Sharing the cost of the polluted river: a class of bilateral compensation methods

Abstract

Water pollution is harmful to people’s health and causes great damage to ecology. Cleaning up pollutants in cross-boundary river requires joint action of agents along the river. In this paper, we deal with the polluted river cost sharing problem in a cooperative situation. We first introduce a class of bilateral compensation methods (shortly the BC method), which are generalizations of the famous cost allocations, such as Upstream Equal Sharing method and Downstream Equal Sharing method. Then, we investigate the relationship between the BC method and the Shapley value of a special responsibility cost game. Also, we illustrate the stability of the BC method by studying the concavity of the corresponding cost game. Moreover, we provide two axiomatic characterizations of the BC method.

Appendix

Proof of Lemma 3.2

The proof proceeds by checking the validity of Eq. (6). Firstly, we will prove that the Shapley value of the ordinary game $(N,V^{C^k},\alpha )$ has three forms.

For notation convenience, given any agent $k\in N$, we denote ${\underline{U}}(k)=\{i|i<k\}$ and ${\underline{D}}(k)=\{i|i>k\}$ as the set of k’s strict upstream agents and the set of k’s strict downstream agents, respectively. Notice that for every $k\in N$, it holds that

$$\begin{aligned}{} & {} \{S|S\subset N\} =\{S|S\ni k\}\cup \{S|S\not \ni k\} \\{} & {} =\{S|S\ni k\}\cup \{S|S\subset {\underline{U}}(k)\} \cup \{S|S\subset {\underline{D}}(k)\} \\{} & {} \cup \{S |S\not \ni k,S\cap {\underline{U}}(k)\ne \emptyset , S\cap {\underline{D}}(k)\ne \emptyset \}. \end{aligned}$$

Then, by Eq. (4), the ordinary game $(N,V^{C^k},\alpha )$ can be simplified as

$$\begin{aligned} V^{C^{k}} (S) =\left\{ \begin{aligned}&c_k,{} & {} S \ni k; \\&\alpha _kc_k,{} & {} S \subset {\underline{U}}(k); \\&(1-\alpha _k)c_k,{} & {} S \subset {\underline{D}}(k);\\&c_k,{} & {} S\not \ni k, S\cap {\underline{U}}(k)\ne \emptyset , S\cap {\underline{D}}(k)\ne \emptyset . \end{aligned} \right. \end{aligned}$$

(24)

Therefore, for all agents $i,i'<k$ and all $S\subset N{\setminus } \{i,i'\}$,

$$\begin{aligned} V^{C^{k}} (S\cup \{i\} ) =V^{C^{k}} (S\cup \{i'\} ) =\left\{ \begin{aligned}&c_k,{} & {} S \ni k; \\&\alpha _kc_k,{} & {} S \subset ({\underline{U}}(k)\setminus \{i,i'\}); \\&c_k,{} & {} S \subset {\underline{D}}(k);\\&c_k,{} & {} S\not \ni k, S\cap ({\underline{U}}(k)\setminus \{i,i'\})\ne \emptyset , S\cap {\underline{D}}(k)\ne \emptyset . \end{aligned} \right. \end{aligned}$$

(25)

This yields that for the game $(N,V^{C^{k}},\alpha )$, all the agents in ${\underline{U}}(k)$ are symmetry agents³. Hence, by Symmetry of Shapley value,⁴ it holds that $Sh_i(N,V^{C^k},\alpha )=Sh_{i'}(N,V^{C^k},\alpha )$ for all $i,i'<k$. Similarly, we can prove that all agents in ${\underline{D}}(k)$ are symmetric, and $Sh_j(N,V^{C^k},\alpha )=Sh_{j'}(N,V^{C^k},\alpha )$ for all $j,j'>k$, where for any $j,j'>k$ and $S\subset N{\setminus } \{j,j'\}$, it holds

$$\begin{aligned} V^{C^{k}} (S\cup \{j\} )=V^{C^{k}} (S\cup \{j'\} ) =\left\{ \begin{aligned}&c_k,{} & {} S \ni k; \\&c_k,{} & {} S \subset {\underline{U}}(k); \\&(1-\alpha _k)c_k,{} & {} S \subset ({\underline{D}}(k)\setminus \{j,j'\});\\&c_k,{} & {} S\not \ni k, S\cap {\underline{U}}(k)\ne \emptyset ,\\{} & {} {}&and\ S\cap ({\underline{D}}(k)\setminus \{j,j'\})\ne \emptyset . \end{aligned} \right. \end{aligned}$$

(26)

Given this, we calculate the Shapley value of $(N,V^{C^k},\alpha )$ by distinguishing if (1) $i<k$, (2) $i>k$ or (3) $i=k$ for any $i\in N$.

Case (1) When $i<k$, we show that $Sh_i(N, V^{C^{k}},\alpha )=\frac{\alpha _kc_k}{k}$.

For the ordinary game $(N,V^{C^k},\alpha )$ with $i<k$ and $S\subset N{\setminus } \{i\}$, by Eqs (24) and (25), we have

$$\begin{aligned} V^{C^k} (S\cup \{i\} )-V^{C^k} (S) =\left\{ \begin{aligned}&\alpha _kc_k,{} & {} S \subset {\underline{D}}(k);\\&0,{} & {} otherwise. \end{aligned} \right. \end{aligned}$$

(27)

Therefore, by Eqs (5) and (27), for every $i<k$,

$$\begin{aligned} \begin{aligned}&Sh_i(N,V^{C^{k}},\alpha )\\&=\sum _{S\subset N\setminus \{i\}} \frac{(n-|S|-1)!|S|!}{n!} \Big (V^{C^k} (S\cup \{i\} )-V^{C^k} (S)\Big ) \\&=\sum _{S\subset N\setminus \{i\}, S\subset {\underline{D}}(k)} \frac{(n-|S|-1)!|S|!}{n!} \Big (V^{C^k} (S\cup \{i\} )-V^{C^k} (S)\Big )\\& \quad +\sum _{S\subset N\setminus \{i\}, S\not \subset {\underline{D}}(k)} \frac{(n-|S|-1)!|S|!}{n!} \Big (V^{C^k} (S\cup \{i\} )-V^{C^k} (S)\Big ) \\&=\sum _{S\subset {\underline{D}}(k)} \frac{(n-|S|-1)!|S|!}{n!}\cdot \alpha _kc_k+0 \\&=\sum _{0\le |S|\le n-k}\left( {\begin{array}{c}n-k\\ |S|\end{array}}\right) \frac{(n-|S|-1)!|S|!}{n!}\cdot \alpha _kc_k\\&=\frac{\alpha _kc_k}{k}, \end{aligned} \end{aligned}$$

(28)

where the last equality holds by using induction on k with $k\in \{1,\cdots ,n\}$.

Case (2) When $i>k$, we show that $Sh_i(N, V^{C^{k}},\alpha )=\frac{(1-\alpha _k)c_k}{n-k+1}$.

Analogously, for the ordinary game $(N,V^{C^k},\alpha )$ with $i>k$ and $S\subset N\setminus \{i\}$, we have

$$\begin{aligned} V^{C^k} (S\cup \{i\} )-V^{C^k} (S) =\left\{ \begin{aligned}&(1-\alpha _k)c_k,{} & {} S \subset {\underline{U}}(k);\\&0,{} & {} otherwise. \end{aligned} \right. \end{aligned}$$

(29)

Hence, by Eqs (5) and (29), it holds that for every $i>k$,

$$\begin{aligned} \begin{aligned}&Sh_{i}(N,V^{C^{k}},\alpha )\\&=\sum _{S\subset N\setminus \{i\}} \frac{(n-|S|-1)!|S|!}{n!} \Big (V^{C^k} (S\cup \{i\} )-V^{C^k} (S)\Big ) \\&=\sum _{S\subset N\setminus \{i\}, S\subset {\underline{U}}(k)} \frac{(n-|S|-1)!|S|!}{n!} \Big (V^{C^k} (S\cup \{i\} )-V^{C^k} (S)\Big ) \\& \quad +\sum _{S\subset N\setminus \{i\}, S\not \subset {\underline{U}}(k)} \frac{(n-|S|-1)!|S|!}{n!} \Big (V^{C^k} (S\cup \{i\} )-V^{C^k} (S)\Big )\\&=\sum _{S\subset {\underline{U}}(k)} \frac{(n-|S|-1)!|S|!}{n!}\cdot (1-\alpha _k)c_k+0 \\&=\sum _{0\le |S|\le k-1}\left( {\begin{array}{c}k-1\\ |S|\end{array}}\right) \frac{(n-|S|-1)!|S|!}{n!}\cdot (1-\alpha _k) c_k\\&=\frac{(1-\alpha _k)c_k}{n-k+1}. \end{aligned} \end{aligned}$$

(30)

Here, the last equality can be proved by using induction on k with $k\in \{1,\cdots ,n\}$ and we omit it.

Case (3) When $i=k$, we show that $Sh_i(N,V^{C^{k}},\alpha )=c_k-\frac{(k-1)\alpha _kc_k}{k}-\frac{(n-k)(1-\alpha _k)c_k}{n-k+1}$. By Efficiency, we have

$Sh_k(N,V^{C^{k}},\alpha )+\sum \limits _{i<k} Sh_i(N,V^{C^{k}},\alpha ) +\sum \limits _{i>k}Sh_i(N,V^{C^{k}},\alpha )=c_{k}.$

This together with Eqs (28) and (30) implies

$$\begin{aligned} Sh_k(N,V^{C^{k}},\alpha )=c_k-\frac{(k-1)\alpha _kc_k}{k}-\frac{(n-k)(1-\alpha _k)c_k}{n-k+1}. \end{aligned}$$

This completes the proof. $\square$