Weighted Shapley Value

Jean-Baptiste-Camille Corot (1796-1875) was a French painter.

In various applications of cooperative games it seems to be natural to assume that the players are given some “a priori measures” of importance, called weights. For example, when we deal with a problem of cost allocation among investment projects, the weights could be associated to the profitability of the different projects. In a question of allocating travel costs among various places visited, the weights could be the number of day spent at each one (Nowak and Radzik \cite[p.389]{now95}).

The (symmetric) Shapley value has been the most prominent solution or value for transferable utility coalitional form games. It has been characterized using variou axioms. In many applications, however, the assumption that every player has an equal power may not appropriate. Players might represent constituencies of different sizes; players might have different bargaining abilities. To take care of this difficulty, weighted generalizations of the Shapley value has been proposed. (Chun \cite[p.183-184]{chu91}).

Each weighted Shapley value associates a positive weight with each player. These weights are the proportions in which the players share in unanimity games. The (symmetric) Shapley value is the special case where all the weights are the same. We can also extend the notion of “weights” to “weight systems” enabling a weight of zero for some players (Kalai and Samet \cite[p.206]{kal87}).

We recall that the (symmetric) Shapley value $\boldsymbol{\phi}$ is the linear function $\boldsymbol{\phi}:\mbox{TU}(N)\rightarrow\mathbb{R}^{|N|}$ such that, for each unanimity game $u_{S}$,
\[\phi_{i}(u_{S})=\left\{\begin{array}{ll}
1/|S| & \mbox{if $i\in S$}\\ 0 & \mbox{if $i\not\in S$}.\end{array}\right .\]
Intuitively, the members of $S$ split equally the one unit between them and players outside $S$ do not contribute anything to the coalition. Since $\{u_{S}\}_{S\subseteq N}$ is a basis to $\mbox{TU}(N)$ and $\boldsymbol{\phi}$ is linear, $\boldsymbol{\phi}$ is defined for all games. A weighted Shapley value generalizes the Shapley value by allowing different ways to split one unit between the members of $S$ in $u_{S}$. We prescribe a vector of positive weights ${\bf w}=(w_{i})_{i\in N}$ and, in each unanimity game $u_{S}$, players split proportionally to their weights. We want to allow some players to have weight zero. This means that if they split one unit with players who have positive weights, they get zero. But then we have to specify how these zero-weight players split one unit when no positive-weight players is with them. This brings us to the lexicographic definition of a weight system (Kalai and Samet \cite[p.207]{kal87}).

A weight system $\omega$ is a pair $({\bf w},\Sigma )$ where ${\bf w}\in\mathbb{R}_{++}^{|N|}$ and $\Sigma =(S_{1},\cdots ,S_{m})$ is an ordered partition of $N$, where the sets $S_{1},\cdots ,S_{m}$ are referred to as the social classes of players in the grand coalition $N$. A weight systme $\omega ({\bf w},\Sigma )$ is called {\bf simple} if $\Sigma =(N)$. The weighted Shapley value with weight system $\omega =({\bf w},\Sigma )$ is the linear map $\boldsymbol{\phi}^{\omega}:\mbox{TU}(N)\rightarrow\mathbb{R}^{|N|}$ which is defined for each unanimity game $u_{S}$ as follows. Let $k=\max\{j:S_{j}\cap S\neq\emptyset\}$ and denote $\bar{S}=S_{k}\cap S$. Then
\[\phi^{\omega}_{i}(u_{S})=\left\{\begin{array}{ll}
{\displaystyle \frac{w_{i}}{\sum_{j\in\bar{S}}w_{j}}} & \mbox{if $i\in\bar{S}$}\\
0 & \mbox{if $i\not\in\bar{S}$},\end{array}\right .\]
where $\phi^{\omega}_{i}$ denotes the $i$th coordinate of $\boldsymbol{\phi}$. This says that the weights of players in $S_{i}$ are $0$ with respect to players in $S_{j}$ with $j>i$. The positive weights of players in $S_{i}$ are used only for games $u_{S}$ such that no player from $S_{j}$ with $j>i$ is in $S$. Observe that $\boldsymbol{\phi}^{\omega}$ is the (symmetric) Shapley value if and only if $\omega =({\bf w},(N))$ and ${\bf w}$ is proportional to the vector ${\bf 1}=(1,1,\cdots ,1)$ (Kalai and Samet \cite[p.207-208]{kal87}).

Next, we investigate another approach by considering the weight system ${\bf w}=(w_{i})_{i\in N}$. Suppose that, for each player $i$, we are given a weight $w_{i}>0$. These weights can be interpreted as “a-priori measures of importance”. We would now desire that in any unanimity game the worth be distributed among the players in proportion to their weights. In a unanimity game, every player has exactly the same (marginal) contribution. Therefore, to obtain the ${\bf w}$-proportional allocation, we need to weight the marginal contribution of each player by his weight $w_{i}$. A ${\bf w}$-potential $P^{\bf w}$ is a function $P^{\bf w}:\mbox{TU}(N) \rightarrow\mathbb{R}$ such that the following conditions are satisfied:
\[P^{\bf w}(\emptyset ,v)=0\mbox{ and }\sum_{i\in N}w_{i}\cdot P^{\bf w}_{i}(N,v)=v(N),\]
where $P^{\bf w}_{i}(N,v)=P^{\bf w}(N,v)-P^{\bf w}(N\setminus\{i\},v)$. (Hart and Mas-Colell \cite[p.603]{har89}).

Theorem. (Hart and Mas-Colell \cite[p.603]{har89}). For every collection ${\bf w}=(w_{i})_{i\in N}$ of positive weights, there exists a unique ${\bf w}$-potential function $P^{\bf w}$. Moreover, the resulting solution function, associating the payoff vector $(w_{i}\cdot P^{\bf w}_{i}(N,v))_{i\in N}$ to the game $(N,v)$, coincides with the ${\bf w}$-weighted Shapley value. Finally, $P^{\bf w}$ can be computed recursively by the formula
\[P^{\bf w}(N,v)=\frac{\displaystyle v(N)+\sum_{i\in N}w_{i}\cdot P^{\bf w}(N\setminus\{i\},v)}{\displaystyle \sum_{i\in N}w_{i}}. \sharp\]

The preservation of difference principle also applies here. It just becomes
\[\frac{1}{w_{i}}x_{i}(T)-\frac{1}{w_{j}}x_{j}(T)=d_{ij}=
\frac{1}{w_{i}}x_{i}(T\setminus\{j\})-\frac{1}{w_{j}}x_{j}(T\setminus\{i\})\]
according to (\ref{har89eqe}). Thus the differences between the normalized payoffs are preserved (Hart and Mas-Colell \cite[p.604]{har89}).

Proposition. (Hart and Mas-Colell \cite[p.604]{har89}). For every collection of positive weights ${\bf w}=(w_{i})_{i\in N}$, the corresponding ${\bf w}$-weighted Shapley value is a consistent solution function. $\sharp$

Next, we generalize the standard solution for two-person games. Let ${\bf w}=(w_{i})_{i\in N}$ be positive weights. A solution function
$\boldsymbol{\phi}$ is $w$-proportional for two-person cooperative games if and only if
\[\phi_{i}(\{i,j\},v)=v(i)+\frac{w_{i}}{w_{i}+w_{j}}\cdot\left [v(\{i,j\})-v(i)-v(\{j\})\right ]\]
for all $i\neq j$.

Theorem. (Hart and Mas-Colell \cite[p.605]{har89}). Let $\boldsymbol{\phi}$ be a solution function and ${\bf w}$ be a collection of positive weights. Then $\boldsymbol{\phi}$ is the ${\bf w}$-weighted Shapley value if and only if the following conditions are satisfied:

$\boldsymbol{\phi}$ is consistent;
$\boldsymbol{\phi}$ is ${\bf w}$-proportional for two-person games. $\sharp$

A solution function $\boldsymbol{\phi}$ is monotonic if and only if, for any two games $(N,u)$ and $(N,v)$, we have that $u(N)>v(N)$ and $u(S)=v(S)$ for all $S\neq N$ imply $\phi_{i}(N,u)>\phi_{i}(N,v)$ for all $i\in N$. This means that if the (grand coalition’s) total payoff increases, but nothing else changes, then every player should gen an increase in his own payoff. Note that, for two-person games, monotonicity is just $\phi_{i}(\{i,j\},u)>\phi_{i}(\{i,j\},v)$ and $\phi_{j}(\{i,j\},u)>\phi_{j}(\{i,j\},v)$ when $u(i)=v(i)$, $u(\{j\})=v(\{j\})$ and $u(\{i,j\})>v(\{i,j\})$ (Hart and Mas-Colell \cite[p.605]{har89}).

Theorem. (Hart and Mas-Colell \cite[p.605]{har89}). Let $\boldsymbol{\phi}$ be a solution function. Then there exist positive weights ${\bf w}$ such that $\boldsymbol{\phi}$ is the ${\bf w}$-weighted Shapley value if and only if the following conditions are satisfied:

$\boldsymbol{\phi}$ is consistent;
for two-person games, $\boldsymbol{\phi}$ is efficient and TU-invariant, and monotonic. $\sharp$

Probabilistic Definition of Weighted Shapley Values.

Let $\Pi (S)$ denote the set of all permutations of players in the coalition $S$. For an ordered parttition $\Sigma =(S_{1},\cdots ,S_{m})$ of $N$, $\Pi_{\Sigma}$ is the set of all permutations for $N$ in which all the palyers of $S_{i}$ precede those of $S_{i+1}$ for $i=1,\cdots ,m-i$. Each $\pi$ in $\Pi_{\Sigma}$ can be described as $\pi=(\pi_{1},\cdots ,\pi_{m})$ where $\pi_{i}\in \Pi (S_{i})$ for $i=1,\cdots ,m$ (Kalai and Samet \cite[p.209]{kal87}).

Let $S\subseteq N$ and ${\bf w}\in\mathbb{R}_{++}^{S}$. We associate with ${\bf w}$ a probability distribution $\mathbb{P}_{\bf w}$ over $\Pi (S)$. For $\pi =(i_{1},\cdots ,i_{|S|})$ in $\Pi (S)$, we define
\[\mathbb{P}_{\bf w}(\pi )=\prod_{j=1}^{|S|}\frac{w_{i_{j}}}
{\sum_{k=1}^{j}w_{i_{k}}}=\frac{w_{i_{1}}}{w_{i_{1}}}\cdot\frac{w_{i_{2}}}
{w_{i_{1}}+w_{i_{2}}}\cdot\cdots\frac{w_{i_{|S|}}}{w_{i_{1}}+\cdots +w_{i_{|S|}}}.\]
One way to obtain this probability distribution is by arranging the players of $S$ in an order, starting from the end, such that the probability of adding a player to the begining of a partially created line is the ratio between his/her weight and the total weight of the players of $S$ that are not yet in the line (Kalai and Samet \cite[p.209]{kal87}).

For each weight system $\omega =({\bf w},\Sigma )$ with $\Sigma =(S_{1},\cdots ,S_{m})$, we associate a probability distribution $\mathbb{P}_{\omega}$ over $\Pi (N)$ as follows. The distribution $\mathbb{P}_{\omega}$ vanishes outside $\Pi_{\Sigma}$, and for
$\pi =(\pi_{1},\cdots ,\pi_{m})$ in $\Pi_{\Sigma}$,
\[\mathbb{P}_{\omega}=\prod_{i=1}^{m}\mathbb{P}_{{\bf w}^{S_{i}}}(\pi_{i}),\]
where ${\bf w}^{S_{i}}$ is the projection of ${\bf w}$ on $\mathbb{R}^{S_{i}}$. Let us recall the marginal vector $m_{i}(v,\pi )$ defined in (\ref{ga33}).

\begin{equation}{\label{kal87t200}}\tag{1}\mbox{}\end{equation}

Theorem \ref{kal87t200}. (Kalai and Samet \cite[p.209]{kal87}). For each cooperative game $(N,v)$, player $i\in N$, and weight system $\omega$, we have
\[\phi^{\omega}_{i}(v)=\mathbb{E}_{\mathbb{P}_{\omega}}\left [m_{i}(v,\cdot)\right ],\]
where $\boldsymbol{\phi}^{\omega}$ is the weighted Shapley value and the right hand side is the expected contribution of player $i$with respect to the probability distribution $\mathbb{P}_{\omega}$. $\sharp$

Given a set $M=\{1,2,\cdots ,m\}$, we define
\begin{equation}{\label{ga34}}\tag{2}
f(x,y_{1},y_{2},\cdots ,y_{m})=\sum_{\{S:S\subseteq M\}}(-1)^{|S|}\frac{1}{x+\sum_{i\in S}y_{i}},
\end{equation}
where $x>0$ and $y_{i}\geq 0$ for all $i\in M$. Then, we have $f(x,y_{1},\cdots ,y_{m})\geq 0$.
Given each player $i$ a weight $w_{i}>0$, the asymmetric Shapley value for players $i$ is given by
\begin{equation}{\label{hsi02eq2}}\tag{3}
\phi_{i}^{\bf w}(v)=\sum_{\{S:i\in S,S\subseteq N\}}\left [
\sum_{\{T:T\subseteq N\setminus S\}}(-1)^{|T|}\frac{w_{i}}
{\sum_{j\in S}w_{j}+\sum_{j\in T}w_{j}}\right ]\cdot\left [v(S)-v(S-\setminus\{i\})\right ].
\end{equation}
Without loss of generality, we may assume $w_{1}+\cdots +w_{n}=1$. It is trivial that when $w_{1}=w_{2}=\cdots =w_{n}$, the asymmetric Shapley value (\ref{hsi02eq2}) is exactly the same as the (symmetric) Shapley value (\ref{peteq25}). We now observe the coefficients of the asymmetric Shapley value. Fixed $i\in N$ for each $i\in S$ and $S\subseteq N$, we denote the coefficient in (\ref{hsi02eq2}) by
\[p^{S}_{i}=\sum_{\{T:T\subseteq N\setminus S\}}(-1)^{|T|}\frac{w_{i}}{\sum_{j\in S}w_{j}+\sum_{j\in T}w_{j}}.\]
By (\ref{ga34}), we have $p^{S}_{i}\geq 0$. Moreover, by some combinatorial calculations, we get
\[\sum_{\{S:i\in S,S\subseteq N\}}p^{S}_{i}=1.\]
Hence, for each fixed $i\in N$, $\{p^{S}_{i}:i\in S,S\subseteq N\}$ is a probability distribution over the collection of coalitions containing $i$. Since $\{S:i\in S,S\subseteq N\}$ is a proper subset of $2^{N}$, it is clear that the probability distribution $\{p^{S}_{i}:i\in S,S\subseteq N\}$ is a conditional probability distribution of some probability distribution over $2^{N}$, say $\{p^{S}:S\subseteq N\}$. The closed form for the probability distribution $\{p^{S}:S\subseteq N\}$ is given below (Hsiao \cite[p.413-414]{hsi02}).

Theorem. (Hsiao \cite[p.415]{hsi02}). For each $S\subseteq N$ with $S\neq\emptyset$, the probability distribution $\{p^{S}:S\subseteq N\}$ has the following closed form:
\begin{equation}{\label{hsi02eq4}}\tag{4}
p^{S}=\sum_{\{T:T\subseteq N\setminus S\}}(-1)^{|T|}\cdot\frac{1}
{\sum_{j\in S}w_{j}+\sum_{j\in T}w_{j}}\cdot\frac{1-p^{\emptyset}}
{\sum_{\{K:K\subseteq N,K\neq\emptyset\}}(-1)^{|K|+1}\cdot\frac{1}{\sum_{j\in K}w_{j}}}. \sharp
\end{equation}

Formula (\ref{hsi02eq4}) simply reflects the relationship between $p^{S}$ and $p^{\emptyset}$. We do not assume that $p^{\emptyset}$ is known. However, in the real world, if no player definitely wants to play the game, we will not study the game. Therefore, we may assume $p^{\emptyset}\neq 0$. Hsiao \cite[p.416]{hsi02}).

An Axiomatic Characterization of the Weighted Shapley Value.

Given a cooperative game $(N,v)$, a coalition $S$ is said to be a coalition of partners in the game $(N,v)$ if and only if, for each $T\subset S$ with $T\neq S$ and each $U\subseteq N\setminus S$, we have $v(U\cup T)=v(U)$. We consider the solution function
$\boldsymbol{\phi}:\mbox{TU}(N)\rightarrow\mathbb{R}^{N}$, which associates to every cooperative game $(N,v)\in \mbox{TU}(N)$ a vector $\boldsymbol{\phi}=(\phi_{i}(v))_{i\in N}$. The real number $\phi_{i}(v)$ represents the payoff to player $i$ in the cooperative game $(N,v)$. We consider the following axioms imposed on $\boldsymbol{\phi}$.

(KS1: Efficiency). $\sum_{i\in N}\phi_{i}(v)=v(N)$;
(KS2: Aditivity). $\boldsymbol{\phi}(v_{1}+v_{2})=\boldsymbol{\phi}(v_{1})+\boldsymbol{\phi}(v_{2})$;
(KS3: Positivity). If $(N,v)$ is monotonic, then $\boldsymbol{\phi}(v)\geq {\bf 0}$;
(KS4: Dummy player). If $i$ is a null player of the cooperative game $(N,v)$ defined in (\ref{ga37}), then $\phi_{i}(v)=0$;
(KS5: Partnership). If $S$ is a coalition of partners in the cooperative game $(N,v)$, then $\phi_{i}(v)=\phi_{i}(\sum_{i\in S}\phi_{i}(v)\cdot u_{S})$ for each $i\in S$.

Axioms (KS1)-(KS4) are standard in various aximatizations of the Shapley value. Now we examine axiom (KS5). A coalition of partners $S$ in the cooperative game $(N,v)$ behaves in a certain sense like one individual in the game $(N,v)$, since all its subcoalitions are completely powerless. In this sense, $S$ behaves internally the same in $(N,v)$ as in $u_{S}$. One can expect that $S$ will take its share in the game $(N,v)$ as one individual and then bargain over this share. We also see that $\sum_{i\in S}\phi_{i}(v)\cdot u_{S}$ is an unanimity game in which the members of $S$ bargain over $\sum_{i\in S}\phi_{i}(v)$ which is what they received together in $\boldsymbol{\phi}(v)$. It shows that $\phi_{i}(\sum_{i\in S}\phi_{i}(v)\cdot u_{S})$ is what player $i$ receives as a result of this bargaining. This should be exactly what he received in $(N,v)$ (Kalai and Samet \cite[p.211]{kal87}).

Theorem. (Kalai and Samet \cite[p.211]{kal87}). A solution function $\boldsymbol{\phi}$ satisfies axioms (KS1)–(KS5) if and only if there exists a weight system $\omega$ such that $\boldsymbol{\phi}$ is the weighted Shapley value $\boldsymbol{\phi}^{\omega}$. $\sharp$

The family of all weighted Shaply value $\boldsymbol{\phi}^{\omega}$ for simple weight system $\omega$ can also be characterized by slightly changing the positivity axiom. We replace now axiom (KS3) by the following one

(KS3′: Positivity). If $(N,v)$ is monotonic and there are no dummy players in $(N,v)$, then $\boldsymbol{\phi}(v)>{\bf 0}$.

Theorem. (Kalai and Samet \cite[p.213]{kal87}). A solution function $\boldsymbol{\phi}$ satisfies axioms (KS1), (KS2), (KS3′), (KS4) and (KS5) if and only if there exists a simple weight system $\omega =({\bf w},(N))$ such that $\boldsymbol{\phi}= \boldsymbol{\phi}^{\omega}$. $\sharp$

In the next result, we show that the weighted Sahpley values can be approximated by simple weighted Shapley values.

Theorem. (Kalai and Samet \cite[p.214]{kal87}). For each weight system $\omega =({\bf w},(S_{1},\cdots ,S_{m}))$, there exists a sequence of simple weight systems $\omega_{t}=({\bf w}_{t},(N))$ such that, for each cooperative game $(N,v)$, $\boldsymbol{\phi}^{{\omega}_{t}}(v) \rightarrow\boldsymbol{\phi}^{\omega}(v)$ when $t\rightarrow\infty$.

Proof. Let $0<\epsilon <1$ and, for each $t$, define $1\leq l\leq m$ and $i\in S_{t}$, $w_{i}^{(t)}=\epsilon^{t(m-l+1)}\cdot w_{i}$ and define $\omega_{t}=({\bf w}_{t},(N))$, where ${\bf w}_{t}=(w_{i}^{(t)})_{i\in N}$. It is easy to see that, for each $S$, $\boldsymbol{\phi}^{{\omega}_{t}}(u_{S})\rightarrow\boldsymbol{\phi}^{\omega}(u_{S})$ and since $\boldsymbol{\phi}^{\omega}$ and $\boldsymbol{\phi}^{{\omega}_{t}}$ are linear, we have $\boldsymbol{\phi}^{{\omega}_{t}}(v)\rightarrow\boldsymbol{\phi}^{\omega}(v)$ when $t\rightarrow\infty$ for each $(N,v)$. $\blacksquare$

We are going to present new axiomatic characterizations of the (symmetric) Shapley value and of weighted Shapley values without imposing the additivity axiom. The main axiom is coalitional strategic equivalence. It requires that an improvement of technology performed by some coalition would not affect the payoffs of the players that do not belong to the coalition (Chun \cite[p.183-184]{chu91}).

Given a cooperative game $(N,v)$, for $\pi =(i_{1},\cdots ,i_{|N|})\in \Pi (N)$ and ${\bf w}\in\mathbb{R}_{++}^{N}$, we define
\[\mathbb{P}_{\bf w}(\pi )=\prod_{k=1}^{n}\frac{w_{i_{k}}}{\sum_{t=1}^{k}w_{i_{t}}}.\]
According to Theorem \ref{kal87t200}, the weighted Shapley value is defined below. Given ${\bf w}\in\mathbb{R}_{++}^{N}$ satisfying $\sum_{i\in N}w_{i}=1$, the weighted Shapley value with weights ${\bf w}$, denoted by $\boldsymbol{\phi}^{\bf w}$, is defined by
\begin{equation}{\label{ga42}}\tag{5}
\phi_{i}^{\bf w}(v)=\sum_{\pi\in \Pi (N)}\mathbb{P}_{\bf w}(\pi )\cdot m_{i}(v,\pi )
\end{equation}
for all $v\in \mbox{TU}(N)$ and all $i\in N$. We see that if $w_{i}=1/n$ for all $i$, then $\boldsymbol{\phi}^{\bf w}$ is the (symmetric) Shapley value $\boldsymbol{\phi}$. Weighted Shapley values attempt to define a fair way of dividing up the worth of the grand coalition by assigning to each player a weighted average of the marginal contributions he/she makes to all possible coalitions, with weights determined by the weights vector ${\bf w}$ and the size of coalitions (Chun \cite[p.184]{chu91}).

Let $\boldsymbol{\phi}$ be a solution function defined on $\mbox{TU}(N)$. Now we introduce the following axioms.

(C1: fficiency). $\sum_{i\in N}\phi_{i}(v)=v(N)$;
(C2: Positivity). If $v$ is monotonic and there are no null players defined in (\ref{ga38}), then $\boldsymbol{\phi}>{\bf 0}$. Alternatively, we could state the axiom as: for all $i\in N$, if $\Delta_{i}v>0$, then $\phi_{i}(v)>0$;
(C3: Homogeneity). For all $\alpha\in\mathbb{R}$, we have $\boldsymbol{\phi}(\alpha v)=\alpha\boldsymbol{\phi}(v)$;
(C4: Partnership). For all S$\subseteq N$, if $S$ is a partnership in $v$, then $\phi_{i}(v)=\phi_{i}(\sum_{i\in S}\phi_{i}(v)\cdot\cdot u_{S})$ for all $i\in S$, where $u_{S}$ is the unanimity game on $S$;
(C5: Coalitional Strategic Equivalence). For all $T\subseteq N$ and for all $\alpha\in\mathbb{R}$, if $v_{2}=v_{1}+\alpha u_{T}$, then $\boldsymbol{\phi}(v_{1})=\boldsymbol{\phi}(v_{2})$ for all $i\in N\setminus T$.

Efficiency, positivity and homogeneity are standard axioms used in various axiomatic studies. Partnership, introduced above, requires that if sub-coalitions of $S$ are powerless, then it should not make any difference whether players of $S$ reveive their individual payoffs in $v$, or they altogether receive their group payoff in $v$ and then determine their individual payoffs later. Coaltional strategic equivalence
requires that adding a constant to the worths of all coalitions containing a given coalition $T$ does not affect the payoffs of the players that do not belong to the coalition $T$ (Chun \cite[p.185-186]{chu91}).

\begin{equation}{\label{chur1}}\tag{6}\mbox{}\end{equation}

Remark \ref{chur1}. (Chun \cite[p.186]{chu91}) Kalai and Samet \cite{kal87} characterized weighted Shapley values by using the five axioms: efficiency, positivity, partnership, dummy and additivity. As shown by them, positivity and additivity together imply homogeneity. Also, as shown by Chun \cite[Lemma 1]{chu89}, dummy and additivity toghther imply coalitional strategic equivalence. Altogether, we conclude that weighted Shapley values satisfy the five axioms mentioned above. $\sharp$

\begin{equation}{\label{chu91t1}}\tag{7}\mbox{}\end{equation}

Theorem \ref{chu91t1}. (Chun \cite[p.186]{chu91}). A solution function $\boldsymbol{\phi}$ satisfies axioms (C1)–(C5) if and only if it is a weighted Shapley value.

Proof. From Remark \ref{chur1}, we only need to show that if a value satisfies the five axioms, then it is a weighted Shapley value. $\blacksquare$

Remark. (Chun \cite[p.188]{chu91}) As shown by Chun \cite[Lemma 2]{chu89}, coalitional strategic equivalence is implied by Young’s \cite{you85} marginality (i.e., for all $i\in N$, if $\Delta_{i}v_{1}=\Delta_{i}v_{2}$, then $\phi_{i}(v_{1})=\phi_{i}(v_{2})$). Since the weighted Shapley values satisfy marginaliy, coalitional strategic equivalence in Theorem \ref{chu91t1} can be replaced by marginality. On the other hand, the symmetric Shapley value is characterized by using efficiency, triviality, coalitonal strategic equivalence, and fair ranking. By weakening fair ranking to weighted fair ranking (i.e., there exists ${\bf w}\in\mathbb{R}_{++}^{N}$ such that, for all $T\subseteq N$, if $v_{1}(S)=v_{2}(S)$ for all $S\neq T$, then $w_{j}\phi_{i}(v_{1})>w_{i}\phi_{j}(v_{1})$ implies $w_{j}\phi_{i}(v_{2})>w_{i}\phi_{j}(v_{2})$ for all $i,j\in T$), it is possible to characterize weighted Shapley values. The proof is similar to that of Chun \cite{chu89}. $\sharp$

Next we investigate whether the Shapley value could be singled out by imposing an additional axiom.

(C6: Symmetry). For all $i\in N$ and for all permutations $\pi$ of $N$, we have $\phi_{\pi (i)}(\pi v)=\phi_{i}(v)$.

Symmetry requires that a value be a symmetrical function of players. Although it is clear that the only weighted Shapley value satisfying axiom (C6) is the (symmetric) Shapley value, it is remarkable that the Shapley value can be characterized as follows (Chun \cite[p.188]{chu91}).

\begin{equation}{\label{chu91t2}}\tag{8}\mbox{}\end{equation}

Theorem \ref{chu91t2}. (Chun \cite[p.188]{chu91}). A solution function $\boldsymbol{\phi}$ satisfies efficiency, symmetry and coalitional strategic equivalence if and only if it is the Shapley value.

Proof. The proof can be obtained by mimicking the proof of Young \cite{you85}. $\blacksquare$

Note that, in the original characterization of the Shapley value, Shapley uses four axioms, efficiency, symmetry, dummy and additivity, whereas Young \cite{you85} uses three axioms, efficiency, symmetry and marginality. As shown by Chun \cite{chu89}, coalitional strategic equivalence is implied by dummy and additivity, and also by marginality. Consequently, Theorem \ref{chu91t2} provides the missing link between these two existing results (Chun \cite[p.188]{chu91}).

We consider another axiomatization of the weighted Shapley values. With a given partition $\Sigma =\{S_{1},\cdots ,S_{m}\}$ of $N$, we associate three relations: $=_{\Sigma}$, $<_{\Sigma}$, and $\leq_{\Sigma}$.

For $i,j\in N$, we put $i=_{\Sigma}j$ if and only if $i,j\in S_{k}$ for some $k$ with $S_{k}\in\Sigma$.
We write $i<_{\Sigma}j$ when, for some $k,d\in N$, $k<d$, $i\in S_{k}$, and $j\in S_{d}$. Also we shall write $i>_{\Sigma}j$ if and only if $j<_{\Sigma}i$.
The relation $\leq_{\Sigma}$ is written when $=_{\Sigma}$ or $<_{\Sigma}$ holds. Also we shall write $i\geq_{\Sigma}j$ if and only if $j\geq_{\Sigma}i$.
f $i\in N$ and $T$ is a nonempty set of players, then $i\geq_{\Sigma}T$ (resp. $i<_{\Sigma}T$) means that $i\geq_{\Sigma}j$ (resp. $i<_{\Sigma}j$) for all $j\in T$.

For any nonempty coalition $T$ in $N$, we define
\[\mbox{Max}_{\Sigma}(T)=\{i\in T:i\geq_{\Sigma}T\}\]
and
\[\Pi_{\Sigma}(T)=\left\{(i_{1},\cdots ,i_{|T|}):i_{k}\in T\mbox{ for each $k$ and }
i_{1}\leq_{\Sigma}i_{2}\leq_{\Sigma}\cdots\leq_{\Sigma}i_{|T|}\right\}.\]
Therefore $\mbox{Max}_{\Sigma}(T)$ is the set of maximal elements in $T$ with respect to the relation $\leq_{\Sigma}$ while $\Pi_{\Sigma}(T)$ is the set of all such orderings of the players in the coalition $T$ which are “nondecreasing” with respect to the same relation (Nowak and Radzik \cite[p.391]{now95}).

For a solution function $\boldsymbol{\phi}$, taking into account the weight system $\omega =({\bf w},\Sigma )$, we now require $\boldsymbol{\phi}$ to satisfy the following axioms.

(NR1: Efficiency). For any game $v$, we have $\sum_{i\in N}\phi_{i}(v)=v(N)$.
(NR2: Linearity). The solution function $\boldsymbol{\phi}$ is linear.
(NR3: Null Player). If $i$ is a null player defined in (\ref{ga39}), then $\phi_{i}(v)=0$;
(NR4: $\omega$-Mutual Dependence). let $v$ be a game and $i$ and $j$ be two different players. If $v(S\cup\{i\})=v(S)=v(S\cup\{j\})$ for all $S\subset N\setminus\{i,j\}$, then we call the players $i$ and $j$ mutually dependent and assume that
\begin{equation}{\label{noweq21}}\tag{9}
\phi_{i}(v)/w_{i}=\left\{\begin{array}{ll}
\phi_{j}(v)/w_{j} & \mbox{if $i=_{\Sigma}j$}\\
0 & \mbox{if $i<_{\Sigma}j$}.
\end{array}\right .
\end{equation}

If players $i$ and $j$ are mutually dependent, then $i$ becomes a null player defined in (\ref{ga39}) when $j$ is excluded from the game. The same concerns $j$ if $i$ is out of the game. Axiom (NR4) says that if players $i$ and $j$ belong to the same social class in $N$,
then their values (payoffs) are in the same proportion as the weights corresponding to them. If $i$ is a member of a lower class compared with the class of $j$, the mentioned allocation rule is not applied and $\phi_{i}(v)=0$. (Nowak and Radzik \cite[p.391-392]{now95}).

Given a coalition $S$ in $N$, we denote by $\Pi (S)$ the set of all permutations of players in $S$. Given a weight system $\omega =((w_{1},\cdots ,w_{n}),(S_{1},\cdots ,S_{m}))$, if, for $i\in S_{t}$ and every $S\subset N\setminus\{i\}$, the following condition is satisfied:
\begin{equation}{\label{noweq23}}\tag{10}
\left\{\begin{array}{ll}
\bigcup_{r=1}^{t-1}S_{r}\subseteq S\subset\bigcup_{r=1}^{t}S_{r}\mbox{ and }
S\neq\bigcup_{r=1}^{t}S_{r}, & \mbox{if $t\geq 2$}\\
S\subset S_{t}\mbox{ and }S\neq S_{t}, & \mbox{if $t=1$}
\end{array}\right .
\end{equation}
then we define the value $p_{i}^{S}$ as
\[p_{i}^{S}=\sum_{(k_{s+1},\cdots ,k_{u})\in\Pi (S_{t}\setminus S),k_{s+1}=i}
\left [\prod_{j=s+1}^{u}\frac{w_{k_{j}}}{\sum_{d=1}^{j}w_{k_{d}}}\right ],\]
where $s=|S|$, $u=s+|S_{t}\setminus S|$. If $S$ does not satisfy $(\ref{noweq23})$, we define $p_{i}^{S}=0$
(Nowak and Radzik \cite[p.392]{now95}).

\begin{equation}{\label{nowt21}}\tag{11}\mbox{}\end{equation}

Theorem \ref{nowt21}. (Nowak and Radzik \cite[p.392]{now95}). For every weight system $\omega=({\bf w},\Sigma )$, there exists a unique solution function $\boldsymbol{\phi}$ satisfying axioms (NR1)–(NR4). Moreover, it can be represented the following form
\begin{equation}{\label{noweq22}}\tag{12}
\phi_{i}(v)=\sum_{\{S:S\subset N\setminus\{i\}\}}p_{i}^{S}\cdot (v(S\cup\{i\})-v(S)). \sharp
\end{equation}

We now discuss some special cases of the above result

Corollary. (Nowak and Radzik \cite[p.393]{now95}). Given a weight system $\omega=({\bf w},\Sigma )$, for any unanimity game $u_{S}$ with $\emptyset\neq S\subseteq N$, the solution function $\boldsymbol{\phi}$ determined in Theorem \ref{nowt21} gives
\[\phi_{i}(u_{S})=\left\{\begin{array}{ll}
w_{i}/\sum_{k\in\mbox{\em Max}_{\Sigma}(S)}w_{k}, &
\mbox{if $i\in\mbox{\em Max}_{\Sigma}(S)$}\\ 0, & \mbox{otherwise}
\end{array}\right .\]

The interpretation of this corollary is clear. Only the players from the highest class $\mbox{Max}_{\Sigma}(S)$ in $S$ get positive payoffs in $u_{S}$ and they split one unit proportionally to their weights. If $\Sigma =\{N\}$ and ${\bf w}=(\delta ,\cdots ,\delta )$ for some $\delta >0$, then the corresponding solution function $\boldsymbol{\psi}$ is the (symmetric) Shapley value (Nowak and Radzik \cite[p.393]{now95}).

Corollary. (Nowak and Radzik \cite[p.393]{now95}). For every weight system $\omega =({\bf w},\Sigma )$ with $\Sigma =\{\{1\},\{2\},\cdots ,\{n\}\}$, there exists a unique solution function $\boldsymbol{\phi}$ satisfying the axioms (NR1)–(NR4), and it is of the form
\[\phi_{i}(v)=v(\{1,2,\cdots ,i\})-v(\{1,2,\cdots ,i-1\})\]
for $i\geq 2$ and $\phi_{1}(v)=v(\{1\})$. $\sharp$

Corollary. (Nowak and Radzik \cite[p.393]{now95}). For every weight system $\omega =({\bf w},\Sigma )$ with ${\bf w}=(\delta ,\cdots ,\delta )$ for some $\delta >0$, there exists a unique solution function $\boldsymbol{\phi}$ satisfying the axioms {(NR1)– (NR4), and it is of the form $(\ref{noweq22})$, where, for $i\in S_{t}$ and $S\subset N\setminus\{i\}$ such that $(\ref{noweq23})$ holds, we have
\[p_{i}^{S}=\frac{1/|S_{t}\setminus S|}{\left (\begin{array}{c}
|S_{t}|\\ |S_{t}\cap S|\end{array}\right )}\]
and $p_{i}^{S}=0$ for any other coalition $S$. $\sharp$

If the solution function $\boldsymbol{\phi}$ is given in (\ref{noweq22}) having the efficiency axiom (NR1), then for every player $i\in N$, we have
\begin{equation}{\label{noweq25}}\tag{13}
\sum_{\{S:S\subset N\setminus\{i\}\}}p_{i}^{S}=1.
\end{equation}
The weighted Shapley value $\boldsymbol{\phi}$ determined in Theorem \ref{nowt21} satisfies the efficient axiom (NR1) and the constants $p_{i}^{S}$ given there are nonnegative. Because (\ref{noweq25}) also takes place for any player $i$, $\boldsymbol{\phi}$ is a probabilistic value. Its interpretation is natural. Let $i\in N$ view his participation in a cooperative game $(N,v)$ as merely joining a coalition $S$, and then receiving as a reward his marginal contribution $v(S\cup\{i\})-v(S)$ to the coalition $S$. If $p_{i}^{S}$ is interpreted as a chance that player $i$ joins $S$, then $\phi_{i}(v)$ is his/her expected payoff from the game. We consider the random order value $\widehat{\boldsymbol{\phi}}$ defined in (\ref{ga42}). One can consider $\mathbb{P}(\pi )$ as follows
\[\mathbb{P}(\pi )=\left\{\begin{array}{ll}
\prod_{j=1}^{n}\left [w_{i_{j}}/
\sum_{d\in\mbox{\scriptsize Max}_{\Sigma}(\{i_{1},\cdots ,i_{j}\})}w_{d}\right ], &
\mbox{if $\pi\in \Pi (N)$}\\
0, & \mbox{otherwise},
\end{array}\right .\]
where $\pi =(i_{1},\cdots ,i_{|N|})$ (Nowak and Radzik \cite[p.394]{now95}).

The random order solution function $\widehat{\boldsymbol{\phi}}$ induced by $\{p_{\pi}\}$ described above was introduced by Kalai and Samet \cite{kal87} as an equivalent definition of the weighted Shapley value for a given weight system by referring to Theorem \ref{kal87t200}. The value $\boldsymbol{\phi}$ determined axiomatically in Theorem \ref{nowt21} coincides with $\widehat{\boldsymbol{\phi}}$. (It follows quickly from the simple fact that $\boldsymbol{\phi}$ and $\widehat{\boldsymbol{\phi}}$ are equal for unanimity games.) Thus, the axioms (NR1)–(NR4), and especially the $\omega$-mutual dependence axiom (NR4), provide a new look at the random order definition of the weighted Shapley value (Nowak and Radzik \cite[p.395]{now95}).

A natural question arises as to whether the weighted Shapley value can be determined if the classical symmetry or equal treatment axioms (ref. Appendix A in Aumann and Shapley \cite{aum74b}) are somehow modified to take into account the given weight system $\omega$. Let $\boldsymbol{\phi}$ be a solution function defined on $\mbox{TU}(N)$. Two natural counterparts of those classical axioms can be formulated as follows.

($\widehat{\mbox{NR}}4$: $\omega$-Symmetry). For any permutation $\pi$ of $N$ and for every $i\in N$,
\[\phi_{i}(\pi v)/w_{i}=\left\{\begin{array}{ll}
\phi_{\pi (i)}(v)/w_{\pi (i)} & \mbox{if $i=_{\Sigma}\pi (i)$}\\
0 & \mbox{if $i<_{\Sigma}\pi (i)$},
\end{array}\right .\]
where $\pi v$ is defined by the equality, $\pi v(S)=v(\pi (S))$ for all $S\subseteq N$.
($\widetilde{\mbox{NR}}4$: $\omega$-Equal Treatment). Let $i,j\in N$ and $i\neq j$. If $v(S\cup\{i\})=v(S\cup\{j\})$ for all $S\subseteq N\setminus\{i,j\}$, then (\ref{noweq21}) holds

We will show that considering Axioms $\widehat{\mbox{NR}}4$ or $\widetilde{\mbox{NR}}4$ is not the right approach to the weighted Shapley value. (Nowak and Radzik \cite[p.395]{now95}).

Theorem. (Nowak and Radzik \cite[p.395]{now95}). Let $\omega =({\bf w},\Sigma )$ be a weight system, where ${\bf w}\neq (\delta ,\cdots ,\delta )$ for any $\delta >0$ or $\Sigma\neq\{N\}$. Then there is no solution function $\boldsymbol{\phi}$ satisfying the efficiency axiom (NR1), linearity axiom (NR2), null player axiom (NR3), and $\omega$-symmetry axiom (\widehat{\mbox{NR}}4) or $\omega$-equal treatment axiom (\widetilde{\mbox{NR}}4). $\sharp$

As noted by Young \cite{you85}, the Shapley value can also be determined when two classical axioms, the “additivity” and “null player”, are replaced with some very natural postulate saying that the value of a player depends on his marginal contributions. We now state a similar result for the weighted Shapley values. For this, we formulate an axiom closely corresponding to the “strong monotonicity” postulate in Young \cite{you85}. Let $\boldsymbol{\phi}$ be a solution function defined on $\mbox{TU}(N)$. We consider the following axiom.

(NR5: Marginal Contribution). Let $(N,v_{1}),(N,v_{2})\in \mbox{TU}(N)$. If, for some player $i\in N$, we have $v_{1}(S\cup\{i\})-v_{1}(S)=v_{2}(S\cup\{i\})-v_{2}(S)$ for all $S\subseteq N\setminus\{i\}$, then $\phi_{i}(v_{1})=\phi_{i}(v_{2})$.

We now drop the linearity and null player axioms and state the following extension of Theorem 2 in Young \cite{you85} (Nowak and Radzik \cite[p.395]{now95}).

Theorem. (Nowak and Radzik \cite[p.396]{now95}). Let $\omega =({\bf w},\Sigma )$ be a weight system. A solution function $\boldsymbol{\psi}$ satisfies the efficiency axiom (NR1), $\omega$-mutual dependence axiom (NR4), and marginal
contributions axiom (NR5) if and only if $\boldsymbol{\psi}$ is the weighted Shapley value defermined (for given $\omega$) in Theorem \ref{nowt21}. $\sharp$

We now drop the assumption that the weight system $\omega$ is given exogenously and consider the class $\Omega$ of all weighted Shapley values. Kalai and Samet \cite{kal87} first characterized the class $\Omega$ axiomatically using among other things the additivity, null player axioms, and a postulate called “partnership consistency”. We provide a new axiomatization of $\Omega$, which is different from that of Kalai and Samet \cite{kal87} in two main respects. We do not impose the additivity and accept the marginal contributions axiom of Young \cite{you85} and, in contrast to the partnership consistency axiom of Kalai and Samet \cite{kal87}, we do not employ unanimity games in the approach. Let $\boldsymbol{\phi}$ be a solution function defined on $\mbox{TU}(N)$. We consider the following axioms.

(NR6: Positivity). If $v$ is monotonic, i.e., $v(T)\geq v(S)$ for each $S\subseteq T$, then $\phi_{i}(v)\geq 0$ for every $i\in N$.
(NR7: Mutual Dependence). Assume that $(N,v_{1}),(N,v_{2})\in \mbox{TU}(N)$ and $i,j\in N$ with $i\neq j$, are mutually dependent players. Then $\phi_{i}(v_{1})\phi_{j}(v_{2})=\phi_{j}(v_{1})\phi_{i}(v_{2})$.

When $\phi_{j}(v_{1}),\phi_{j}(v_{2})\neq 0$, axiom (NR7) says that if $i$ and $j$ are mutually dependent in both games $(N,v_{1})$ and $(N,v_{2})$, then the proportion of their payoffs $\phi_{i}(v_{1})/\phi_{j}(v_{1})$ and $\phi_{i}(v_{2})/\phi_{j}(v_{2})$ in these two games are equal (Nowak and Radzik \cite[p.396]{now95}).

\begin{equation}{\label{nowt24}}\tag{14}\mbox{}\end{equation}

Theorem \ref. {nowt24}. (Nowak and Radzik \cite[p.396]{now95}). A solution function $\boldsymbol{\phi}$ satisfies the efficiency axiom (NR1), marginal contributions axiom (NR5), positivity axiom (NR6), and mutual dependence axiom (NR7) if and only if there
exists a weight system $\omega =({\bf w},\Sigma )$ such that $\boldsymbol{\phi}$ is equal to the weighted Shapley value associated with $\omega$. $\sharp$

Theorem \ref{nowt24} remains true if we replace the marginal contributions axiom (NR5) by the linearity axiom (NR2) and null player axiom (NR3) (Nowak and Radzik \cite[p.396]{now95}).

Weighted Shapley Value of Dual Game.

The dual game of a cooperative game $(N,v)$ is denoted by $(N,v^{*})$ and is defined by $v^{*}(S)=v(N)-v(N\setminus S)$ for each $S\subseteq N$. The transformation $v\mapsto v^{*}$ is a one-to-one map from $\mbox{TU}(N)$ onto itself. In particular, the family $\{u_{S}^{*}\}_{S\subseteq N}$ is a basis for $\mbox{TU}(N)$, where
\[u_{S}^{*}(T)=\left\{\begin{array}{ll}
1 & \mbox{if $T\cap S\neq\emptyset$}\\ 0 & \mbox{if $T\cap S=\emptyset$}.
\end{array}\right .\]
The game $u_{S}^{*}$ is called as the representation game of the coalition $S$. The game $u_{S}^{*}$ has a natural interpretation as a cost-game in which $u_{S}^{*}(T)$ is the cost incurred by $T$. The presence of any number of members of $S$ in $T$ incurs a unit cost (Kalai and Samet \cite[p.214]{kal87}).

For a weight system $\omega =({\bf w},(S_{1},\cdots ,S_{m}))$, we define a linear function $\widehat{\boldsymbol{\phi}}^{\omega}:\mbox{TU}(N)\rightarrow\mathbb{R}^{N}$ on the basis $\{u_{S}^{*}\}_{S\subseteq N}$ as follows. For a given coalition $S$, we denote $k=\max\{j:S_{j}\cap S\neq\emptyset\}$ and let $\bar{S}=S\cap S_{k}$. Then, we define
\[\widehat{\phi}^{\omega}_{i}(u_{S}^{*})=\left\{\begin{array}{ll}
{\displaystyle \frac{w_{i}}{\sum_{j\in\bar{S}}w_{j}}}, & \mbox{if $i\in\bar{S}$}\\
0, & \mbox{if $i\not\in\bar{S}$}.\end{array}\right .\]
An equivalent random permutation approach is defined for $\widehat{\boldsymbol{\phi}}^{\omega}$. For a permutation $\pi$, we denote by $\pi^{*}$ the reverse permutation. For a given probability distribution $\mathbb{P}$ over $\Pi (N)$, we define $\mathbb{P}^{*}$ by $\mathbb{P}^{*}(\pi )=\mathbb{P}(\pi^{*})$ (Kalai and Samet \cite[p.215]{kal87}).

Theorem. (Kalai and Samet \cite[p.215]{kal87}). For each cooperative game $(N,v)$, player $i$, and weight system $\omega$, we have
$\widehat{\phi}^{\omega}_{i}(v)=\mathbb{E}_{\mathbb{P}_{\omega}^{*}}\left [m_{i}(v,\cdot)\right ]$. $\sharp$

Theorem. (Kalai and Samet \cite[p.215]{kal87}). For each cooperative game $(N,v)$ and weight system $\omega$, we have $\widehat{\boldsymbol{\phi}}^{\omega}(v)=\boldsymbol{\phi}^{\omega}(v^{*})$.

Proof. Consider the game $v=u_{S}^{*}$. Then $v^{*}=(u_{S}^{*})^{*}=u_{S}$. By definition of $\boldsymbol{\phi}^{\omega}$ and
$\widehat{\boldsymbol{\phi}}^{\omega}$, we see that $latex \widehat{\boldsymbol{\phi}}^{\omega}(v)=\boldsymbol{\phi}^{\omega}
(v^{*})$, i.e., $\widehat{\boldsymbol{\phi}}^{\omega}(u_{S}^{*})=\boldsymbol{\phi}^{\omega}(u_{S})$. Now let $v=\sum_{S\subseteq N}c_{S}u_{S}^{*}$. Then
\[\widehat{\boldsymbol{\phi}}^{\omega}(v)=\sum_{S\subseteq N}c_{S}
\widehat{\boldsymbol{\phi}}^{\omega}(u_{S}^{*})=\sum_{S\subseteq N}
c_{S}\boldsymbol{\phi}^{\omega}(u_{S})=\boldsymbol{\phi}^{\omega}
\left (\sum_{S\subseteq N}c_{S}u_{S}\right )=\boldsymbol{\phi}^{\omega}(v^{*}).\]
This completes the proof. $\blacksquare$

Part of the reasoning of the partnership axiom (KS5) is that a coalition of partners can be treated in a certain sense as one individual. We call a coalition of partners as a $p$-type coalition. An axiomatic characterization of the family $\{\widehat{\boldsymbol{\phi}}^{\omega}\}$ is obtained by changing axiom (KS5). We say that a coalition $S$ is of $p^{*}$-type in the cooperative game $(N,v)$ if and only if, for each $U\supseteq S$ and $T\subseteq S$, $T\neq S$, $v(U\setminus T)=v(U)$. As in the case of $p$-type cialitions, a $p^{*}$-type coalition can be considered as one individual represented by several players. But in the $p^{*}$-type case, any nonempty sub-coalition of players has the same effect on the cost as the coalition of all players, while in the $p$-type case, all the proper sub-coalitions of players are powerless. A $p^{*}$-type coalition is called as a coalition of representatives. Let us introduce axiom (KS5′).

(KS5′). If $S$ is a $p^{*}$-type coalition in $(N,v)$, then $\phi_{i}(v)=\phi_{i}(\sum_{i\in S}\phi_{i}(v)\cdot u_{S}^{*})$ for each $i\in S$.

Axiom (KS5′) is analogous to axiom (KS5). It requires that if $S$ is a $p^{*}$-type coalition in the cooperative game $(N,v)$, then the cost shared by each one of its members can be computed by letting the players in $S$ bargain over the splitting of the total cost shared by $S$ in $\widehat{\boldsymbol{\phi}}^{\omega}(v)$. Clearly by the nature of the $p^{*}$-type coalition, this bargaining is represented by the game $u_{S}^{*}$ (Kalai and Samet \cite[p.215]{kal87}).

Theorem. (Kalai and Samet \cite[p.216]{kal87}). A solution function $\boldsymbol{\phi}$ satisfies axioms (KS1)–(KS4)} and (KS5′) if and only if there exists a weight system $\omega$ satisfying $\boldsymbol{\phi}=\widehat{\boldsymbol{\phi}}^{\omega}$. $\sharp$

\begin{equation}{\label{kal87t61}}\tag{15}\mbox{}\end{equation}

Theorem \ref{kal87t61}. (Kalai and Samet \cite[p.216]{kal87}). Let $|N|\geq 3$. If $\omega =({\bf w},(N))$ and $latex \omega^{\prime}=
({\bf w}’,(N))$ are two simple weight systems and $\widehat{\boldsymbol{\phi}}^{\omega}(v)=\boldsymbol{\phi}^{{\omega}^{\prime}}(v)$ for each game $(N,v)$, then both ${\bf w}$ and ${\bf w}’$ are multiples of the vector $(1,1,\cdots ,1)$ and thus both $\widehat{\boldsymbol{\phi}}^{\omega}(v)$ and $\boldsymbol{\phi}^{{\omega}^{\prime}}(v)$ are the Shapley values. $\sharp$

We can also obtain a characterization of the (symmetric) Shapley value, one which does not use the symmetry axiom.

\begin{equation}{\label{kal87t6}}\tag{16}\mbox{}\end{equation}

Theorem \ref{kal87t6}. (Kalai and Samet \cite[p.217]{kal87}). For $|N|\geq 3$, a solution function $\boldsymbol{\phi}$ satisfies axioms (KS1)–KS5) and (KS5′) if and only if it is the Shapley value. $\sharp$

For $|N|=2$, there exists a transformation $\omega\mapsto\omega^{*}$ of simple weight systems satisfying $\widehat{\boldsymbol{\phi}}^{\omega}=\boldsymbol{\phi}^{{\omega}^{*}}$. Indeed, it is easy to see that if, for $\omega =({\bf w},(N))$, we set $\omega^{*}=({\bf w}^{*},(N))$, where ${\bf w}^{*}=(w_{2},w_{1})$, then $latex \widehat{\boldsymbol{\phi}}^{\omega}=
\boldsymbol{\phi}^{{\omega}^{*}}$. We state now the extension of Theorem \ref{kal87t6} to general weight systems (Kalai and Samet \cite[p.217]{kal87}).

Theorem. (Kalai and Samet \cite[p.218]{kal87}). If $\omega =({\bf w},(S_{1},\cdots ,S_{m}))$ and $\omega^{\prime}=({\bf w}’,(T_{1},\cdots ,T_{k}))$ are weigh systems for which $\widehat{\boldsymbol{\phi}}^{\omega}=\boldsymbol{\phi}^{{\omega}^{\prime}}$, then we have the following properties:

$m=k$;
$S_{i}=T_{m+1-i}$ for $i=1,\cdots ,m$;
if $|S_{i}|>3$, then ${\bf w}^{S_{i}}$ and ${\bf w}_{m+1-i}^{\prime}$ are proportional to $(1,1,\cdots ,1)$;
f $|S_{i}|=2$, then ${\bf w}^{S_{i}}$ is proportional to ${\bf w}^{*}_{T_{m+1-i}}$. $\sharp$

Here, we show how a $p$-type coalition can be practically defined as one player, thereby reducing the size of the game. Let us fix a coalition $S_{0}$ with more than one player. Consider the set $\bar{N}$ which consists of all the players of $N$ except that all the players in $S_{0}$ are replaced by a single player denoted by $s$, i.e., $\bar{N}=(N\setminus S)\cup\{s\}$. For any game $(N,v)$, we define a game $(\bar{N},\bar{v})$ by
\[\bar{v}(S)=\left\{\begin{array}{ll}
v(S) & \mbox{if $s\not\in S$}\\ v((S\setminus\{s\})\cup S_{0}) &
\mbox{if $s\in S$}.\end{array}\right .\]
Let $\omega ({\bf w},(S_{1},\cdots ,S_{m}))$ be a weight system for $(N,v)$, and let $k=\max\{j:S_{j}\cap S_{0}\neq\emptyset\}$. The weight system $\bar{\omega}=(\bar{\bf w},(\bar{S}_{1},\cdots ,\bar{S}_{m}))$ for $(\bar{N},\bar{v})$ is defined as follows. For each $i\neq s$, $\bar{w}_{i}=w_{i}$ and $\bar{w}_{s}=\sum_{i\in S_{i}}w_{i}$. For each $j\neq k$, $\bar{S}_{j}=S_{j}\setminus S_{0}$ and $\bar{S}_{k}=(S_{k}\setminus S_{0})\cup\{s\}$ (Kalai and Samet \cite[p.219]{kal87}).

\begin{equation}{\label{kal87t9}}\tag{17}\mbox{}\end{equation}

Theorem \ref{kal87t9}. (Kalai and Samet \cite[p.220]{kal87}). If $S_{0}$ is a $p$-type coalition in the cooperative game $(N,v)$, then, for each $i\neq s$, $\phi^{\bar{\omega}}_{i}(\bar{v})=\phi^{\omega}_{i}(v)$ and $latex \phi^{\bar{\omega}}_{s}(\bar{v})=
\sum_{i\in S_{0}}\phi^{\omega}_{i}(v)$. Similarly, if $S_{0}$ is a $p^{*}$-type coalition in $(N,v)$, then, for each $i\neq s$, $\phi^{\bar{\omega}^{*}}_{i}(\bar{v})=\widehat{\phi}^{\omega}_{i}(v)$ and $\phi^{\bar{\omega}^{*}}_{s}(\bar{v})=\sum_{i\in S_{0}}\widehat{\phi}^{\omega}_{i}(v)$. $\sharp$

The following corollary follows from Theorem \ref{kal87t9}. It is important for applications in which the players themselves are, or are representing, groups of individuals. Such is the case for example when the players are parties, cities, or management boards. The use of the symmetric Shapley value seems to be unjustified in certain cases of this type, because the players represent constituencies of different sizes. A natural candidate for a solution is the weighted Shapley value where the players are weighted by the size of the constituences they stand for. The following corollary shows that such a procedure is justified in the two special cases described below (Kalai and Samet \cite[p.221]{kal87}).

Corollary. (Kalai and Samet \cite[p.221]{kal87}). Let $(N,v)$ be a game with $|N|=n$ in which each player $i$ is a set of individuals $M_{i}$ with $m_{i}$ members. Consider the set of individuals $\bar{N}=\bigcup_{i\in N}M_{i}$ and the cooperative games $(N,v_{1})$ and $(N,v_{2})$ defined on $\bar{N}$ as follows. For each $S\subseteq\bar{N}$, we defined
\[v_{1}(S)=v(\{i:M_{i}\subseteq S\})\mbox{ and }v_{2}(S)=v(\{i:M_{i}\cap S\neq\emptyset\}).\]
Let $\omega$ be the simple weight system $\omega =((m_{1},\cdots ,m_{n},(\bar{N}))$. Then, for each $i$, we have
\[\phi^{\omega}_{i}(v)=\sum_{k\in M_{i}}\phi_{k}(v_{1})\mbox{ and }
\widehat{\phi}^{\omega}_{i}(v)=\sum_{k\in M_{i}}\phi_{k}(v_{2}),\]
where $\boldsymbol{\phi}$ is the (symmetric) Shapley value. $\sharp$

Weighted Shapley Values and the Core.

The cores of convex games and market games with a continuum of players always contain the Shapley value. Here, the main result is that the set of all weighted Shapley value of a cooperative game contains the core of the game. Convex games have, in a sense, large cores, which “explains” why they contain the Shapley value. In the case of market games with a continuum of players, it is homogeneity of the games and the diagonal property of the Shapley value that guarantee this fact (Monderer et al. \cite[p.27]{mon}).

For $S\subseteq N$, we denote $S^{c}=N\setminus S$. If $S\subseteq T$, then ${\bf x}^{T|S}$ is the projection of ${\bf x}^{T}$ on $\mathbb{R}^{S}$. For ${\bf x},{\bf y}\in\mathbb{R}^{N}$, we write ${\bf x}\geq {\bf y}$ if $x_{i}\geq y_{i}$ for all $i\in N$, we write ${\bf x}>{\bf y}$ if ${\bf x}\geq {\bf y}$ and ${\bf x}\neq {\bf y}$, and we write ${\bf x}\gg {\bf y}$ if $x_{i}>y_{i}$ for all $i\in N$. For ${\bf x}\in \mathbb{R}^{N}$ and $S\subseteq N$, we write ${\bf x}(S)$ for $\sum_{i\in S}x_{i}$ and ${\bf x}(S)=0$ for $S=\emptyset$. For a finite set $U$, $\Delta (U)$ is the unit simplex in $\mathbb{R}^{U}$. We denote by $\mbox{int}(\Delta (U))$ the relative interior of $\Delta (U)$ (Monderer et al. \cite[p.29]{mon}).

The structure of the core of a convex game is described as follows. Let $\Sigma$ be the set of all partitions of $N$. Elements of $\Sigma$ are of the form $\sigma =(S_{1},\cdots ,S_{k})$ with $\bigcup_{h=1}^{k}S_{h}=N$ and $S_{i}\cap S_{j}=\emptyset$ for $i\neq j$. Let ${\cal C}(v)$ be the core of cooperative game $(N,v)$. For each $\sigma$ in $\Sigma$, we define
\[F_{\sigma}(v)=\left\{{\bf x}\in {\cal C}(v):{\bf x}\left (\bigcup_{j=1}^{h}S_{j}\right )=v\left (\bigcup_{j=1}^{h}S_{j}\right )
\mbox{ for all }1\leq h\leq k\right\}.\]
If $(N,v)$ is convex, then $F_{\sigma}(v)$ is a nonempty face of ${\cal C}(v)$ of dimension $n-k$ at most. In particular, for $\sigma$ with $k=n$, $F_{\sigma}(v)$ consists of one point. This point is a contribution vector, and ${\cal C}(v)$ is the convex hull of all contribution vectors. If $v$ is strictly convex, then $F_{\sigma}(v)$ is of dimemnsion $n-k$ and all faces $F_{\sigma}(v)$ are distinct. Finally, if the core ${\cal C}(v)$ contains all the contribution vectors of $v$, then $(N,v)$ is convex. If, in addition, all the contribution vectors are distinct, then the game is strictly convex (Monderer et al. \cite[p.30]{mon}).

For a vector ${\bf w}\in\mathbb{R}_{++}^{N}$, i.e., ${\bf w}\gg {\bf 0}$, the positively weighted value $\boldsymbol{\phi}^{\bf w}:\mbox{TU}(N)\rightarrow\mathbb{R}^{N}$ is the unique positive operator satisfying, for each unanimity game $u_{S}$,
\begin{equation}{\label{moneq31}}\tag{18}
\phi_{i}^{\bf w}(u_{S})=\left\{\begin{array}{ll}
w_{i}/{\bf w}(S), & \mbox{if $i\in S$}\\ 0, & \mbox{if $i\not\in S$}.
\end{array}\right .
\end{equation}
Since $\boldsymbol{\phi}^{\bf w}(\cdot)$ is linear, the expression (\ref{moneq31}) defines $\boldsymbol{\phi}^{\bf w}(v)$ for all $v\in \mbox{TU}(N)$. If $w_{i}=1/n$ for all $i\in N$, then $\boldsymbol{\phi}^{\bf w}$ is the Shapley value (Monderer et al. \cite[p.30]{mon}).

Owen \cite{owe72} has shown that the positively weighted Shapley values are random order valus. For a weight ${\bf w}$, we define $\mathbb{P}_{\bf w}$ as follows. Let $X_{i}$, for $i\in N$, be independent random variables distributed over $[0,1]$ such that, for every $0\leq t\leq 1$ and $i\in N$, $\bar{\mathbb{P}}(X_{i}\leq t)=t^{w_{i}}$. For $\pi\in\Pi (N)$, we define
\begin{equation}{\label{moneq32}}\tag{19}
\mathbb{P}_{\bf w}(\pi )=\bar{\mathbb{P}}(X_{i}<X_{j}\mbox{ for all }i<_{\pi}j,i,j\in N).
\end{equation}
Then, for every $v\in \mbox{TU}(N)$, $\boldsymbol{\phi}^{\bf w}(v)=\boldsymbol{\psi}^{v}(\mathbb{P}_{\bf w})$. Note that both $\boldsymbol{\psi}^{\bf w}(\cdot)$ and $\mathbb{P}_{(\cdot)}$ are positively homogeneous of degree one. Therefore, we can restrict the attention to vector ${\bf w}$ in $\mbox{int}(\Delta (N))$, that will be called weight vectors (Monderer et al. \cite[p.30]{mon}).

We generalize now the notion of a weight vector to enable some players to have zero weight. The value corresponding to the generalized weights will be called weighted values. These values are axiomatized by Kalai and Samet \cite{kal87}. The following consideration will lead to this generalization (Monderer et al. \cite[p.30]{mon}).

When zero-weight players are allowed, we cannot use directly (\ref{moneq31}) to define a value, since for $S$ which contains only zero-weight players (\ref{moneq31}) is not defined. We need therefore to assign secondary weights to the zero-weight players. These new secondary weights may themselves assign zeros for some of the players and we have to assign also weights to these doubly zero-weighted players, and so on. We are naturally lead to the following definition.

Definition. (Monderer et al. \cite[p.31]{mon}). A generalized weight vector is a $2k$-tupke, $1\leq k\leq n$, $(S_{1},\cdots ,S_{k},{\bf w}^{S_{1}},\cdots ,{\bf w}^{S_{k}})$ satisfying $(S_{1},\cdots ,S_{k})\in\Sigma$ and ${\bf w}^{S_{h}}\in\mbox{int}(\Delta (S_{h}))$ for $h=1,\cdots ,k$. $\sharp$

The interpretation of the generalized weight vectors is as follows. The players in $S_{k}$ are the non-zero weight players with weights given by ${\bf w}^{S_{k}}$, while the rest of the players are zero-weight players. Among the zero-weight players the “heaviest” are the members of $S_{k-1}$ with weights ${\bf w}^{S_{k-1}}$. All players in $\bigcup_{h\leq k-1}S_{h}$ are zero-weight players relative to players in $S_{k-1}$, etc. Note that every weight vector ${\bf w}$ can be naturally identified with the generalized weight vector $(N,{\bf w})$ (Monderer et al. \cite[p.31]{mon}).

Given the generalized weight vector $(S_{1},\cdots ,S_{k},w^{S_{1}},\cdots , w^{S_{k}})$, we can find the relative weights of players in each coalition $S$. For a given $S$, let $m=\max\{h:S_{h}\cap S\neq\emptyset\}$. Define ${\bf w}^{S}$ by
\[w_{i}^{S}=\left\{\begin{array}{ll}
\frac{w_{i}^{S_{m}}}{{\bf w}^{S_{m}}(S\cap S_{m})} &
\mbox{if $i\in S\cap S_{m}$}\\ 0 & \mbox{if $i\in S\setminus S_{m}$}.
\end{array}\right .\]
Thus $S\cap S_{m}$ consists of all the “heaviest” players in $S$. The relative weights of these players are determined by their weights in $S_{m}$. The rest of the players have zero weight in $S$ (Monderer et al. \cite[p.31]{mon}).

Denote ${\bf w}=({\bf w}^{S})_{S\subseteq N}$. Then ${\bf w}$ is a vector in $\prod_{S\subseteq N}\Delta (S)$ and it is easy to verify that is satisfies
\begin{equation}{\label{moneq41}}\tag{20}
w_{i}^{S}=w_{i}^{T}/{\bf w}^{T}(S)
\end{equation}
for each $i\in S\subseteq T$ such that ${\bf w}^{T}(S)>0$.

Definition. (Monderer et al. \cite[p.31]{mon}). A vector ${\bf w}\in\prod_{S\subseteq N}\Delta (S)$ which satisfies
(\ref{moneq41}) is called a {\bf weight system}. The set of all weight system is denoted by ${\cal W}$. $\sharp$

We saw that each generalized weight vector corresponds to a weight system. It is easy to see that this correspondence is one to one. We show now that it is also onto ${\cal W}$. Let ${\bf w}\in {\cal W}$ and define $\sigma ({\bf w})\in\Sigma$ as follows. Let $T_{1}=\{i\in N:w_{i}^{N}>0\}$ and, for $h\geq 2$, we define $T_{h}$ to be the set
\[\left\{i\in\left (\cup_{j=1}^{h-1}T_{j}\right )^{c}:w_{i}^{(\cup_{j=1}^{h-1}T_{j})^{c}}>0\right\}\]
when this set is nonempty. Let $T_{k}$ be the last nonempty set so defined, then clearly $(T_{1},\cdots ,T_{k})$ is an ordered partition. Now, for each $1\leq h\leq k$, let $S_{h}=T_{k-h+1}$ and $\sigma ({\bf w})=(S_{1},\cdots ,S_{k})$. It is easy to see that ${\bf w}$ is the weight system that corresponds to the generalized weight vector $(S_{1},\cdots ,S_{k},{\bf w}^{S_{1}},\cdots ,{\bf w}^{S_{k}})$ (Monderer et al. \cite[p.31]{mon}).

For a given $\sigma\in\Sigma$, we denote by ${\cal W}_{\sigma}$ the set of all weight systems ${\bf w}$ for which $\sigma ({\bf w})=\sigma$. Note that, for ${\bf w}\in {\cal W}$, we see that ${\bf w}\in {\cal W}_{(N)}$ if and only if ${\bf w}^{N}\gg {\bf 0}$ and that ${\cal W}_{(N)}$ is dense in ${\cal W}$ (Monderer et al. \cite[p.31]{mon}).

Definition. (Monderer et al. \cite[p.32]{mon}). For a given ${\bf w}\in {\cal W}$, we define the weighted value $\boldsymbol{\phi}^{\bf w}$ as the linear function $\boldsymbol{\phi}^{\bf w}:\mbox{TU}(N)\rightarrow\mathbb{R}^{N}$ which is defined, for each unanimity game $u_{S}$, by
\[\phi_{i}^{\bf w}(u_{S})=\left\{\begin{array}{ll}
w_{i}^{S} & \mbox{if $i\in S$}\\ 0 & \mbox{otherwise}.
\end{array}\right .\]
\end{Def}

Note that, for ${\bf w}\in {\cal W}_{(N)}$, the weighted value $\boldsymbol{\phi}^{\bf w}$ coincides with the positively weighted value $\boldsymbol{\phi}^{{\bf w}^{N}}$ (Monderer et al. \cite[p.32]{mon}).

Weighted value are also random order values. The probability distribution $\mathbb{P}_{\bf w}$ in $\Delta ({\cal R})$ which defines $\boldsymbol{\phi}^{\bf w}$ is described as follows.

Definition. (Monderer et al. \cite[p.32]{mon}). We say that an order $R$ of $N$ is {\bf consistent} with $\sigma=(S_{1},\cdots ,S_{k})$ in $\Sigma$ if, for each $1\leq h\leq k$, each player in $S_{h}$ precedes each player in $S_{h+1})$. $\sharp$

For each ${\bf w}\in {\cal W}$, we define a probability measure $\mathbb{P}_{\bf w}$ over all orders of $N$, the support of which is the set of all orders which are consistent with $\sigma ({\bf w})$. Now let $(S_{1},\cdots ,S_{k},{\bf w}^{S_{1}},\cdots ,{\bf w}^{S_{k}})$ be the generalized weight vector which corresponds to ${\bf w}$. Since ${\bf w}^{S_{h}}\in\mbox{int}(\Delta (S_{h}))$ for all $1\leq h\leq k$, we can define, for each such $h$, a probability distribution $\mathbb{P}_{{\bf w}^{S_{h}}}$ on the orders of $S_{h}$ in the same way $\mathbb{P}_{{\bf w}^{N}}$ was defined in (\ref{moneq32}). We define now
\[\mathbb{P}_{\bf w}(R)=\prod_{h=1}^{k}\mathbb{P}_{{\bf w}^{S_{h}}}(R_{h}),\]
where $R_{h}$ is the order on $S_{h}$ induced by $R$. Then $\mathbb{P}_{\bf w}$ is the probability distribution for which $\boldsymbol{\phi}^{\bf w}(v)=\boldsymbol{\phi}^{v}(\mathbb{P}_{\bf w})$ for eahc game $v$. Notice that in all the orders which are consistent with $\sigma ({\bf w})$, the non zero-weight players (those in $S_{k}$) are preceded by all other players, all the players in $S_{k-1}$ are preceded by the players in $\bigcup_{h\leq k-2}S_{h}$ etc. ((Monderer et al. \cite[p.32]{mon}).

\begin{equation}{\label{monta}}\tag{21}\mbox{}\end{equation}

Theorem \ref{monta}. (Monderer et al. \cite[p.32]{mon}). For every game $v$, each element in the core of $v$ is the weighted value
of $v$ for some weight system. That is, $C(v)\subseteq\boldsymbol{\phi}^{\cal W}(v)$. $\sharp$

Definition. (Monderer et al. \cite[p.32]{mon}). The anti-core $AC(v)$ of the game $v$ is defined to be the set of all ${\bf x}\in\mathbb{R}^{N}$ for which ${\bf x}(N)=v(N)$ and ${\bf x}(S)\leq v(S)$ for all $S\subseteq N$. $\sharp$

The anti-core is a natural solution concept for games that model cost allocation problems. Note that $AC(v)=-C(-v)$ for all $v$. Therefore, Theorem \ref{monta} and the linearity property of the weighted values imply the following result.

\begin{equation}{\label{montb}}\tag{22}\mbox{}\end{equation}

Theorem \ref{montb}. (Monderer et al. \cite[p.32]{mon})}. For every game $v$, each element in the anti-core of $v$ is the weighted
value of $v$ for some weight system. That is, $AC(v)\subseteq\boldsymbol{\phi}^{\cal W}(v)$.$\sharp$

\begin{equation}{\label{montc}}\tag{23}\mbox{}\end{equation}

Theorem \ref{montc}. (Monderer et al. \cite[p.33]{mon}). A game $v$ is convex if and only if $C(v)=\boldsymbol{\phi}^{\cal W}(v)$. Moreover, $v$ is strictly convex if and only if $\boldsymbol{\phi}^{(\cdot)}(v)$ is a homeomorphism between ${\cal W}$ and $C(v)$. In this case, for each $\sigma\in\Sigma$, $\boldsymbol{\phi}^{(\cdot)}(v)$ maps homeomorphically ${\cal W}_{\sigma}$ onto the relative interior of the face $F_{\sigma}(v)$ of $C(v)$. $\sharp$

\begin{equation}{\label{moncc}}\tag{24}\mbox{}\end{equation}

Corollary \ref{moncc}. (Monderer et al. \cite[p.33]{mon}). The set ${\cal W}$ of all weight systems of $N$ is homeomorphic to a $|N|-1$ dimensional ball. $\sharp$

Weber (1988) proved that, for every game $v$, $C(v)\subseteq\boldsymbol{\psi}^{v}(\Delta ({\cal R}))$, where $\boldsymbol{\psi}^{v}(\Delta ({\cal R}))=\{\boldsymbol{\phi}^{\mathbb{P}}:\mathbb{P}\in\Delta ({\cal R})\}$ is the set of all random order values of $v$. Since weighted values are random order values $\boldsymbol{\phi}^{\cal W}(v)\subseteq\boldsymbol{\psi}^{v}(\Delta ({\cal R}))$ and therefore Theorem \ref{monta} implies Weber’s result. Moreover, the set of all weighted values has the dimension of ${\cal W}$ which according to Corollary \ref{moncc} is $n-1$, where $n=|N|$. The set of all random order values can be shown to have the dimension $2^{n-1}(n-2)+1$. Thus Theorem \ref{monta} shows that a much “thiner” set of values is required in order to cover the core of each game. Note also that $\boldsymbol{\phi}^{\cal W}(v)$ is not a convex set in general, and therefore it is strictly contained in the set $\boldsymbol{\psi}^{v}(\Delta ({\cal R}))$ (Monderer et al. \cite[p.33]{mon}).

For every $i\in N$ and for every ${\bf w}_{1},{\bf w}_{2}\in {\cal W}$, we write ${\bf w}>_{i}{\bf u}$ if $w_{i}^{S}\geq u_{i}^{S}$ for every $S$ which contains $i$ and ${\bf w}^{S}={\bf u}^{S}$ for all $S\subseteq N \setminus\{i\}$ (Monderer et al. \cite[p.33]{mon}).

Definition. (Monderer et al. \cite[p.33]{mon}). We say that $\boldsymbol{\phi}^{(\cdot)}(v)$ is increasing if, for each $i$, for each ordered partition $\sigma$ and for each ${\bf w},{\bf u}\in {\cal W}_{\sigma}$ such that ${\bf w}>_{i}{\bf u}$,
\begin{equation}{\label{moneq52}}\tag{25}
\phi_{i}^{\bf w}(v)\geq\phi_{i}^{\bf u}(v).
\end{equation}
$\boldsymbol{\phi}^{(\cdot)}(v)$ is strictly increasing if the inequalities in (\ref{moneq52}) are strict.
\end{Def}

Theorem. (Monderer et al. \cite[p.33]{mon}). The game $v$ is (resp. strictly) convex if and only if $\boldsymbol{\phi}^{(\cdot)}(v)$ is (resp. strictly) increasing. $\sharp$

Weight systems can be used to define a different family of values which are called dual weighted values.

Definition. (Monderer et al. \cite[p.37]{mon}). For ${\bf w}\in {\cal W}$, the dual weighted value $(\boldsymbol{\phi}^{*})^{\bf w}$ is defined by $(\boldsymbol{\phi}^{*})^{\bf w}(v)=\boldsymbol{\phi}^{\bf w}(v^{*})$, where $v^{*}$ is the dual game of $v$ defined by $v^{*}(S)=v(N)-v(N\setminus S)$ for each $S\subseteq N$. $\sharp$

An axiomatization of the whole family of the dual weighted values was given by Kalai and Samet \cite{kal87}. Like weighted values, dual weighted values are also random order values. The probability distribution $\mathbb{P}_{\bf w}^{*}$ which determines $(\boldsymbol{\phi}^{*})^{\bf w}$ assigns to each order $R$ the probability $\mathbb{P}_{\bf w}^{*}(R^{*})$ where $\mathbb{P}_{\bf w}$ is the probability distribution described above which defies $\boldsymbol{\phi}^{\bf w}$ and $R^{*}$ is the order $R$ reversed (Monderer et al. \cite[p.37]{mon}).

The family of the weighted values and of the dual weighted values intersect. Thus, for example, the random order values that are defined by probability distributions on ${\cal R}$ that are concentrated on a single order in ${\cal R}$ belong to both families. However, Kalai and Samet \cite{kal87} proved that for $n\geq 3$, the Shapley value is the unique element in the intersection of the positively weighted values and the dual positively weighted values (Monderer et al. \cite[p.38]{mon}).

Note that, for every game $v$, $C(v)=AV(v^{*})$. Therefore, we get the analogoues of Theorems \ref{monta} and \ref{montb} as follows.

Theorem. (Monderer et al. \cite[p.38]{mon}). For every game $v$, each element in the core of $v$ is the dual weighted value of $v$ for some weight system. That is, $C(v)\subseteq (\boldsymbol{\phi}^{*})^{\cal W}(v)$. $\sharp$

Theorem. (Monderer et al. \cite[p.38]{mon}). For every game $v$, each element in the anti-core of $v$ is the dual weighted value of $v$ for some weight system. That is, $AC(v)\subseteq (\boldsymbol{\phi}^{*})^{\cal W}(v)$. $\sharp$

Observe further that a game $v$ is (resp. strictly) convex if and only if $-v^{*}$ is (resp. strictly) convex, and that, for each game $v$, $\boldsymbol{\phi}^{\cal W}(-v)=-\boldsymbol{\phi}^{\cal W}(v)$. Hence Theorem \ref{montc} implies the next result.

\begin{equation}{\label{momtc*}}\tag{26}\mbox{}\end{equation}

Theorem \ref{momtc*}. (Monderer et al. \cite[p.38]{mon}). A game $v$ is convex if and only if $latex C(v)=
(\boldsymbol{\phi}^{*})^{\cal W}(v)$. Moreover, $v$ is strictly convex if and only if $(\boldsymbol{\phi}^{*})^{(\cdot)}(v)$ is a homeomorphism between ${\cal W}$ and $C(v)$. In this case, for each $\sigma\in\Sigma$, $(\boldsymbol{\phi}^{*})^{(\cdot)}(v)$ maps homeomorphically ${\cal W}_{\sigma}$ onto the relative interior of the face $F_{\sigma^{*}}^{v}$ of $C(v)$, where $\sigma^{*}$ is ordered partition $\sigma$ reversed. $\sharp$

Theorems \ref{montc} and \ref{montc*} imply that, for convex game $v$, $\boldsymbol{\phi}^{\cal W}(v)=(\boldsymbol{\phi}^{*})^{\cal W}(v)$.

Definition. (Monderer et al. \cite[p.38]{mon}). We say that $(\boldsymbol{\phi}^{*})^{(\cdot)}(v)$ is increasing if, for each $i$, for each ordered partition $\sigma$ and for each ${\bf w},{\bf u}\in {\cal W}_{\sigma}$ such that ${\bf w}>_{i}{\bf u}$,
\begin{equation}{\label{moneq71}}\tag{27}
(\phi^{*})_{i}^{\bf w}(v)\geq (\phi^{*})_{i}^{\bf u}(v).
\end{equation}
$(\boldsymbol{\phi}^{*})^{(\cdot)}(v)$ is {\bf strictly increasing} if the inequalities in (\ref{moneq71}) are strict.
\end{Def}

Theorem. (Monderer et al. \cite[p.38]{mon}). The game $v$ is (resp. strictly) convex if and only if $(\boldsymbol{\phi}^{*})^{(\cdot)}(v)$ is (resp. strictly) increasing. $\sharp$