reduced to individual rationality: an allocation ${\varvec{x}}$ is in the R-core $\mathcal {C}^{R}$ if and only if

$$\begin{aligned} \sum _{i\in N}x_{i}&=v(N) \end{aligned}$$

(16)

$$\begin{aligned} \forall S\in NB,\sum _{i\in S}x_{i}&\ge v^{R}(S) \end{aligned}$$

(17)

$$\begin{aligned} \forall i\in N,&x_{i}\le b_{i} \end{aligned}$$

(18)

$$\begin{aligned} \forall S\in \overset{\circ }{\mathcal {N}},\sum _{i\in S}x_{i}&\le b(S)-(c(N)-c(N\backslash S)) \end{aligned}$$

(19)

where the constraints (17) contain the individual rationality constraints. We consider the linear program (LP5):

$$\begin{aligned} \max _{{\varvec{x}}}\left\{ \left. \sum _{i\in N}x_{i}\right| \forall S\in \bar{\mathcal {N}}, \quad \sum _{i\in S}x_{i}\le b(S)-(c(N)-c(N\backslash S)) \text{ and } \forall S\in NB,\sum _{i\in S}x_{i}\ge v^{R}(S)\right\} \end{aligned}$$

The R-core $\mathcal {C}^{R}$ is non-empty if and only if (LP5) is feasible and reaches a value higher than v(N). We first note that such a program would always be feasible under Assumption 5. Second, as in the proof of Proposition 2, we can show that the constraints (17) are never binding under Assumption 5.

Let ${\varvec{x^{*}}}$ be an optimal solution to the above linear program and assume there exists $i_{0}\in N$ such that:

$$\begin{aligned} x_{i_{0}}^{*}<b_{i_{0}}+\min _{T\in \bar{\mathcal {N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

(20)

Then we can increase $x_{i_{0}}^{*}$ by some $\epsilon >0$ such that:

$$\begin{aligned} x_{i_{0}}^{*}+\epsilon <b_{i_{0}}+\min _{T\in \bar{\mathcal {N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

(21)

Such an increase improves on the objective. We shall show that it also leads to a feasible solution. First, it is straightforward to see that the constraints (17) are met. We concentrate on the remaining constraints.

Let $S\in \bar{\mathcal {N}}$ with at least two communities, such that $i_{0}\in S$. Because $S\in \bar{\mathcal {N}}$, $S\backslash \{i_{0}\}$ also pertains to $\bar{\mathcal {N}}$. By feasibility of ${\varvec{x^{*}}}$, we have:

$$\begin{aligned} \sum _{i\in S\backslash \{i_{0}\}}x_{i}^{*}\le b(S\backslash \{i_{0}\})-(c(N)-c((N\backslash S)\cup \{i_{0}\})) \end{aligned}$$

(22)

Summing inequalities (21) and (22), we get:

$$\begin{aligned} \sum _{i\in S}x_{i}^{*}+\epsilon <b(S)-(c(N)-c((N\backslash S)\cup \{i_{0}\}))+\min _{T\in \bar{\mathcal {N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{i\in S}x_{i}^{*}+\epsilon <b(S)-(c(N)-c(N\backslash S)) \end{aligned}$$

Therefore, all the constraints involving $x_{i_{0}}$ are met. This contradicts the optimality of ${\varvec{x^{*}}}$. Hence, inequality (20) cannot hold by contradiction. We have:

$$\begin{aligned} x_{i_{0}}^{*}\ge b_{i_{0}}+\min _{T\in \bar{\mathcal {N}}:i_{0}\in T}\left\{ c(N\backslash T)-c((N\backslash T)\cup \{i_{0}\})\right\} \end{aligned}$$

Besides, as established in the proof of Proposition 2:

$$\begin{aligned} \forall S\in \mathcal {\bar{N}}:i_{0}\in S,c(N\backslash S)-c((N\backslash S)\cup \{i_{0}\})\ge -\max _{j\in N\backslash \{i_{0}\}}c_{ji_{0}} \end{aligned}$$

so $x_{i_{0}}^{*}\ge b_{i_{0}}-\max _{j\in N\backslash \{i_{0}\}}c_{ji_{0}}$ and $\forall S\in \{T\subset N|b(T)<c(T)\}$, $\sum _{i\in T}x_{i}^{*}\ge b(T)-\sum _{i\in T}\max _{j\in N\backslash \{i\}}c_{ji}$. Hence, using Assumption 5, $\sum _{i\in T}x_{i}^{*}\ge v^{R}(T)$.

The constraints (17) can then be removed from the linear program (LP5) without changing its value. This leads us back to the linear program (LP2) and the proof of Proposition 2 applies.

Appendix H: Proof of Corollary 4

The cost of the project on a graph depends on the minimal degree of this graph. For any $S\subseteq N$, we denote by $\underline{d}(S)$ the minimal degree of the graph induced by S on G. Rewriting the condition $I({\varvec{C}})\ge 1$, we get the following condition on $\delta$:

$$\begin{aligned} \delta \le \bar{\delta }({\varvec{G}})=\frac{1}{max_{{\varvec{\chi }}}\left\{ \sum _{S\in \bar{\mathcal {N}}}\chi _{S}(\underline{d}(N){-}\underline{d}(N\backslash S))|\forall i\in N,\sum _{S:i\in S}\chi _{S}=1,\chi _{S}\ge 0\right\} {-}\underline{d}(N)} \end{aligned}$$

We want to show $\bar{\delta }({\varvec{G}})>0$. Let $h\in \mathcal {H}$ be an optimal host in N and $j\in \overset{\circ }{\mathcal {N}}(h)$.²¹ Consider the following partition: $\{\overset{\circ }{\mathcal {N}}(h),S_{j},N\backslash (\overset{\circ }{\mathcal {N}}(h)\cup S_{j})\}$, where $S_{j}=\overset{\circ }{\mathcal {N}}(j)\backslash \overset{\circ }{\mathcal {N}}(h)$ is the strict neighborhood j from which we withdraw members of $\overset{\circ }{\mathcal {N}}(h)$ . A feasible solution ${\varvec{\chi '}}$ associated with this partition is defined as follows:

• $\chi '_{\mathcal {\overset{\circ }{\mathcal {N}}}(h)}=1$;

• $\chi '_{S_{j}}=1$;

• $\chi '_{N\backslash (\overset{\circ }{\mathcal {N}}(h)\cup S_{j})}=1$;

• $\chi '_{S}=0$ for all other coalitions

We compute the value of this linear program at this feasible solution. First, we know that $|\overset{\circ }{\mathcal {N}}(h)|=\underline{d}(N)$. Hence community j has at most $\underline{d}(N)-1$ neighbors in $\overset{\circ }{\mathcal {N}}(h)$. The withdrawal of its neighbors in $S_{j}$ therefore leads to a graph with a degree of at least $\underline{d}(N)-1$. Hence, $\underline{d}(N)-1\ge \underline{d}(N\backslash S_{h})$, which implies that $\underline{d}(N)-\underline{d}(N\backslash S_{h})\ge 1$. Second, we have $\underline{d}(N\backslash \mathcal {\overset{\circ }{\mathcal {N}}}(h))=0$; hence, $\underline{d}(N)-\underline{d}(N\backslash \overset{\circ }{\mathcal {N}}(h))=\underline{d}(N)$. Finally, as we have $h\in S_{j}$ by construction, the minimal degree of $\overset{\circ }{\mathcal {N}}(h)\cup S_{h}$ is at most $\underline{d}(N)$; hence $\underline{d}(N)-\underline{d}(\overset{\circ }{\mathcal {N}}(h)\cup S_{h})\ge 0$. The value associated with the feasible solution ${\varvec{\chi '}}$ is $\underline{d}(N)+1$, hence the optimal value of the linear program defining $\bar{\delta }({\varvec{G}})$ can only be higher than it. Therefore, $\bar{\delta }({\varvec{G}})>0$.

Appendix I: Code (software R)

1.1 NIMBY problems of graphs

1.2 NIMBY problem on a French administrative unit

The GIS data used is the GEOFLA Communes database. It is publicly available at http://professionnels.ign.fr/geofla.