Cooperative Games - Non-Transferable Utility Games

Alfred Elmore (1815–1881) was an English painter.

The topics are

Axiomatizations of the NTU Values \ref{a}
NTU Games with Coalition Structure
The Cores

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

Axiomatizations of the NTU Values.

The objects of study are the $n$-person cooperative games, in which utility is not (necessarily) transferable. Such a game is described by a
set of players and a set of feasible outcomes for each coalition. In general, the players may not be able to make side payments to each other in such a way that the total utility gains are equal to the total utility losses. These games are thus called non-transferable utility games, or NTU games for short (Hart \cite[p.1295]{har85}).

Let $N$ be a finite set of players and let $S$ be a coalition in $N$. We write ${\bf 1}^{S}$ for the indicator of $S$, i.e., the member of $\mathbb{R}^{N}$ whose $i$th coordinate is $1$ or $0$ according as $i$ is or is not in $S$. We call ${\bf x}$ positive if $x_{i}>0$ for all $i\in N$. If $\boldsymbol{\lambda}$ and ${\bf x}$ are in $\mathbb{R}^{N}$, we define $\boldsymbol{\lambda}{\bf x}$ in $\mathbb{R}^{N}$ by $(\boldsymbol{\lambda}{\bf x})_{i}=\lambda_{i}x_{i}$, and denote the scalar product $\sum_{i\in N}x_{i}y_{i}$ by $\langle {\bf x},{\bf y}\rangle$. We write ${\bf x}\geq {\bf y}$ if $x_{i}\geq y_{i}$ for all $i\in N$ and ${\bf x}>{\bf y}$ if $x_{i}>y_{i}$ for all $i\in N$. Let $A,B\subseteq\mathbb{R}^{N}$ and $\boldsymbol{\lambda},{\bf x}\in\mathbb{R}^{N}$. We write $A+B=\{{\bf a}+{\bf b}:{\bf a}\in A\mbox{ and }{\bf b}\in B\}$ and $\boldsymbol{\lambda}A=\{\boldsymbol{\lambda}{\bf a}:{\bf a}\in A\}$. We denote by $\partial A=\mbox{cl}(A)\cap (\mbox{cl}(A))^{c}$ frontier of $A$. Let $A$ and $B$ be closed subsets of $\mathbb{R}^{N}$. Then $A+B$ is the closure of $\{{\bf a}+{\bf b}:{\bf a}\in A,{\bf b}\in B\}$.

If $A$ is convex, we call it {\bf smooth} if and only if it has a unique supporting hyperplane at each point of its frontier.
We call $A$ {\bf comprehensive} if and only if ${\bf x}\in A$ and ${\bf x}\geq {\bf y}$ imply ${\bf y}\in A$.

(Hart \cite[p.1297]{har85} and Aumann \cite[p.600]{aum85}).

\begin{equation}{\label{aum85d1}}\tag{1}\mbox{}\end{equation}

Definition \ref{aum85d1}. (Aumann \cite[p.601]{aum85}) A NTU game on the set $N$ of players is a function $V$ that assigns to each coalition $S$ a proper subset $V(S)$ of $\mathbb{R}^{S}$ satisfying the following conditions.

(i) $V(S)$ is convex and comprehensive nonempty proper subset of $\mathbb{R}^{S}$.

(ii) $V(N)$ is smooth. This condition is a substantive restriction from the intuitive viewpoint.

(iii) If ${\bf x},{\bf y}\in\partial V(N)$ and ${\bf x}\geq {\bf y}$, then ${\bf x}={\bf y}$. This condition says that $\partial V(N)$ has no
“level” segments, i.e., segments parallel to a coordinate hyperplane. A verbal statement is that the weak and strong Pareto optimality are equivalent.

(iv) For each coalition $S$, there is a payoff vector ${\bf x}\in\mathbb{R}^{N}$ such that $V(S)\times \{{\bf 0}^{N\setminus S}\}\subseteq V(N)+\{{\bf x}\}$. This condition says that if one thinks of $V(S)$ as embedded in $\mathbb{R}^{N}$ by assigning ${\bf 0}$ to players outside $S$, then $V(S)$ is included in some translate of $V(N)$. It can be thought of as an extremely weak kind of monotonicity. $\sharp$

The smoothness of $V(N)$ in condition (ii) implies that, for every ${\bf x}\in\partial V(N)$, there exists a unique normalized vector $\boldsymbol{\lambda}\in\mathbb{R}^{N}$ such that $\boldsymbol{\lambda}\cdot{\bf x}\geq \boldsymbol{\lambda}\cdot{\bf y}$ for all ${\bf y}\in\mbox{cl}(V(N))$. The normalization we will use is $\max_{i\in N}|\lambda_{i}|=1$ (so that ${\bf 1}^{N}=(1,1,\cdots ,1)$ is normalized); let $\boldsymbol{\lambda}(V(N),{\bf x})$ denote this unique vector. Note that $\boldsymbol{\lambda}$ must be positive by the comprehensiveness and condition (iii). We have many other definitions for NTU games. (Hart \cite[p.1297]{har85}).

\begin{equation}{\label{har85d1}}\tag{2}\mbox{}\end{equation}

Definition \ref{har85d1}. (Hart \cite[p.1297]{har85}). The NTU game on $N$ is a set-valued function $V$ that assigns to every $S\subseteq N$ a subset $V(S)$ of $\mathbb{R}^{S}$ satisfying the following conditions.

For every $S\subseteq N$, the set $V(S)$ is a closed, convex and comprehensive nonempty subset of $\mathbb{R}^{S}$.
The set $V(N)$ is smooth.
If ${\bf x},{\bf y}\in\partial V(N)$ and ${\bf x}\geq {\bf y}$, then ${\bf x}={\bf y}$. $\sharp$

\begin{equation}{\label{pel85d21}}\tag{3}\mbox{}\end{equation}

Definition \ref{pel85d21}. (Peleg \cite[p.204]{pel85}). An NTU game is a pair $(N,V)$, where $N$ is a coalition and $V$ is a function that assigns to each coalition $S\subseteq N$ a nonempty subset $V(S)$ of $\mathbb{R}^{S}$ such that the following conditions are satisfied:

$V(S)$ is closed and comprehensive;
$V(S)\cap ({\bf x}^{S}+\mathbb{R}_{+}^{S})$ is bounded for every ${\bf x}^{S}\in\mathbb{R}^{S}$ (this guarantees that $V(S)$ is a proper subset of $\mathbb{R}^{S}$);
if ${\bf x}^{S},{\bf y}^{S}$ and ${\bf x}^{S}\geq {\bf y}^{S}$, then ${\bf x}^{S}={\bf y}^{S}$. $\sharp$

\begin{equation}{\label{gad48}}\tag{4}\mbox{}\end{equation}

Definition \ref{gad48}. (Levy and McLean \cite[p.341]{lev91}). An NTU game on $N$ is a collection $\{V(S):\emptyset\neq S\subseteq N\}$ where $V(S)$ is a nonempty, convex, closed, proper and comprehensive subset of $\mathbb{R}^{S}$ satisfying the following conditions:

$V(N)$ is smooth;
if ${\bf x},{\bf y}\in\partial V(N)$ and ${\bf x}\geq {\bf y}$, then ${\bf x}={\bf y}$;
for each $S\subseteq N$, there exists an ${\bf x}$ such that $V(S) \times\{{\bf 0}^{N\setminus S}\}\subseteq V(N)+{\bf x}$;
for each $S\subseteq N$, the set of extreme points of $V(S)$ is bounded. $\sharp$

We adopt the following notations.

We denote by $\Gamma^{N}$ the set of all NTU games on $N$ defined by Definition \ref{har85d1}.
We denote by $\Gamma_{0}^{N}$ the set of all NTU games on $N$ defined by Definition \ref{aum85d1}.
We denote by $\Gamma_{1}^{N}$ the set of all NTU games on $N$ defined by Definition \ref{gad48}.
We denote by $\Gamma_{2}^{N}$ the set of all NTU games on $N$ defined by Definition \ref{pel85d21}.

A TU game on $N$ is a function $v$ that assigns to each coalition $S$ a real number $v(S)$ with $v(\emptyset )=0$. The NTU game $V$ corresponding to a TU game $v$ is defined by
\begin{equation}{\label{ga45}}\tag{5}
V(S)=\left\{{\bf x}\in\mathbb{R}^{S}:\sum_{i\in S}x_{i}\leq v(S)\right\},
\end{equation}
i.e., the game $V$ defined in (\ref{ga45}) satisfying all conditions in Definitions \ref{aum85d1} and \ref{har85d1}.

If $T$ is a coalition, the unanimity game $u_{T}$ is a TU game. The NTU game $U_{T}$ corresponding to $u_{T}$ is also called the unanimity game on $T$. For every coalition $T$ in $N$ and every real constant $c$, let $u_{T,c}$ be the TU game given by
\[u_{T,c}(S)=\left\{\begin{array}{ll}
c & \mbox{if $T\subseteq S$}\\ 0 & \mbox{otherwise}.
\end{array}\right .\]
for all $S\subseteq N$. We denote by $U_{T,c}$ the corresponding NUT game of $u_{T,c}$, which is called a unanimity game on $T$; it models the situation where each coalition $S$ can arbitrarily divide among its member the amount $c$ if it contains all the players of $T$, or nothing if it does not (Hart \cite[p.1298]{har85}).

Let $X$ denote the product $\prod_{S\subseteq N}\mathbb{R}^{S}$. An element ${\bf x}=({\bf x}^{S})_{S\subseteq N}$, where ${\bf x}^{S}\in\mathbb{R}^{S}$ for each $S\subseteq N$, is called a {\bf payoff configuration}. It assigns to every coalition $S$ a {\bf payoff vector} ${\bf x}^{S}=(x^{S}_{i})_{i\in S}$. Note that a NTU game $(N,V)$ may be regarded as a (rectangular) subset of $X$, namely $\prod_{S\subseteq N}V(S)$. Operations on games are defined like the corresponding operations on sets for each coalition separately. Thus, for NTU games $(N,V)$ and $(N,W)$, $V\subseteq W$ means $V(S)\subseteq W(S)$ for all $S\subseteq N$, $(V+W)(S)=V(S)+W(S)$, $\bar{V}(S)=\mbox{cl}(V(S))$, and if $\boldsymbol{\lambda}\in\mathbb{R}^{N}$, the game $\boldsymbol{\lambda}V$ is defined by
\[(\boldsymbol{\lambda}V)(S)=\boldsymbol{\lambda}^{S}V(S)=\left\{(\lambda_{i}x_{i})_{i\in S}:{\bf x}=(x^{i})_{i\in S}\in V(S)\right\}.\]
Finally, $\partial V$ stands for $\prod_{S\subseteq N}\partial V(S)\subseteq X$ (Hart \cite[p.1298]{har85} and Aumann \cite[p.601]{aum85}).

An Axiomatization of Shapley Correspondence.

Given an NTU game $(N,V)$ in $\Gamma_{0}^{N}$, for each positive $\boldsymbol{\lambda}\in\mathbb{R}^{N}$, we define
\begin{equation}{\label{aum85eq42}}\tag{6}
v_{\lambda}(S)=\sup_{{\bf x}\in V(S)}\left\langle\boldsymbol{\lambda}^{S},{\bf x}\right\rangle .
\end{equation}
We say that the TU game $(N,v_{\lambda})$ in (\ref{aum85eq42}) is well-defined if and only if the right side of (\ref{aum85eq42}) is finite for all $S$. A Shapley value of the NTU game $(N,V)$ is a point ${\bf y}\in\bar{V}(N)=\mbox{cl}(V(N))$ such that, for some positive $\boldsymbol{\lambda}\in\mathbb{R}^{N}$, the TU game $(N,v_{\lambda})$ is defined and $\boldsymbol{\lambda}{\bf y}=\boldsymbol{\phi}(v_{\lambda})$, where $\boldsymbol{\phi}(v_{\lambda})$ is the Shapley value of game $(N,v_{\lambda})$. The
set of all Shapley values of $(N,V)$ is denoted by $\boldsymbol{\lambda}_{0}(V)$. The set of games $(N,V)$ for which $\boldsymbol{\lambda}_{0}(V)\neq\emptyset$ is denoted by $\widehat{\Gamma}_{0}^{N}$. The correspondence from $\widehat{\Gamma}_{0}^{N}$ to $\mathbb{R}^{N}$ is called the {\bf Shapley correspondence} (Aumann \cite[p.602]{aum85}).

In general, a value correspondence $\boldsymbol{\phi}$ is a correspondence that associates with each NTU game $(N,V)$ in $\widehat{\Gamma}_{0}^{N}$ a set $\boldsymbol{\phi}(V)$ of payoff vectors satisfying the following axioms for all games $(N,U)$, $(N,V)$ and $(N,W)$ in $\widehat{\Gamma}_{0}^{N}$:

(A1: Non-Emptiness). $\boldsymbol{\phi}(V)\neq\emptyset$;
(A2: Efficiency). $\boldsymbol{\phi}(V)\subseteq\partial V(N)$;
(A3: Conditional Additivity). If $U=V+W$, then $(\boldsymbol{\phi}(V)+\boldsymbol{\phi}(W))\cap\partial U(N)\subseteq\boldsymbol{\phi}(U)$;
(A4: Unanimity). If $U_{T}$ is the unanimity game on a coalition $T$, then $\boldsymbol{\phi}(U_{T})=\{{\bf 1}_{T}/|T|\}$;
(A5: Closure Invariance). $\boldsymbol{\phi}(\bar{V})=\boldsymbol{\phi}(V)$, where $\bar{V}(S)=\mbox{cl}(V(S))$;
(A6: Scale Covariance). If $\boldsymbol{\lambda}\in\mathbb{R}^{N}$ is positive, then $latex \boldsymbol{\phi}
(\boldsymbol{\lambda}V)=\boldsymbol{\lambda}\boldsymbol{\phi}(V)$;
(A7: Independence of Irrelevant Alternatives(IIA)). If $V(N)\subseteq W(N)$ and $V(S)=W(S)$ for $S\neq N$, then $(\boldsymbol{\phi}(W)\cap V(N))\subseteq\boldsymbol{\phi}(V)$.

For a fixed value correspondence $\boldsymbol{\phi}$, we call ${\bf x}$ a {\bf value} of $(N,V)$ if ${\bf x}\in\boldsymbol{\phi}(V)$. The interpretation of the above axioms are given below.

The efficiency axiom (A2) says that all values are Pareto optimal.
Suppose that ${\bf y}$ and ${\bf z}$ are values of $V$ and $W$, respectively. We cannot in general expect ${\bf y}+{\bf z}$ to be a value of $V+W$, because it need not be Pareto optimal there. The conditional aditivity axiom (A3) says that if ${\bf y}+{\bf z}$ does happen to be a Pareto optimal in $V+W$, then it is a value of $V+W$, i.e., that additivity obtains whenever it does not contradict efficiency.
The unanimity axiom (A4) says that the unanimity game on $T$ has a unique value, which provides that the coalition $T$ split the available amount equally.
The closure invariance axiom (A5) is a conceptually harmless technical assumption; we simply do not distinguish between a convex set and its closure.
If the payoffs are in utilities, then the scale invariance axiom (A6) says that representing the same real outcome by different utility functions does not affect the value in real terms.
The independence of irrelevant alternatives axiom (A7) says that a value ${\bf y}$ of a game $W$ remains a value when one removes outcomes other than ${\bf y}$ (“irrelevant alternative”) from the set $W(N)$ of all feasible outcomes, without changing $W(S)$ for coalitions $S$ other than the all player coalition. (Aumann \cite[p.602-603]{aum85}).

Whether or not the IIA axiom (A7) is reasonable depends on how we view the value. If we view it as an expected ot average outcome, then IIA is not very convincing. By removing parts of the feasible set, we decrease the range of possible outcomes, and so the average may change even if it itself remains feasible. But in NTU games, viewing the value as an average is fraught with difficulty even without IIA, because the convexity of $V(N)$ implies that, in general, an average will not be Pareto optimal. An alternative is to view the value as a grooup decision or arbitrated outcome, i.e., a reasonable compromise in view of all the possible alternatives open to the players. In that case, IIA does sound quite convincing and even compelling. Therefore, we offer an axiomatic treatment that avoids using it. Let us nevertheless examine the consequences of omitting this axiom. (Aumann \cite[p.603]{aum85}).

Theorem. (Aumann \cite[p.604]{aum85}). Let $(N,V)$ be an NTU game in $\Gamma_{0}^{N}$. The Shapley correspondnece is the maximal correspondence from $\widehat{\Gamma}_{0}^{N}$ to $\mathbb{R}^{N}$ satisfying axioms {\em (A1)}-{\em (A6)}. More explicitly, if $\boldsymbol{\lambda}_{0}$ and $\boldsymbol{\phi}$ satisfies axioms {\em (A1)}-{\em (A6)}, then $\boldsymbol{\phi}(V)\subseteq \boldsymbol{\lambda}_{0}(V)$ for all games $V$ in $\widehat{\Gamma}_{0}^{N}$. $\sharp$

Lemma. (Aumann \cite[p.605]{aum85}). Let $(N,V)$ be an NTU game in $\Gamma_{0}^{N}$. The following statements hold true.

(i) We have $\boldsymbol{\lambda}_{0}(V)\subseteq\partial V(N)$.

(ii) Let ${\bf y}\in\boldsymbol{\lambda}_{0}(V)$ and let $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf y})$. Then, the TU game $(N,v_{\lambda})$ is well-defined and $\boldsymbol{\lambda}{\bf y}=\boldsymbol{\phi}(v_{\lambda})$. $\sharp$

\begin{equation}{\label{aum85p73}}\tag{7}\mbox{}\end{equation}

Proposition \ref{aum85p73}. (Aumann \cite[p.605]{aum85}). Let $(N,V)$ be an NTU game in $\Gamma_{0}^{N}$. The Shapley correspondence $\boldsymbol{\lambda}$ from $\widehat{\Gamma}_{0}^{N}$ to $\mathbb{R}^{N}$ satisfies the axioms (A1)– (A7). $\sharp$

\begin{equation}{\label{aum85l86}}\tag{8}\mbox{}\end{equation}

Lemma \ref{aum85l86}. (Aumann \cite[p.605]{aum85}). Let $(N,V)$ be an NTU game in $\Gamma_{0}^{N}$. The following statements hold true.

(i) We have $\boldsymbol{\phi}(V)\subseteq\boldsymbol{\lambda}_{0}(V)$ for each $V\in\widehat{\Gamma}_{0}^{N}$.

(ii) If $\boldsymbol{\phi}$ is a value correspondence, i.e., it satisfies axioms (A1)–(A7), then $\boldsymbol{\lambda}_{0}(V)\subseteq\boldsymbol{\phi}(V)$ for all $V\in\widehat{\Gamma}_{0}^{N}$. $\sharp$

\begin{equation}{\label{aum85t}}\tag{9}\mbox{}\end{equation}

Theorem \ref{aum85t}. (Aumann \cite[p.603]{aum85}). For games in $\Gamma_{0}^{N}$, there is a unique value correspondence, and it is the Shapley correspondence.

Proof. The result follows from Proposition \ref{aum85p73} and Lemma \ref{aum85l86}. $\blacksquare$

The definition of “Shapley value” explicitly takes $\boldsymbol{\lambda}$ positive; and the non-levelness condition (iii) in Definition \ref{aum85d1} assures that whatever emerges from the axioms will be associated with a positive $\boldsymbol{\lambda}$. Recall that a verbal statement of condition (iii) is that weak and strong Pareto optimality are equivalent. One can avoid the non-levelness conditions by strengthing the efficiency axiom (A2) to read as follows:

(A2′: Strong Efficiency). If ${\bf y}\in\boldsymbol{\phi}(V)$, then $\{{\bf x}\in\partial V(N):{\bf x}\leq {\bf y}\}=\{{\bf y}\}$.

It says that ${\bf y}$ is in the relative (to $\partial V(N)$) interior of the strongly Pareto optimal set, or equivalently that $\boldsymbol{\lambda}(V(N),{\bf y})$ is positive. If we replace axiom (A2) by axiom (A2′), then one can simply drop condition (iii) in Definition \ref{aum85d1}, and theorems remain true (Aumann \cite[p.609]{aum85}).

The domain $\widehat{\Gamma}_{0}^{N}$ of the axioms is the set of all games that possess at least one Shapley value. This might be considered an estheic drawback, since in this way the Shapley value enters into its own axiomatic characterization. If one wishes to avoid this, one can replace $\widehat{\Gamma}_{0}^{N}$ by any family $\Delta$ with following properties:

(a) Every game in $\Delta$ has a Shapley value.
(b) All games corresponding to TU games are in $\Delta$.
(c) If $V\in\Delta$ and $\boldsymbol{\lambda}$ is a positive vector in $\mathbb{R}^{N}$, then $\boldsymbol{\lambda}V\in\Delta$.
(d) The game obtained from any $V$ in $\Delta$ by replacing $V(N)$ by any one of its supporting half-spaces is also in $\Delta$.
(e) $V\in\Delta$ if and only if $\bar{V}\in\Delta$.

For example, we may take $\Delta$ to be the family of all games $V$ such that, for all coalitions $S$, the set of extreme points of $\bar{V}(S)$ is bounded, or equivalently, $\bar{V}(S)$ is the sum of a compact set and a cone. We can verify that the family $\Delta$ defined above does satisfy properties (a)-(e), and so can replace $\widehat{\Gamma}_{0}^{N}$ as the domain of the value correspondence. We adopted $\widehat{\Gamma}_{0}^{N}$ as the domain because it is the largest on which the axioms “work”, and is thus the most useful from the point of view of applications. The conditional additivity axiom (A3) can be replaced by the following pair of axioms:

(A3a’: Conditional Sure-Thing). If $U=\frac{1}{2}V+\frac{1}{2}W$, then $\boldsymbol{\phi}(V)\cap\boldsymbol{\phi}(W)\cap\partial U(N)\subseteq\boldsymbol{\phi}(U)$;
(A3b’: Translation Covariance). For all ${\bf x}\in\mathbb{R}^{N}$, $\boldsymbol{\phi}(V+{\bf x})=\boldsymbol{\phi}(V)+{\bf x}$.

In the above discussion, we suggest that the value of a game may be viewed as a group decision, compromise, or arbitrated outcome, that is reasonable in view of the alternatives open to the players and coalitions (rather than an outcome that is itself in some sense stable, such as a core point). In these terms, axiom (A3a’) says the following: Suppose that ${\bf y}$ is a reasonable compromise both in the game $V$ and in the game $W$. Suppose further that one of the games $V$ and $W$ will be played: at present it is not yet known which one, but it is common knowledge that the probabilities are half-half. Then ${\bf y}$ is a reasonable compromise to their mutual advantage (Aumann \cite[p.610-611]{aum85}).

An Axiomatization of Harsanyi’s Solution.

A solution function on $\Gamma^{N}$ is a set-valued function $\boldsymbol{\Psi}$ that assigns to each game $(N,V)$ in $\Gamma^{N}$ a set of payoff configurations $\boldsymbol{\Psi}(V)\subseteq X$. An element of $\boldsymbol{\Psi}(V)$ is called a solution of $(N,V)$. A solution ${\bf x}$ of a game $(N,V)$ is a payoff configuration, which specifies a feasible (and even efficient) outcome ${\bf x}^{S}$ for the coalition $S$. One may view ${\bf x}^{S}$ as the payoff vector (chosen from their feasible set $V(S)$) that the members of $S$ agree upon. If coalition $S$ “forms”, $x^{S}_{i}$ is the amount that player $i$ in $S$ will receive (Hart \cite[p.1298 and p.1299]{har85}).

\begin{equation}{\label{har85d2}}\tag{10}\mbox{}\end{equation}

Definition \ref{har85d2}. (Hart \cite[p.1300]{har85}). A payoff configuration ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ is a
Harsanyi solution of an NTU game $(N,V)$ in $\Gamma^{N}$ when there exists a vector $\boldsymbol{\lambda}\in\mathbb{R}^{N}$ and real numbers $\{\xi^{T}\}_{T\subseteq N}$ such that the following conditions are satisfied:

(H1: Efficiency). for each $S\subseteq N$, ${\bf x}^{S}\in\partial V(S)$;
(H2: Utilitariansim). $\boldsymbol{\lambda}\cdot {\bf x}^{N}\geq\boldsymbol{\lambda}\cdot{\bf y}$ for all ${\bf y}\in V(N)$;
(H3: Equity). for each $S\subseteq N$ and each $i\in S$, we have $\lambda_{i}x^{S}_{i}=\sum_{\{T:T\subseteq S,i\in T\}}\xi^{T}$. $\sharp$

Let ${\cal H}(V)$ denote the set of all Harsanyi solutions of the game $(N,V)$. Then ${\cal H}$ is called the Harsanyi solution function, which is a set-valued function defined on $\Gamma^{N}$. Assume that $\boldsymbol{\lambda}={\bf 1}^{N}=(1,1,\cdots ,1)$. Then the payoff vector of every coalition is efficient. For the grand coalition $N$, it is moreover utilitarian, maximizing the sum of payoffs over the feasible set $V(N)$. And finally, the payoffs $x^{S}_{i}$ of each member of any coalition $S$ is the sum of the “dividends” $\xi^{T}$ that player $i$ has accumulated from all subcoalitions $T$ of $S$ to which $i$ belongs; the dividend $\xi^{T}$ being the same for all members of $T$, the vectors ${\bf x}^{S}$ are said to be equitable. In the general case, the payoffs of each player are appropriately rescaled so that $\boldsymbol{\lambda} ={\bf 1}^{N}$, and then the above three criteria apply. For games $(N,V)$ in $\Gamma^{N}$, the axioms (H1)–(H3) can be restated as follows: ${\bf x}\in{\cal H}(V)$ if and only if the following axioms are satisfied:

(H4) ${\bf x}\in\partial V$;
(H5) there exist real numbers $\{\xi^{T}\}_{T\subseteq N}$ such that $\lambda_{i}x^{S}_{i}=\sum_{\{T:T\subseteq S,i\in T\}}\xi^{T}$ for each $i\in S\subseteq N$, where $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf x}^{N})$.

Indeed, by axiom (H1), ${\bf x}^{N}\in\partial V(N)$ implies that there exists a unique normalized $\boldsymbol{\lambda}= \boldsymbol{\lambda}(V(N),{\bf x}^{N})$ satisfying axiom (H2). The normalization does not matter, since axiom (H3) is homogeneous in $\boldsymbol{\lambda}$ and the $\xi^{T}$’s. For a TU game $(N,v)$, let $(S,v^{S})$ denote its subgame. For $i\in S\subseteq N$, $\phi_{i}(v^{S})$ will denote the Shapley value of player $i$ in the game $(S,v^{S})$. Then axiom (H5) is equivalent to the following axiom:

(H5′) for all $i\in S\subseteq N$, $\lambda_{i}x^{S}_{i}=\phi_{i}(S,v)$, where $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf x}^{N})$ and
\[v^{S}(T)=\left\langle\boldsymbol{\lambda}^{T},{\bf x}^{T}\right\rangle=\sum_{i\in T}\lambda_{i}x^{T}_{i}\]
for all $T\subseteq S$.

This may be checked by noting that $v(S)=\sum_{T\subseteq S}|T|\xi^{T}$ (see also Imai \cite{ima83}). Axioms (H5) and (H5′) depend on the game $(N,V)$ only through $\boldsymbol{\lambda}$. Therefore, if $(N,V)$ and $(N,W)$ are games in $\Gamma^{N}$, then ${\bf x}\in\partial W$ and $\boldsymbol{\lambda}(V(N),{\bf x}^{N})=\boldsymbol{\lambda}(W(N),{\bf x}^{N})$ imply ${\bf x}\in {\cal H}(W)$ (Hart \cite[p.1300-1301]{har85}).

The Harsanyi solutions of a given game $(N,V)$ in $\Gamma^{N}$ are constructed as follows. For every positive vector $\boldsymbol{\lambda}$ in $\mathbb{R}^{N}$, we define inductively
\begin{equation}{\label{ga46}}\tag{11}
\xi^{S}_{\lambda}=\max_{{\bf z}^{S}_{\lambda}(t)\in V(S)}t,
\end{equation}
where ${\bf z}^{S}_{\lambda}(t)=(z^{S}_{i}(t))_{i\in S}$ is given by
\[z^{S}_{i}(t)=\frac{1}{\lambda_{i}}\left (t+\sum_{\{T:T\subseteq S,i\in T\}}\xi^{T}_{\lambda}\right ).\]
Let ${\bf x}^{S}_{\lambda}={\bf z}^{S}_{\lambda}(\xi^{S}_{\lambda})$. Then ${\bf x}_{\lambda}=({\bf x}^{S}_{\lambda})_{S\subseteq N}$ is a Harsanyi solution of $V$ if and only if axiom (H2) is satisfied, i.e., $latex \boldsymbol{\lambda}=
\boldsymbol{\lambda}(V(N),{\bf x}^{N}_{\lambda})$. Since $V(S)$ is nonempty and comprehensive, there exists $t$ small enough such
that ${\bf z}^{S}_{\lambda}(t)\in V(S)$. Moreover, since $V(S)$ is closed, the maximum in (\ref{ga46}) is indeed attained. Note that ${\bf z}^{S}_{\lambda}(t)\in V(S)$ if and only if $t\leq\xi^{S}_{\lambda}$ and ${\bf x}^{S}_{\lambda}\in\partial V(S)$. (Hart \cite[p.1301]{har85}).

An NTU game $(N,V)$ is {\bf inessential} if and only if there exists ${\bf a}\in\mathbb{R}^{N}$ such that ${\bf a}^{S}=(a^{S}_{i})_{i\in S}\in\partial V(S)$ for all $S$. If ${\bf a}={\bf 0}$, i.e., ${\bf 0}\in\partial V(S)$ for all $S$, then $(N,V)$ is called zero-inessential, which means that the zero payoff vector is efficient for all coalitions.

Proposition. (Hart \cite[p.1301]{har85}). We have the following properties.

(i) Let $(N,V)$ be a zero-inessential game in $\Gamma^{N}$, i.e., ${\bf 0}\in\partial V$. Then ${\cal H}(V)=\{{\bf 0}\}$.

(ii) If $(N,V)$ is an inessential game, then $(N,V)$ has a unique Harsanyi solution ${\bf x}$ given by ${\bf x}^{S}={\bf a}^{S}$ for all $S$. $\sharp$

Proposition. (Hart \cite[p.1301]{har85}). Let $(N,V)$ be an NTU game in $\Gamma^{N}$ with $V(N)$ a half-space, i.e.,
\[V(N)=\left\{{\bf y}\in\mathbb{R}^{N}:\left\langle\boldsymbol{\mu},{\bf y}\right\rangle\leq c\right\}\]
for some positive vector $\boldsymbol{\mu}$ in $\mathbb{R}^{N}$ and $c\in\mathbb{R}$. Then $(N,V)$ has a unique Harsanyi solution. $\sharp$

Proposition. (Hart \cite[p.1302]{har85}). Let $(N,V)$ be an NTU game in $\Gamma^{N}$ and correspond to a TU game $(N,v)$. Then $(N,V)$ has a unique Harsanyi solution ${\bf x}$ given by ${\bf x}^{S}_{i}=\phi_{i}(v^{S})$ for all $i\in S\subseteq N$, where $\phi_{i}$ is the Shapley value for the subgame $(S,v^{S})$. $\sharp$

Let ${\bf h}(v)$ denote the unique Harsanyi solution of a game $V$ that corresponds to a TU game $v$, i.e., ${\cal H}(V)=\{{\bf h}(v)\}$. Note that ${\bf h}$, as a function from the space of TU games into the set $X$ of payoff configurations, is a linear function; it assigns to every coalition the Shapley value of the subgame restricted to that coalition (Hart \cite[p.1302]{har85}).

Let $\boldsymbol{\Psi}$ be a solution function defined on $\Gamma^{N}$. We consider the following axioms for NTU games $(N,V)$, $(N,W)$ and $(N,U)$ in $\Gamma^{N}$.

(HA1:Efficiency). $\boldsymbol{\Psi}(V)\subseteq\partial V$;
(HA2: Scale Covariance). $\boldsymbol{\Psi} (\boldsymbol{\lambda}V)=\boldsymbol{\lambda}\boldsymbol{\Psi}(V)$ for all positive $\boldsymbol{\lambda}$ in $\mathbb{R}^{N}$;
(HA3: Conditional Additivity). If $U=V+W$, then $(\boldsymbol{\Psi}(V)+\boldsymbol{\Psi}(W))\cap\partial U\subseteq\boldsymbol{\Psi}(U)$, where $\partial U$ stands for $\prod_{S\subseteq N}\partial U(S)\subseteq X$;
(HA4: Independence of Irrelevant Alternatives) (IIA). If $V\subseteq W$, then $latex \boldsymbol{\Psi}(W)\cap V\subseteq
\boldsymbol{\Psi}(V)$;
(HA5: Unanimity Games). For every nonempty coaliton $T\subseteq N$ and every real number $c$, we define ${\bf z}\equiv {\bf z}_{T,c}\in X$ by ${\bf z}^{S}=(c/|T|){\bf 1}_{T}$ if $T\subseteq S$ and ${\bf z}^{S}=0$ otherwise; then $\boldsymbol{\Psi}(U_{T,c})=\{{\bf z\}}$;
(HA6: Zero-Inessential Games). If ${\bf 0}\in\partial V$, then ${\bf 0}\in\boldsymbol{\Psi}(V)$.

The interpretations of the above axioms are given below.

Axiom (HA1) says that every solution ${\bf x}\in\boldsymbol{\Psi}(V)$ satisfies Pareto efficiency for all coalitions, which is just ${\bf x}^{S}\in\partial V(S)$ by the comprehensiveness.
Axiom (HA2) says that if the payoffs of the players are (independently) scaled, all solutions will also be accordingly rescaled.
For axiom (HA3), let ${\bf x}\in\boldsymbol{\Psi}(V)$ and ${\bf y}\in\boldsymbol{\Psi}(W)$ be solutions of $V$ and $W$, respectively. If ${\bf z}={\bf x}+{\bf y}$ is efficient fo r$U$, i.e., ${\bf z}^{S}\in\partial U(S)$ for all $S$, then ${\bf z}$ is a solution of $U$.
For axiom (HA4), let ${\bf x}\in\boldsymbol{\Psi}(W)$ be a solution of $W$. If $V$ is a game such that $V(S)\subseteq W(S)$ and ${\bf x}^{S}\in V(S)$ for all $S$, then ${\bf x}$ is also a solution of $V$. Note that axiom (HA4) may be equivalently stated in a weaker form as follows: Let $T$ be such that $V(T)\subseteq W(T)$ and $V(S)=W(S)$ for all $S\neq T$; then $\boldsymbol{\Psi}(W)\cap V \subseteq\boldsymbol{\Psi}(V)$.
Axiom (HA5) says that each unanimity game $U_{T,c}$ has a unique solution ${\bf z}\equiv {\bf z}_{T,c}$; the payoff vector ${\bf z}^{S}$ of a coalition $S$ that contains $T$ assigns equal shares $c/|T|$ to all members of $T$, and zero to the rest; if $S$ does not
contain $T$, everyone gets zero.
Axiom (HA6) means that, for all coalitions, ${\bf 0}$ is feasible, whereas no positive vector is feasible; in particular, $V(i)=\{x^{i}\in\mathbb{R}^{\{i\}}:x^{i}\leq 0\}$. For such
a game, the payoff configuration ${\bf 0}$ is a solution. (Hart \cite[p.1298-1299]{har85}).

Proposition. (Hart \cite[p.1302]{har85}). The Harsanyi solution function ${\cal H}$ defined on $\Gamma^{N}$ satisfies axioms (HA1)}–(HA6). $\sharp$

Theorem. (Hart \cite[p.1299]{har85}). There exists a unique solution function defined on $\Gamma^{N}$ satisfying axioms (HA1)–(HA6). Moreover, it is the Harsanyi solution function. $\sharp$

If we disregard axiom (HA6) on zero-inessentiall games, then the remaining axioms (HA1)–(HA5) no longer fully characterize the Harsanyi solution. However, we have the following result.

\begin{equation}{\label{har85ta}}\tag{12}\mbox{}\end{equation}

Theorem \ref{har85ta}. (Hart \cite[p.1303]{har85}). The Harsanyi solution function ${\cal H}$ is the maximal (relative to set inclusion) solution function on $\Gamma^{N}$ satisfying axioms (HA1)–(HA5). It says that if $\boldsymbol{\Psi}$ is a solution function defined on $\Gamma^{N}$ satisfying axioms (HA1)–(HA5), then $\boldsymbol{\Psi}(V)\subseteq {\cal H}(V)$ for all $V\in\Gamma^{N}$. $\sharp$

A payoff configuration ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ is a Shapley solution} of a game $(N,V)$ in $\Gamma^{N}$ if and only if there exists a vector $\boldsymbol{\lambda}\in\mathbb{R}^{N}$ such that the following conditions are satisfied:

(S1). for all ${\bf y}\in V(S)$ and all $S\subseteq N$, $\langle\boldsymbol{\lambda}^{S},{\bf x}^{S}\rangle\geq\langle\boldsymbol{\lambda}^{S},{\bf y}\rangle$;
(S2). for all $i\in N$, $\lambda_{i}x^{N}_{i}=\phi_{i}(v)$ where $v(S)=\langle\boldsymbol{\lambda}^{S},{\bf x}^{S}\rangle$ for all $S\subseteq N$ and $\phi_{i}(v)$ is the Shapley value for the TU game $(N,v)$.

We shall denote the set of all Shapley solutions of the game $(N,V)$ by $\boldsymbol{\lambda}(V)$ and $\boldsymbol{\lambda}$ is called the Shapley solution function which is defined on $\Gamma^{N}$. The comparisons between the Shapley and Harsanyi solutions are given below.

Both are efficient, i.e., for ${\cal H}(V)$ and axiom (H1), ${\bf x}\in\partial V$; for $\boldsymbol{\lambda}$, this follows from condition (S1).
${\cal H}(V)$ satisfies the utilitariansim axiom (H2) for the grand coalition $N$ only, whereas $\boldsymbol{\lambda}$ satisfies it for all coalitions from condition (S1).
As for equity, $\boldsymbol{\lambda}$ requires it for $N$ only from condition (S2), and ${\cal H}(V)$ for all coalitions from axiom (H3) or (H5′).
When there are only two players, i.e., $|N|=2$, the two solutions are easily seen to coincide (for coalistions $S$ consisting of a single player, both (H5′) and (S1) impose no restrictions). In case $V$ corresponds to a two-person pure bargaining problem (the only additional requirement being that there exists at least one agreement which is beneficial to both participants; formally, that $V(\{1\})\times V(\{2\})\subseteq\mbox{int}V(N)$, where $N=\{1,2\}$), the two solutions coincide with the Nash bargaining solution (Nash \cite{nas}; ${\bf x}_{\{1\}}$ and ${\bf x}_{\{2\}}$ are the disagreement payoffs, and ${\bf x}^{N}$ the agreement). (Hart \cite[p.1303]{har85}).

Since both the Harsanyi and the Shapley solution are extensions of the Nash solution, it is of interest to consider also their intersection $K$, given by ${\cal HS}(V)={\cal H}(V)\cap \boldsymbol{\lambda}(V)$ for all $V\in\Gamma^{N}$. We call it the Harsanyi-Shapley solution function (Hart \cite[p.1303]{har85}).

\begin{equation}{\label{har85tb}}\tag{13}\mbox{}\end{equation}

Theorem \ref{har85tb}. (Hart \cite[p.1303]{har85}). The Harsanyi-Shapley solution function ${\cal HS}(V)$ is the minimal (relative
to set inclusion) solution function of $\Gamma^{N}$ satisfying axioms (HA1)–(HA5). $\sharp$

Theorems \ref{har85ta} and \ref{har85tb} together state that any solution function $\boldsymbol{\Psi}$ that is defined on $\Gamma^{N}$ and satisfies axioms (HA1)-(HA5) must satisfy
\[{\cal HS}(V)\subseteq\boldsymbol{\Psi}(V)\subseteq {\cal H}(V)\]
for all NTU games $(N,V)\in\Gamma^{N}$. Moreover, the two extreme functions ${\cal HS}(V)$ and ${\cal H}(V)$ also satisfy the axioms (HA1)–(HA5) (Hart \cite[p.1304]{har85}).

A direct characterization of ${\cal HS}(V)$ is as follows. The payoff configuration ${\bf x}\in {\cal HS}(V)$ if and only if both axioms (S1) and (H5′) are satisfied. Indeed, condition (S1) includes axioms (H1) and (H2), and axiom (H5′) includes condition (S2). From this, we further obtain that ${\bf x}\in {\cal HS}(V)$ if and only if there exists a positive vector $\boldsymbol{\lambda}$ in $\mathbb{R}^{N}$ and a TU game $(N,w)$ satisfying
\begin{equation}{\label{har85eq53}}\tag{14}
{\bf h}(w)=\boldsymbol{\lambda}{\bf x}\in\boldsymbol{\lambda}V\subseteq W,
\end{equation}
where $(N,W)\in\Gamma^{N}$ corresponds to $(N,w)$ (note that $w(S)=\langle\boldsymbol{\lambda}^{S},{\bf x}^{S}\rangle$ for all $S$). Thus, ${\cal HS}(V)$ is nonempty if and only if, after appropriate re-scaling, the NTU game $(N,V)$ is contained in an NTU game $(N,W)$ corresponding to a TU game $(N,w)$, and it contains its unique Harsanyi solution ${\bf h}(w)$. For a general game $(N,V)$, the two sets ${\cal H}(V)$ and $\boldsymbol{\lambda}(V)$ may well be incomparable. By Theorem \ref{har85ta}, $\boldsymbol{\lambda}$ cannot satisfy all axioms (HA1)–(HA5). Indeed, we have the following result.

Proposition. (Hart \cite[p.1304]{har85}). For the NTU games in $\Gamma^{N}$, the Shapley solution function $\boldsymbol{\lambda}$ satisfies axioms (HA1)–(HA4) and does not satisfy axioms (HA5) and (HA6). $\sharp$

Let us replace axiom (HA5) by the following axiom.

(HA5′). For every nonempty coalition $T\subseteq N$ and every real number $c$, we have
\[\boldsymbol{\Psi}(U_{T,c})=\left\{{\bf x}\in X:{\bf x}^{N}=\frac{c}{|T|}\cdot{\bf 1}_{T}\mbox{ and }\sum_{i\in S}x^{S}_{i}=u_{T,c}(S)\mbox{ for all }S\neq N\right\}.\]

\begin{equation}{\label{har85tc}}\tag{15}\mbox{}\end{equation}

Theorem \ref{har85tc}. (Hart \cite[p.1304]{har85}). The Shapley solution function $\boldsymbol{\lambda}$ is the minimal (relative to set inclusion) solution function defined on $\Gamma^{N}$ satisfying axioms (HA1)–(HA4) and (HA5′). $\sharp$

The Harsanyi and the Shapley solutions differ in axiom (HA5) versus axiom (HA5′). Note that (HA5) prescribes to a unanimity game the smallest conceivable set of solutions (just one), whereas B5 yields the largest such set (of course, consistent with the Shapley TU value). This leads to the maximality versus minimality requirements in Theorems \ref{har85ta} and \ref{har85tc}, respectively. The fact that the Shapley solution is not unique for unanimity games — the outcome of subcoalitions $S\neq N$ not being uniquely determined — yields, when applying the other axioms, different solutions for other games as well. This nondeterminacy for $S\neq N$ suggests considering, instead of payoff configurations, payoff vectors for the grand coalition $N$ only.

Let $\Gamma_{\Lambda}^{N}$ be the subset of games in $\Gamma^{N}$ that are monotone, i.e., satisfying condition (iv) in Definition \ref{aum85d1}, and have a Shapley solution. A {\bf value function} $\boldsymbol{\phi}$ on $\Gamma_{\Lambda}^{N}$ is a set-valued function that assigns to every game $V$ in $\Gamma_{\Lambda}^{N}$ as subset $\boldsymbol{\phi}(V)$ of $\mathbb{R}^{N}$, where we distinguish between “solution” and “value”; the former is a payoff configuration, and the latter a payoff vector for $N$. Let us consider the Axioms (S1)–(S7) in \S\ref{aum85} without needing the closure invariance axiom, since we only consider games with $V(S)$ is closed set for all $S$. Then we have Theorem \ref{aum85t}. To compare the differences, let $\Gamma_{\cal H}^{N}$ be the subset of games in $\Gamma^{N}$ that have at least one Harsanyi solution, i.e., $\Gamma_{\cal H}^{N}=\{V\in\Gamma^{N}:{\cal H}(V)\neq\emptyset\}$. Let $(HA5”)$ be the axiom (HA5) stated for unanimity games with $c=1$ only, and let (HA6) be the following axiom:

(HA6: Nonemptiness). $\boldsymbol{\phi}(V)\neq\emptyset$.

Then we have the following interesting result.

\begin{equation}{\label{har85td}}\tag{16}\mbox{}\end{equation}

Theorem \ref{har85td}. (Hart \cite[p.1305]{har85}). There exists a unique solution function defined on $\Gamma_{\cal H}^{N}$ satisfying axioms (HA1)–(HA4), (HA5”) and (HA6). Moreover, it is the Harsanyi solution function. $\sharp$

Theorems \ref{aum85t} and \ref{har85td} should be viewed in parallel: the two axiom systems differ only through the consideration of all coalitions in the latter, versus the grand coalition only in the former. It is remarkable that essentially the same axioms, stated in two contexts, characterize both Harsanyi’s and Shapley’s functions. The comparison between the two axiom systems are given below.

Axioms (HA6), (HA1), (HA2) and (HA5”) are similar to (S1), (S2), (S6) and (S4), respectively.
Axiom (HA3) is weaker than (S3), since the sum has to be efficient for all $S$ in (HA3), and only for $N$ in (S3).
Axiom (HA4) is stronger than (S7), since one may decrease the feasible set of any coalition in (HA4), but only that of $N$ in (S7).
Both the Harsanyi and the Shapley solutions satisfy (HA4) (and thus also (S7)); as for conditional additivtiy, the stronger requirement (HA3) holds for the Shapley, but not for the Harsanyi solution.

Finally, some evidences suggest that the Shapley NTU solution (value) is best suited for large games. (Hart \cite[p.1306]{har85} and Hart \cite[p.1308-1309]{har85}).

An Axiomatization of Harsanyi-Shapley Solution.

Let ${\cal U}$ be the universe of players. A coalition is an nonempty finite subset of ${\cal U}$. If $N$ is a coalition, any nonempty proper
subset of $N$ is called a subcoalition of $N$. Here, the player set $N$ is not fixed, i.e., the population is variable. An NTU game is a pair $(N,V)$, where $N$ is a coalition and $V$ is a set-valued function that assigns to every $S\subseteq N$ a closed, convex and comprehensive nonempty subset of $\mathbb{R}^{S}$.

The NTU game $(N,V)$ is called {\bf essential} if and only if
\begin{equation}{\label{cha03eq1}}\tag{17}
\prod_{i\in S}V(i)\subseteq V(S)\mbox{ for each coalition }S\subseteq N.
\end{equation}
The game $(N,V)$ is called smooth when $V(S)$ is smooth for each coalition $S\subseteq N$.

We denote by $\bar{\Gamma}({\cal U})$ the set of all essential and smooth games. For any coalition $S\subseteq N$, the NTU game $(S,V^{S})$ is the subgame of $(N,V)$ which $V^{S}$ is obtained by restricting $V$ to subsets of $S$ only (Chang and Hwang \cite[p.255]{cha03}).

We consider the subset $\bar{\Gamma}^{\prime}({\cal U})$ of $\bar{\Gamma}({\cal U})$. The concepts of value function and solution function are defined below.

Let $\boldsymbol{\Psi}$ be a function defined on $\bar{\Gamma}^{\prime}({\cal U})$ with $\boldsymbol{\Psi}(N,V)\in V(N)$, where $(N,V)\in\bar{\Gamma}^{\prime}({\cal U})$. Such a function is called a value function on $\bar{\Gamma}^{\prime}({\cal U})$. The image of a value function may be interpreted as the final payoff which the players will agree to or an arbitrator will recommend when the grand coalition is formed.
A solution function $\boldsymbol{\Psi}$ on $\bar{\Gamma}^{\prime}({\cal U})$ is a function that assigns to each NTU game $(N,V)\in\bar{\Gamma}^{\prime}({\cal U})$ a payoff configuration $\boldsymbol{\Psi}(N,V)=(\boldsymbol{\Psi}^{S}(N,V))_{S\subseteq N}$, where $\boldsymbol{\Psi}^{S}(N,V)\in V(S)$ for each coalition $S$ in $N$. We may say that $\boldsymbol{\Psi}^{S}(N,V)$ is the payoff vector that the members of $S$ agree upon and $\Psi_{i}^{S} (N,V)$ is the amount that player $i$ of $S$ will receive if the coalition $S$ forms. Another interpretation is to view $\boldsymbol{\Psi}^{S} (N,V)$ as an optimal threat of coalition $S$ against its complement $N\setminus S$ in the bargaining to form the grand coalition. (Chang and Hwang \cite[p.256]{cha03}).

We shall say that $\boldsymbol{\lambda}$ is a comparison vector with respect to the NTU game $(N,V)$ if $\boldsymbol{\lambda}$ is positive and $v_{\lambda}(S)$ defined in (\ref{aum85eq42}) is finite for each $S\subseteq N$. A payoff configuration ${\bf x}=({\bf x}^{S})_{S\subseteq N}\in\prod_{S\subseteq N}V(S)$ is called a {\bf Harsanyi-Shapley solution} if and only if there exists a comparison vector $\boldsymbol{\lambda}$ such that $\boldsymbol{\lambda}^{S}{\bf x}^{S}=\boldsymbol{\phi}(S,v_{\lambda})$ for each $\emptyset\neq S\subseteq N$, where $\boldsymbol{\phi}(S,v_{\lambda})$ is the Shapley value for TU game $(S,v_{\lambda})$. The component ${\bf x}^{N}$ of a Harsanyi-Shapley solution ${\bf x}$ is called a Harsanyi-Shapley value. We see that the Harsanyi-Shapley solution is both a Harsanyi solution and a Shapley solution. Given a Harsanyi solution ${\bf y}=({\bf y}^{S})_{S\subseteq N}$ of $(N,V)$, we also see that $\boldsymbol{\lambda}^{S}{\bf x}^{S}$ is the Shapley value of $(S,v)$, where $v$ is defined by $v(S)=\boldsymbol{\lambda}^{S} {\bf x}^{S}$ for $S\subseteq N$ and $\boldsymbol{\lambda}$ is a positive vector in $\mathbb{R}^{N}$. It is easy to see that ${\bf y}^{S}$ does not necessarily satisfy utilitarianism axiom (H2). However, for a Harsanyi-Shapley solution ${\bf y}=({\bf y}^{S})_{S\subseteq N}$, ${\bf y}^{S}$ satisfies utilitarianism axiom (H2) for each coalition $S$ in $N$, since $\boldsymbol{\lambda}^{S}{\bf y}^{S}$ is the Shapley value of the game $(S,v_{\lambda})$. Hence, a Harsanyi-Shapley solution satisfies efficiency axiom (H1), utilitarianism axiom (H2) and equity axiom (H3) for all coalitions. We denote by $\bar{\Gamma}_{HS}({\cal U})({\cal U})$ the subset of $\bar{\Gamma}({\cal U})$ that have a Harsanyi-Shapley solution. The set of Harsanyi-Shapley solution of an NTU game $(N,V)$ is denoted by ${\cal HS}(N,V)$ and the set of Harsanyi-Shapley values of $(N,V)$ is denoted by $HS(N,V)$ (Chang and Hwang \cite[p.257]{cha03}).

Now we shall consider the unique solution for games in $\bar{\Gamma}_{HS}({\cal U})$. The following results were derived by Aumann \cite{aum85}. (It may be something wrong, since the definitions of NTU games are different.)

Lemma. (Chang and Hwang \cite[p.257]{cha03}). For every $(N,V)\in\bar{\Gamma}({\cal U})$, er have the following properties.

(i) We have $HS(N,V)\subseteq\partial V(N)$.

(ii) Let ${\bf y}\in HS(N,V)$ and $\boldsymbol{\lambda}\in\mathbb{R}^{N}$. Then $\boldsymbol{\lambda}$ is a comparison vector if and only if $\boldsymbol{\lambda}$ is a positive multiple of $\boldsymbol{\lambda}(N(V),{\bf y})$. $\sharp$

Aumann \cite{aum85} mentioned that $(N,V)\in\bar{\Gamma}$ such that $|N|=1$ or $|N|=2$ has a unique Shapley NTU value. Hence, it has a unique HS value (Chang and Hwang \cite[p.257]{cha03}).

Theorem. (Chang and Hwang \cite[p.258]{cha03}). Let $(N,V)\in\bar{\Gamma}_{HS}({\cal U})$. Then, we have
\[|HS(N,V)|=1=|{\cal HS}(N,V)|. \sharp\]

Since $HS(N,V)$ is the set of all Harsanyi-Shapley values of the game $(N,V)$ and $|HS(N,V)|=1$ on $\bar{\Gamma}_{HS}({\cal U})$, we will denote the Harsanyi-Shapley value function to be $HS(N,V)$ when we characterize the Harsanyi-Shapley values on $\bar{\Gamma}_{HS}({\cal U})$ if no confusion arises. Similarly, since $|{\cal HS}(N,V)|=1$ on $\bar{\Gamma}_{HS}({\cal U})$, we will denote the Harsanyi-Shapley solution function to be ${\cal HS}(N,V)$ (Chang and Hwang \cite[p.258]{cha03}).

We are going to characterize the Harsanyi-Shapley values on $\bar{\Gamma}_{HS}({\cal U})({\cal U})$. Given an NTU game $(N,V)$, let $\emptyset\neq T\subset N’\subseteq {\cal U}$ with $|N|=|N’|$. For each bijective function $\pi :N\rightarrow N’$, the NTU game $(N’,\pi V)$ is defined by
\[(\pi V)(T)=\left\{{\bf z}\in\mathbb{R}^{T}:\mbox{there exists $latex {\bf y}\in
V(\pi^{-1}(T))$ such that $z_{i}=y_{\pi^{-1}(i)}$ for all $i\in T$}\right\}.\]
Given the function $\pi$, we denote by $\pi^{*}$ the corresponding transformation from $\mathbb{R}^{N}$ to $\mathbb{R}^{N’}$. For each coalition $S\subseteq N$, $\pi_{S}$ denotes the restriction of $\pi$ to $S$, and $\pi_{S}^{*}$ is the corresponding transformation of $\pi_{S}$. For instance, if $N=\{1,2,3\}$, $N’=\{4,5,6\}$, $S=\{1,2\}$ and $\pi (1)=4,\pi (2)=5,\pi(3)=6$, then $\pi^{*}(x_{1},x_{2},x_{3})=(x_{4},x_{5},x_{6})$ and $\pi_{\{1,2\}}^{*}(x_{1},x_{2})=(x_{4},x_{5})$. Let $(N,V_{0})$ represent the NTU game corresponding to the TU game $(N,v_{0})$, where $v_{0}(S)=0$ for all $S\subseteq N$. Let $N,N’\subseteq {\cal U}$ and let $(N,V)$ and $(N,W)$ be NTU games in $\bar{\Gamma}^{\prime}({\cal U})$. Let $\boldsymbol{\Psi}$ be a value function defined on
$\bar{\Gamma}^{\prime}({\cal U})$. We consider the following axioms.

(HS1: Efficiency). $\boldsymbol{\Psi}(N,V)\in\partial V(N)$;
(HS2: Anonymity). For each bijective function $\pi :N\rightarrow N’$ with $(N’,\pi V)\in\bar{\Gamma}^{\prime}({\cal U})$, we have $\boldsymbol{\Psi}(N’,\pi V)=\pi^{*}(\boldsymbol{\Psi}(N,V))$. Note that anonymity implie symmetry;
(HS3: Scale Covariance). If $\boldsymbol{\lambda}$ is a positive vector in $\mathbb{R}^{N}$ and $(N,\boldsymbol{\lambda}V)\in \bar{\Gamma}^{\prime}({\cal U})$, then $\boldsymbol{\Psi}(N,\boldsymbol{\lambda}V)=\boldsymbol{\lambda}\boldsymbol{\Psi}(N,V)$;
(HS4: Independence of Irrelevant Alternatives)(IIA). Let $V(N)\subseteq W(N)$ and $V(S)=W(S)$ for all $S\neq N$. If $\boldsymbol{\Psi}(N,W)\in V(N)$, then $\boldsymbol{\Psi}(N,V) =\boldsymbol{\Psi}(N,W)$.

We recall an axiomatiztion of the Shapley value on TU games by Young \cite{you85}. Let $\boldsymbol{\phi}$ be a solution function on $\mbox{TU}(N)$. A value function satisfies {\bf independence condition} if and only if, for some $i\in N$, $v(S\cup\{i\})-v(S)=w(S\cup\{i\})-w(S)$ for all $S \subseteq N$ implies $\phi_{i}(N,v)=\phi_{i}(N,w)$. The idea of independence condition is very closed related to monotonicity. It is a notion somewhat stronger than monotonicity. Young \cite{you85} presented that the Shapley value is the unique solution function on $\mbox{TU}(N)$ satisfying independence condition, symmetry and efficiency. In order to extend the idea of independence condition to the context of NTU games, we shall introduce the following notations. Given $i\in S\subseteq N$, we define
\[V_{i}(S)=\left\{{\bf y}\in\mathbb{R}^{S}:{\bf y}^{S\setminus\{i\}}\in V(S\setminus\{i\})\mbox{ and }y_{i}=0\right\}\]
and
\[\Delta_{i}V(S)=(V(S)-V_{i}(S))\cup (V_{i}(S)-V(S)).\]

Lemma. (Chang and Hwang \cite[p.262]{cha03}). Let $(N,V)$ and $(N,W)$ be two NTU games in $\bar{\Gamma}({\cal U})$. Let $i\in S\subseteq N$ and ${\bf t}\in\mathbb{R}^{S}$ such that $t_{i}=0$. If $\Delta_{i}V(S)=\Delta_{i}W(S)+\{{\bf t}\}$, then $V(S)=W(S)+\{{\bf t}\}$ and $V(S\setminus\{i\})=W(S\setminus\{i\})+\{{\bf t}^{S\setminus\{i\}}\}$. $\sharp$

Now, we consider the following axiom.

(HS5: Independence Condition). For each $S\subseteq N$ with $i\in S$, if $\Delta_{i}V(S)=\Delta_{i}W(S)+\{{\bf t}\}$ for some ${\bf t}\in\mathbb{R}^{S}$ with $t_{i}=0$, then $\psi_{i}(N,V)=\psi_{i}(N,W)$.
(HS6: Independence of Irrelevant Expansions)(IIE). Suppose that $V(T)\subseteq W(T)$ for $T\subseteq N$, and $V(S)=W(S)$ for $S\subseteq N$ and $S\neq T$. If $(T,V)$ and $(T,W)$ are in $\bar{\Gamma}^{\prime}$ and $\boldsymbol{\Psi}(T,V)=\boldsymbol{\Psi}(T,W)$, then $\boldsymbol{\Psi}(N,V)=\boldsymbol{\Psi}(N,W)$.
(HS7: Strong Independence of Irrelevant Alternatives)(SIIA). Let the NTU games $(N,V)$, $(N,W)$ and all their subgames be in $\bar{\Gamma}^{\prime}({\cal U})$. Let $V(S)\subseteq W(S)$ for all $S\subseteq N$ and $\boldsymbol{\Psi(S,V)=\boldsymbol{\Psi}(S,W)$ for all $S\subseteq N$ and $S\neq N$. If $\boldsymbol{\Psi}(N,W)\in V(N)$, then $\boldsymbol{\Psi}(N,W)=\boldsymbol{\Psi}(N,V)$.

The idea of independence condition to the context of TU games is that the solution does not change even though the underlying characteristic functions do change. Although the characteristic functions do change from $v$ to $w$, the differences $v(S)-v(S\setminus\{i\})$ and $w(S)-w(S\setminus\{i\})$ remain the same. Axiom (HS5) generalizes the idea of the independence condition to the context of NTU games. It also keeps the differences $\Delta_{i}V(S)$ and $\Delta_{i}W(S)$ the same except by a constant. On the other hand, IIE can be interpreted as follows. $(N,V)$ expands to $(N,W)$ in favor of coalition $T$, since $V(T)\subseteq W(T)$ and $V(S)=W(S)$ for $S\neq T$. If this kind of expansion does not change the bargaining position of coalition $T$ against its complement $N-T$ to form the grand coalition, it is reasonable to expect that the payoff vector of grand coalition will not change. It is quite clear to see that SIIA is very close related to IIA, and SIIA implies IIA. (Chang and Hwang \cite[p.263-264]{cha03}).

Proposition. (Chang and Hwang \cite[p.264]{cha03}). Let $\boldsymbol{\Psi}$ be a value function defined on $\bar{\Gamma}_{HS}({\cal U})$. Then $\boldsymbol{\Psi}$ satisfies SIIA if and only if $\boldsymbol{\Psi}$ satisfies IIA and IIE. $\sharp$

Lemma. (Chang and Hwang \cite[p.263]{cha03}). The Harsanyi-Shapley value on $\bar{\Gamma}_{HS}({\cal U})$ satisfies the axioms (HS1)–(HS6). $\sharp$

\begin{equation}{\label{cha03l10}}\tag{18}\mbox{}\end{equation}

Lemma \ref{cha03l10}. (Chang and Hwang \cite[p.263]{cha03}). Let $\boldsymbol{\Psi}$ be a value function defined on $\bar{\Gamma}_{HS}({\cal U})$ satisfying axioms (HS1), (HS2) and (HS5). If the NTU game $(N,V)$ corresponds to a TU game $(N,v)$, then $\boldsymbol{\Psi}(N,V)$ is the Shapley value of $(N,v)$. $\sharp$

\begin{equation}{\label{cha03l12}}\tag{19}\mbox{}\end{equation}

Lemma \ref{cha03l12}. (Chang and Hwang \cite[p.264]{cha03}). Let $\boldsymbol{\Psi}$ be a value function defined on $\bar{\Gamma}_{HS}({\cal U})$ satisfying axioms (HS1)–(HS6). Then $\boldsymbol{\Psi}$ is equal to the Harsanyi-Shapley value on $\bar{\Gamma}_{HS}({\cal U})$. $\sharp$

\begin{equation}{\label{cha03t13}}\tag{20}\mbox{}\end{equation}

Theorem \ref{cha03t13}. (Chang and Hwang \cite[p.264]{cha03}). There exists a unique value function defined on $\bar{\Gamma}_{HS}({\cal U})$ satisfying axioms (HS1)–(HS6), and it is the Harsanyi-Shapley value.

Proof. The result follows from Lemmas \ref{cha03l10} and \ref{cha03l12}. $\blacksquare$

Let $\bar{\Gamma}_{HS}({\cal U},n)$ be the set of games $(N,V)$ in $\bar{\Gamma}_{HS}({\cal U})$ with $|N|=n$. Recall that $(N,V_{0})$ represents the NTU game corresponding to the TU game $(N,v_{0})$, where $v_{0}(S)=0$ for all $S\subseteq N$. Let $(N,V),(N,U),(N,W)$ be in $\bar{\Gamma}_{HS}({\cal U},n)$. We define $(V+W)(S)=\{{\bf x}+{\bf y}:{\bf x}\in V(S),{\bf y}\in W(S)\}$ for $\emptyset\neq S\subseteq N$. We also consider the following axioms.

(HS8: Unanimity Games). If $U_{T}$ is a unanimity game on $T$, then $\boldsymbol{\Psi}(N,U_{T})=({\bf 1}_{T}/|T|,{\bf 0}_{N\setminus T})$;
(HS9: Conditional Additivity). If $U=V+W$ and $(\boldsymbol{\Psi}(N,V)+\boldsymbol{\Psi}(N,W))\in\partial U(N)$, then $\boldsymbol{\Psi}(N,V)+\boldsymbol{\Psi}(N,W))=\boldsymbol{\Psi}(N,U)$.

The following lemma is proved by Aumann \cite{aum85} (Chang and Hwang \cite[p.269]{cha03}).

Lemma. (Chang and Hwang \cite[p.269]{cha03}). The Harsanyi-Shapley value defined on $\bar{\Gamma}_{HS}({\cal U},n)$ satisfies axioms (HS1), (HS3), (HS4), (HS8) and (HS9). $\sharp$

Lemma. (Chang and Hwang \cite[p.269]{cha03}). Let $\boldsymbol{\Psi}$ be a value function defined on $\bar{\Gamma}_{HS}({\cal U},n)$ that satisfies axioms (HS1), (HS3), (HS4), (HS8) and (HS9). For any $(N,V)\in \bar{\Gamma}_{HS}({\cal U},n)$ corresponding to a TU game $(N,v)$, $\boldsymbol{\Psi}(N,V)$ is equal to the Harsanyi-Shapley value. $\sharp$

\begin{equation}{\label{cha03t24}}\tag{21}\mbox{}\end{equation}

Theorem \ref{cha03t24}. (Chang and Hwang \cite[p.270]{cha03}). There exists a unique value function $\boldsymbol{\Psi}$ defined on
$\bar{\Gamma}_{HS}({\cal U},n)$ satisfying axioms (HS1), (HS3), (HS4), (HS8) and {(HS9). Moreover, it is the Harsanyi-Shapley value on $\bar{\Gamma}_{HS}({\cal U},n)$. $\sharp$

Let $(N,V)\in\bar{\Gamma}_{HS}({\cal U},n)$.

We say that $(N,V)$ satisfies weak monotonicity if and only if, for each coalition $S$, there is a ${\bf x}\in\mathbb{R}^{N}$ such that $V(S)\times\{{\bf 0}^{N\setminus S}\}\subseteq V(N)+\{{\bf x}\}$.
We say that $(N,V)$ satisfies the non-levelness if ${\bf x},{\bf y}\in\partial V(N)$ and ${\bf x}\geq {\bf y}$, then ${\bf x}={\bf y}$.

Let $\bar{\Gamma}_{HS}^{0}({\cal U},n)$ denote the subclass of $\bar{\Gamma}_{HS}({\cal U},n)$ satisfying the conditions of non-levelness and weak monotonicity. If we restrict the class of games on $\bar{\Gamma}_{HS}^{0}({\cal U},n)$, we can obtain Theorem \ref{cha03t24} by deleting axiom (HS4) (Chang and Hwang \cite[p.270]{cha03}).

\begin{equation}{\label{cha03t25}}\tag{22}\mbox{}\end{equation}

Theorem \ref{cha03t25}. (Chang and Hwang \cite[p.270]{cha03}). The Harsanyi-Shapley value is the unique value function satisfying the axioms (HS1), (HS3), (HS8) and (HS9) on $\bar{\Gamma}_{HS}^{0}({\cal U},n)$. $\sharp$

In the sequel, we are going to characterize the Harsanyi-Shapley solution. In fact, the same axioms which we used to characterize the Harsanyi-Shapley value in Theorem \ref{cha03t13} are enough to do it. We only need to rewrite these axioms in terms of payoff configurations instead of payoff vectors. We consider the following axioms for the solution function $\boldsymbol{\Psi}$:

(HSS1: Efficiency). $\boldsymbol{\Psi}(N,V)\subseteq\prod_{S\subseteq N}\partial V(S)$;
(HSS2: Anonymity). For each bijective function $\pi :N\rightarrow N’$ with $(N’,\pi V)\in\bar{\Gamma}^{\prime}({\cal U})$, we have $\boldsymbol{\Psi}_{\pi (S)}(N’,\pi V)=\pi_{S}^{*}(\boldsymbol{\Psi}^{S}(N,V))$ for each $S\subseteq N$;
(HSS3: Scale Covariance). If $\boldsymbol{\lambda}$ is a positive vector in $\mathbb{R}^{N}$ and $(N,\boldsymbol{\lambda}V)\in\bar{\Gamma}^{\prime}({\cal U})$, then $\boldsymbol{\Psi}(N,\boldsymbol{\lambda}V)=\boldsymbol{\lambda}\boldsymbol{\Psi}(N,V)$;
(HSS4: Independence of Irrelevant Alternatives)(IIA). Let $V(S)\subseteq W(S)$ for all $S\neq N$. If $\boldsymbol{\Psi}(N,W)\in\prod_{S\subseteq N}V(S)$, then $\boldsymbol{\Psi}(N,V)=\boldsymbol{\Psi}(N,W)$;
(HSS5: Independence Condition). For each $S\subseteq N$ with $i\in S$, if $\Delta_{i}V(S)=\Delta_{i}W(S)+\{{\bf t}\}$ for some ${\bf t}\in\mathbb{R}^{S}$ with $t_{i}=0$, then $\boldsymbol{\Psi}_{i}(N,V)=\boldsymbol{\Psi}_{i}(N,W)$, where $\boldsymbol{\Psi}_{i}(N,V)=(\Psi^{S}_{i}(N,V))_{i\in S\subseteq N}$;
(HSS6: Independence of Irrelevant Expansions)(IIE). Suppose that $V(T)\subseteq W(T)$ for $T\subseteq N$, and $V(S)=W(S)$ for $S\subseteq N$ and $S\neq T$. If $(T,V)$ and $(T,W)$ are in $\bar{\Gamma}^{\prime}({\cal U})$ and $\boldsymbol{\Psi}(T,V)=\boldsymbol{\Psi}(T,W)$, then $\boldsymbol{\Psi}(N,V)=\boldsymbol{\Psi}(N,W)$.

Theorem. (Chang and Hwang \cite[p.264]{cha03}). There exists a unique solution function defined on $\bar{\Gamma}_{HS}({\cal U})$
satisfying axioms (HSS1)–(HSS6), and it is the Harsanyi-Shapley solution.

Proof. The proof is analog to the proof of Theorem \ref{cha03t13}. $\blacksquare$

Theorems \ref{cha03t24} and \ref{cha03t25} can be extended from the value function to the solution function as well if we modify the axioms of unanimity games and conditional additivity in the context of the solution functuon $\boldsymbol{\Psi}$ as follows.

(HSS7: Unanimity Games). If $U_{T}$ is a unanimity game on $T$, then $\boldsymbol{\Psi}^{S}(N,U_{T})=({\bf 1}_{T}/|T|,{\bf 0}_{S\setminus T})$ if $T\subseteq S$ and $\boldsymbol{\Psi}^{S}(N,U_{T})=0$, otherwise;
(HSS8: Conditional Additivity). If $U=V+W$ and $(\boldsymbol{\Psi}(N,V)+\boldsymbol{\Psi}(N,W))\in\prod_{\emptyset\neq S\subseteq N}\partial U(S)$, then$\boldsymbol{\Psi}(N,V)+\boldsymbol{\Psi}(N,W))=\boldsymbol{\Psi}(N,U)$. (Chang and Hwang \cite[p.271]{cha03}).

An Axiomatization of the Weighted NTU Value.

Here, we consider the NTU games $(N,V)$ in $\Gamma_{1}^{N}$. Let $(N,v)$ be a TU game. We recall that player $i$ is a null player
if $v(S\cup\{i\})=v(S)$ for all $S\subseteq N\setminus\{i\}$. A coalition $S$ is a {\bf partnership} in $v$ if $v(C\cup T)=v(C)$ for each $T\subset S$ with $T\neq S$ and each $C\subseteq N\setminus S$. A coalition is flat in $v$ if $v(S)=\sum_{i\in S}v(i)$. A linear operator $\boldsymbol{\psi}:\mbox{TU}(N)\rightarrow\mathbb{R}^{n}$ is efficient if $v(N)=\sum_{i\in N}\psi_{i}(v)$ for each $v\in \mbox{TU}(N)$. An operator $\boldsymbol{\psi}$ satisfies the null player property if $\psi_{i}(v)=0$ whenever $i$ is a null player in $v$ (Levy and McLean \cite[p.341]{lev91}).

Let $v_{\lambda}(S)$ be a TU game defined in \ref{aum85eq42}. For each ${\bf w}\in\mathbb{R}_{++}^{N}$ and ${\bf x}\in V(N)$, we shall call ${\bf x}$ a ${\bf w}$-NTU value payoff vector if and only if $\boldsymbol{\lambda}{\bf x}=\boldsymbol{\phi}^{\bf w}(v_{\lambda})$, where $\boldsymbol{\phi}^{\bf w}(v_{\lambda})$ is the weighted Shapley value of $(N,v_{\lambda})$. The set of all
${\bf w}$-NTU value payoff vectors will be denoted by $\boldsymbol{\phi}^{\bf w}(V)$. The correspondence that associates this set $\boldsymbol{\phi}^{\bf w}(V)$ with each $V\in\Gamma_{1}^{N}$ is called the ${\bf w}$-Shapley correspondence (Levy and McLean \cite[p.342]{lev91}).

We now introduce the axioms that we shall use to characterize the ${\bf w}$-Shapley correspondence. In the axioms below, $(N,U)$, $(N,V)$ and $(N,W)$ are NTU games and $\boldsymbol{\psi}$ is a correspondence that assigns a subset of $\mathbb{R}^{N}$ to each game $V$ in $\Gamma_{1}^{N}$.

(LM0: Non-emptiness). $\boldsymbol{\psi}(V)\neq\emptyset$;

(LM1: Efficiency). $\boldsymbol{\psi}(V)\subseteq\partial V(N)$;
(LM2: Conditional Additivity). If $U=V+W$, then $(\boldsymbol{\psi}(V)+\boldsymbol{\psi}(W))\cap\partial U(N)\subseteq\boldsymbol{\psi}(U)$;
(LM3: Null Players). for each $\emptyset\neq T\subseteq N$, there exists ${\bf q}^{T}\in\mathbb{R}^{N}$ satisfying $\boldsymbol{\psi}(U_{T})=\{{\bf q}^{T}\}$ and $q_{i}^{T}=0$ if $i\not\in T$;
(LM4: Scale Covariance). if $\boldsymbol{\alpha}\in\mathbb{R}_{++}^{N}$, then $latex \boldsymbol{\psi}(\boldsymbol{\alpha}V)
=\boldsymbol{\alpha}\boldsymbol{\psi}(V)$;
(LM5: Independence of Irrelevant Alternatives). if $V(N)\subseteq W(N)$ and if $W(S)=V(S)$ for all $S\neq N$, then $\boldsymbol{\psi}(W)\cap\boldsymbol{\psi}(N)\subseteq\boldsymbol{\psi}(V)$;
(LM6: Positivity). if $v$ is a monotonic TU game without null players and if $V$ is the NTU game corresponding to $v$, then ${\bf x}>{\bf 0}$ for each ${\bf x}\in\boldsymbol{\psi}(V)$;
(LM7: Partnership). let $T$ be a partnership in the TU game $v$ and let $V$ be the NTU game corresponding to $v$, then there exists ${\bf y}\in\boldsymbol{\psi}\left ((\sum_{i\in T}x_{i})U_{T}\right )$ such that $x_{i}=y_{i}$ for each $i\in T$, where $U_{T}$ is the NTU unanimity game corresponding to the TU unanimity game $u_{T}$.

The interpretations of the above axioms are given below.

Axiom (LM1) requires that solutions be Pareto optimal.
Axiom (LM2) states that if ${\bf x}$ is a solution for $V$, ${\bf y}$ is a solution for $W$ and ${\bf x}+{\bf y}$ is a Pareto optimal in $V+W$, then ${\bf x}+{\bf y}$ is a solution for $V+W$.
Axiom (LM3) is an adaptation of the usual null player axiom for TU games to the NTU framework.
Axiom (LM4) says that solutions of a re-scaled problem are simply re-scaled solutions of the original problem.
Axiom (LM5) states that a solution ${\bf x}$ for a game $(N,W)$ remains a solution when payoff vectors differ from ${\bf x}$ are removed from $W(N)$ while $W(S)$ is left unaltered for $S\neq N$.
Axiom (LM6) requires that every player receive a positive payoff in any monotonic game in which each player makes a positive marginal contribution to at least one coalition.
For Axiom (LM7), if $T$ is a parnership in $v$, then the proper subcoalitions of $T$ are “poweless” in $v$ and so $v$ will “behave” as a single unit in $v$. This same type of behavior on the part of $T$ is also reasonable in any game which is a scalar multiple of $u_{T}$. Axiom LM7 requires that if ${\bf x}$ is a solutuon for $V$ which is corresponding to $v$, then ${\bf x}^{T}$ must be part of some solution for the game $(\sum_{j\in T}x_{j})U_{T}$. (Levy and McLean \cite[p.343]{lev91}).

Theorem. (Levy and McLean \cite[p.343]{lev91}). A correspondence $\boldsymbol{\psi}:\Gamma_{1}^{N}\rightarrow\mathbb{R}^{N}$ satisfies axioms {\em (LM0)}-{\em (LM7)} if and only if there exists ${\bf w}\in\mathbb{R}_{++}^{N}$ satisfying $\boldsymbol{\psi}=\boldsymbol{\phi}^{\bf w}$. $\sharp$

Let $\mathbb{P}$ be the probability distribution on the set of all $|N|!$ permutations of $N$. A linear operator $\boldsymbol{\phi}^{\mathbb{P}}:\mbox{TU}(N)\rightarrow\mathbb{R}^{N}$ is defined by
\[\phi_{i}^{\mathbb{P}}(v)=\sum_{\pi\in\Pi (N)}\mathbb{P}(\pi )\cdot m_{i}(v,\pi ).\]
Then $\boldsymbol{\phi}^{\mathbb{P}}$ is efficient. We call ${\bf x}$ a {\bf $\mathbb{P}$-NTU value payoff vector} for $V$ if and only if ${\bf x}\in V(N)$ and $\boldsymbol{\lambda}{\bf x}=\boldsymbol{\phi}^{\mathbb{P}}(v_{\lambda})$. The set of all $\mathbb{P}$-NTU payoff vectors for $V$ is denoted by $\boldsymbol{\phi}^{\mathbb{P}}(V)$ and defines the $\mathbb{P}$-Shapley correspondence. The characterization of the $\mathbb{P}$-Shapley correspondence will utilize a weaker version of the positivity axiom.

(LM6′: Monotonicity). if $v$ is a monotonic TU game and $V$ is the NTU game corresponding to $v$, then ${\bf x}\geq {\bf 0}$ for each ${\bf x}\in \boldsymbol{\psi}(V)$ (Levy and McLean \cite[p.347]{lev91}).

\begin{equation}{\label{lev91tb}}\tag{23}\mbox{}\end{equation}

Theorem \ref{lev91tb}. (Levy and McLean \cite[p.347]{lev91}). A solution correspondence $\boldsymbol{\psi}$ satisfies axioms
(LM0)–(LM5) and (LM6′) if and only if there exists a probability distribution $\mathbb{P}$ such that $\boldsymbol{\psi}=\boldsymbol{\phi}^{\mathbb{P}}$. $\sharp$

We can dispense with axiom (LM6′) in Theorem \ref{lev91tb} and characterize a non-monotonic, non-symmetric, generalization of the NTU value as follows. Let $\Xi$ be the set of efficient linear operator $\boldsymbol{\xi}:\mbox{TU}(N)\rightarrow\mathbb{R}^{N}$ satisfying the null player property. A vector ${\bf x}\in\mathbb{R}^{N}$ is a $\boldsymbol{\xi}$-NTU value payoff vector for $V$ if and only if ${\bf x}\in V(N)$ and $\boldsymbol{\lambda}{\bf x}=\boldsymbol{\xi}(v_{\lambda})$. The collection of $\boldsymbol{\xi}$-NTU payoff vectors of $V$ is denoted by $\boldsymbol{\phi}^{\xi}(V)$. For example, we let
\[\xi_{i}(v)=\sum_{\pi\in\Pi (N)}c(\pi )\cdot m_{i}(v,\pi )\]
for some collection of numbers $\{c(\pi )\}_{\pi\in\Pi (N)}$ with $\sum_{\pi\in\Pi (N)}c(\pi )=1$. This operator is linear, efficient and satisfies the null player property, but it is not monotonic if at least one of the constants is negative (Levy and McLean \cite[p.349]{lev91}).

Theorem. (Levy and McLean \cite[p.349]{lev91}). A solution correspondence $\boldsymbol{\psi}$ satisfies axioms (LM0)–(LM5) if and only if there exists $\boldsymbol{\xi}\in\Xi$ satisfying $\boldsymbol{\psi}=\boldsymbol{\phi}^{\xi}$. $\sharp$

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

NTU Games with Coalition Structure.

Many political and economic confrontations are characterized by having some prior description of the cooperative structure of the players. These structures, which are influenced either by utility considerations of the players or by ideological aspects in political situations, many reflect binding agreements between the players, pressue group, or even descriptions of channels of negotiation between the players. In each of these cases the structures are assumed to affect the bargaining position of the players, and hence the final allocation of the utility between them (Winter \cite[p.53]{win91}).

Games without side payments are widely recognized as a satisfactory formulation of trading in markets. When one considers international trading, the aspect of integration and cooperation becomes extremely relevant. Governments can intervene in international transactions by issuing regulations and by forming agreements with other countries’ governments. One can find many such examples in economic theory, especially in the European economy. In 1948, Belgium, the Netherlands and Luxemburg formed a customs union which was later merged (in 1958) to the European Economic Community. In 1960, Britian, Norway, Sweden, Denmark, Austria, Switzerland and Portugal formed the European Free Trade Area which changed its form later in 1973 when Britian and Denmark left to join the E.E.C. These variations in
the “economic map” of Europe naturally have influenced firms’ bargaining positions, and thus the outcomes of tradings (Winter \cite[p.53-54]{win91}).

The main idea of the treatment is to apply the idea suggested by Harsanyi \cite{har63} for payoff allocations in NTU games and to generalize them to cases where prior coalition structures are given (Winter \cite[p.54]{win91}).

Preliminaries.

Here, we consider the NTU game $(N,V)$ in $\Gamma^{N}$. Let $N$ be a finite set of players. If $\boldsymbol{\lambda}\in\mathbb{R}^{N}$, then $1/\boldsymbol{\lambda}\in\mathbb{R}^{N}$ is defined by
$(1/\boldsymbol{\lambda})_{i}=1/\lambda_{i}$. According to the smoothness of $V(N)$, let us recall that, for all ${\bf x}\in\partial V(N)$, there exists a unique vector $\boldsymbol{\lambda}\in\mathbb{R}^{N}$ satisfying $\max |\lambda_{i}|=1$ such that $\langle\boldsymbol{\lambda},{\bf x}\rangle\geq\langle\boldsymbol{\lambda},{\bf y}\rangle$ for all ${\bf y}\in V(N)$. By the comprehensiveness and the non-levelness, this $\boldsymbol{\lambda}$ is positive (Winter \cite[p.55]{win91}).

Let us revisit the Harsanyi solutions for NTU games. Given an NTU game $(N,V)$, an Harsanyi solution for $V$ is a payoff configuration
defined inductively as follows. Let $\boldsymbol{\lambda}$ be a positive vector in $\mathbb{R}^{N}$ and defined by induction on the size of coalitions
\begin{align*}
\boldsymbol{\xi}^{S} & =\frac{1}{\boldsymbol{\lambda}^{S}}\max\left\{t:\left ({\bf z}^{S}+\frac{t}{\boldsymbol{\lambda}^{S}}
\right )\in V(S)\right\} (\boldsymbol{\xi}_{\emptyset}={\bf 0})\\
{\bf z}^{S} & =(z^{S}_{i})_{i\in S},\mbox{ where }z^{S}_{i}=\sum_{\{T:i\in T\subset S,T\neq S\}}\xi^{T}_{i}\\
{\bf x}^{S} & =(x^{S}_{i})_{i\in S},\mbox{ where }x^{S}_{i}=\sum_{\{T:i\in T\subset S\}}\xi^{T}_{i}.
\end{align*}
If $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf x}^{N})$, then ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ is an Harsanyi solution for $V$. The scenario in the background of this definition is roughly the following. Players obtain dividends from coalitions of which they are members. When the players negotiate within the coalition $S$ they have already received dividends from all its subcoalitions $T\subset S$ and $T\neq S$. The sum of these dividends, namely ${\bf z}^{S}$, induces a point which may be in $V(S)$ or outside $V(S)$. In any case, the dividends that players of $S$ receive from the coalition $S$ is calculated by moving from ${\bf z}^{S}$ towards the boundary of $V(S)$ in the direction $\boldsymbol{\lambda}$. Finally, if ${\bf x}^{N}$ is utilitarian with respect to $\boldsymbol{\lambda}$, namely, $\langle\boldsymbol{\lambda}, {\bf x}^{N}\rangle\geq\langle\boldsymbol{\lambda},{\bf y}\rangle$
for all ${\bf y}\in V(N)$, then ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ is a Harsanyi solution. Note that if $\boldsymbol{\lambda}=(1,\cdots ,1)$, then for each $S\subset N$, the players of $S$ share equal dividends from $S$. This would be essentially different when a prior organization of the players is considered, as will be seen later (Winter \cite[p.55-56]{win91}).

A Solution for NTU Games with Coalition Structure.

If ${\cal B}$ is a coalition structure on $N$ and $i\in N$, we denote by $S(i)$ the coalition in ${\cal B}$ to which $i$ belongs. The players of $S(i)$ are called $i$’s associates with respect to ${\cal B}$. For $i\in T\subseteq N$, let $a^{T}_{i}$ denote the number of $i$’s associates in $T$ with respect to ${\cal B}$, that is, $a^{T}_{i}=|S(i)\cap T|$ and write ${\bf a}^{T}=(a^{T}_{i})_{i\in T}$.

An NTU game with coalition structure is a pair $(N,{\cal B},V)$. A solution for $(N,{\cal B},V)$ is a payoff configuration which is defined
inductively as follows. Let $\boldsymbol{\lambda}$ be a positive vector in $\mathbb{R}_{++}^{N}$ and defined by induction on the size of coalitions as follows:
\begin{align*}
\boldsymbol{\xi}^{S} & =\frac{1}{\boldsymbol{\lambda}^{S}}\frac{1}{{\bf a}^{S}}
\max\left\{t:\left ({\bf z}^{S}+t\frac{1}{\boldsymbol{\lambda}^{S}}\frac{1}{{\bf a}^{S}}\right )\in V(S)
\right\} (\boldsymbol{\xi}_{\emptyset}={\bf 0})\\
{\bf z}^{S} & =(z^{S}_{i})_{i\in S},\mbox{ where }z^{S}_{i}=\sum_{\{T:i\in T\subset S,T\neq S\}}\xi^{T}_{i}\\
{\bf x}^{S} & =(x^{S}_{i})_{i\in S},\mbox{ where }x^{S}_{i}=\sum_{\{T:i\in T\subset S\}}\xi^{T}_{i}.
\end{align*}
Now if $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf x}^{N})$, then ${\bf x}=({\bf x}^{S})_{S\subset N}$ is a solution of $(N,{\cal B},V)$ (Winter \cite[p.57]{win91}).

We use ${\cal B}^{S}$ and $v^{S}$ to denote the coalition structure ${\cal B}$ restricted to $S\subseteq N$ and the TU game $v$ restricted to $S$. Therefore, $R\in {\cal B}^{S}$ if and only if $R=Q\cap S$ for some $Q\in {\cal B}$. (Winter \cite[p.58]{win91}).

\begin{equation}{\label{win91t1}}\tag{24}\mbox{}\end{equation}

Theorem \ref{win91t1}. (Winter \cite[p.58]{win91}). A payoff configuration ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ is a solution of the NTU game with coalition structure $(N,{\cal B},V)$ if and only if it satisfies the following two conditions:

${\bf x}\in\partial V$, i.e., ${\bf x}$ is efficient.
For $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf x}^{N})$ and for all $i\in S\subseteq N$, we have $\lambda_{i}x^{S}_{i}=\phi_{i}(N,{\cal B}^{S},v^{S})$, where $v^{S}(T)=\boldsymbol{\lambda}^{T}\cdot{\bf x}^{T}=\sum_{i\in T}\lambda_{i}^{T}x^{T}_{i}$. $\sharp$

Any efficient payoff configuration ${\bf x}$ with $\boldsymbol{\lambda}=\boldsymbol{\lambda}(V(N),{\bf x})$ can be translated into a TU game $(N,v)$. In this game, $v(T)$ is the sum of the re-scaled payoffs, (re-scaled by $\boldsymbol{\lambda}$), for players in $T$. The second condition in Theorem \ref{win91t1} says that the re-scaled payoffs for players in $S\subseteq N$ should coincide with the Owen CS value for the pair $(N,{\cal B}^{S},v^{S})$. (Winter \cite[p.58]{win91}).

Proposition (Winter \cite[p.60]{win91}). We have the following properties.

(i) Given a TU game $(N,v)$, if $(N,{\cal B},V)$ is an NTU game with coalition structure such that $(N,V)$ is the NTU game which corresponds to $(N,v)$, then there exists a unique solution ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ for $(N,{\cal B},V)$ and ${\bf x}^{S}=\boldsymbol{\phi}(N,{\cal B}^{S},v^{S})$. In particular, ${\bf x}^{N}=\boldsymbol{\phi}(N,{\cal B},V)$.

(ii) If ${\cal B}$ is a trivial coalition structure, i.e., ${\cal B}=\{N\}$ or ${\cal B}=\{\{1\},\{2\},\cdots ,\{n\}\}$, then a payoff configuration ${\bf x}$ is a solution of $(N,{\cal B},V)$ if and only if ${\bf x}$ is a Harsanyi solution of $(N,V)$. $\sharp$

The existence of solution is not guaranteed for all NTU games with coalition structure (just as not every NTU games has a Harsanyi solution). However, if we take the games where the set of all extreme points of $V(N)$ is compact, then for each of these games there always exists a solution for any coalition structure ${\cal B}$. The proof is parallel to a well-known argument which shows the existence of Harsanyi solutions for such games. If the set of exterme points of $V(N)$ is compact, then the set $Z$ of all $latex \boldsymbol{\lambda}=
\boldsymbol{\lambda}\delta (V(N),{\bf x})$, where ${\bf x}\in\partial V(N)$, and satisfies $\sum_{i\in N}\lambda_{i}=1$ is convex and compact. Now, it is easy to show by induction on the coalitions that for $S\subseteq N$, the function $\boldsymbol{\lambda}\mapsto\boldsymbol{\xi}^{S}$ is continuous for any coalition structure ${\cal B}$ and thus also the function $\boldsymbol{\lambda}\mapsto {\bf x}^{S}$. Since the function which matches each point in $\partial V(N)$ with its supporting hyperplanes is itself continuous, we obtain a function $\boldsymbol{\phi}:Z\rightarrow Z$ which has a fixed point $\boldsymbol{\lambda}^{*}$. The point ${\bf x}=({\bf x}^{S})_{S\subseteq N}$ which is obtained from $\boldsymbol{\lambda}^{*}$ by the inductive definition is a solution (Winter \cite[p.60]{win91}).

The Axiom Approach.

We consider the function $\boldsymbol{\psi}$ which assigns to each NTU game with coalition structure $(N,{\cal B},V)$ a subset
$\boldsymbol{\psi}(N,{\cal B},V)$ of the set of all payoff configurations $X$. The function $\boldsymbol{\psi}$ which assigns to each $(N,{\cal B},V)$ the set of all solutions of $(N,{\cal B},V)$ is called the solution function and is denoted by $\boldsymbol{\phi}$. We begin to state the axioms characterizing the solution function $\boldsymbol{\phi}$ from all functions $\boldsymbol{\psi}$. In the following axioms, $\boldsymbol{\psi}$ denotes an arbitrary function, $(N,V)$, $(N,W)$, $(N,U)$ are NTU games in $\Gamma^{N}$,
and ${\cal B}$ is an arbitrary coalition structure.

(GCS1: Efficiency). $\boldsymbol{\psi}(N,{\cal B},V)\subseteq\partial V$;
(GCS2: Scale Covariance). $\boldsymbol{\psi}({\cal B},\boldsymbol{\lambda}V)=\boldsymbol{\lambda}\boldsymbol{\psi}(N,{\cal B},V)$ for all positive vectors $\boldsymbol{\lambda}$ in $\mathbb{R}^{N}$;
(GCS3: Conditional Additivity). If $U=V+W$ then $(\boldsymbol{\psi}(N,{\cal B},V)+\boldsymbol{\psi}(N,{\cal B},W))\cap\partial U\subseteq\boldsymbol{\psi}({\cal B},U)$;
(GCS4: Independence of Irrelevant Alternatives). If $V\subseteq W$ then $latex \boldsymbol{\psi}(N,{\cal B},W)\cap V\subseteq
\boldsymbol{\psi}(N,{\cal B},V)$;
(GCS5: Inessential Games). If ${\bf 0}\in\partial V$ then ${\bf 0}\in\boldsymbol{\psi}(N,{\cal B},V)$;
(GCS6: Unanimity Games). Let $(N,U_{T,c})$ be an NTU game which corresponds to some TU unanimity game $(N,u_{T,c})$. Then $\boldsymbol{\psi}(N,{\cal B},U_{T,c})=\{{\bf z}\}$, where
\[z^{S}_{i}=\left\{\begin{array}{ll}
c/(a^{T}_{i}k_{T}), & i\in T\\ 0, & i\not\in T
\end{array}\right .\]
if $T\subseteq S$ and $z^{S}_{i}=0$ otherwise (Winter \cite[p.61]{win91}).

Axioms (GCS1)–(GCS5) were already formulated in Hart \cite{har85} as axioms characterizing the Harsanyi solution in the case of NTU games without the coalition structure. Therefore, we shall concentrate on axiom (GCS6). Let $U_{T,c}$ be an NTU unanimity game on $T\subseteq N$, where $T$ is called the domain of the game $U_{T,c}$ or the set of non-dummy players. Now we take a coalition structure ${\cal B}$. According to axiom (GCS6), $\boldsymbol{\psi}$ yields a unique payoff configuration for $(N,{\cal B},U_{T,c})$ which is given as follows. If $S$ does not contain the domain of the game, then the payoff for the member if $S$ are zero. However, if $S$ contains all non-dummy players of $U_{T,c}$, then the players of $S$ share payoffs according to the following procedure. First, all components of ${\cal B}$ which have representatives in the domain $T$ share the amount $c$ equally between themselves, getting $c/k_{T}$ each. Next, if $i$ is a non-dummy player in one of these components of ${\cal B}$, he gets from his non-dummy associates an equal share of the total amount, namely, $c/(a^{T}_{i}k_{T})$, while his dummy associates get zero (Winter \cite[p.62]{win91}).

Theorem. (Winter \cite[p.62]{win91}). The only function satisfying axioms (GCS1)–(GCS6) is the solution function $\boldsymbol{\phi}$. $\sharp$

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

The Cores.

Sometimes the games without side payments are also regarded as the NTU games. However, there is a little difference between the games without side payment and the NTU games in the literature. Let $N$ the set of players and let $S$ be a coalition in $N$.

For the games without side payments, $V(S)$ is assumed as a subset of $\mathbb{R}^{N}$.
For the NTU games, $V(S)$ is assumed as a subset of $\mathbb{R}^{S}$.

Here we are going to discuss the cores of games without side payments and the NTU games.

The Cores of Games without Side Payments.

For the games with side payments, we have made the implicit assumption that utility is freely transferable among members of a coalition. This is, in particular, possible in the presence of an “ideal money”, i.e., a commodity whose utility is directly proportional to quantity, and independent of any other assets, or liabilities, which a player may have. Whatever the realism of such a situation, its principal effect is to simplify enormously the description of the game. In fact, a single number $v(S)$ is fully adequate to describe the coalition’s possibilities: it can obtain this total amount in some manner, and then make side payments of money to satisfy any complaints that its members might make. In general, unfortunately, the situation is not this simple. Even if money is available in large amounts, we find that the several players’ utility for money is not linear (e.g., the utility of the $1$st dollar is much greater than that of the $1001$st), and not always independent of their other assets. In some cases, side payments may even be forbidden (e.g., by antitrust or antikickback laws). This being so, we must represent each coalition’s posibilities, not by a single number, but rather by a set: the set of all points which the coaliiton can obtain for its members (Owen \cite[p.367]{owe}).

A characteristic function is a pair $(N,V)$, where $N$ is a finite set and $V$ is a function that carries each subset $S$ of $N$ into a
subset $V(S)$ of $\mathbb{R}^{N}$ such that the following conditions are satisfied:

(a) $V(S)$ is closed and convex;
(b) $V(\emptyset )=\mathbb{R}^{N}$;
(c) if ${\bf x}\in V(S)$ and $y_{i}\leq x_{i}$ for all $i\in S$, then ${\bf y}\in V(S)$;
(d) if $S\cap T=\emptyset$, then $V(S)\cap V(T)\subseteq V(S\cup T)$.

A game in characteristic function form without side payment is a triple $(N,V,H)$, where $(N,V)$ is a characteristic function and $H$ is a convex compact polyhedral subset of $\mathbb{R}^{N}$. Condition (d) is kind of supper additivity. It says that if a certain outcome can be achieved by the disjoint coalitions $S$ and $T$ when acting separately, then it can also be achieved by them when acting in concert. We shall say that $(N,V,H)$ is an ordinary game if the following condition is satisfied:

(e) ${\bf x}\in V(N)$ if and only if there is a ${\bf y}\in H$ satisfying ${\bf x}\leq {\bf y}$.

Fix a game $(N,V,H)$. A payoff vector ${\bf x}$ is said to dominate a payoff vector ${\bf y}$ via $S$ if and only if ${\bf x}\in v(S)$ and $x_{i}>y_{i}$ for all $i\in S$. The payoff vector ${\bf x}$ is said to dominate ${\bf y}$ if and only if there is an $S$ such that ${\bf x}$ dominates ${\bf y}$ via $S$. If $R$ is an arbitrary set of payoff vectors, we define the $R$-core ${\cal C}(V,R)$ to be the set of all members of $R$ not dominated by any other member of $R$. (Aumann \cite[p.542-543]{aum61})

It is easy to show that, for each $i\in N$, there is a real number $\xi_{i}$ such that $V(i)=\{{\bf x}:x_{i}<\xi_{i}\}$, where $\xi_{i}$
can be $\pm\infty$. A payoff vector ${\bf x}$ is called individually rational if and only if $x_{i}\geq\xi_{i}$ for all $i\in N$. A payoff vector ${\bf x}$ is called group rational if and only if there is no ${\bf y}\in H$ satisfying $y_{i}>x_{i}$ for all $i\in N$. We shall denote by $E$ the set of group rational payoff vectors in $H$, and by $\widehat{A}$ the set of individual rational payoff vectors in $H$. We set $A=E\cap\widehat{A}$ and $\widehat{E}=H$. We shall consider the $R$-core for $R=E,\widehat{E},A$ and $\widehat{A}$. For
two-person games, all these cores turn out to be equal to $A$ If $B\subset\mathbb{R}^{N}$, we denote by ${\cal D}(B)$ the set of all members ${\bf x}$ of $B$ for which there is no ${\bf y}$ such that $y_{i}>x_{i}$ for all $i\in N$. For example, $A={\cal D}(\widehat{A})$ and $E={\cal D}(\widehat{E})$ (Aumann \cite[p.543-544]{aum61}).

Theorem. (Aumann \cite[p.545]{aum61}). Let $B$ be a compact polyhedron in $\mathbb{R}^{N}$ and let ${\bf y}\in {\cal D}(B)$.
If there is a ${\bf z}\in B$ which dominates ${\bf y}$, then there is also a ${\bf w}\in {\cal D}(B)$ which dominates ${\bf y}$. $\sharp$

Corollary. (Aumann \cite[p.546]{aum61}). We have the following properties.

(i) If $B$ is a compact polyhedron, then ${\cal C}({\cal D}(B))={\cal C}(B)\cap {\cal D}(B)$.

(ii) We have ${\cal C}(V,E)={\cal C}(V,\widehat{E})\cap E$ and ${\cal C}(V,A)={\cal C}(V,\widehat{A})\cap A$. $\sharp$

Theorem. (Aumann \cite[p.546]{aum61}). If $(N,V,H)$ is an ordinary game, then we have
\[{\cal C}(V,E)={\cal C}(V,\widehat{E})={\cal C}(V,A)={\cal C}(V,\widehat{A}). \sharp\]

If $(N,V,H)$ is an ordinary game, the {\bf core} of $(N,V,H)$ is denoted and defined by
\[{\cal C}(V)={\cal C}(V,E)\cap{\cal C}(V,\widehat{E})\cap{\cal C}(V,A)\cap{\cal C}(V,\widehat{A}).\]
Let $G_{1}=(N_{1},V_{1},H_{1})$ and $G_{2}=(N_{2},V_{2},H_{2})$ be games whose player sets $N_{1}$ and $N_{2}$ are disjoint. The composition $G$ of $G_{1}$ and $G_{2}$ is the game each play of which consists of a play of $G_{1}$ and a play of $G_{2}$, played without any interconnection. Formally, we define $G=(N,V,H)$, where $N=N_{1}\cup N_{2}$, $H=H_{1}\times H_{2}$, and for each $S\subseteq N$, $V(S)=V_{1}(S\cap N_{1})\times V_{2}(S\cap N_{2})$. Let $R_{1}\subseteq\mathbb{R}^{N_{1}}$ and $R_{2}\subseteq\mathbb{R}^{N_{2}}$, and set $R=R_{1}\times R_{2}$. Then it is easily seen that ${\cal C}(V,R)={\cal C (V_{1},R_{1})\times {\cal C}(V_{2},R_{2})$. Furthermore, if $G_{1}$ and $G_{2}$ are ordinary, then so is $G$. It follows that, in this case, the core of $G$ is the cartesian product of the cores of $G_{1}$ and $G_{2}$ (Aumann \cite[p.547-548]{aum61}).

We consider another definition. A game without side payment is a triple $(N,V,H)$, where $H$ is a compact subset of $\mathbb{R}^{N}$ and $V$ is a function which assigns to each $S\subseteq N$ a nonempty subset of $\mathbb{R}^{N}$ satisfying the following conditions:

(i) $V(S)$ is a closed and convex subset of $\mathbb{R}^{N}$;

(ii) if ${\bf x}\in V(S)$ and $y_{i}\leq x_{i}$ for all $i\in S$, then ${\bf y}\in V(S)$;

(iii) if $S\cap T=\emptyset$, then $V(S)\cap V(T)\subseteq V(S\cup T)$;

(iv) ${\bf x}\in V(N)$ if and only if ${\bf x}\leq {\bf y}$ for some ${\bf y}\in H$.

Condition (ii) is a “comprehensiveness” condition; that is, if $S$ can obtain ${\bf x}$, then it can a fortiori obtain the smaller amounts ${\bf y}$. Condition (iii) is superadditivity in a new setting. Condition (iv) guarantees that $V(N)$ is not too large, which, together with condition (iii), letting $T=N\setminus S$, guarantees that $V(S)$ is not too large. We say that ${\bf x}$ belongs to the core of $(N,V,H)$, denoted by ${\cal C}(V)$, if and only if the following conditions are satisfied:

${\bf x}\in V(N)$;
there is no $\emptyset\neq S\subseteq N$ and ${\bf y}\in V(S)$ such that $y_{i}>x_{i}$ for all $i\in S$. (Owen \cite[p.368,372]{owe}).

Theorem. (Owen \cite[p.373]{owe}). A sufficient condition for the game $(N,V,H)$ to have a nonempty core is that the inclusion $\bigcap_{j=1}^{m}V(S_{j})\subseteq V(N)$ hold for any balanced collection $\{S_{1},\cdots ,S_{m}\}$. $\sharp$

Another game without side payment and without considering the set $H$ is given below. A characteristic function game without side payment will be defined by a pair $(N,V)$ satisfying the following conditions:

(i) $V(S)$ is a closed nonempty subset of $\mathbb{R}^{N}$ for every $S\subseteq N$ and $V(N)$ is convex;
(ii) If ${\bf x}=(x_{1},\cdots ,x_{n})\in V(S)$ and $j\not\in S$, then $x_{j}=0$;
(iii) If $S\neq\emptyset$, ${\bf x}\in V(S)$, ${\bf y}\leq {\bf x}$ and $y_{j}=0$ for all $j\not\in S$, then ${\bf y}\in V(S)$;
(iv) $V(\emptyset )=\{{\bf 0}\}$;
(v) $V(i)=\{{\bf x}:x_{i}\leq 0, x_{j}=0\mbox{ if }j\neq i\}$;
(vi) $V(S)\cap\mathbb{R}_{+}^{N}$ is bounded and nonempty for all $S\subseteq N$.

We will define the vector ${\bf x}^{(S)}\in\mathbb{R}^{N}$ such that $x_{i}^{(S)}=x_{i}$ if $i\in S$ and $x_{i}^{(S)}=0$ if $i\not\in S$. We will say that ${\bf x}^{(S)}\in\mbox{ri}(V(S))$ if and only if ${\bf x}^{(S)}+{\bf y}^{(S)}\in V(S)$ for some vector ${\bf y}>{\bf 0}$. The core of a game $(N,V)$ is the set ${\cal C}(V)$ defined by
\[{\cal C}(V)=\left\{{\bf x}\in V(N):{\bf x}^{(S)}\not\in\mbox{ri}(V(S))\mbox{ for all }S\subseteq N\right\}.\]
The vector ${\bf y}$ is said to dominate ${\bf x}$ if there exists a coalition $S$ such that ${\bf y}^{(S)}\in V(S)$ and $y_{i}>x_{i}$ for all $i\in S$. A von Neumann-Morgenstern solution for a game $(N,V)$ is a set $X\subseteq V(N)$ such that no element of $X$ dominates another and every vector ${\bf y}\in V(N)\setminus X$ is dominated by some element in $X$.

A game $(N,V)$ is a {\bf cardinal convex game} if and only if, for all $S,T\subseteq N$, we have $V(S)+V(T)\subseteq V(S\cup T)+V(S\cap T)$.
A game $(N,V)$ is a {\bf ordinal convex game} if and only if, for any $S,T\subseteq N$ and any vector ${\bf x}$ such that ${\bf x}^{(S)}\in V(S)$ and ${\bf x}^{(T)}\in V(S)$, either ${\bf x}^{(S\cup T)}\in V(S\cup T)$ or ${\bf x}^{(S\cap T)}\in V(S\cap T)$ (Sharkey \cite[p.102]{sha}).

Let us recall that a collection of nonempty coalitions $\{S_{1},\cdots ,S_{k}\}$ is said to be balanced if and only if there exists a strictly positive weight vector ${\bf w}=(w_{1},\cdots ,w_{k})$ such that, for every $j=1,\cdots ,n$, we have $\sum_{j\in S_{i}}w_{i}=1$. The game $(N,V)$ is {\bf balanced} if and only if, for every balanced collection $\{S_{1},\cdots ,S_{k}\}$, we have
\[\sum_{i=1}^{k}w_{i}V(S_{i})\subseteq V(N).\]
It is possible to define a subgames $(S,V^{S})$ in an obvious manner for $S\subseteq N$. Since $V(S)=V^{S}(S)$ need not be convex in general, we replace $V(S)$ by its convex hull in each subgame $(S,V^{S})$. Then, the game $(N,V)$ is said to be {\bf totally balanced} if and only if every subgame is balanced. (Sharkey \cite[p.103]{sha}).

Theorem. (Sharkey \cite[p.104]{sha}). A cardinal convex game $(N,V)$ is totally balanced. $\sharp$

For any $S,T\subseteq N$ and any vectors ${\bf x}\in V(S)$, ${\bf y}\in V(T)$, we define the vector ${\bf p}$ such that
\[p_{i}=\left\{\begin{array}{ll}
\max\{x_{i},y_{i}\}, & \mbox{if $i\in S\cap T$}\\
0, & \mbox{if $i\not\in S\cap T$}
\end{array}\right .\]
and the vector ${\bf z}={\bf x}^{(S\setminus T)}+{\bf y}^{(T\setminus S)}+{\bf p}$. Then the game $(N,V)$ is a strong cardinal convex game if, for any ${\bf x}\in V(S)$ and ${\bf y}\in V(T)$, there exist vectors ${\bf u}\in V(S\cup T)$ and ${\bf v}\in V(S\cap T)$ such that ${\bf x}+{\bf y}={\bf u}+{\bf v}$ and either ${\bf u}\geq {\bf z}$ or ${\bf v}\geq {\bf p}$. (Sharkey \cite[p.103]{sha}).

Let $S$ be a coalition in $N$. Recall that the symbol ${\bf x}^{S}$ represents an $|S|$-vector with components corresponding to the players $i\in S$.

Theorem. (Sharkey \cite[p.105]{sha}). Suppose that $(N,V)$ is a strong cardinal convex game and that ${\bf x}^{S}$ is contained in the core of the subgame $(S,V)$. Then there exists a vector ${\bf x}$ in the core of $(N,V)$ such that $x_{i}^{S}=x_{i}$ for all $i\in S$. $\sharp$

Corollary. (Sharkey \cite[p.105]{sha}). In a strong cardinal convex game $(N,V)$, the core and the von Neumann-Morgenstern solution coincide. $\sharp$

The Cores of NTU Games.

Here, we consider the NTU game $(N,V)$ in $\Gamma_{2}^{N}$. Let ${\bf x}\in V(N)$ and $\emptyset\neq S\subseteq N$. The reduced game with respect to $S$ and ${\bf x}$ is the game $(S,V_{\bf x}^{S})$ with
\[V_{\bf x}^{S}(S)=\left\{{\bf y}^{S}:({\bf y}^{S},{\bf x}^{N\setminus S})\in V(N)\right\}\]
and
\[V_{\bf x}^{S}(T)=\bigcup_{Q\subseteq N\setminus S}\left\{{\bf y}^{T}:({\bf y}^{T},{\bf x}^{Q})\in V(T\cup Q)\right\}\mbox{ for }
T\subseteq S\mbox{ with }T\neq S.\]
In the reduced game, the players of $S$ are allowed to choose only payoff vectors ${\bf y}^{S}$ that are compatible with ${\bf x}^{N\setminus S}$, the fixed payoff distribution to the members of $N\setminus S$. On the other hand, proper subcoalitions $T$ of $S$ may count on the cooperation of subsets $Q$ of $N\setminus S$, provided that in the resulting payoff vectors for $T\cup Q$ each member of $Q$ gets exactly $x_{i}$. (Hence, $T$ has to ensure the feasibility of ${\bf x}^{Q}$ but not that of ${\bf x}^{N\setminus S}$.) Thus, the reduced game $(S,V_{\bf x}^{S})$ describes the following situation. Suppose that all the mmebers of $N$ agree that the memebrs of $N\setminus S$ will get ${\bf x}^{N\setminus S}$. Further, assume that the members of $N\setminus S$ continue to cooperate with the members of $S$ (subject to the goregoing agreement). Then $V_{\bf x}^{S}$ describes the possible payoffs that various coalitions of members of $S$ may obtain. However, it is assumed that $S$ will choose some payoff vector in $V_{\bf x}^{S}(S)$. Thus, the sets $V_{\bf x}^{S}(T)$, for $T\subset S$ with $S\neq T$, serve only to determine the final choice in $V_{\bf x}^{S}(S)$ (Peleg \cite[p.205]{pel85}).

Proposition. (Peleg \cite[p.205]{pel85}). Given an NTU game $(N,V)$ in $\Gamma_{2}^{N}$, let ${\bf x}\in V(N)$ and $\emptyset\neq S\subseteq N$. Then the reduced game $(S,V_{\bf x}^{S})$ is an NTU game in $\Gamma_{2}^{N}$. $\sharp$

Given an NTU game $(N,V)$ in $\Gamma_{2}^{N}$, let ${\bf x}\in V(N)$. A coalition $S$ in $N$ can {\bf improve upon} ${\bf x}$ if and only if there exists ${\bf y}^{S}\in V(S)$ such that $y_{i}^{S}>x_{i}^{S}$ for all $i\in S$. We say that ${\bf x}$ is in the core of $(N,V)$, denoted by ${\cal C}(N,V)$, if and only if no coalition can improve upon ${\bf x}$. We denote by $\widehat{\Gamma}_{2}^{N}$ the set of all NTU games $(N,V)$ in $\Gamma_{2}^{N}$ with nonempty core. For ${\bf x}\in V(N)$, we say that ${\bf x}$ is Pareto optimal if and only if there is no ${\bf y}\in V(N)$ such that $y_{i}\geq x_{i}$ for all $i$ and $y_{i}>x_{i}$ for at least one $i$. Because of comprehensiveness and condition (iii) in Definition \ref{pel85d21}, we see that ${\bf x}\in V(N)$ is Pareto optimal if and only if ${\bf x}\in\partial V(N)$. Clearly, we have ${\cal C}(N,V)\subseteq\partial V(N)$; that is, every payoff vector in the core is Pareto optimal (Peleg \cite[p.206]{pel85}).

Proposition. (Peleg \cite[p.206]{pel85}). Given an NTU game $(N,V)$ in $\Gamma_{2}^{N}$, let ${\bf x}\in V(N)$ and $\emptyset\neq S\subseteq N$. Then ${\bf x}$ is Pareto optimal if and only if ${\bf x}^{S}$ is Pareto optimal in the reduced game $(S,V_{\bf x}^{S})$. $\sharp$

Proposition. (Peleg \cite[p.206]{pel85})} .Given an NTU game $(N,V)$ in $\Gamma_{2}^{N}$, let ${\bf x}\in V(N)$ and $\emptyset\neq S\subseteq N$. If ${\bf x}\in {\cal C}(N,V)$, then ${\bf x}^{S}\in {\cal C}(S,V_{\bf x}^{S})$. $\sharp$

We define $\zeta (N)=\{\{i,j\}:i,j\in N, i\neq j\}$

Proposition. (Peleg \cite[p.207]{pel85}). Given an NTU game $(N,V)$ in $\Gamma_{2}^{N}$, let ${\bf x}\in V(N)$. If, for every $S\in\zeta (N)$, ${\bf x}^{S}\in {\cal C}(S,V_{\bf x}^{S})$, then ${\bf x}\in {\cal C}(N,V)$. $\sharp$

For $i\in N$, we denote $v_{i}=\sup\{x_{i}:x_{i}\in V(i)\}$. By the comprehensiveness and condtion (ii) in Definition \ref{pel85d21}, $v_{i}$ is well-defined. For ${\bf x}\in V(N)$, we say that ${\bf x}$ is individually rational if and only if $x_{i}\geq v_{i}$ for all $i\in N$. We see that if ${\bf x}\in {\cal C}(N,V)$, then ${\bf x}$ is individually rational (Peleg \cite[p.207]{pel85}).

A {\bf solution function} defined on $\widehat{\Gamma}_{2}^{N}$ is a function $\boldsymbol{\psi}$ that assigns to each NTU game $(N,V)$ in $\widehat{\Gamma}_{2}^{N}$ a subset $\boldsymbol{\psi}(N,V)$ of $V(N)$. We consider the following axioms.

(P1: Nonemptiness). $\boldsymbol{\psi}(N,V)\neq\emptyset$ for every $(N,V)\in\widehat{\Gamma}_{2}^{N}$;
(P2: Individual Rationality). if, for every $(N,V)\in\widehat{\Gamma}_{2}^{N}$, every payoff vector in $\boldsymbol{\psi}(N,V)$ is individually rational.
(P3: Reduced Game Property). if $(N,V)\in\widehat{\Gamma}_{2}^{N}$, $\emptyset\neq S\subseteq N$ and ${\bf x}\in\boldsymbol{\psi}(N,V)$, then $(S,V_{\bf x}^{S})\in\widehat{\Gamma}_{2}^{N}$ and ${\bf x}^{S}\in\boldsymbol{\psi}(S,V_{\bf x}^{S})$;
(P4: Converse Reduced Game Property). if $(N,V)\in\widehat{\Gamma}_{2}^{N}$, ${\bf x}\in V(N)$ and, for every $S\in\pi (N)$, $(S,V_{\bf x}^{S})\in\widehat{\Gamma}_{2}^{N}$ and ${\bf x}^{S}\in\boldsymbol{\psi}(S,V_{\bf x}^{S})$, then ${\bf x}\in \boldsymbol{\psi}(N,V)$.

Axiom (P3) is a condition of self-consistency: if $(N,V)\in\widehat{\Gamma}_{2}^{N}$ and ${\bf x}\in\boldsymbol{\psi}(N,V)$, that is, ${\bf x}$ is prescribed as a solution to $(N,V)$, then, for every $\emptyset\neq S\subseteq N$, ${\bf x}^{S}$ is a solution for the reduced game $(S,V_{\bf x}^{S})$. Axiom (P4) focuses on pairs of players: if, at a given payoff distribution ${\bf x}$, every pair of players is “in equilibrium”, then ${\bf x}$ is in the solution (Peleg \cite[p.207-208]{pel85}).

Theorem. (Peleg \cite[p.208]{pel85}). There is a unique solution on $\widehat{\Gamma}_{2}^{N}$ that satisfies axioms (P1)–(P4), and it is the core. $\sharp$

Let $M$ be an infinite set of players. We denote by $\bar{\Gamma}_{2}^{N}$ the subset of $\Gamma_{2}^{N}$ with ${\cal C}(N,V)\neq\emptyset$ for any finite subset $N$ of $M$.

Theorem. (Peleg \cite[p.211]{pel85}). The core is the unique solution function defined on $\bar{\Gamma}_{2}^{N}$ satisfying axioms (P1)–(P3). $\sharp$

Cooperative Games — Non-Transferable Utility Games

Axiomatizations of the NTU Values.

An Axiomatization of Shapley Correspondence.

An Axiomatization of Harsanyi’s Solution.

An Axiomatization of Harsanyi-Shapley Solution.

An Axiomatization of the Weighted NTU Value.

NTU Games with Coalition Structure.

Preliminaries.

A Solution for NTU Games with Coalition Structure.

The Axiom Approach.

The Cores.

The Cores of Games without Side Payments.

The Cores of NTU Games.

Hsien-Chung Wu

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Axiomatizations of the NTU Values.

An Axiomatization of Shapley Correspondence.

An Axiomatization of Harsanyi’s Solution.

An Axiomatization of Harsanyi-Shapley Solution.

An Axiomatization of the Weighted NTU Value.

NTU Games with Coalition Structure.

Preliminaries.

A Solution for NTU Games with Coalition Structure.

The Axiom Approach.

The Cores.

The Cores of Games without Side Payments.

The Cores of NTU Games.

Hsien-Chung Wu

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