Characterizations of weighted and equal division values

Abstract

New and recent axioms for cooperative games with transferable utilities are introduced. The non-negative player axiom requires to assign a non-negative payoff to a player that belongs to coalitions with non-negative worth only. The axiom of addition invariance on bi-partitions requires that the payoff vector recommended by a value should not be affected by an identical change in worth of both a coalition and the complementary coalition. The nullified solidarity axiom requires that if a player who becomes null weakly loses (gains) from such a change, then every other player should weakly lose (gain) too. We study the consequence of imposing some of these axioms in addition to some classical axioms. It turns out that the resulting values or set of values have all in common to split efficiently the worth achieved by the grand coalition according to an exogenously given weight vector. As a result, we also obtain new characterizations of the equal division value.

Appendix: Logical independence of the axioms in the characterizations

We focus on non-trivial cases, i.e., if $n>1$ or $n>2$. In each of the following proofs, we exhibit a value that satisfies all of the axioms in one of our characterizations except for the one that is named. Details are provided for the toughest cases.

1.1 For Theorem 1:

• Not efficiency: the null value;

• Not linearity: the value $\varphi$ defined by

  $$\begin{aligned} \varphi _{i}(v)= & {} \left( v(i)-v(N\setminus i)\right) \cdot v(N)\quad \text {and}\quad \nonumber \\ \varphi _{1}(v)= & {} \bigg ( 1-\sum _{i\in N\setminus 1} [v(i)-v(N\setminus i)]\bigg ) \cdot v(N) \end{aligned}$$

• for all $v\in {\mathbb {V}}$ and $i\in N\setminus \left\{ 1\right\}$;

• Not the nullifying player axiom: Sh-value;

• Not addition invariance on bi-partitions: the value $\varphi$ defined by (4) in Remark 3.

1.2 For Corollary 1:

• Not efficiency: the null value;

• Not linearity: the value $\varphi$ defined by

  $$\begin{aligned} \varphi _{i}(v)= & {} \left( v(i)-v(N\setminus i)\right) \cdot v(N)\quad \text {and}\quad \nonumber \\ \varphi _{1}(v)= & {} \bigg ( 1-\sum _{i\in N\setminus 1} [v(i)-v(N\setminus i)]\bigg ) \cdot v(N) \end{aligned}$$

• for all $v\in {\mathbb {V}}$ and $i\in N\setminus \left\{ 1\right\}$;

• Not the nullifying player axiom: Sh-value;

• Not self-duality: the value $\varphi$ defined by (4) in Remark 3.

1.3 For Theorem 2:

• Not the nullifying player axiom: Sh-value;

• Not weak covariance: any value $\varphi \in {\mathcal {W}}^{+}\setminus \{\text {ED}\}$;

• Not addition invariance on bi-partitions: the value $\varphi$ defined by $\varphi _{i}(v)=v(i)$ for all $v\in {\mathbb {V}}$ and $i\in N$.

1.4 For Corollary 2:

• Not the nullifying player axiom: Sh-value;

• Not weak covariance: any value $\varphi \in {\mathcal {W}}^{+}\setminus \{\text {ED}\}$;

• Not additivity: note that a game v is additive but not symmetric if there exists a weight vector $(c_{1},\ldots ,c_{n})\in {\mathbb {R}}^{n}$ with not all identical coordinates and such that $v=\sum _{i\in N}c_{i}u_{i}.$ Let A be the class of all games on N that are additive but not symmetric. Define the value $\varphi$ by

  $$\begin{aligned} \varphi _{i}\left( v\right) =\left\{ \begin{array}{ll} v(i), &{} v\in A,\\ \text {ED}_{i}\left( v\right) , &{} v\in {\mathbb {V}}\setminus A \end{array} \right. \qquad \text {for all }v\in {\mathbb {V}}\text { and }i\in N. \end{aligned}$$

• Note that for all $a,c\in {\mathbb {R}}$, $v\in A$ if and only if $(a\cdot v+\mathbf{c })\in A$, $a\ne 0$, i.e., the class of all additive but not symmetric games on N is closed under the “$(a\cdot v+\mathbf{c })$-operation”, provided that $a\ne 0$. If $a=0$ then $(a\cdot v+\mathbf{c })=\mathbf{c }$ but in this case, for all $i\in N$, $\text {ED} _{i}(\mathbf{c })=c=\mathbf{c }(i)$. As a consequence, $\varphi$ satisfies weak covariance. For any additive game, observe that $v^{D}=v$, so that $v\in A$ if and only if $v^{D}\in A$. In particular, we have $v(i)=v(N)-v(N\setminus i)=v^{D}(i)$. This implies that $\varphi$ satisfies self-duality. It is also easy to check that $\varphi$ satisfies the nullifying player axiom. Finally, let $v\in A$, i.e., $v=\sum _{j\in N}c_{j}\cdot u_{j}$ with $c_{i}\ne c_{j}$ for some $i,j\in N$. For any given $c\in {\mathbb {R}} \setminus \{0\}$, both games $c\cdot e_{N}$ and $v-c\cdot e_{N}$ are not additive, and thus not in A. It follows that, for all $i\in N$, $\varphi _{i}(v-c\cdot e_{N})=(v(N)-c)/n$ and $\varphi _{i}(c\cdot e_{N})=c/n$. Therefore, $\varphi _{i}(v-c\cdot e_{N})+\varphi _{i}(c\cdot e_{N})=v(N)/n$ for all $i\in N$, i.e., all players get the same payoff in the sum of the two games. But $\varphi _{i}(v-c\cdot e_{N}+c\cdot e_{N})=\varphi _{i}(v)=v(i)=c_{i}$ for all $i\in N$ which implies that not all players get the same payoff in game $v-c\cdot e_{N}+c\cdot e_{N}$, proving that $\varphi$ does not satisfy additivity;

• Not addition invariance on bi-partitions: the value $\varphi$ defined by $\varphi _{i}(v)=v(i)$ for all $v\in {\mathbb {V}}$ and $i\in N$.

1.5 For Theorem 3:

• Not efficiency: the value $\varphi$ defined by $\varphi _i(v)=v(i)$ for all $v\in {\mathbb {V}}$ and all $i\in N$; Not linearity: the value $\varphi$ defined by

  $$\begin{aligned} \varphi _{i}\left( v\right) =\left\{ \begin{array}{ll} \dfrac{v(i)^{2}}{\sum _{j\in N}v(j)^{2}}\cdot v(N)&{} \quad \text {if }\sum _{j\in N}v(j)^{2} \ne 0,\\ \text {ED}_{i}\left( v\right) &{}\quad \text { if }\sum _{j\in N}v(j)^{2}=0 \end{array} \right. \end{aligned}$$

  (5)

• for all $v\in {\mathbb {V}}$ and $i\in N$;

• Not the nullifying player axiom: Sh-value;

• Not the null player in a productive environment axiom: any value $\varphi \in {\mathcal {W}}\setminus {\mathcal {W}}^{+}$.

1.6 For Theorem 4:

• Not efficiency: the null value;

• Not linearity: the value given by (5);

• Not the non-negative player axiom: Sh-value.

1.7 For Theorem 5:

• Not efficiency: the null value;

• Not linearity: let $\omega \in {\mathbb {R}}^{N}$ be such that $\sum _{i\in N}\omega _{i}=0$ and $\omega _{i}\ne 0$ for some $i\in N$. Construct the value $\varphi$ defined by $\varphi _{i}(v)=\text {ED}_{i}(v)+\omega _{i}$ for all $v\in {\mathbb {V}}$ and $i\in N$;

• Not nullified solidarity: Sh-value.

1.8 For Theorem 6:

• Not efficiency: any value $\varphi \in {\mathcal {W}}^{+}\setminus \{\text {ED}\}$;

• Not weak covariance: for some $i\in N$, the value $\varphi ^{(i)}$ defined by $\varphi _{j}^{(i)}(v)=v(i)$ for all $v\in {\mathbb {V}}$ and all $j\in N$;

• Not nullified solidarity: Sh-value.

1.9 For Theorem 7 (a):

• Not the equal treatment axiom: any value $\varphi \in {\mathcal {W}}^{+}\setminus \{\text {ED}\}$;

• Not efficiency: the null value;

• Not additivity: value given by (5);

• Not the non-negative player axiom: Sh-value.

1.10 For Theorem 7 (b):

• Not the equal treatment axiom: any value $\varphi \in {\mathcal {W}}^{+}\setminus \{\mathrm{ED}\}$;

• Not efficiency: the null value;

• Not additivity: Suppose that $n\ge 3$. Let $w\in {\mathbb {V}}$ be such that no two distinct players are substitutes,

  $$\begin{aligned} w(N)>0,\quad \text {and}\quad w(N\setminus i)=0\quad \mathrm{for all}i\in N. \end{aligned}$$

  (6)

• Let $\omega \in {\mathbb {R}}_{+}^{n}$ such that $\sum _{i\in N}\omega _{i}=1$ and $\omega _{i}\ne \omega _{j}$ for some $i,j\in N$. Define the value $\varphi$ by $\varphi _{i}(w)={\mathrm{WD}}_{i}^{\omega }\left( w\right)$ and $\varphi _{i}(v)=\mathrm{ED}_{i}(v)$ if $v\in {\mathbb {V}}\setminus \{w\}$. Since w does not contain any pair of substitute players, $\varphi$ satisfies the equal treatment axiom. It is also obvious that $\varphi$ satisfies efficiency. Regarding nullified solidarity, let $v\in {\mathbb {V}}\setminus \{w\}$. Since condition (6) implies that w does not contain any null player, we have $v^{\mathbf{N}i}\ne w$ for all $i\in N$, so that nullified solidarity is satisfied when the considered game is $v\in {\mathbb {V}}\setminus \{w\}$. Now, let us test nullified solidarity starting with game w. By (6), we have $w^{\mathbf{N} i}(N)=w(N\setminus i)=0$ for all $i\in N$. Therefore,

  $$\begin{aligned} \varphi _{i}(w)={\text {WD}}_{i}^{\omega }\left( w\right) \ge 0=\mathrm{ED} _{i}(w^{\mathbf{N}i})=\varphi _{i}(w^{\mathbf{N}i}), \end{aligned}$$

  but also

  $$\begin{aligned} \varphi _{j}(w)={\text {WD}}_{j}^{\omega }\left( w\right) \ge 0=\text {ED} _{j}(w^{\mathbf{N}j})=\varphi _{j}(w^{\mathbf{N}i}), \end{aligned}$$

• for all $j\in N\setminus i$, which shows that $\varphi$ satisfies nullified solidarity. Finally, by considering two games $v^{1}$ and $v^{2}$ such that $v^{1}\ne \mathbf{0}$, $v^{2}\ne \mathbf{0}$ and $v^{1}+v^{2}=w$, it is easy to see that $\varphi$ does not satisfies additivity;

• Not nullified solidarity: Sh-value.