Shapley Value For Multi-choice Cooperative Games

James Edward Buttersworth (1817-1894) was an English painter.

The multi-choice cooperative games was discussed in \S\ref{mul}. We are going to present the Shapley value for multi-choice games. Now we are ready to consider the power indices of the players. Instead of defining the power index of a game as a vector, we define the power index or value of a game as a matrix. Let $\boldsymbol{\phi}:\mbox{TU}(N)\rightarrow M_{m\times n}$ be the function such that
\[\boldsymbol{\phi}(v)=\left [\begin{array}{ccc}
\phi_{11}(v) & \cdots & \phi_{1n}(v)\\
\phi_{21}(v) & \cdots & \phi_{2n}(v)\\
\vdots & \vdots & \vdots\\
\phi_{m1}(v) & \cdots & \phi_{mn}(v)
\end{array}\right ]=(\boldsymbol{\phi}_{1}(v),\cdots ,
\boldsymbol{\phi}_{n}(v)),\]
where $\boldsymbol{\phi}_{i}(v)=(\phi_{1i}(v),\phi_{2i}(v),\cdots ,\phi_{mi}(v))^{T}$. Here $\phi_{ji}(v)$ is the power index or the value of player $i$ when he/she takes action $\sigma_{j}$ in game $v$ (Hsiao and Raghavan \cite[p.243-244]{hsi93}).

Since we do not assume that $\sigma_{2}$ is, say, twice as powerful as action $\sigma_{1}$, and since we do not assume that the difference between $\sigma_{k-1}$ and $\sigma_{k}$ is the same as the difference between $\sigma_{k}$ and $\sigma_{k+1}$, etc., giving weights (discrimination) to actions is necessary. We denote by $w_{k}$ the weight of action $\sigma_{k}$. It is reasonable to assume that $w_{0}=0$ and $w_{1}\leq w_{2}\leq\cdots\leq w_{m}$. Here ${\bf w}=(w_{0},w_{1},\cdots ,w_{m})$ is also called a weight of $\beta$ (Hsiao and Raghavan \cite[p.244]{hsi93}).

Axiom 1. Suppose the weight ${\bf w}=(w_{0},w_{1},\cdots w_{m})$ is given. If $v$ is of the form
\[v({\bf y})=\left\{\begin{array}{ll}
c>0 & \mbox{if ${\bf y}\geq {\bf x}$}\\ 0 & \mbox{otherwise}
\end{array}\right .\]
then $\phi_{x_{i},i}(v)$ is proportional to $w_{x_{i}}$.

Axiom 1 states that, for binary-valued ($0$ or $c$) games that stipulate a minimal exertion players, the reward for a player using the minila exertion level is proportional to the weight of his minimal level action (where ${\bf x}$ may be regarded as a minimal exertion) (Hsiao and Raghavan \cite[p.244]{hsi93}).

Definition. (Hsiao and Raghavan \cite[p.244]{hsi93}). A vector ${\bf x}^{*}\in\beta^{n}$ is called a carrier of $v$, if $v({\bf x}^{*}\wedge {\bf x})=v({\bf x})$ for all ${\bf x}\in\beta^{n}$. We call $\bar{\bf x}$ a {\bf minimal carrier} of $v$ if $\sum_{i}\bar{x}_{i}=\min\{\sum_{i}x_{i}:{\bf x}\mbox{ is a carrier of }v\}$. $\sharp$

Let $({\bf x}|x_{i}=k)$ denote a vector ${\bf x}$ with the $i$th coordinate $x_{i}=k$.

Definition. (Hsiao and Raghavan \cite[p.244]{hsi93}). Player $i$ us said to be a {\bf dummy player} if $v(({\bf x}|x_{i}=k))=v(({\bf x}|x_{i}=0))$ for all ${\bf x}\in\beta^{n}$ and for all $k\in\beta$. $\sharp$

Let ${\bf x}^{*}$ be a carrier of $v$ such that $x_{i}^{*}=0$. In this case, player $i$ is obviously a dummy player of $v$. Conversely, if player $i$ is a dummy player of $v$, then the action vector whose $i$th component is $0$ and all whose other components are $m$ is a carrier of $v$. The following axiom is a version of the usual efficiency axiom that combines the carrier and the notion of dummy player.

Axiom 2. If ${\bf x}^{*}$ is a carrier of $v$ then, for ${\bf m}=(m,m,\cdots ,m)$, we have
\[\sum_{x_{i}^{*}\neq 0}\phi_{x_{i}^{*},i}(v)=v({\bf m}).\]
(Let us recall that if $S$ is a carrier then, for the traditional game $v$, we have $\sum_{i\in S}\phi_{i}(v)=v(S)=v(S\cap N)=v(N)$).
Axiom 3. $\boldsymbol{\phi}(v_{1}+v_{2})=\boldsymbol{\phi}(v_{1})+\boldsymbol{\phi}(v_{2})$, where
$(v_{1}+v_{2})({\bf x})=v_{1}({\bf x})+v_{2}({\bf x})$.
Axiom 4. Given $\bar{\bf x}\in\beta^{n}$ if $v({\bf x})=0$, whenever ${\bf x}\not\geq\bar{\bf x}$, then, for each $i\in N$, $\phi_{k,i}(v)=0$ for all $k<\bar{x}_{i}$.

Axiom 4 states that, in games that stipulate a minimal exertion from players, those who fail to meet this minimal level cannot be rewarded (Hsiao and Raghavan \cite[p.245]{hsi93}).

Proposition. (Hsiao and Raghavan \cite[p.245]{hsi93}). If ${\bf x}$ and ${\bf y}$ are carriers of $(\beta^{n},v)$, then ${\bf x}\wedge {\bf y}$ is a carrier of $v$. If a game has a carrier then it has a unique minimal carrier. $\sharp$

Proposition. (Hsiao and Raghavan \cite[p.245]{hsi93}). Let $v$ be defined by
\[v({\bf y})=\left\{\begin{array}{ll}
c\neq 0 & \mbox{if ${\bf y}\geq {\bf x}$}\\ 0 & \mbox{otherwise}.
\end{array}\right .\]
Then ${\bf x}$ is the minimal carrier of $v$. $\sharp$

Theorem. (Hsiao and Raghavan \cite[p.245]{hsi93}). If $w_{0}=0$, $w_{1},\cdots ,w_{m}$ are given, then there exists a unique
function $\boldsymbol{\phi}:\mbox{TU}(N)\rightarrow M_{m\times n}$ satisfying axioms 1,2,3 and 4. $\sharp$

Definition. (Hsiao and Raghavan \cite[p.248]{hsi93}). Given $\beta^{n}$ and $w_{0}=0$, $w_{1},\cdots ,w_{m}$, for any ${\bf x}\in\beta^{n}$, we define $\parallel {\bf x}\parallel_{\bf w}=\sum_{i=1}^{n}w_{x_{i}}$. For $j\in N$, we define $latex M_{j}({\bf x})=
\{i\in N:x_{i}\neq m\mbox{ for }i\neq j\}$. $\sharp$

\begin{equation}{\label{hsi93t2}}\mbox{}\end{equation}

Theorem \ref{hsi93t2}. (Hsiao and Raghavan \cite[p.248]{hsi93}). Suppose $w_{0}=0$, $w_{1},\cdots ,w_{m}$ are given, and $\boldsymbol{\phi}:\mbox{TU}(N)\rightarrow M_{m\times n}$ satisfies axioms 1,2,3 and 4. Then $\boldsymbol{\phi}(v)$ is given by
\[\phi_{ij}(v)=\sum_{k=1}^{i}\sum_{{\bf 0}\neq {\bf x}\in\beta^{n},x_{j}=k}
\left [\sum_{\{T:T\subseteq M_{j}({\bf x})\}}(-1)^{|T|}\cdot\frac{w_{x_{j}}}{\parallel {\bf x}\parallel_{\bf w}+
\sum_{r\in T}(w_{x_{r}+1}-w_{x_{r}})}\right ]\cdot\left [v({\bf x})-v({\bf x}-{\bf b}_{\{j\}})\right ]. \sharp\]

We shall show that the value $\boldsymbol{\phi}$ given in Theorem \ref{hsi93t2} is an extension of the traditional Shapley value. Given $(\beta^{n},v)$ with $\beta =\{0,1\}$. Now, for every $j\in N$, we have
\[\phi_{1j}(v)=\sum_{{\bf x}\in\beta^{n},x_{j}=1}\sum_{\{T:T\subseteq M_{j}({\bf x})\}}(-1)^{|T|}\cdot\frac{1}
{|T|+\sum_{i=1}^{n}x_{i}}\cdot\left[v({\bf x})-v({\bf x}-{\bf b}_{\{j\}})\right ].\]
Since $\beta^{n}\sim {\cal N}$, we can rewrite the above formula as follows:
\[\phi_{1j}(v)=\sum_{\emptyset\neq S\subseteq N,j\in S}\sum_{\{T:T\subseteq N\setminus S,j\not\in T\}}(-1)^{|T|}\cdot\frac{1}
{|T|+|S|}\cdot\left[v(S)-v(S\setminus\{j\})\right ].\]
Now, given $j\in S\subseteq N$ and $T\subseteq N\setminus S$,
\begin{align*}
\sum_{\{T:T\subseteq N\setminus S,j\not\in T\}}\frac{1}{|S|+|T|} & =\sum_{|T|=0}^{n-|S|}(-1)^{T}\left (\begin{array}{c}
n-|S|\\ |T|\end{array}\right )\cdot\frac{1}{|S|+|T|}\\
& =\sum_{|T|=0}^{n-|S|}(-1)^{|T|}\left (\begin{array}{c}n-|S|\\ |T|\end{array}\right )\cdot\int_{0}^{1}x^{|S|+|T|-1}dx\\
& =\int_{0}^{1}x^{|S|-1}\cdot\sum_{|T|=0}^{n-|S|}(-1)^{|T|}
\left (\begin{array}{c} n-|S|\\ |T|\end{array}\right )x^{|T|}dx\\
& =\int_{0}^{1}x^{|S|-1}(1-x)^{n-|S|}dx\\
& =\frac{(|S|-1)!(n-|S|)!}{n!}.
\end{align*}
Therefore
\[\phi_{1j}(v)=\sum_{\emptyset\neq S\subseteq N,j\in S}\frac{(|S|-1)!(n-|S|)!}{n!}\left [v(S)-v(S\setminus\{j\}(\right ].\]
Since $\beta =\{0,1\}$, the index $1$ in $\phi_{ij}$ is redundant and we obtain the traditional Shapley value (Hsiao and Raghavan \cite[p.252]{hsi93}).

We discuss further the axioms of the extended Shapley value. Given any permutation $\pi :N\rightarrow N$, we define $\pi ({\bf x}){\bf y}$, where $y_{j}=x_{i}$, if and only if $\pi (i)=j$, for all ${\bf x}\in\beta^{n}$. We also define $(\pi\circ v)(\pi ({\bf x}))=v({\bf x})$ for all permutations $\pi$ and all $v\in \mbox{TU}(N)$.

Axiom 5. For any permutation $\pi$, $\phi_{k,\pi (i)}(\pi\circ v)=\phi_{k,i}(v)$, i.e., $latex \boldsymbol{\phi}_{\pi (i)}(\pi\circ v)=
\boldsymbol{\phi}_{i}(v)$, for allpermutations $\pi$

Axiom 5 means that if player $i$ in game $v$ changes his/her name to $\pi (i)$ and playes the new game $\pi\circ v$ and if the player does not change his/her action then this power index does not change. If we restrict $\beta$ to $\{0,1\}$, then axiom 5 is exactly the same as the traditional symmetry axiom (Hsiao and Raghavan \cite[p.252]{hsi93}).

Corollary. (Hsiao and Raghavan \cite[p.253]{hsi93}). If $\boldsymbol{\phi}$ satisfies Axioms 1,2,3 and 4, then $\boldsymbol{\phi}$ satisfies Axioms 5,2,3 and 4. $\sharp$

Axioms 5,2,3 and 4 do not imply Axioms 1,2,3 and 4 in general. However, when $\beta =\{0,1\}$, Axioms 5,2,3 and 4 imply Axioms 1,2,3 and 4. Given a nondecreasing multi-choice cooperative game $(\beta^{n},v)$, and its Shapley value $\phi_{ij}(v)$, we natually expect that $0=\phi_{0j}(v)\leq\phi_{1j}(v)\leq\cdots\phi_{mj}(v)$ for all $j\in N$ (Hsiao and Raghavan \cite[p.254]{hsi93}).

Definition. (Hsiao and Raghavan \cite[p.311]{hsi92}). Let ${\bf x},{\bf y}\in\beta^{n}$. We say that ${\bf x}$ is strictly greater than ${\bf y}$, denoted by ${\bf x}>{\bf y}$, if there exists a $j\in N$ such that $x_{j}>y_{j}$. A multi-choice cooperative game $(\beta^{n},v)$ is said to be strictly increasing if $v({\bf x})>v({\bf y})$ whenever ${\bf x}>{\bf y}$. $\sharp$

Theorem. (Hsiao and Raghavan \cite[p.311]{hsi92}). Given a nondecreasing multi-choice cooperative game $(\beta^{n},v)$ and its Shapley value $\phi_{ij}(v)$, we have $0=\phi_{0j}(v)\leq\phi_{1j}(v)\leq\cdots\leq\phi_{mj}(v)$ for all $j\in N$. In case, the multi-choice cooperative game is strictly increasing, then we have $0=\phi_{0j}(v)<\phi_{1j}(v)<\cdots <\phi_{mj}(v)$ for all $j\in N$. $\sharp$

Example. (Hsiao and Raghavan \cite[p.254]{hsi93}) Player 1 is a doctor who can also do an assistant’s job. Player 2 is trained as a doctor’s assistant. He has acted in movies as a doctor. Now they are about to perform an operation on a patient. Player 1 is not able to do both a doctor’s job and an assistant’s job at the same time. We may characterize the game by a traditional cooperative game with the characteristic function $v(\emptyset )=0$, $v(\{1\})=0=v(\{2\})$, i.e., fail, and $v(\{1,2\})=1$, i.e., succeed. The traditional Shapley value is $\phi_{1}(v)=\frac{1}{2}=\phi_{2}(v)$. But the Shapley value is by no means fair in this case. Consider the multi-choice cooperative game which allows players to have the following three actions. $\sigma_{0}$: do nothing; $\sigma_{1}$: do an assistant’s job only; $\sigma_{2}$: do one’s best to maximize the result of the operation, i.e., do an assistant’s job or a doctor’s job so that the result of the operation is maximized. Then $v(0,0)=v(0,1)=v(0,2)=v(1,0)=v(1,1)=v(1,2)=v(2,0)=0$ and $v(2,1)=v(2,2)=1$. Note that we do not take $\sigma_{2}$ to be the action that the player does a doctor’s job. In that case, we must take $v(2,2)=0$ since none does the assistant’s job and this destroys the nondecreasing structure of the original game. Note also that, for fairness both players are modeled as having the same number of available actions in the game, but since a doctor’s job requires more training than an assistant’s job, the action vector $(2,2)$ is a carrier for the game. We can give weights to $\sigma_{0},\sigma_{1}$ and $\sigma_{2}$, and find the extended Shapley value. Giving weights to the actions is an art depending on the market conditions and the bargaining ability of the players. Suppose $w_{0}=0$, $w_{1}=1$ and $w_{2}=2$. The extended Shapley value satisfies $\phi_{11}(v)=0$, $\phi_{21}9v)=\frac{2}{3}$ and $\phi_{12}(v)=\phi_{22}(v)=\frac{1}{3}$ which means that when player 1 takes action $\sigma_{2}$, he shares $\frac{2}{3}$ of the payoff, and player 2 shares $\frac{1}{3}$ of the payoff for both actions $\sigma_{1}$ and $\sigma_{2}$. This is so because his actions $\sigma_{1}$ and $\sigma_{2}$ have the same contribution to the operation. Let us remark that this example is indeed a nondecreasing multi-choice cooperative game. $\sharp$

Monotonicity and Dummy Free Property.

Given a multi-choice cooperative game $(\beta^{n},v)$ and $\boldsymbol{\phi}(v)=(a_{ij})_{m\times n}$. Suppose we allow a dummy player, say $(n+1)$ to join the game. Then we have a new game $(\beta^{n+1},v^{D})$ such that $v^{D}(({\bf x}|x_{n+1}=i))=v({\bf x})$, for all ${\bf x}\in\beta^{n}$ and all $i\in\beta$, is called a dummy extension of $(\beta^{n},v)$ (Hsiao and Raghavan \cite[p.303-304]{hsi92}).

Suppose $\boldsymbol{\phi}(v^{D})=(b_{ij})_{m\times n+1}$. It is clear that $b_{i,n+1}=0$ for all $i\in\beta$. Now we could ask whether $a_{ij}=b_{ij}$ for all $i\in\{1,2,\cdots ,m\}$ and all $j\in N$. A solution of a multi-choice cooperative game is said to be dummy free of players if $a_{ij}=b_{ij}$ for all $i\in\{1,2,\cdots ,m\}$ and all $j\in N$; otherwise, the solution is said to be dummy dependent of players. We shall show that the extended Shapley value is dummy free of players (Hsiao and Raghavan \cite[p.304]{hsi92}).

Proposition. (Hsiao and Raghavan \cite[p.304]{hsi92}). A multi-choice cooperative game $(\beta^{n},v)$ can be written as
\[v=\sum_{{\bf x}\in\beta^{n},{\bf x}\neq {\bf 0}}a_{\bf x}v^{\bf x},\]
where
\[a_{\bf x}=\sum_{S\subseteq S({\bf x})}(-1)^{|S|}v({\bf x}-{\bf b}(S))\mbox{ and }v^{\bf x}({\bf z})=\left\{\begin{array}{ll}
1 & \mbox{if ${\bf z}\geq {\bf x}$}\\ 0 & \mbox{otherwise}.
\end{array}\right .\]

Theorem. (Hsiao and Raghavan \cite[p.305]{hsi92}). The Shapley value for a multi-choice cooperative game is dummy free of players. $\sharp$

If a solution of a game is dummy dependent of a player, then the dummy player acts like a catalyst, and those who are not dummy will invite him to join the game if he can make their income bigger. Conversely, they will reject the dummy player joining the game if he will make their income smaller (Hsiao and Raghavan \cite[p.306]{hsi92}).

Definition. (Hsiao and Raghavan \cite[p.306]{hsi92}). Given $(\beta^{n},v)$ and given ${\bf x}\in\beta^{n}$ with ${\bf x}\neq {\bf 0}$, let $A_{i}({\bf x})=\{j:x_{j}=i\}$. We call the action $\sigma_{i}$ a dummy action if $v({\bf x})=v({\bf x}-{\bf b}(T))$ for all $T\subseteq A_{i}({\bf x})$ and all ${\bf x}\in\beta^{n}$. $\sharp$

Given $(\beta^{n},v)$, where $\beta =\{0,1,2,\cdots ,m\}$, and given $r\in\beta$, allow players to have one more choice $\sigma_{r’}$ such that $\sigma_{r’}$ has a level which is in between $\sigma_{r}$ and $\sigma_{r+1}$. When $r=m$ we assume $r’=m+1$. Thus, we have a new action space $\beta_{*}^{n}$, where $\beta_{*}=\{0,1,2,\cdots ,r,r’,r+1,\cdots ,m\}$. Let $(\beta_{*}^{n},v^{A})$ be the game such that $v^{A}({\bf x})=v({\bf x})$ whenever $\beta_{*}^{n}\cap\beta^{n}=\beta^{n}$. If $\sigma_{r’}$ is a dummy action of $v^{A}$, then we call $(\beta_{*}^{n},v^{A})$ a dummy action extension of $(\beta^{n},v)$ (Hsiao and Raghavan \cite[p.306]{hsi92}).

After extending $\beta^{n}$ to $\beta_{*}^{n}$, we encounter a notational difficulty with $\beta_{*}=\{0,1,\cdots ,r,r’,r+1,\cdots ,m\}$, where $r’$ denotes an action whose level is no lower than $r$ and no higher than $r+1$. Here $r’$ is just a number but not necessarily a natural number. We decide to leave $r’$ alone, and make the following modification: Given ${\bf x}\in\beta_{*}^{n}$ with $A_{r’}({\bf x})=\{j:x_{j}=r’\}$, for any $S\subseteq A_{r’}({\bf x})$, we have ${\bf x}-{\bf b}(S)={\bf y}$, where $y_{i}=x_{i}$ if $i\not\in S$ and $y_{i}=r$ (one level lower than $r’$) if $i\in S$. Similarly, for any $T\subseteq A_{r+1}({\bf x})$, we have ${\bf x}-{\bf }(T)={\bf z}$, where $z_{i}=x_{i}$ if $i\not\in T$ and $z_{i}=r’$ if $i\in T$ (Hsiao and Raghavan \cite[p.306]{hsi92}).

Suppose $\boldsymbol{\phi}$ is a solution of $(\beta^{n},v)$, and $\boldsymbol{\phi}(v)=(a_{ij})_{m\times n}$. Suppose, after a dummy action extension, that we have
\[\boldsymbol{\phi}(v^{A})=\left [\begin{array}{ccc}
b_{11} & \cdots & b_{1n}\\
\vdots & \cdots & \vdots\\
b_{r’1} & \cdots & b_{r’n}\\
\vdots & \cdots & \vdots\\
b_{m1} & \cdots & b_{mn}
\end{array}\right ].\]
Then the solution $\boldsymbol{\phi}$ is said to be dummy free of action if $a_{ij}=b_{ij}$ for all $i\in\{1,2,\cdots ,m\}$ and all $j\in N$. Otherwise, $\boldsymbol{\phi}$ is said to be dummy dependent of action (Hsiao and Raghavan \cite[p.307]{hsi92}).

Theorem. (Hsiao and Raghavan \cite[p.307]{hsi92}). The Shapley value for multi-choice cooperative game is dummy free of action, and the value for a player using the dummy action is same as the value using the action, one step lower. $\sharp$

Since the Shapley value for multi-choice cooperative game is dummy free of action, in many cases, we can insert some dummy actions to make the calculations of the Shapley value easier. For example, if $w_{0}=0$, $w_{1}=1$, $w_{2}=3$ and $\beta =\{0,1,2\}$, then we may insert a dummy action $\sigma_{1′}$ such that $w_{0}=0$, $w_{1}=1$, $w_{1′}=2$, $w_{2}=3$ and $\beta_{*}=\{0,1,1’2\}$. Then $\beta_{*}$ has the property that when the level of action increases, the weights also increase at a fixed rate, and this property can make the calculation easier. See formula (\ref{hsi92eq41}) below (Hsiao and Raghavan \cite[p.311]{hsi92}).

Theorem. (Hsiao and Raghavan \cite[p.311]{hsi92}). Given $(\beta^{n},v)$, if $w_{0}=0$, $w_{1}=1$, $w_{2}=2,\cdots$, $w_{m}=m$, then the Shapley value for multi-choice cooperative game is given by
\begin{equation}{\label{hsi92eq41}}
\phi_{ij}(v)=\sum_{k=1}^{i}\sum_{{\bf x}\in\beta^{n},{\bf x}\neq {\bf 0},x_{j}=k}
\frac{k\cdot|M_{j}({\bf x})|!}{\prod_{t=0}^{|M_{j}({\bf x})}\left [
t+\sum_{r=1}^{n}x_{r}\right ]}\left [v({\bf x})-v({\bf x}-{\bf b}(\{j\}))\right ]. \sharp
\end{equation}

Transferable Utility Invariance.

Given $\beta =\{0,1,\cdots ,m\}$, a matrix $A=(a_{ij})_{m\times n}$ and ${\bf x}\in\beta^{n}$, let ${\bf x (A)=\sum_{j=1}^{n}a_{x_{j},j}$, where $a_{0j}=0$ for all $j\in N$ (Hsiao \cite[p.427]{hsi95}).

Definition. (Hsiao \cite[p.427]{hsi95}). Given two mutli-choice cooperative games $(\beta^{n},v)$ and $(\beta^{n},w)$, $v$ and $w$ are said to be {\bf strategically equivalent} if there exists a matrix $A=(a_{ij})_{m\times n}$ and a constant $k$ such that
\[v({\bf x})=k\cdot w({\bf x})+{\bf x}(A)\]
for all ${\bf x}\in\beta^{n}$. A solution of multi-choice cooperative games $\boldsymbol{\phi}$ is said to be transferable utility invariant if $\boldsymbol{\phi}(v)=k\cdot\boldsymbol{\phi}(w)+A$, whenever $v$ and $w$ are strategically equivalent. $\sharp$

Theorem. (Hsiao \cite[p.429]{hsi95}). The Shapley value for multi-choice cooperative game is transferable utility invariant. $\sharp$

\subsection{Independence of Non-Essential Players}

Definition. (Hsiao \cite[p.429]{hsi95}). A multi-choice cooperative game $(\beta^{n},v)$ is called a non-essential game if
\[v({\bf x})=\sum_{j=1}^{n}v(({\bf 0}|x_{j}),\]
where $({\bf 0}|x_{j})$ is an action tuple where player $j$ uses action $\sigma_{x_{j}}$ and all the other players use action $\sigma_{0}$. $\sharp$

Definition. (Hsiao \cite[p.430]{hsi95}). Player $j$ in the game $(\beta^{n},v)$ is called a non-essential player if
\[v({\bf x})=v(({\bf x}|x_{j}=0))+v({\bf 0}|x_{j}))\]
for all ${\bf x}\in\beta^{n}$, where ${\bf x}$ and $({\bf x}|x_{j}=0)$ are action tuple such that player $j$ uses action $\sigma_{x_{j}}$ in ${\bf x}$ and uses $\sigma_{0}$ in $({\bf x}|x_{j}=0)$, and all the other players use the same actions in both ${\bf x}$ and $({\bf x}|x_{j}=0)$. $\sharp$

Definition. (Hsiao \cite[p.430]{hsi95}). Given a multi-choice cooperative game $(\beta^{n},v)$, allow a new player, say $(n+1)$, to join the game, then we have a new action space $\beta^{n+1}$. Let $v^{0}(({\bf x}|x_{n+1}=0))=v({\bf x})$ for all ${\bf x}\in\beta^{n}$ and all $({\bf x}|x_{n+1}=0)=(x_{1},\cdots ,x_{n},0)\in\beta^{n+1}$. Assign each $v^{0}(({\bf 0}|x_{n+1}=k))$, $k\neq 0$, a value not necessarily zero. Then we can define a new game $(\beta^{n+1},v^{0})$ such that $n+1$ is a non-essential player in $(\beta^{n+1},v^{0})$. We call $(\beta^{n+1},v^{0})$ a non-essential extension of $(\beta^{n},v)$. A solution of $(\beta^{n},v)$ is said to be independent of non-essential players if $\phi_{ij}(v)=\phi_{ij}(v^{0})$ for all $i\in\beta$ and $j\in N$. Otherwise, $\boldsymbol{\phi}$ is said to be dependent of non-essential players. $\sharp$

It is obvious that a dummy extension is a special case of non-essential extension and a dummy free of players is a special case of independence of non-essential players (Hsiao \cite[p.430]{hsi95}).

Theorem. (Hsiao \cite[p.430]{hsi95}). The Shapley value for multi-choice cooperative games is independent of non-essential players. $\sharp$

Example. (Hsiao \cite[p.431]{hsi95}) A harvest car (harvester) needs two persons to operate. One person has to drive the car, and the other one has to put away the crop that is harvested by the car. Nobody can do both jobs at the same time. A company has $3$ harvest cars, and the company wants to hire temporary employees to operate the cars. In a small town, there are $3$ girls who are able to drive the harvest cars and $2$ boys who are able to drive the cars and also can put away teh crop that is harvestsed by the cars. The company will pay \$100 for operating a car. Suppose none of the girls is strong enough to put away the crop that is harvested by the cars. If all $3$ girls and $2$ boys work together, and if they share the income by the Shapley value for multi-choice cooperative game, then we can find their shares in the following way: Each player is allowed to take one of the three actions: $\sigma_{0}$, do nothing; $\sigma_{1}$, drive a car; and $\sigma_{2}$, do one’s best to maximize the payoff. Suppose the first three players are girls and the last two players are boys. Then we can characterize the game by $v({\bf 0})=0$, $\cdots$, $v({\bf 2})=200$. Let $\phi_{i,g}(v)$ be the value of a girl when she takes action $\sigma_{i}$ and $\phi_{i,b}(v)$ be the value of a boy when he takes the action $\sigma_{i}$, and suppose $w_{i}=i$; then we have $\phi_{1,g}(v)=\phi_{2,g}(v)=170/21$ and $\phi_{1,b}(v)=35/21$ an $\phi_{2,b}(v)=1845/21$. If the calculation is to be done by a computer, the property of transferable utility invariance allows us to multiply each value by a suitable number so that the computer will not round of the value. $\sharp$

In the above example, by the idea of transferable utility invariance, the company may encourage the players to work in the following way: The company takes \$20 from \$100, and announces to the players that one can have \$10 whenever he or she wants to work, and pay \$80 for operating a car. By the idea that the Shaplet value is independent of non-essential players, the company may also encourage the players to work in the following way: The company hires a boy and pays him $\phi_{2,b}(v)$, and announces to the other players that the boy will do his best to help the other players and will not share the payoff. Of course, the company will pay \$200 less $\phi_{2,b}(v)$ for the grand coalition (Hsiao \cite[p.431]{hsi95}).

Games under Precedence Constraints.

In \S\ref{fai92}, we present the Shapley value for games under precedence constraints. Now we consider a trivially ordered set $N$ of players, i.e., ${\cal C}=2^{N}$. In this model, we assume that each player $i\in N$ may take one in a finite set $A_{i}$ of actions: $A_{i}=\{a_{i0},a_{i1},\cdots ,a_{in_{i}}\}$. $A_{i}$ is assumed to be hierarchical ordered $a_{i0}<a_{i1}<\cdots <a_{in_{i}}$ with the interpretation that $a_{i0}$ indicates “no action” on the part of $i$ while $a_{n_{i}}$ is the strongest action $i$ may take. If $|A_{i}|=m+1$ for all $i\in N$, this model reduces to the multi-choice model discussed above. Now we are going to discuss this model by using the approach presented in \S\ref{fai92} (Faigle and Kern \cite[p.261]{fai92}).

The equivalent of a “coalition” in this setting is just the choice of an action by each individual player. In other words, a coalition corresponds to a vector ${\bf x}\in I\!\! N^{N}$ such that, for each $i\in N$, $0\leq x_{i}\leq n_{i}$. A multi-choice cooperative game in the present context is a function $v$ that assigns a real number $v({\bf x})$ to each feasible coalition ${\bf x}\in I\!\! N^{N}$ and satisfies $v({\bf 0})=0$ (Faigle and Kern \cite[p.261]{fai92}).

It is natural to study multi-choice games within the framework of \S\ref{fai92} as follows. Taking the disjoint union
\[N’=\bigcup_{i\in N}(A_{i}\setminus\{a_{i0}\})\]
of all nontrivial actions, we may view $N’$ as a set of pseudo-players that are partially ordered by the ordering induced on each $A_{i}\setminus\{a_{i0}\}$. Thus any two actions associated with different players will be incomparable in this ordering (Faigle and Kern \cite[p.261]{fai92}).

There is a one-to-one correspondence between the feasible coalition vectors ${\bf x}\in I\!\! N^{N}$ introduced above and the feasible coalitions on $N’$ in the sense of \S\ref{fai92}; the nonzero components of such an ${\bf x}\in I\!\! N^{N}$ may be interpreted as the maximal elements of a feasible coalition of $N’$ and conversely, while zero vector ${\bf 0}\in I\!\! N^{N}$ corresponds to the empty feasible coalition of $N’$. Moreover, each multi-choice game corresponds naturally with a game $G^{P’}$, where $P’=(N’,<)$ is the precedence structure among the non-trivial actions (Faigle and Kern \cite[p.261]{fai92}).

A simple game $\zeta_{S}\in G^{P’}$ now has the following interpretation. We choose a subset $T\subseteq N$ of players. Each player $t\in T$ is to take an action $a_{tj}$ that is at least as strong as the prescribed action $a_{tj_{t}}$ (i.e., $j\geq j_{t}$, where $j_{t}$ is given). The actions $a_{tj_{t}}$ may thus be thought of as “threshold actions” for the game $\zeta_{S}$, where $S$ consists of all those actions that do not properly exceed the corresponding threshold. A choice of actions by the set $N$ of all players is rewarded by “1” if it passes all thresholds, and by “0” otherwise (Faigle and Kern \cite[p.262]{fai92}).

Let $\boldsymbol{\phi}^{\prime}$ be the Shapley value defined on the class $G^{P’}$. Because of the special form of the precedence structure $P’$, formula (\ref{fai92eq10}) can be made more explicit with the following result (Faigle and Kern \cite[p.262]{fai92}).

Proposition. (Faigle and Kern \cite[p.261]{fai92}). Let $n=\prod_{i\in N}n_{i}$ be the number of actions in $N’$. Then
\[|{\cal R}(N’)|=\left (\begin{array}{c}n\\ n_{1}\end{array}\right )\left (\begin{array}{c}n-n_{1}\\ n_{2}\end{array}\right )
\left (\begin{array}{c}n-n_{1}-n_{2}\\ n_{3}\end{array}\right )\cdots 1. \sharp\]

Recall from \S\ref{fai92} that $\phi^{\prime}_{a_{ij}}(v)$ may be viewed as the expected marginal contribution of player $i$’s action $a_{ij}$ to the multi-choice game $v$, assuming that the actions are evaluated in a feasible random order. To obtain an allocation rule for individual players, it appears natural to go one step further and to assign to each player $i\in N$ the value
\[\widehat{\phi}_{i}(v)=\sum_{j=1}^{n_{i}}\phi^{\prime}_{a_{ij}}(v)\]
as his Shapley worth, i.e., the expected marginal contribution of his total set of actions. The Shaply worth of a simple multi-choice game $\zeta_{S}$ thus assigns to each player $i\in N$ the expected marginal contribution of $i$’s actions in the game $\zeta_{S}$. It is extended linearly from the class of simple games to the class of all multi-choice cooperatuve games (Faigle and Kern \cite[p.262]{fai92}).

We finally want to comment on the relationship between the Shapley value $\boldsymbol{\phi}^{\prime}$ for multi-choice games and the Shapley value introduced above. Hence, we assume from nown on that all players in $N$ have identical sets of actions, i.e., $A_{i}=A$ for all $i,j\in N$. The action $a_{ij}$ thus just indicates that player $i$ engages in an activity at level $j$. Consider the simple game $\zeta_{S}in G^{P’}$ with underlying set $T\subseteq N$ of players to whom threshold actions are assigned. The next result states that the hierarchical strength of a threshold action in the game $\zeta_{S}$ depends monotonically on its activity level (Faigle and Kern \cite[p.262]{fai92}).

\begin{equation}{\label{fai92p10}}\mbox{}\end{equation}

Proposition \ref{fai92p10}. (Faigle and Kern \cite[p.262]{fai92}). Let $a_{il}$ and $a_{jk}$ be the threshold actions for the players $i,j\in T$ in the game $\zeta_{S}$. Then the associated hierarchical strength satisfies

$h_{S}(a_{il})=h_{S}(a_{jk})$ if $l=k$;
$h_{S}(a_{il})>h_{S}(a_{jk})$ if $l>k$. $\sharp$

Recall from Axiom FK3 in \S\ref{fai92} that the Shapley value $\boldsymbol{\phi}^{\prime}$ distributes the value of a simple game according to the hierarchical strenghth of the maximal actions in the relevant coalition. Interpreted in the present multi-choice games,
Proposition \ref{fai92p10} implies that the Shapley worth $\widehat{\boldsymbol{\phi}}$ rewards a player in monotonical dependence on the level of the minimal action he must take if the (simple) multi-choice game is to be successful (Faigle and Kern \cite[p.263]{fai92}).

The Shapley value $\boldsymbol{\phi}^{\prime\prime}$ introduced by Hsiao and Raghavan \cite{hsi93} for the set $N’$ of actions only achieves the monotonicity property above by direct construction as follows. Numbers $0=w_{0}<w_{1}<\cdots <w_{m}$ are assumed to be given from the outside with the understanding that $w_{l}$ reflects the power of any action $a_{il}$ at level $l$. In a simple multi-choice game $\boldsymbol{\phi}^{\prime\prime}$ now is defined to assign the value $w_{t}/W$ to any action $a_{ik}$ that is not weaker than the given threshold $a_{it}$ for player $i$, where $W$ is the sum of the weights of all threshold actions. Thus $\boldsymbol{\phi}^{\prime\prime}$ depends on the chosen set $\{w_{1},\cdots ,w_{m}\}$ of weights and extends ideas behind weighted Shapley values (Faigle and Kern \cite[p.263]{fai92}).

The value $\boldsymbol{\phi}^{\prime}$ is a purely combinatorial parameter, depending only on the order structure of the set of actions. It naturally generalizes to a model in which the actions individual players may take are not necessarily linearly ordered and where the “threshold” of a player may consist of several actions to be taken (Faigle and Kern \cite[p.264]{fai92}).

Multi-choice Values.

The above multi-choice games consider that all players have the same number of activity levels. Here we allow for different numbers of activity levels for different players. Suppose each player $i\in N$ has $m_{i}$ levels at which he can actively participate. Let ${\bf m}=(m_{1},\cdots ,m_{n})$ be the vector that describes the number of activity levels for every player. We set $M_{i}=\{0,1,\cdots ,m_{i}\}$ as the action space of player $i$, where the action $0$ measn not participating. Let $M=\prod_{i\in N}M_{i}$ be the product set of actions spaces. A characteristic function $v:M\rightarrow\mathbb{R}$ which assigns to each coalition ${\bf s}=(s_{1},\cdots ,s_{n})$ the worth that the players can obtain when each player $i$ plays at activity level $s_{i}\in M_{i}$ with $v({\bf 0})=0$. A multi-choice game is given by a triple $(N,{\bf m},v)$. Let us denote the class of multi-choice games with player set $N$ and activity level ${\bf m}$ by $MC^{N,{\bf m}}$, and the class of all multi-choice games by $MC$ (Klijn et al. \cite[p.522]{kli99}).

For $i\in N$, let $M_{i}^{+}\equiv M_{i}\setminus\{0\}$. Further, let $M^{+}=\bigcup_{i\in N}(\{i\}\times M_{i}^{+})$. A solution on $MC$ is a map $\boldsymbol{\Psi}$ assigning to each multi-choice game $(N,{\bf m},v)$ and element $\boldsymbol{\Psi}(N,{\bf m},v)\in\mathbb{R}^{M^{+}}$. The analogue of unanimity games for multi-choice games are minimal effort games $(N,{\bf m},u_{\bf s})\in MC^{N,{\bf m}}$, where ${\bf s}\in\prod_{i\in N}M_{i}$, defined by
\[u_{\bf s}({\bf t})=\left\{\begin{array}{ll}
1 & \mbox{if $t_{i}\geq s_{i}$ for all $i\in N$}\\
0 & \mbox{otherwise}.
\end{array}\right .\]
for all ${\bf t}\in\prod_{i\in N}M_{i}$. One can prove that the minimal effort games form a basis of the space $MC^{N,{\bf m}}$, and that, for a multi-choice game $(N,{\bf m},v)$, it holds that
\[v=\sum_{{\bf s}\in M}\Delta_{v}({\bf s})u_{\bf s},\]
where the $\Delta_{\bf v}({\bf s})$ are the extended dividends defined by
\[\Delta_{v}({\bf 0})=0\mbox{ and }\Delta_{v}({\bf s})=v({\bf s})-
\sum_{{\bf t}\leq {\bf s},{\bf t}\neq {\bf s}}\Delta_{v}({\bf t})\mbox{ for }{\bf s}\neq {\bf 0}\]
(Klijn et al. \cite[p.523]{kli99}).

Now we can go on to the extension of the Shapley value of Derks and Peteres \cite{der93}). For a multi-choice game $(N,{\bf m},v)\in MC^{N,{\bf m}}$, the value $\boldsymbol{\Theta}(N,{\bf m},v)$ is defined by
\begin{equation}{\label{kli99eq1}}
\Theta_{ij}(N,{\bf m},v)=\sum_{{\bf s}\in M,s_{i}\geq j}\frac{\Delta_{v}({\bf s})}{\sum_{k\in N}s_{k}}
\end{equation}
for all $(i,j)\in M^{+}$. So, the dividend $\Delta_{v}({\bf s})$ is divided equally among the necessary levels. In fact, this value can be seen as the vector of average marginal contributions of the pairs $(i,j)\in M^{+}$. Let us point this out formally. For this, we may suppose that $M^{+}\neq\emptyset$. An order for a multi-choice game $(N,{\bf m},v)$ is a bijection $\pi :M^{+}\rightarrow\{1,\cdots ,\sum_{i\in N}m_{i}\}$. The subset $\pi^{-1}(\{1,\cdots ,k\})$ of $M^{+}$, which is present after $k$ steps according to $\pi$, is denoted by $S^{\pi ,k}$. The marginal vector ${\bf w}^{\pi}\in\mathbb{R}^{M^{+}}$ corresponding to $\pi$ is defined by
\[w_{ij}^{\pi}=v(\rho (S^{\pi ,\pi (i,j)}))-v(\rho (S^{\pi ,\pi (i,j)-1}))\]
for all $(i,j)\in M^{+}$, where $\rho$ is the map that assigns to every subset $S\subseteq M^{+}$ the maximal feasible coalition $\rho (S)$ that is a “subset” of $S$, i.e., for $S\subseteq M^{+}$, $\rho (S)=(t_{1},\cdots ,t_{n})$ with
\[t_{i}=\left\{\begin{array}{ll}
\max\{k\in M_{i}^{+}:(i,1),\cdots ,(i,k)\in S\} & \mbox{if $(i,1)\in S$}\\
0 & \mbox{otheriwse}.
\end{array}\right .\]
Now, we define
\[\Lambda_{ij}(N,{\bf m},v)=\frac{1}{(\sum_{k\in N}m_{k})!}\cdot\sum_{\pi}w_{ij}^{\pi}\]
for all $(i,j)\in M^{+}$. The number $\Lambda_{ij}(N,{\bf m},v)$ is the average marginal contribution of the pair $(i,j)\in M^{+}$ to the maximal feasible coalition. In fact, the number $\Lambda_{ij}(N,{\bf m},v)$ is equal to the Shapley value of player $(i,j)$ in the conventional TU-game $(M^{+},\bar{v})$, where the characteristic function $\bar{v}$ is defined by $\bar{v}(T)=v(\rho (T))$ for all $T\subseteq M^{+}$. A multi-choice game $(N,{\bf m},v)$ is said to be convex if $v({\bf s}\vee {\bf t})+v({\bf s}\wedge {\bf t})\geq v({\bf s})+v({\bf t})$ for all ${\bf s},{\bf t}\in\prod_{i\in N}M_{i}$, where $({\bf s}\vee {\bf t})_{i}=\max\{s_{i},t_{i}\}$ and $({\bf s}\wedge {\bf t})_{i}=\min\{s_{i},t_{i}\}$. One can prove that a multi-choice game $(N,{\bf m},v)$ is convex if and only if the TU-game $(M^{+},\bar{v})$ is convex (Klijn et al. \cite[p.523-524]{kli99}).

It is not difficult to see that, for a minimal effort game $(N,{\bf m},u_{\bf s})$, we have
\begin{equation}{\label{kli99eq4}}
\Theta_{ij}(N,{\bf m},u_{\bf s})=\Lambda_{ij}(N,{\bf m},u_{\bf s})
=\left\{\begin{array}{ll}
1/\sum_{i\in N}s_{i} & \mbox{if $j\leq s_{i}$}\\ 0 & \mbox{otherwise}.
\end{array}\right .
\end{equation}
for all $(i,j)\in M^{+}$. Derivation of formula (\ref{kli99eq4}) is straightforward for $\Theta_{ij}(N,{\bf m},u_{\bf s})$ by using formula
(\ref{kli99eq1}). To see the equality for $\Lambda_{ij}(N,{\bf m},u_{\bf s})$ first note that, for all $\pi$ and all $(i,j)\in M^{+}$, we have $w_{ij}^{\pi}\in\{0,1\}$. Now, if $j>s_{i}$, then $w_{ij}^{\pi}=0$. If $j\leq s_{i}$, then note that the number of $\pi$ for which $w_{ij}^{\pi}=1$, or, equivalently
\begin{equation}{\label{kli99eq5}}
S^{\pi ,\pi (i,j)}\supseteq S=\{(1,1),\cdots ,(1,s_{1}),\cdots ,(n,1),\cdots ,(n,s_{n})\}
\end{equation}
does not depend upon $(i,j)\in S$; the numbers of permutations of $M^{+}$ with $(i,j)$ last element of $S$ is the same for every $(i,j)\in S$. Hence, the number of permutations for which (\ref{kli99eq5}) holds is the same for all $(i,j)\in S$ and is therefore equal to $(\sum_{k\in N}m_{k})!/(\sum_{k\in N}s_{k})$. This implies that indeed formula (\ref{kli99eq4}) holds true for $\Lambda_{ij}(N,{\bf m},u_{\bf s})$. From formula (\ref{kli99eq4}) and the linearity of both $\boldsymbol{\Phi}$ and $\boldsymbol{\Theta}$, it follows that
$\boldsymbol{\Phi}=\boldsymbol{\Theta}$ (Klijn et al. \cite[p.524]{kli99}).

Let $\boldsymbol{\Psi}$ be a solution on $MC$. We consider the following axioms.

Axiom KSZ1: (Efficiency). For all games $(N,{\bf m},v)\in MC$, we have $\sum_{i\in N}\sum_{j=1}^{m_{i}}\Psi_{ij}(N,{|bf m},v)=v({\bf m})$;
Axiom KSZ2: (Strong Monotonicity). If all games $(N,{\bf m},v_{1}),(N,{\bf m},v_{2})\in MC$, $(i,j)\in M^{+}$ such that, for all ${\bf s},{\bf t}\in\prod_{i\in N}M_{i}$ with $s_{i}=j$ and $t_{k}=s_{k}$ if $k\neq i$ and $t_{i}=s_{i}-1$, $v_{1}({\bf s})-v_{1}({\bf t})\geqv_{2}({\bf s})-v_{2}({\bf t})$, then $\Psi_{ij}(N,{\bf m},v_{1})\geq\Psi_{ij}(N,{\bf m},v_{2})$;
Axiom KSZ3: (Veto Property). If all games $(N,{\bf m},v)\in MC$ and all $i_{1},i_{2}\in N$, whenever $j_{1}\in M_{i_{1}}^{+}$ and $j_{2}\in M_{i_{2}}^{+}$ are veto levels, then $\Psi_{i_{1}j_{1}}(N,{\bf m},v)=\Psi_{i_{2}j_{2}}(N,{\bf m},v)$, where $j\in M_{i}^{+}$ is a veto levelif $v({\bf s})=0$ for all ${\bf s}\in\prod_{i\in N}M_{i}$ with $s_{i}<j$.

The strong monotonicity says that if, for two games $(N,{\bf m},v_{1}),(N,{\bf m},v_{2})\in MC$ and a player $i\in N$, it holds that the marginal contribution of level $j\in M_{i}^{+}$ in the game $(N,{\bf m},v_{1})$ is not smaller than the marginal contribution in the game $(N,{\bf m},v_{2})$, then the payoff to level $j\in M_{i}^{+}$ in the game $(N,{|bf m},v_{1})$ is not smaller than the payoff in the game $(N,{\bf m},v_{2})$. The veto property says that, for a game $(N,{\bf m},v)\in MC$, the payoffs to all players $i\in N$ and levels $j\in M_{i}^{+}$ that have veto power (i.e., a level of player $i$ less than $j$ yields worth $0$, independent of the levels of the other players) should be equal (Klijn et al. \cite[p.526]{kli99}).

Theorem. (Klijn et al. \cite[p.526]{kli99}). A solution $\boldsymbol{\Psi}$ satisfies Axioms KSZ1–KSZ3 if and only if $\boldsymbol{\Psi}=\boldsymbol{\Theta}$. $\sharp$

Axiom KSZ4: (Additivity). For all games $(N,{\bf m},v_{1}),(N,{\bf m},v_{2})\in MC$, we have $\boldsymbol{\Psi}(v_{1}+v_{2})=\boldsymbol{\Psi}(v_{1})+\boldsymbol{\Psi}(v_{2})$;
Axiom KSZ5: (Dummy Property). If, for all games $(N,{\bf m},v)\in N$ and all $i\in N$, $j\in M_{i}^{+}$ is a dummy level, then $\Psi_{ij}(N,{\bf m},v)=0$, where $j\in M_{i}^{+}$ is a dummay level if $v({\bf s}_{-i},j-1)=v({\bf s}_{-i},l)$ for all ${\bf s}_{-i}\in\prod_{k\in N\setminus\{i\}}M_{k}$ and all $j\leq l\leq m_{i}$.

Theorem. (Klijn et al. \cite[p.527]{kli99}). A solution $\boldsymbol{\Psi}$ satisfies Axioms KSZ1, KSZ3–5 if and only if $\boldsymbol{\Psi}=\boldsymbol{\Theta}$. $\sharp$

With a slight abuse of notation, we write $(N,{\bf m}’,v)$ for the restriction of the game $(N,{\bf m},v)$ to the activity levels ${\bf m}’\in M$.

Axiom KSZ6: (Equal Loss Property). For all games $(N,{\bf m},v)\in MC$ all $(i,k)\in M^{+}$ and $k\neq m_{i}$, we have $\Psi_{ik}(N,{\bf m},v)- \Psi_{ik}(N,{\bf m}-{\bf e}^{\{i\}},v)=\Psi_{im_{i}}(N,{\bf m},v)$;
Axiom KSZ7: (Upper Balanced Contributions Property). For all games $(N,{\bf m},v)\in MC$ and all $(i,m_{i}),(j,m_{j})\in M^{+}$ with $i\neq j$, we have $\Psi_{im_{i}}(N,{\bf m},v)-\Psi_{im_{i}}(N,{\bf m}-{\bf e}^{\{j\}},v)=\Psi_{jm_{j}}(N,{\bf m},v)-\Psi_{jm_{j}}(N,{\bf m}-{\bf e}^{\{i\}},v)$.0

The equal loss property says that whenever a player gets availabale a higher activity level that payoff for all original levels changes with an
amount equal to the payoff for the highest level in the new situation. The upper balanced contributions property says that, for every pair $i,j$ of different players, the change in payoff for the highest level of player $i$ when player $j$ gets available a higher activity level is equal to the change in payoff for the highest level of player $j$ when player $i$ gets available a higher activity level (Klijn et al. \cite[p.527-528]{kli99}).

Theorem. (Klijn et al. \cite[p.528]{kli99}). A solution $\boldsymbol{\Psi}$ satisfies Axioms KSZ1, KSZ6–7 if and only if $\boldsymbol{\Psi}=\boldsymbol{\Theta}$. $\sharp$

Axiom KSZ8: (Lower Balanced Contributions Property). For all games $(N,{\bf m},v)\in MC$ and all $(i,1),(j,1)\in M^{+}$ with $i\neq j$, we have $\Psi_{i1}(N,{\bf m},v)-\Psi_{i1}(N,{\bf m}-m_{j}{\bf e}^{\{j\}},v)=\Psi_{j1}(N,{\bf m},v)-\Psi_{j1}(N,{\bf m}-m_{i}{\bf e}^{\{i\}},v)$.

Theorem. (Klijn et al. \cite[p.529]{kli99}). A solution $\boldsymbol{\Psi}$ satisfies Axioms KSZ1, KSZ6 and KSZ8 if and only if $\boldsymbol{\Psi}=\boldsymbol{\Theta}$. $\sharp$

Axiom KSZ9: (Strong Balanced Contributions Property). For all games $(N,{\bf m},v)\in MC$ and all $(i,k_{i}),(j,k_{j})\in M^{+}$ with $i\neq j$, we have $\Psi_{ik_{i}}(N,{\bf m},v)-\Psi_{ik_{i}}(N,{\bf m}-(m_{j}-k_{j}+1){\bf e}^{\{j\}},v)=\Psi_{jk_{j}}(N,{\bf m},v)-\Psi_{jk_{j}}(N,{\bf m}-(m_{i}-k_{i}+1){\bf e}^{\{i\}},v)$.

Axiom KSZ9 is stronger than Axioms KSZ7 and KSZ8. If we take $k_{i}=m_{i}$ and $k_{j}=m_{j}$ in Axiom KSZ9, we get KSZ7, and if we take $k_{i}=k_{j}=1$ in Axiom KSZ9, we get Axiom KSZ8.

Axiom KSZ10: (Weak Equal Loss Property). For all games $(N,{\bf m},v)\in MC$ with ${\bf m}=m_{i}{\bf e}^{\{i\}}$ for some $i\in N$ and all $(i,k)\in M^{+}$ with $k\neq m_{i}$, we have $\Psi_{ik}(N,{\bf m},v)-\Psi_{ik}(N,{\bf m}-{\bf e}^{\{i\}},v)=\Psi_{im_{i}}(N,{\bf m},v)$;

Theorem. (Klijn et al. \cite[p.530]{kli99}). A solution $\boldsymbol{\Psi}$ satisfies Axioms KSZ1, KSZ9–1 if and only if $\boldsymbol{\Psi}=\boldsymbol{\Theta}$. $\sharp$