Benefit Allocation in Shared Water-Saving Management Contract Projects Based on Modified Expected Shapley Value

Abstract

Water-Saving Management Contract (WSMC) is an innovative business model to reduce water consumption and improve water-use efficiency. The Shared Water-Saving Model, as a primary operating model in WSMC projects, is relatively widely used compared with other patterns. However, the lack of a water-saving benefit allocation scheme is one of the critical obstacles frustrating the implementation and promotion of the Shared Water-Saving Model. To correct this deficiency, a modified expected Shapley value method is developed, in which alliance revenues are characterized as uncertain variables due to the lack of historical data. Firstly, risk, input and effort are identified as key influencing factors to improve the deficiencies in the distribution of benefits based on the contribution. Secondly, equity criteria, including symmetry, efficiency and additivity, are redefined based on correction factors to measure the fairness of the allocation scheme. Thirdly, we prove that the allocation result obtained by the proposed method satisfies the equity criteria and is unique. Finally, a water-saving project launched by Handan City located in North China is studied to illustrate the applicability of the proposed method. The result shows that the modified expected Shapley value method significantly enhances the cooperative relationship and can therefore be used as an effective tool for the fair and reasonable distribution of benefits in Shared WSMC projects.

Appendix

Proof of Theorem 1

We denote by S_ij the set containing neither i nor j, by S_i the set including j but not i, and by S_j the set including i but not j.□

Symmetry

Assume that i and j are interchangeable. Then, the modified excepted Shapley value of player i is

$$ \begin{array}{@{}rcl@{}} \tilde{\varphi}_{i}(v)&=&\underset{S\subseteq {N\backslash i}}{\sum}\ \frac{|S|!(|N|-|S|-1)!}{|N|!}\{E[v(S\cup i)]-E[v(S)]\}+\left( \beta_{i}-\frac{1}{|N|}\right)E[v(N)]\\ &=&\underset{{S_{ij}}\subseteq N}{\sum}\ \frac{|S_{ij}|!(|N|-|S_{ij}|-1)!}{|N|!}\{E[v(S_{ij}\cup i)]-E[v(S_{ij})]\}\\ &&+\underset{{S_{i}}\subseteq N}{\sum}\ \frac{|S_{i}|!(|N|-|S_{i}|-1)!}{|N|!}\{E[v(S_{i}\cup i)]-E[v(S_{i})]\}+\left( \beta_{i}-\frac{1}{|N|}\right)E[v(N)]\\ &=&\underset{{S_{ij}}\subseteq N}{\sum}\ \frac{|S_{ij}|!(|N|-|S_{ij}|-1)!}{|N|!}\{E[v(S_{ij}\cup i)]-E[v(S_{ij})]\}\\ &&+\underset{\{{S_{ij}\cup j\}}\subseteq N}{\sum}\ \frac{({|S_{ij}|+1})!(|N|-|S_{ij}|-2)!}{|N|!}\{E[v(S_{ij}\cup j\cup i)]-E[v(S_{ij}\cup j)]\}\\ &&+\left( \beta_{i}-\frac{1}{|N|}\right)E[v(N)]. \end{array} $$

Similarly, the modified excepted Shapley value of player j is

$$ \begin{array}{@{}rcl@{}} \tilde{\varphi}_{j}(v)&=&\underset{{S_{ij}}\subseteq N}{\sum}\ \frac{|S_{ij}|!(|N|-|S_{ij}|-1)!}{|N|!}\{E[v(S_{ij}\cup j)]-E[v(S_{ij})]\}\\ &&+\underset{\{{S_{ij}\cup i\}}\subseteq N}{\sum}\ \frac{({|S_{ij}|+1})!(|N|-|S_{ij}|-2)!}{|N|!}\{E[v(S_{ij}\cup i\cup j)]-E[v(S_{ij}\cup i)]\}\\&&+\left( \beta_{j}-\frac{1}{|N|}\right)E[v(N)]. \end{array} $$

Since players i and j are interchangeable, we have E[v(S_ij ∪ j)] = E[v(S_ij ∪ i)], β_i = β_j. Therefore, $\tilde {\varphi }_{i}=\tilde {\varphi }_{j}$. It follows that $\tilde {\varphi }$ satisfies symmetry.

Efficiency

It is clear that Definition 3.2 is equivalent to Eq. 11

$$ \tilde{\varphi_{i}}(N,v)=\underset{\pi}{\sum}\ \frac{1}{|N|!}\{E[v(S_{\pi}^{i}\cup i)]-E[v(S_{\pi}^{i})]\}+\left( \beta_{i}-\frac{1}{|N|}\right)E[v(N)]. $$

(11)

where $S_{\pi }^{i}$ is the set of players preceding i in the ordering π. Let T be a carrier of v; then, Eq. 11 is

$$\tilde{\varphi_{i}}(T,v)=\underset{\pi}{\sum}\ \frac{1}{|T|!}\{E[v(S_{\pi}^{i}\cup i)]-E[v(S_{\pi}^{i})]\}+\left( \beta_{i}-\frac{1}{|T|}\right)E[v(T)].$$

Therefore,

$$ \begin{array}{@{}rcl@{}} \underset{i\in T}{\sum}\ \tilde{\varphi}_{i}(v)&=&\tilde{\varphi}_{1}(v)+\tilde{\varphi}_{2}(v)+...+\tilde{\varphi}_{t}(v)\\ &=&\underset{\pi}{\sum}\ \frac{1}{|T|!}\ \{E[v(S_{\pi}^{1}\cup 1)]-E[v(S_{\pi}^{1})]\}+\left( \beta_{1}-\frac{1}{|T|}\right)E[v(T)]\\ &&+\underset{\pi}{\sum}\ \frac{1}{|T|!}\ \{E[v(S_{\pi}^{2}\cup 2)]-E[v(S_{\pi}^{2})]\}\\ &&+\left( \beta_{2}-\frac{1}{|T|}\right)E[v(T)]+...+\underset{\pi}{\sum}\ \frac{1}{|T|!}\ \{E[v(S_{\pi}^{t}\cup t)]-E[v(S_{\pi}^{t})]\}\\ &&+\left( \beta_{t}-\frac{1}{|T|}\right)E[v(T)]\\ &=&\frac{1}{|T|!}\{E[v(S_{\pi_{1}}^{1}\cup 1)]-E[v(S_{\pi_{1}}^{1})]+E[v(S_{\pi_{2}}^{1}\cup 1)]-E[v(S_{\pi_{2}}^{1})]\\ &&+...+E[v(S_{\pi_{t}}^{1}\cup 1)]-E[v(S_{\pi_{t}}^{1})]\}+\left( \beta_{1}-\frac{1}{|T|}\right)E[v(T)]\\ &&+\frac{1}{|T|!}\{E[v(S_{\pi_{1}}^{2}\cup 2)]-E[v(S_{\pi_{1}}^{2})]+E[v(S_{\pi_{2}}^{2}\cup 2)]-E[v(S_{\pi_{2}}^{2})]\\ &&+...+E[v(S_{\pi_{t}}^{2}\cup 2)]-E[v(S_{\pi_{t}}^{2})]\}+\left( \beta_{2}-\frac{1}{|T|}\right)E[v(T)]\\ &&+...+\frac{1}{|T|!}\{E[v(S_{\pi_{1}}^{t}\cup t)]-E[v(S_{\pi_{1}}^{t})]+E[v(S_{\pi_{2}}^{t}\cup t)]-E[v(S_{\pi_{2}}^{t})]\\ &&+...+E[v(S_{\pi_{t}}^{t}\cup t)]-E[v(S_{\pi_{t}}^{t})]\}+\left( \beta_{t}-\frac{1}{|T|}\right)E[v(T)]\\ &=&\underset{\pi}{\sum}\ \frac{1}{|T|!}\ \{E[v(S_{\pi}^{1}\cup 1)]-E[v(S_{\pi}^{1})]+E[v(S_{\pi}^{2}\cup 2)]-E[v(S_{\pi}^{2})]\\ &&+...+E[v(S_{\pi}^{t}\cup t)]-E[v(S_{\pi}^{t})]\}+\underset{i\in T}{\sum}\ \beta_{i}-|T|\frac{1}{|T|}\\ &=&\frac{1}{|T|!}|T|!E[v(T)]\\ &=&E[v(T)]. \end{array} $$

It follows that $\tilde {\varphi }$ satisfies efficiency.

Additivity

Based on Definition 3.2, we have

$$ \begin{array}{@{}rcl@{}} \tilde{\varphi}_{i}(v+w)&=&\sum\limits_{{S}\subseteq {N\backslash i}}\ \frac{|S|!(|N|-|S|-1)!}{|N|!}\{E[(v+w)(S\cup i)]-E[(v+w)(S)]\}\\&&+\left( \beta_{i}-\frac{1}{|N|}\right)E[(v+w)(N)]\\ \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=&\sum\limits_{{S}\subseteq {N\backslash i}}\ \frac{|S|!(|N|-|S|-1)!}{|N|!}\{E[(v)(S\cup i)]-E[(v)(S)]\}+\left( \beta_{i}-\frac{1}{|N|}\right)E[(v)(N)]\\ &&+\sum\limits_{{S}\subseteq {N\backslash i}}\ \frac{|S|!(|N|-|S|-1)!}{|N|!}\{E[(w)(S\cup i)]-E[(w)(S)]\}+\left( \beta_{i}-\frac{1}{|N|}\right)E[(w)(N)]\\ &=&\tilde{\varphi}_{i}(v)+\tilde{\varphi}_{i}(w). \end{array} $$

It follows that $\tilde {\varphi }$ satisfies additivity.

Proof of Lemma 1

Take i and j in T,and choose $S_{ij}\subseteq N$ such that

$$cv_{T}(S_{ij}\cup i)=cv_{T}(S_{ij}\cup j)=0.$$

Then, we have

$$E[cv_{T}(S_{ij}\cup i)]-E[cv_{T}(S_{ij})]=E[cv_{T}(S_{ij}\cup j)]-E[cv_{T}(S_{ij})].$$

Hence, by Axiom 1,

$$ |T|\{\tilde{\varphi}_{i}(cv_{T})-\left( \beta_{i}-\frac{1}{|T|}\right)E[cv_{T}(T)]\}=c. $$

(12)

Simplifying Eq. 12, we can obtain

$$ \tilde{\varphi_{i}}(cv_{T})=\beta_{i}c. $$

(13)

Take i∉T. Both T and T ∪ i are carriers of cv_T, and cv_T(T) = cv_T(T ∪ i). By Axiom 2,

$$ \begin{array}{@{}rcl@{}} \underset{j\in T}{\sum}\tilde{\varphi}_{j}(cv_{T})=cv_{T}(T)&=&cv_{T}(T\cup i)\\ &=&\underset{j\in {T\cup i}}{\sum}\tilde{\varphi}_{j}(cv_{T})\\ &=&\underset{j\in T}{\sum}{\tilde{\varphi}}_{j}(cv_{T})+\tilde{\varphi}_{i}(cv_{T}). \end{array} $$

Hence,

$$ \tilde{\varphi}_{i}(cv_{T})=0, $$

(14)

by Eqs. 13 and 14, achieves Eq. 8. This completes the proof.□

Proof of Lemma 2

Take any $S\subseteq N$, and substitute Eq. 10 into Eq. 9. Then, by Eq. 7, the right-hand side of Eq. 9 can be simplified to

$$ \begin{array}{@{}rcl@{}} \underset{T\subseteq N}{\sum}c_{T}v_{T}(S)&=&\underset{\begin{array}{l}T {\subseteq} N\\ T {\subseteq} S \end{array}}{\sum}c_{T}v_{T}(S)+\underset{\begin{array}{l}T {\subseteq} N\\ T {\varsubsetneq} S \end{array}}{\sum}c_{T}v_{T}(S)\\ &=&\underset{T \subseteq S}{\sum}c_{T}\\ &=&\underset{T \subseteq S}{\sum}\underset{U\subseteq T}{\sum}(-1)^{|T|-|U|}E[v(U)]\\ &=&\underset{U \subseteq S}{\sum}\left[ \sum\limits_{|T|=|U|}^{|S|}(-1)^{|T|-|U|}\dbinom{|S|-|U|}{|T|-|U|} \right]E[v(U)]. \end{array} $$

The expression in brackets vanishes except for |S| = |U|,so we are left with the identity $\underset { \begin {array}{l}T {\subseteq } N\\ T {\neq } \emptyset \end {array}}{\sum } c_{T}v_{T}(S)=E[v(S)]$. This completes the proof. □

Proof of Theorem 2

We regard an uncertain coalition game v(S) as a collection of 2^|N|− 1 numbers $(v(S))_{S\subseteq N}$. We begin by showing that for any uncertain game v(S), there is a unique collection $(c_{T})_{T\subseteq N}$ of real numbers such that $E[v(S)]={\sum }_{T\subseteq N}c_{T}v_{T}(S)$. That is, we show that $(v_{T})_{T\subseteq N}$ is an algebraic basis for the space of games. Since the collection $(v_{T})_{T\subseteq N}$ of uncertain games contains 2^|N|− 1 numbers, it suffices to show that these uncertain games are linearly independent. Suppose that ${\sum }_{T\subseteq N}\alpha _{T}v_{T}(S)=0$ for any $S\subseteq N$. We need to show that α_T = 0 for all T. Suppose to the contrary that there exists some coalition $T^{\prime }$ with $\alpha _{T^{\prime }}\neq 0$. Then, we can choose such a coalition $T^{\prime }$ for which α_T = 0 for all $T\subset T^{\prime }$, in which case

$$ \begin{array}{@{}rcl@{}} \underset{T\subseteq N}{\sum}\alpha_{T}v_{T}(S)&=&\underset{ \begin{array}{l}T {\subseteq} N\\ T {\subset} T^{\prime} \end{array}}{\sum}\alpha_{T}v_{T}(S)+\alpha_{T^{\prime}}v_{T^{\prime}}(S)+\underset{ \begin{array}{l} T {\subseteq} N\\ T {\nsubseteq} T^{\prime} \end{array}}{\sum}\alpha_{T}v_{T}(S)\\ &=&\underset{ \begin{array}{l}T {\subseteq} N\\ T {\subset} T^{\prime} \end{array}}{\sum}\alpha_{T}v_{T}(T^{\prime})+\alpha_{T^{\prime}}v_{T^{\prime}}(T^{\prime})+\underset{ \begin{array}{l}T {\subseteq} N\\ T {\nsubseteq} T^{\prime} \end{array}}{\sum}\alpha_{T}v_{T}(T^{\prime})\\ &=&\alpha_{T^{\prime}}\\ &\neq&0. \end{array} $$

This is a contradiction, so the value of E[v] is determined uniquely.

We can therefore apply Lemma 2 to the representation of Lemma 1, and by Axiom 3, we can obtain the unique value $\tilde {\varphi }_{i}$ of player i :

$$ \begin{array}{@{}rcl@{}} \tilde{\varphi}_{i}(v)&=&\tilde{\varphi}_{i}(E[v])\\ &=&\tilde{\varphi}_{i}\left( \underset{ \begin{array}{l}T {\subseteq} N\\ T {\neq} \emptyset \end{array}}{\sum} c_{T}v_{T}\right)\\ &=&\underset{ \begin{array}{l}T {\subseteq} N\\ T {\neq} \emptyset \end{array}}{\sum}\tilde{\varphi}_{i}(c_{T}v_{T})\\ &=&\left\{\begin{array}{cl} \displaystyle \underset{ \begin{array}{l}T {\subseteq} N\\ T {\neq} \emptyset \end{array}}{\sum}\beta_{i}c_{T} &\text{if }\displaystyle i\in T,\\ \displaystyle 0 &\text{if }\displaystyle i\notin T. \end{array} \right. \end{array} $$

Therefore, this completes the proof.□