Weakly differentially monotonic solutions for cooperative games

Abstract

The principle of differential monotonicity for cooperative games states that the differential of two players’ payoffs weakly increases whenever the differential of these players’ marginal contributions to coalitions containing neither of them weakly increases. Together with the standard efficiency property and a relaxation of the null player property, differential monotonicity characterizes the egalitarian Shapley values, i.e., the convex mixtures of the Shapley value and the equal division value for games with more than two players. For games that contain more than three players, we show that, cum grano salis, this characterization can be improved by using a substantially weaker property than differential monotonicity. Weak differential monotonicity refers to two players in situations where one player’s change of marginal contributions to coalitions containing neither of them is weakly greater than the other player’s change of these marginal contributions. If, in such situations, the latter player’s payoff weakly/strictly increases, then the former player’s payoff also weakly/strictly increases.

Appendices

Proof of Theorem 3

1.1 Preamble: squeezing water from stone

Proving characterizations for a parametrized classes of solutions with few rather weak axioms is like squeezing water from stone. Their proofs tends to be lengthy and rather technical. Our proof is no exception in this respect. It is by induction on the number of non-vanishing Harsanyi dividends of a game for non-singleton coalitions and can be divided into three major parts. While the first two parts provide the induction basis, the third one the induction step.

The first part (Claim 1) shows the theorem for inessential games, which includes the derivation of the parameter $\alpha \in \left[ 0,1\right]$ from the solution $\varphi .$ Its proof largely mimics the proof of Yokote and Casajus (2017, Theorem 2) . Since the latter result involves a related but different average dummy player property and in order to keep our paper self-contained, we present the full proof. The second part (Claims 2, 3, and 4) and the third part extend part one to general games. Their proof is an adaptation of the proof of Casajus and Yokote (2017, Theorem 2) to the egalitarian Shapley values and the use of the average dummy property instead of the null player property.

1.2 The proof

It is well-known that any value $\mathrm {Sh}^{\alpha },$$\alpha \in \left[ 0,1\right]$ satisfies E. Since DMo implies DMo$^{-}$, Casajus and Huettner (2013, Theorem 4) entails that any $\mathrm {Sh}^{\alpha }$ also obeys DMo$^{-}$. By (4) and the fact that $\mathrm {Sh}$ meets D, any $\mathrm {Sh}^{\alpha }$ meets AD. Let $\left| N\right| >3$ and let the solution $\varphi$ meet E, AD, and DMo$^{-}$.

For $v\in \mathbb {V},$ set

$$\begin{aligned} \mathcal {T}_{>1}\left( v\right) :=\left\{ T\subseteq N\mid \left| T\right| >1\text { and }\lambda _{T}\left( v\right) \ne 0\right\} . \end{aligned}$$

We show $\varphi =\mathrm {Sh}^{\alpha }$ for some $\alpha \in \left[ 0,1\right]$ by induction on $\left| \mathcal {T}_{>1}\left( v\right) \right| .$ For this purpose, we “reduce” $\left| \mathcal {T}_{>1}\left( v\right) \right|$ without changing $v\left( N\right)$ by the following construction: For $T\subseteq N,$$\left| T\right| >1$, let $\bar{u}_{T}\in \mathbb {V}$ be given by

$$\begin{aligned} \bar{u}_{T}:=u_{T}-\sum _{\ell \in T}\frac{u_{\left\{ \ell \right\} } }{\left| T\right| }. \end{aligned}$$

(A.1)

Note that $\mathrm {Sh}_{i}^{\alpha }\left( \bar{u}_{T}\right) =0$ for all $\alpha \in \left[ 0,1\right]$ and $i\in N.$ For $T\in \mathcal {T}_{>1}\left( v\right) ,$ let $v_{T}\in \mathbb {V}$ be given by

$$\begin{aligned} v_{T}:=v-\lambda _{T}\left( v\right) \cdot \bar{u}_{T}. \end{aligned}$$

(A.2)

By construction, (*) $\left| \mathcal {T}_{>1}\left( v_{T}\right) \right| =\left| \mathcal {T}_{>1}\left( v\right) \right| -1$ and (**) $v\left( N\right) =v_{T}\left( N\right) .$

Induction basis: We show $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right)$ for some $\alpha \in \left[ 0,1\right]$ and all $v\in \mathbb {V}$ such that $\left| \mathcal {T} _{>1}\left( v\right) \right| \le 1$ by a number of claims and subclaims.

If $\left| \mathcal {T}_{>1}\left( v\right) \right| =0$ for $v\in \mathbb {V},$ then $v\in \mathbb {\bar{V}}$, i.e., v is inessential.

Claim 1, C1: There exists some $\alpha \in \left[ 0,1\right]$ such that $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right)$ for all $v\in \mathbb {\bar{V}}.$

One can easily check that there is a bijection $\mathbb {R}^{N}\rightarrow \mathbb {\bar{V}}$, $x\mapsto v_{x},$ where $v_{x}$ is given by $v_{x}\left( S\right) =\sum _{\ell \in S}x_{\ell }$ for all $S\subseteq N.$ Abusing notation, we identify $\mathbb {\bar{V}}$ with $\mathbb {R}^{N}$ and write x instead of $v_{x}.$ By D, we have $\mathrm {Sh}\left( x\right) =x$ for all $x\in \mathbb {R}^{N}$ and therefore

$$\begin{aligned} \mathrm {Sh}_{i}^{\alpha }\left( x\right) \overset{\text {(4)}}{=}\alpha \cdot x_{i}+\left( 1-\alpha \right) \cdot \left| N\right| ^{-1}\cdot \sum _{\ell \in N}x_{\ell }\qquad \text {for all }\alpha \in \left[ 0,1\right] \text {, }x\in \mathbb {R}^{N}\text { and }i\in N. \end{aligned}$$

(A.3)

Set $n:=\left| N\right| .$ For $\lambda \in \mathbb {R}$ and $x\in \mathbb {R}^{N},$ we define $\lambda \cdot x\in \mathbb {R}^{N}$ by $\left( \lambda \cdot x\right) _{\ell }=\lambda \cdot x_{\ell }$ for all $\ell \in N.$ Further, for $i,j\in N,$$i\ne j,$ we define $e^{ij}\in \mathbb {R}^{N}$ by $e_{i}^{ij}=1,$$e_{j}^{ij}=-1,$ and $e_{\ell }^{ij}=0$ for all $\ell \in N{\setminus }\left\{ i,j\right\} .$ Moreover, for $\mu \in \mathbb {R},$ we define $e^{\mu }\in \mathbb {R}^{N}$ by $e_{\ell }^{\mu }=\frac{\mu }{n}$ for all $\ell \in N.$

For all $i\in N,$$j\in N{\setminus }\left\{ i\right\} ,$ and $\mu \in \mathbb {R}$, let the mapping $g_{ij}^{\mu }:\mathbb {R}\rightarrow \mathbb {R}$ be given by

$$\begin{aligned} g_{ij}^{\mu }\left( \lambda \right) :=\varphi _{i} ( e^{\mu }+\lambda \cdot e^{ij}) -\frac{\mu }{n}\quad \text {for all }\lambda \in \mathbb {R}. \end{aligned}$$

(A.4)

Note that for $\varphi =\mathrm {Sh}^{\alpha },$$\alpha \in \left[ 0,1\right] ,$ we have $g_{ij}^{\mu }\left( \lambda \right) =\alpha \cdot \lambda$ for all $\lambda \in \mathbb {R}.$ In the following, we use the mappings $g_{ij}^{\mu }$ in order to derive the parameter $\alpha$ from $\varphi .$ We proceed by a number of subclaims. First, we show that $g_{ij}^{\mu }$ does not depend on the choice of $j\in N{\setminus }\left\{ i\right\} .$

Claim C1a. For $i\in N$ and $\lambda ,\mu \in \mathbb {R},$ we have $\varphi _{i} ( e^{\mu }+\lambda \cdot e^{ij}) =\varphi _{i}( e^{\mu }+\lambda \cdot e^{ik})$ for all $j,k\in N{\setminus }\left\{ i\right\} .$

For $j=k,$ nothing is to show. Let now $j\ne k\$and $\ell \in N{\setminus } \left\{ i,j,k\right\} .$ Player $\ell$ is a dummy player in $e^{\mu } +\lambda \cdot e^{ij}\in \mathbb {\bar{V}}$ and in $e^{\mu }+\lambda \cdot e^{ik}\in \mathbb {\bar{V}}$ with $\left( e^{\mu }+\lambda \cdot e^{ij}\right) _{\ell }=\frac{\mu }{n}=\frac{1}{n}\cdot \left( e^{\mu }+\lambda \cdot e^{ij}\right) \left( N\right)$ and $\left( e^{\mu }+\lambda \cdot e^{ik}\right) _{\ell }=\frac{\mu }{n}=\frac{1}{n}\cdot \left( e^{\mu } +\lambda \cdot e^{ik}\right) \left( N\right) .$ By AD, we have ($\dagger$) $\varphi _{\ell }\left( e^{\mu }+\lambda \cdot e^{ij}\right) =\varphi _{\ell }\left( e^{\mu }+\lambda \cdot e^{ik}\right) .$ Since i and $\ell$ are symmetric in $\lambda \cdot e^{ij}-\lambda \cdot e^{ik},$ players i and $\ell$, $e^{\mu }+\lambda \cdot e^{ij},$ and $e^{\mu }+\lambda \cdot e^{ik}$ satisfy the hypothesis of DMo$^{-}.$ Hence, DMo$^{-}$ and ($\dagger$) imply $\varphi _{i}\left( e^{\mu }+\lambda \cdot e^{ij}\right) =\varphi _{i}\left( e^{\mu }+\lambda \cdot e^{ik}\right)$ . $\square$

For $i\in N$ and $\mu \in \mathbb {R},$ we define $g_{i}^{\mu }:\mathbb {R} \rightarrow \mathbb {R}$ by

$$\begin{aligned} g_{i}^{\mu }:=g_{ij}^{\mu }\qquad \text {for all }\lambda \in \mathbb {R}\text { and some }j\in N{\setminus }\left\{ i\right\} . \end{aligned}$$

(A.5)

By (A.4) and C1a, $g_{ij}^{\mu }$ does not depend on the choice of $j\in N{\setminus }\left\{ i\right\} .$ Hence, $g_{i}^{\mu }$ is well-defined. By AD, we have $g_{i}^{\mu }\left( 0\right) =0.$ Next, we show that $g_{i}^{\mu }$ does not depend on the choice of $i\in N.$

Claim C1b. For all $i,j\in N,$$i\ne j$ and $\lambda ,\mu \in \mathbb {R},$ we have $g_{i}^{\mu }\left( \lambda \right) =g_{j}^{\mu }\left( \lambda \right) .$

For $k\in N{\setminus }\left\{ i,j\right\} ,$ we have

$$\begin{aligned} g_{i}^{\mu }\left( \lambda \right) +g_{k}^{\mu }\left( -\lambda \right)&\overset{\text {(A.4),(A.5)}}{=}\varphi _{i} ( e^{\mu }+\lambda \cdot e^{ik}) -\frac{\mu }{n}+\varphi _{k} ( e^{\mu }-\lambda \cdot e^{ki}) -\frac{\mu }{n}\nonumber \\&=\varphi _{i} ( e^{\mu }+\lambda \cdot e^{ik}) -\frac{\mu }{n}+\varphi _{k} ( e^{\mu }+\lambda \cdot e^{ik}) -\frac{\mu }{n}, \end{aligned}$$

(A.6)

where the second equation drops from $\lambda \cdot e^{ik}=-\lambda \cdot e^{ki}.$ By AD, we have $\varphi _{\ell }( e^{\mu }+\lambda \cdot e^{ik}) =\frac{\mu }{n}$ for all $\ell \in N{\setminus } \{ i,k \} .$ Hence, E entails $\varphi _{i} ( e^{\mu } +\lambda \cdot e^{ik}) +\varphi _{k} ( e^{\mu }+\lambda \cdot e^{ik}) =\frac{2\mu }{n}.$ Together with (A.6), we obtain $g_{i}^{\mu } ( \lambda ) +g_{k}^{\mu } ( -\lambda ) =0.$ Analogously, one shows $g_{j}^{\mu } ( \lambda ) +g_{k}^{\mu } ( -\lambda ) =0,$ which concludes the proof. $\square$

For $\mu \in \mathbb {R},$ we define $g^{\mu }:\mathbb {R}\rightarrow \mathbb {R}$ by $g^{\mu }=g_{i}^{\mu }$ for some $i\in N.$ By C1b, $g^{\mu }$ is well-defined. In the following, we show certain properties of the mappings $g^{\mu }$ and their relation to $\varphi .$ For later use, we first show that $g^{\mu }$ is odd.

Claim C1c. For all $\lambda ,\mu \in \mathbb {R},$ we have $g^{\mu }\left( \lambda \right) =-g^{\mu }\left( -\lambda \right) .$

For $i,j\in N,$$i\ne j,$ we have

$$\begin{aligned} g^{\mu }\left( \lambda \right) +g^{\mu }\left( -\lambda \right)&\overset{\text {(A.4),(A.5)}}{=}\varphi _{i}\left( e^{\mu }+\lambda \cdot e^{ij}\right) -\frac{\mu }{n}+\varphi _{j}\left( e^{\mu }-\lambda \cdot e^{ji}\right) -\frac{\mu }{n}\nonumber \\&=\varphi _{i}\left( e^{\mu }+\lambda \cdot e^{ij}\right) -\frac{\mu }{n}+\varphi _{j}\left( e^{\mu }+\lambda \cdot e^{ij}\right) -\frac{\mu }{n}, \end{aligned}$$

(A.7)

where the second equation drops from $\lambda \cdot e^{ij}=-\lambda \cdot e^{ji}.$ By AD, we have $\varphi _{\ell }\left( e^{\mu }+\lambda \cdot e^{ij}\right) =0$ for all $\ell \in N{\setminus }\left\{ i,j\right\} .$ Hence, E entails $\varphi _{i}\left( e^{\mu }+\lambda \cdot e^{ij}\right) +\varphi _{j}\left( e^{\mu }+\lambda \cdot e^{ij}\right) =\frac{2\mu }{n}.$ Together with (A.7), this proves the claim. $\square$

For later use, we show a technical relation between the mappings $g^{\mu }$ and $\varphi .$ Note that in view of AD, average players in $x\in \mathbb {R}_{\mu }^{N}$, i.e., players i with $x_{i}=\frac{\mu }{n}$ are of particular interest. For $\mu \in \mathbb {R},$ set $\mathbb {R}_{\mu } ^{N}=\left\{ x\in \mathbb {R}^{N}\mid \sum _{\ell \in N}x_{\ell }=\mu \right\}$ and $\mathbb {\bar{R}}_{\mu }^{N}:=\left\{ x\in \mathbb {R}_{\mu }^{N} \mid \text {there exist some }i\in N\right. \left. \text {such that }x_{i}=\frac{\mu }{n}\right\} .$ For $x\in \mathbb {\bar{R}}_{\mu }^{N},$ we set $C_{\mu }\left( x\right) :=\left\{ i\in \mathbb {N}_{n}\mid x_{i}\ne \frac{\mu }{n}\right\} .$

Claim C1d. For all $\mu \in \mathbb {R},$$x\in \mathbb {\bar{R} }_{\mu }^{N},$ and $i\in N$, we have $\varphi _{i}\left( x\right) =g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}.$

We proceed by induction on $\left| C_{\mu }\left( x\right) \right| .$

Induction base: For $\left| C_{\mu }\left( x\right) \right| =0,$AD entails $\varphi _{i}\left( x\right) =\frac{\mu }{n}=g^{\mu }\left( 0\right) +\frac{\mu }{n}=g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}$ for all $i\in N.$ Note that $\left| C_{\mu }\left( x\right) \right| \ne 1$ for all $x\in \mathbb {\bar{R}}_{\mu }^{N}$. If $\left| C_{\mu }\left( x\right) \right| =2,$ then there are $i,j\in N,$$i\ne j$ such that $x=e^{\mu }+\left( x_{i}-\frac{\mu }{n}\right) \cdot e^{ij}=e^{\mu }+\left( x_{j} -\frac{\mu }{n}\right) \cdot e^{ji}.$ By (A.4) and (A.5 ), we have $\varphi _{i}\left( x\right) =g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}$ and $\varphi _{j}\left( x\right) =g^{\mu }\left( x_{j}-\frac{\mu }{n}\right) +\frac{\mu }{n}.$ Moreover, for $k\in N{\setminus }\left\{ i,j\right\} ,$AD implies $\varphi _{k}\left( x\right) =\frac{\mu }{n}=g^{\mu }\left( 0\right) +\frac{\mu }{n}=g^{\mu }\left( x_{k}-\frac{\mu }{n}\right) +\frac{\mu }{n}.$

Induction hypothesis: Let the claim hold for all $x\in \mathbb {\bar{R}}_{\mu }^{N}$ such that $\left| C_{\mu }\left( x\right) \right| \le t,$$t\in \mathbb {N}.$

Induction step: Let $x\in \mathbb {\bar{R}}_{\mu }^{N}$ be such that $\left| C_{\mu }\left( x\right) \right| =t+1>2.$ Suppose $\varphi _{i}\left( x\right) \ne g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}$ for some $i\in \mathbb {N}_{n}.$ By AD, $i\in C_{\mu }\left( x\right) .$ Let $j,k\in C_{\mu }\left( x\right) {\setminus }\left\{ i\right\} ,$$j\ne k,$ and $y=x-\left( x_{j}-\frac{\mu }{n}\right) \cdot e^{jk}.$ Note that $y\in \mathbb {\bar{R}}_{\mu }^{N},$$\left| C_{\mu }\left( y\right) \right| \le t,$ and $\left| C_{\mu }\left( y\right) \right| \ne 1.$ By the induction hypothesis, we have $\varphi _{i}\left( y\right) =g^{\mu }\left( y_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}=g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}.$ By assumption, there exists $\ell \in N{\setminus } C_{\mu }\left( x\right)$ such that $x_{\ell }=\frac{\mu }{n}.$ Hence, we obtain $\varphi _{i}\left( x\right) -\varphi _{i}\left( y\right) \ne g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}-g^{\mu }\left( x_{i}-\frac{\mu }{n}\right) -\frac{\mu }{n}=0$ and $\varphi _{\ell }\left( x\right) -\varphi _{\ell }\left( y\right) =0,$ where the latter drops from AD. Since i and $\ell$ are symmetric in $x-y,$x,  y,  i,  and j satisfy the hypothesis of DMo$^{-}.$ Hence, this contradicts DMo$^{-}$. $\square$

For later use, we show crucial properties of the mappings $g^{\mu },$ where linearity is of particular importance.

Claim C1e. For all $\mu \in \mathbb {R},$ the mapping $g^{\mu }:\mathbb {R}\rightarrow \mathbb {R}$ is linear and monotonic.

We show that the mapping $g^{\mu }$ is additive and monotonic. Then, linearity drops from Aczél (1966, Theorem 1).

Additivity: Let $a,b\in \mathbb {R}.$ Let $i,j,k\in N$ and $x\in \mathbb {R}_{\mu }^{N}$ be such that $i\ne j,$$j\ne k,$$k\ne i$,

$$\begin{aligned} x_{i}=\frac{\mu }{n}+a,\!\quad x_{j}=\frac{\mu }{n}+b,\!\quad x_{k}=\frac{\mu }{n}-a-b,\!\quad x_{\ell }=\frac{\mu }{n}\!\qquad \text {for all }\ell \in \mathbb {N} _{n}{\setminus }\left\{ i,j,k\right\} . \end{aligned}$$

Since $n>3,$$x\in \mathbb {\bar{R}}_{\mu }^{N}.$ By C1d, we have

$$\begin{aligned} \varphi _{i}\left( x\right) =g^{\mu }\left( a\right) +\frac{\mu }{n} ,\quad \varphi _{j}\left( x\right) =g^{\mu }\left( b\right) +\frac{\mu }{n},\quad \text {and}\quad \varphi _{k}\left( x\right) =g^{\mu }\left( -a-b\right) +\frac{\mu }{n}. \end{aligned}$$

Further, by AD, we have $\varphi _{\ell }\left( x\right) =\frac{\mu }{n}$ for all $\ell \in N{\setminus }\left\{ i,j,k\right\} .$ Hence, we obtain

$$\begin{aligned}&0\overset{\mathbf{E }}{=}\varphi _{i}\left( x\right) +\varphi _{j}\left( x\right) +\varphi _{k}\left( x\right) -\frac{3\mu }{n}=g^{\mu }\left( a\right) +g^{\mu }\left( b\right) +g^{\mu }\left( -a-b\right) \overset{\mathbf{C1c }}{=}g^{\mu }\left( a\right) \\&\quad +g^{\mu }\left( b\right) -g^{\mu }\left( a+b\right) . \end{aligned}$$

That is, the mapping g is additive.

Monotonicity: Let $a,b\in \mathbb {R}$ and $i,j,k\in N$ be such that $i\ne j,$$j\ne k,$$k\ne i,$ and $a\ge b.$ For $x=e^{\mu }+a\cdot e^{ij}$ and $y=e^{\mu }+b\cdot e^{ij},$ we have $x_{i}-y_{i}=a\ge b=x_{k}-y_{k}.$ Moreover, by AD, $\varphi _{k}\left( x\right) =\varphi _{k}\left( y\right) =\frac{\mu }{n}.$ Hence, we obtain

$$\begin{aligned} g^{\mu }\left( a\right) \overset{\text {(A.4),(A.5)} }{=}\varphi _{i}\left( x\right) \overset{\mathbf{DMo }^{-}}{\ge }\varphi _{i}\left( y\right) \overset{\text {(A.4),(A.5)}}{=}g^{\mu }\left( b\right) . \end{aligned}$$

That is, the mapping $g^{\mu }$ is monotonic. $\square$

For $\mu \in \mathbb {R},$ set $\alpha ^{\mu }:=g^{\mu }\left( 1\right) .$ The next claim already establishes C1 for all $x\in \mathbb {R}_{\mu }^{N}.$

Claim C1f. For all $\mu \in \mathbb {R}$ and $x\in \mathbb {R}_{\mu }^{N},$ we have $\varphi \left( x\right) =\alpha ^{\mu }\cdot x+\left( 1-\alpha ^{\mu }\right) \cdot e^{\mu }.$

Case (i): For $x\in \mathbb {\bar{R}}_{\mu }^{N}$ and $i\in N,$ we obtain

$$\begin{aligned} \varphi _{i}\left( x\right) \overset{\mathbf{C1d }}{=}g_{i}^{\mu }\left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}\overset{\mathbf{C1e }}{=}\alpha ^{\mu }\cdot \left( x_{i}-\frac{\mu }{n}\right) +\frac{\mu }{n}=\alpha ^{\mu }\cdot x_{i}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}. \end{aligned}$$

Case (ii): Let $x\in \mathbb {R}_{\mu }^{N} {\setminus }\mathbb {\bar{R}}_{\mu }^{N}.$ Suppose $\varphi \left( x\right) \ne \alpha ^{\mu }\cdot x+\left( 1-\alpha ^{\mu }\right) \cdot e^{\mu }.$ By E, we have $\sum _{\ell \in N}\varphi _{\ell }\left( x\right) =\mu =\sum _{\ell \in N}\left[ \alpha ^{\mu }\cdot x_{\ell }+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}\right] .$ Hence, there are $i,j\in N$ such that $\varphi _{i}\left( x\right) >\alpha ^{\mu }\cdot x_{i}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}$ and $\varphi _{j}\left( x\right) <\alpha ^{\mu }\cdot x_{j}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}.$ Let $k,\ell \in N{\setminus }\left\{ i,j\right\} ,$ and $y=x-\left( x_{k}-\frac{\mu }{n}\right) \cdot e^{k\ell }.$ Note that $x_{i}=y_{i}$ and $x_{j}=y_{j}.$ Further, note that $y_{k}=\frac{\mu }{n}$ and therefore $y\in \mathbb {\bar{R} }_{\mu }^{N}.$ By Case (i), we obtain $\varphi \left( y\right) =\alpha ^{\mu }\cdot y+\left( 1-\alpha ^{\mu }\right) \cdot e^{\mu }.$ Moreover, we have

$$\begin{aligned} \varphi _{i}\left( x\right) -\varphi _{i}\left( y\right) >\alpha ^{\mu }\cdot x_{i}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}-\left[ \alpha ^{\mu }\cdot y_{i}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}\right] =0 \end{aligned}$$

and

$$\begin{aligned} \varphi _{j}\left( x\right) -\varphi _{j}\left( y\right) <\alpha ^{\mu }\cdot x_{j}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}-\left[ \alpha ^{\mu }\cdot y_{j}+\left( 1-\alpha ^{\mu }\right) \cdot \frac{\mu }{n}\right] =0. \end{aligned}$$

Since i and j are symmetric in $x-y,$x,  y,  i,  and j satisfy the hypothesis of DMo$^{-}.$ Hence, this contradicts DMo$^{-}$. $\square$

Now, we are ready to prove C1.

Case (a): Suppose $\alpha ^{\mu }=0$ for all $\mu \in \mathbb {R}.$ By C1f, we obtain $\varphi _{i}\left( x\right) =\frac{1}{n}\cdot \sum _{\ell \in N}x_{\ell }\overset{\text {(A.3)} }{=}\mathrm {Sh}^{0}\left( x\right)$ for all $x\in \mathbb {R}^{N}.$

Case (b): Suppose $\alpha ^{\bar{\mu }}\ne 0$ for some $\bar{\mu }\in \mathbb {R}.$ By C1e, we have $\alpha ^{\mu }>0.$ Set $\alpha :=\alpha ^{\bar{\mu }}.$ We show that

$$\begin{aligned} \varphi _{i}\left( x\right) =\alpha \cdot x_{i}+\frac{1-\alpha }{n}\cdot \sum _{\ell \in N}x_{\ell }\qquad \text {for all }x\in \mathbb {R}^{N}\text { and }i\in N. \end{aligned}$$

(A.8)

Suppose there exists some $x\in \mathbb {R}^{N}$ such that (A.8) fails for some $i\in N$. By C1f, $x\notin \mathbb {R}_{\bar{\mu }}^{N}.$ By E and w.l.o.g., there exists $j\in N{\setminus }\left\{ i\right\}$ such that

$$\begin{aligned} \varphi _{i}\left( x\right) >\alpha \cdot x_{i}+\frac{1-\alpha }{n}\cdot \sum _{\ell \in N}x_{\ell } \end{aligned}$$

(A.9)

and

$$\begin{aligned} \varphi _{j}\left( x\right) <\alpha \cdot x_{j}+\frac{1-\alpha }{n}\cdot \sum _{\ell \in N}x_{\ell }. \end{aligned}$$

(A.10)

Let

$$\begin{aligned} X:=\frac{1-\alpha }{n}\cdot \left[ \left( \sum _{\ell \in N}x_{\ell }\right) -\bar{\mu }\right] . \end{aligned}$$

(A.11)

Further, let $k\in N{\setminus }\left\{ i,j\right\}$ and let $y\in \mathbb {R}^{N}$ be given by

$$\begin{aligned} y_{i}=x_{i}+\frac{X}{\alpha },\quad y_{j}=x_{j}+\frac{X}{\alpha },\quad y_{k}=-x_{i}-x_{j}-\frac{2\cdot X}{\alpha }+\bar{\mu },\quad \text {and}\quad y_{\ell }=0 \end{aligned}$$

(A.12)

for all $\ell \in N{\setminus }\left\{ i,j,k\right\} .$ Since $y\in \mathbb {R}_{\bar{\mu }}^{N},$ by C1f, we have $\varphi \left( y\right) =\alpha \cdot y+\left( 1-\alpha \right) \cdot e^{\bar{\mu }}.$ By (A.9), (A.10), (A.11), and (A.12), we obtain

$$\begin{aligned} \varphi _{i}\left( x\right) -\varphi _{i}\left( y\right) >\alpha \cdot x_{i}+\frac{1-\alpha }{n}\cdot \sum _{\ell \in N}x_{\ell }-\left( \alpha \cdot x_{i}+X+\frac{1-\alpha }{n}\cdot \bar{\mu }\right) =0 \end{aligned}$$

and

$$\begin{aligned} \varphi _{j}\left( x\right) -\varphi _{j}\left( y\right) <\alpha \cdot x_{j}+\frac{1-\alpha }{n}\cdot \sum _{\ell \in N}x_{\ell }-\left( \alpha \cdot x_{j}+X+\frac{1-\alpha }{n}\cdot \bar{\mu }\right) =0. \end{aligned}$$

Since i and j are symmetric in $x-y,$x,  y,  i,  and j satisfy the hypothesis of DMo$^{-}.$ Therefore, this contradicts DMo$^{-}$. Hence, $\varphi \left( x\right) \overset{\text {(A.3)} }{=}\mathrm {Sh}^{\alpha }\left( x\right)$ for all $x\in \mathbb {R}^{N}.$

Finally, we have

$$\begin{aligned} \alpha \overset{\text {(A.8)}}{=}-\varphi _{j}\left( e^{ij}\right) \overset{\mathbf{AD }}{\le }-\left( e^{ij}\right) _{j}=1\qquad \text {for }i\in N\text { and }j\in N{\setminus }\left\{ i\right\} , \end{aligned}$$

which concludes the proof of C1. $\square$

If $\left| \mathcal {T}_{>1}\left( v\right) \right| =1$ for $v\in \mathbb {V},$ then there are $\delta ^{v}\in \mathbb {R}^{N}$ and $\beta ^{v}\in \mathbb {R}$, and $T^{v}\subseteq N,$$\left| T^{v}\right| >1$ such that $\beta ^{v}\ne 0$ and

$$\begin{aligned} v=\beta ^{v}\cdot \bar{u}_{T^{v}}+\sum _{\ell \in N}\delta _{\ell }^{v}\cdot u_{\left\{ \ell \right\} }. \end{aligned}$$

(A.13)

Set

$$\begin{aligned} R^{v}:=\left\{ i\in N\mid v\left( \left\{ i\right\} \right) \ne \frac{1}{\left| N\right| }\cdot v\left( N\right) \right\} . \end{aligned}$$

Note that players $i\in N{\setminus }\left( R^{v}\cup T^{v}\right)$ are average dummy players, i.e., dummy players with $v\left( \left\{ i\right\} \right) =\frac{1}{\left| N\right| }\cdot v\left( N\right)$ for which AD implies $\varphi \left( v\right) =v\left( \left\{ i\right\} \right) .$

We now show that $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right)$ for all $v\in \mathbb {V}$ with $\left| \mathcal {T}_{>1}\left( v\right) \right| =1$ by a number of claims. First, we deal with games in which there exists a average dummy player.

Claim 2, C2: For all $v\in \mathbb {V}$ with $\left| \mathcal {T}_{>1}\left( v\right) \right| =1$ and such that $R^{v}\cup T^{v}\ne N,$ we have $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right) .$

By C1, we have

$$\begin{aligned} \varphi \left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) \overset{\text {(4),(A.1)}}{=}\mathrm {Sh}^{\alpha }\left( v\right) . \end{aligned}$$

(A.14)

For $i\in N{\setminus }\left( R^{v}\cup T^{v}\right) ,$ we have

$$\begin{aligned} \varphi _{i}\left( v\right) \overset{\mathbf{AD }}{=}\frac{v\left( N\right) }{\left| N\right| }\overset{\text {(4),(A.1)}}{=}\mathrm {Sh}_{i}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) \overset{\mathbf{C1 }}{=}\varphi _{i}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) . \end{aligned}$$

(A.15)

Since all players in $N{\setminus } T^{v}$ are pairwise symmetric in $-\beta ^{v}\cdot \bar{u}_{T^{v}}$, v, $v-\beta ^{v}\cdot \bar{u}_{T^{v}},$ and $i,\ell \in N{\setminus } T^{v}$ satisfy the hypothesis of DMo$^{-}.$ Hence, we have

$$\begin{aligned} \varphi _{\ell }\left( v\right) \overset{\text {(A.15)},\mathbf DMo ^{-}}{=}\varphi _{\ell }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) \qquad \text {for all }\ell \in N{\setminus } T^{v}. \end{aligned}$$

(A.16)

Since any two players in $T^{v}$ are pairwise symmetric in $-\beta ^{v} \cdot \bar{u}_{T^{v}}$, v, $v-\beta ^{v}\cdot \bar{u}_{T^{v}},$ and $k,\ell \in T^{v}$ satisfy the hypothesis of DMo$^{-},$ which implies that we have

$$\begin{aligned} \varphi _{k}\left( v\right) \gtrless \varphi _{k}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) \quad \text {if and only if}\quad \varphi _{\ell }\left( v\right) \gtrless \varphi _{\ell }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v} }\right) \end{aligned}$$

(A.17)

for all $k,\ell \in T^{v}.$ By E, (A.16), and (A.17 ), we finally have $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right)$. $\square$

Next, we handle games in which there are average players but which are not dummy players.

Claim 3, C3: For all $v\in \mathbb {V}$ with $\left| \mathcal {T}_{>1}\left( v\right) \right| =1$ such that $R^{v}\cup T^{v}=N$ and $\left| T^{v}{\setminus } R^{v}\right| \ge 1,$ we have $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right) .$

Suppose $\varphi \left( v\right) \ne \mathrm {Sh}\left( v\right)$. By E, there are $i,j\in N$ such that

$$\begin{aligned} \varphi _{i}\left( v\right) >\mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {and}\qquad \varphi _{j}\left( v\right) <\mathrm {Sh}_{j}^{\alpha }\left( v\right) . \end{aligned}$$

(A.18)

Case (i): Suppose $i,j\in R^{v}{\setminus } T^{v}$ or $i,j\in T^{v}.$ By (A.13), we have $v-\beta ^{v}\cdot \bar{u}_{T^{v} }\in \mathbb {\bar{V}}.$ Hence, C1 implies

$$\begin{aligned} \varphi \left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) . \end{aligned}$$

(A.19)

By (A.18) and (A.19), we further have

$$\begin{aligned} \varphi _{i}\left( v\right) -\varphi _{i}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) >\mathrm {Sh}_{i}^{\alpha }\left( v\right) -\mathrm {Sh} _{i}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh} _{i}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}\right) \overset{\text {(4),(A.1)}}{=}0 \end{aligned}$$

and

$$\begin{aligned} \varphi _{j}\left( v\right) -\varphi _{j}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) <\mathrm {Sh}_{j}^{\alpha }\left( v\right) -\mathrm {Sh} _{j}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh} _{j}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}\right) \overset{\text {(4),(A.1)}}{=}0. \end{aligned}$$

Since i and j are symmetric in $-\beta ^{v}\cdot \bar{u}_{T^{v}},$v,  $v-\beta ^{v}\cdot \bar{u}_{T^{v}},$i,  and j satisfy the hypothesis of DMo$^{-}.$ Hence, this contradicts DMo$^{-}$.

Case (ii): Suppose, w.l.o.g., $i\in R^{v}{\setminus } T^{v}$ and $j\in T^{v}.$

Case (ii-a): Suppose $j\in T^{v}{\setminus } R^{v}.$ Let $w:=v-\beta ^{v}\cdot \bar{u}_{T^{v}}-\beta ^{v}\cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ j\right\} \right) \cup \left\{ i\right\} }.$ By C2, we have

$$\begin{aligned} \varphi \left( w\right) =\mathrm {Sh}^{\alpha }\left( w\right) . \end{aligned}$$

(A.20)

By (A.18) and (A.20), we further have

$$\begin{aligned} \varphi _{i}\left( v\right) -\varphi _{i}\left( w\right) >\mathrm {Sh} _{i}^{\alpha }\left( v\right) -\mathrm {Sh}_{i}^{\alpha }\left( w\right) =\mathrm {Sh}_{i}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}+\beta ^{v} \cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ j\right\} \right) \cup \left\{ i\right\} }\right) \overset{\text {(4),(A.1)}}{=}0 \end{aligned}$$

and

$$\begin{aligned} \varphi _{j}\left( v\right) -\varphi _{j}\left( w\right) <\mathrm {Sh} _{j}^{\alpha }\left( v\right) -\mathrm {Sh}_{j}^{\alpha }\left( w\right) =\mathrm {Sh}_{j}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}+\beta ^{v} \cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ j\right\} \right) \cup \left\{ i\right\} }\right) \overset{\text {(4),(A.1)}}{=}0. \end{aligned}$$

Since i and j are symmetric in $-\beta ^{v}\cdot \bar{u}_{T^{v}}-\beta ^{v}\cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ j\right\} \right) \cup \left\{ i\right\} },$v,  w,  i,  and j satisfy the hypothesis of DMo$^{-}.$ Hence, this contradicts DMo$^{-}$.

Case (ii-b): Suppose $j\in T^{v}\cap R^{v}.$ By assumption, there exists $k\in T^{v}{\setminus } R^{v}$ such that $k\ne i$ and $k\ne j.$ By C1, we have

$$\begin{aligned}&\varphi _{j}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh} _{j}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) \qquad \text {and}\nonumber \\&\varphi _{k}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh}_{k}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) . \end{aligned}$$

(A.21)

By (A.18) and (A.21), we have

$$\begin{aligned} \varphi _{j}\left( v\right) -\varphi _{j}\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) <\mathrm {Sh}_{j}^{\alpha }\left( v\right) -\mathrm {Sh} _{j}^{\alpha }\left( v-\beta ^{v}\cdot \bar{u}_{T^{v}}\right) =\mathrm {Sh} _{j}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}\right) \overset{\text {(4),(A.1)}}{=}0. \end{aligned}$$

Since j and k are symmetric in $-\beta ^{v}\cdot \bar{u}_{T^{v}},$v,  $v-\beta ^{v}\cdot \bar{u}_{T^{v}},$i,  and k satisfy the hypothesis of DMo$^{-}.$ Hence, DMo$^{-}$ entails

$$\begin{aligned} \varphi _{k}( v) -\varphi _{k}( v-\beta ^{v}\cdot \bar{u}_{T^{v}}) <0. \end{aligned}$$

Since

$$\begin{aligned} \varphi _{k} ( v-\beta ^{v}\cdot \bar{u}_{T^{v}}) \overset{\text {(A.21)}}{=}\mathrm {Sh}_{k}^{\alpha } ( v-\beta ^{v} \cdot \bar{u}_{T^{v}}) \overset{\text {(4),(A.1)} }{=}\mathrm {Sh}_{k}^{\alpha }\left( v\right) , \end{aligned}$$

we obtain

$$\begin{aligned} \varphi _{k}\left( v\right) <\mathrm {Sh}_{k}^{\alpha }\left( v\right) . \end{aligned}$$

(A.22)

Let $z:=v-\beta ^{v}\cdot \bar{u}_{T^{v}}-\beta ^{v}\cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ k\right\} \right) \cup \left\{ i\right\} }.$ By (A.18), (A.22), and C2, we have

$$\begin{aligned} \varphi _{i}\left( v\right) -\varphi _{i}\left( z\right) >\mathrm {Sh} _{i}^{\alpha }\left( v\right) -\mathrm {Sh}_{i}^{\alpha }\left( z\right) =\mathrm {Sh}_{i}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}+\beta ^{v} \cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ k\right\} \right) \cup \left\{ i\right\} }\right) \overset{\text {(4),(A.1)}}{=}0 \end{aligned}$$

(A.23)

and

$$\begin{aligned} \varphi _{k}\left( v\right) -\varphi _{k}\left( z\right) <\mathrm {Sh} _{k}^{\alpha }\left( v\right) -\mathrm {Sh}_{k}^{\alpha }\left( z\right) =\mathrm {Sh}_{k}^{\alpha }\left( \beta ^{v}\cdot \bar{u}_{T^{v}}+\beta ^{v} \cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ k\right\} \right) \cup \left\{ i\right\} }\right) \overset{\text {(4),(A.1)}}{=}0. \end{aligned}$$

(A.24)

Since $i\in R^{v}{\setminus } T^{v}$ and $k\in T^{v}{\setminus } R^{v},$i and k are symmetric in $-\beta ^{v}\cdot \bar{u}_{T^{v}}-\beta ^{v}\cdot \bar{u}_{\left( T^{v}{\setminus }\left\{ k\right\} \right) \cup \left\{ i\right\} },$v,  z,  i,  and k satisfy the hypothesis of DMo$^{-}.$ Hence, (A.23) and (A.24) contradict DMo$^{-}$.$\square$

Finally, we deal with games in which there are no average players.

Claim 4, C4: For all $v\in \mathbb {V}$ with $\left| \mathcal {T}_{>1}\left( v\right) \right| =1$ such that $R^{v}\cup T^{v}=N$ and $\left| T^{v}{\setminus } R^{v}\right| =0,$ we have $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right) .$

By assumption, we have $R^{v}=N.$ Suppose $\varphi \left( v\right) \ne \mathrm {Sh}\left( v\right)$. By E, there are $i,j\in N$ such that

$$\begin{aligned} \varphi _{i}\left( v\right) >\mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {and}\qquad \varphi _{j}\left( v\right) <\mathrm {Sh}_{j}^{\alpha }\left( v\right) . \end{aligned}$$

(A.25)

Let $k\in N{\setminus }\left\{ i,j\right\}$, $\ell \in N{\setminus }\left\{ i,j,k\right\} ,$ and $q\in \mathbb {V}$ be given by

$$\begin{aligned} q:=\left( v\left( \left\{ k\right\} \right) -\frac{v\left( N\right) }{\left| N\right| }\right) \cdot \left( u_{\left\{ k\right\} }-u_{\left\{ \ell \right\} }\right) . \end{aligned}$$

(A.26)

By (A.13), we have

$$\begin{aligned} v-q=\beta ^{v}\cdot \bar{u}_{T^{v}}+\frac{v\left( N\right) }{\left| N\right| }\cdot u_{\left\{ k\right\} }+\left( \delta _{\ell }^{v} +\delta _{k}^{v}-\frac{v\left( N\right) }{\left| N\right| }\right) \cdot u_{\left\{ \ell \right\} }+\sum _{h\in N{\setminus }\left\{ k\right\} }\delta _{h}^{v}\cdot u_{\left\{ h\right\} }. \end{aligned}$$

Hence, we have $\left| \mathcal {T}_{>1}\left( v-q\right) \right| =1,$$T^{v-q}=T^{v},$ and

$$\begin{aligned} \left( v-q\right) \left( \left\{ k\right\} \right) =\frac{v\left( N\right) }{\left| N\right| }=\frac{\left( v-q\right) \left( N\right) }{\left| N\right| }, \end{aligned}$$

where the latter implies $k\notin R^{v-q}.$ Note that q is constructed in a way such that k is an average player in $v-q.$

If $k\notin T^{v},$ then $v-q$ satisfies the hypothesis of C2 and we obtain

$$\begin{aligned} \varphi \left( v-q\right) =\mathrm {Sh}^{\alpha }\left( v-q\right) . \end{aligned}$$

(A.27)

If $k\in T^{v},$ then $v-q$ satisfies the hypothesis of C3 and we also obtain (A.27). By (A.25) and (A.27), we have

$$\begin{aligned} \varphi _{i}\left( v\right) -\varphi _{i}\left( v-q\right) >\mathrm {Sh} _{i}^{\alpha }\left( v\right) -\mathrm {Sh}_{i}^{\alpha }\left( v-q\right) =\mathrm {Sh}_{i}^{\alpha }\left( q\right) \overset{\text {(4),(A.26)}}{=}0 \end{aligned}$$

(A.28)

and

$$\begin{aligned} \varphi _{j}\left( v\right) -\varphi _{j}\left( v-q\right) <\mathrm {Sh} _{j}^{\alpha }\left( v\right) -\mathrm {Sh}_{j}^{\alpha }\left( v-q\right) =\mathrm {Sh}_{j}^{\alpha }\left( q\right) \overset{\text {(4),(A.26)}}{=}0. \end{aligned}$$

(A.29)

Since i and j are symmetric in $-q,$v,  $v-q,$i and j satisfy the hypothesis of DMo$^{-}.$ Hence, (A.28) and (A.29) together contradict DMo$^{-}$.$\square$

Note that the induction basis (see page 8) is proved by C1, C2, C3, and C4.

Induction hypothesis: Let the claim hold for all $v\in \mathbb {V}$ such that $\left| \mathcal {T}_{>1}\left( v\right) \right| \le t,$$t\in \mathbb {N},$$t\ge 1.$

Induction step: Let now $v\in \mathbb {V}$ be such that $\left| \mathcal {T}_{>1}\left( v\right) \right| =t+1.$ There exist $S,T\in \mathcal {T}_{>1}\left( v\right)$ such that $S\ne T.$ By (3), (A.2), (*) (see page 8), and the induction hypothesis, we have

$$\begin{aligned} \varphi \left( v_{S}\right) =\mathrm {Sh}^{\alpha }\left( v_{S}\right) =\mathrm {Sh}^{\alpha }\left( v\right) =\mathrm {Sh}^{\alpha }\left( v_{T}\right) =\varphi \left( v_{T}\right) . \end{aligned}$$

(A.30)

Case (i): $S\cap T\ne \emptyset .$ W.l.o.g., $S{\setminus } T\ne \emptyset .$ Let $i\in S\cap T$ and $j\in S{\setminus } T.$ By (A.30) and DMo$^{-}$, we have

$$\begin{aligned} \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in S,\\ \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in T,\\ \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{j}\left( v\right) \gtrless \mathrm {Sh}_{j}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in N{\setminus } T, \end{aligned}$$

and therefore

$$\begin{aligned} \varphi _{\ell }\left( v\right) \gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in N. \end{aligned}$$

(A.31)

Case (ii): $S\cup T\ne N.$ W.l.o.g., $S{\setminus } T\ne \emptyset .$ Let $i\in N{\setminus }\left( S\cup T\right)$ and $j\in S{\setminus } T.$ By (A.30) and DMo$^{-}$, we have

$$\begin{aligned} \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in N{\setminus } S,\\ \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in N{\setminus } T,\\ \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{j}\left( v\right) \gtrless \mathrm {Sh}_{j}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in S, \end{aligned}$$

and therefore

$$\begin{aligned} \varphi _{\ell }\left( v\right) \gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in N. \end{aligned}$$

(A.32)

Case (iii): $S\cap T=\emptyset$ and $S\cup T=N.$ Hence, $\mathcal {T}_{>1}\left( v\right) =\left\{ S,T\right\} .$ Let $i\in S$, $j\in T,$ and $w\in \mathbb {V}$ be given by

$$\begin{aligned} w=v_{S}-\lambda _{S}\left( v\right) \cdot u_{\left( S{\setminus }\left\{ i\right\} \right) \cup \left\{ j\right\} }+\frac{\lambda _{S}\left( v\right) }{\left| S\right| }\cdot \sum _{\ell \in \left( S{\setminus } \left\{ i\right\} \right) \cup \left\{ j\right\} }u_{\left\{ \ell \right\} }. \end{aligned}$$

(A.33)

By construction, we have $\mathcal {T}_{>1}\left( w\right) =\left\{ \left( S{\setminus }\left\{ i\right\} \right) \cup \left\{ j\right\} ,T\right\}$ and (****) $v\left( N\right) =w\left( N\right) .$ In view of Case (i), we have (*****) $\varphi \left( w\right) =\mathrm {Sh}\left( w\right) .$

Since i and j are symmetric in $v-w,$v,  w,  i,  and j satisfy the hypothesis of DMo$^{-}.$ Hence, by DMo$^{-}$ and (A.30), we have

$$\begin{aligned} \varphi _{j}\left( v\right)&\gtrless \varphi _{j}\left( w\right) \overset{\text {(*****)}}{=}\mathrm {Sh}_{j}^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \varphi _{i}\left( w\right) \overset{\text {(*****)}}{=}\mathrm {Sh} _{i}^{\alpha }\left( v\right) ,\\ \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in S,\\ \varphi _{\ell }\left( v\right)&\gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{j}\left( v\right) \gtrless \mathrm {Sh}_{j}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in T, \end{aligned}$$

and therefore

$$\begin{aligned} \varphi _{\ell }\left( v\right) \gtrless \mathrm {Sh}_{\ell }^{\alpha }\left( v\right) \quad \text {if and only if}\quad \varphi _{i}\left( v\right) \gtrless \mathrm {Sh}_{i}^{\alpha }\left( v\right) \qquad \text {for all }\ell \in N. \end{aligned}$$

(A.34)

Finally, (A.31), (A.32), (A.34), and E imply $\varphi \left( v\right) =\mathrm {Sh}^{\alpha }\left( v\right)$.$\square$

Counterexample to Theorem 3 for $\left| N\right| =2$

Theorem 3 fails for $\left| N\right| =2.$ Let $N=\left\{ 1,2\right\} .$ Consider the solution $\varphi ^{\heartsuit }:\mathbb {V}\rightarrow \mathbb {R}^{2}$ given by

$$\begin{aligned} \left( \varphi _{1}^{\heartsuit }\left( v\right) ,\varphi _{2}^{\heartsuit }\left( v\right) \right) =\left\{ \begin{array} [c]{ll} \left( \mathrm {Sh}_{1}\left( v\right) ,\mathrm {Sh}_{2}\left( v\right) \right) , &{} \mathrm {Sh}_{1}\left( v\right) \ge 0,~\mathrm {Sh}_{2}\left( v\right) \ge 0,\\ \left( \mathrm {Sh}_{1}\left( v\right) +\dfrac{\mathrm {Sh}_{2}\left( v\right) }{2},\dfrac{\mathrm {Sh}_{2}\left( v\right) }{2}\right) , &{} \mathrm {Sh}_{1}\left( v\right)>0,~\mathrm {Sh}_{2}\left( v\right)<0,~v\left( N\right) \ge 0,\\ \left( \dfrac{\mathrm {Sh}_{1}\left( v\right) }{2},\mathrm {Sh}_{2}\left( v\right) +\dfrac{\mathrm {Sh}_{1}\left( v\right) }{2}\right) , &{} \mathrm {Sh}_{1}\left( v\right)>0,~\mathrm {Sh}_{2}\left( v\right)<0,\text { }v\left( N\right)<0,\\ \left( \mathrm {Sh}_{1}\left( v\right) ,\mathrm {Sh}_{2}\left( v\right) \right) , &{} \mathrm {Sh}_{1}\left( v\right) \le 0~\mathrm {Sh}_{2}\left( v\right) \le 0,\\ \left( \mathrm {Sh}_{1}\left( v\right) +\dfrac{\mathrm {Sh}_{2}\left( v\right) }{2},\dfrac{\mathrm {Sh}_{2}\left( v\right) }{2}\right) , &{} \mathrm {Sh}_{1}\left( v\right)<0\text {, }\mathrm {Sh}_{2}\left( v\right)>0,~v\left( N\right) \ge 0,\\ \left( \dfrac{\mathrm {Sh}_{1}\left( v\right) }{2},\mathrm {Sh}_{2}\left( v\right) +\dfrac{\mathrm {Sh}_{1}\left( v\right) }{2}\right) , &{} \mathrm {Sh}_{1}\left( v\right)<0\text {, }\mathrm {Sh}_{2}\left( v\right) >0,~v\left( N\right) <0 \end{array} \right. \end{aligned}$$

for all $v\in \mathbb {V}.$ One can easily check that $\varphi ^{\heartsuit } \ne \mathrm {Sh}^{\alpha }$ for all $\alpha \in \left[ 0,1\right]$ and that $\varphi ^{\heartsuit }$ inherits E, AD, and DMo$^{-}$ from $\mathrm {Sh}$.

Non-redundancy of Theorem 3 for $\left| N\right| >3$

Our characterization is non-redundant for $\left| N\right| >3$. The value $\varphi ^{\mathbf{E }}$ given by $\varphi _{i}^{\mathbf{E } }\left( v\right) =v\left( \left\{ i\right\} \right)$ for all $v\in \mathbb {V}$ and $i\in N$ satisfies AD and DMo$^{-}$ but not E. The strictly positively weighted division values (Béal et al. 2016, Theorem 2) with non-uniform weights satisfy E and DMo$^{-}$ but not AD. For $v\in \mathbb {V},$ let $D_{0}\left( v\right) :=\left\{ i\in N\mid i~\text {is a dummy player in }v\right\} .$ The value $\varphi ^{\mathbf{DMo }^{-}}$ given by

$$\begin{aligned} \varphi _{i}^{\mathbf{DMo }^{-}}\left( v\right) =\left\{ \begin{array} [c]{ll} &{} \dfrac{v\left( N\right) -\sum _{\ell \in D_{0}\left( v\right) }v\left( \left\{ \ell \right\} \right) }{\left| N{\setminus } D_{0}\left( v\right) \right| }, \quad i\in N{\setminus } D_{0}\left( v\right) ,\\ &{} v\left( \left\{ i\right\} \right) , \quad D_{0}\left( v\right) \end{array} \right. \qquad \text {for all }v\in \mathbb {V}\text { and }i\in N \end{aligned}$$

satisfies E and AD but not DMo$^{-}$.