Pure Bargaining Problems with a Coalition Structure

Abstract

We consider here pure bargaining problems endowed with a coalition structure such that each union is given its own utility. In this context we use the Shapley rule in order to assess the main options available to the agents: individual behavior, cooperative behavior, isolated unions behavior, and bargaining unions behavior. The latter two respectively recall the treatment given by Aumann–Drèze and Owen to cooperative games with a coalition structure. A numerical example illustrates the procedure. We provide criteria to compare any pair of behaviors for each agent, introduce and axiomatically characterize a modified Shapley rule, and determine its natural domain, that is, the set of problems where the bargaining unions behavior is the best option for all agents.

Appendix

Here we collect additional information that might disturb the reading of the article: mainly, detailed proofs of some results.

1.1 Axiomatic Characterization of the Shapley Rule

Let us consider the following properties for a sharing rule $f:E_{n+1}\longrightarrow \mathbb {R}^n$.

• 1. Group rationality: $\sum _{i\in N}f_i[u]=u_N$ for every $u\in E_{n+1}$.

• 2. Individual rationality: if $\Delta (u)\ge 0$ then $f_i[u]\ge u_i$ for all $i\in N$.

• 3. Symmetry: if $u_i=u_j$ then $f_i[u]=f_j[u]$.

• 4. Additivity: $f[u+v]=f[u]+f[v]$ for all $u,v\in E_{n+1}$.

We shall show that this set of properties characterizes the Shapley rule $\varphi$, defined by Eq. (1). We will assume in this section that $n\ge 2$, since for $n=1$ we find that, exceptionally, dim $E_{n+1}=1$, and hence group rationality suffices to characterize $\varphi$.

The following lemma, whose proof is straightforward, will be applied.

Lemma 8.1

• (a) A basis ${\mathcal{B}_0}=\{u_0^1,u_0^2,\dots ,u_0^n,u_0^N\}$ of $E_{n+1}$ is given by

  • $u_0^k=(1,\dots ,1,\overset{{\underset{\smile }{k}}}{2},1,\dots ,1|n+1)$  for  $k=1,2,\dots ,n$,  and

  • $u_0^N=(1,1,\dots ,1|n+1)$

• (b) All $u_0^k$ for $k=1,2,\dots ,n$ are additive, whereas $u_0^N$ is superadditive.

• (c) Moreover, if $u=(u_1,u_2,\dots ,u_n|u_N)\in E_{n+1}$ then it can be uniquely written as a linear combination of the members of ${\mathcal{B}_0}$ as

  $$\begin{aligned} u=\sum _{k=1}^n\left( u_k-\frac{u_N}{n+1}\right) u_0^k + \Delta (u)u_0^N. \end{aligned}$$

  (3)

Then we have:

Theorem 8.2

(Theorem 2.3) There is one and only one sharing rule on $E_{n+1}$ that satisfies properties (1)–(4). It is the Shapley rule $\varphi$.

Proof

(Existence) It suffices to check that the Shapley rule satisfies (1)–(4), and this follows at once from Eq. (1).

(Uniqueness) We shall see that if f satisfies (1)–(4) then $f=\varphi$. To this end, we will use the basis ${\mathcal{B}_0}$. For any $\lambda \in \mathbb {R}$, from (1) and (2) it follows that, for any $i\in N$ and $k=1,2,\dots ,n$,

$$\begin{aligned} f_i[\lambda u_0^k]= {\left\{ \begin{array}{ll} 2\lambda & \quad \text{ if } i=k,\\ \lambda & \quad \text{ if } i\ne k, \end{array}\right. } \end{aligned}$$

and, from (1) and (3),

$$\begin{aligned} f_i[\lambda u_0^N]=\lambda \frac{n+1}{n}\quad \text{ for } \text{ all } i\in N. \end{aligned}$$

Then, for any $i\in N$ and any $u\in E_{n+1}$, from (4) and using Eq. (3) we obtain

$$\begin{aligned} f_i[u]=2\left( u_i-\frac{u_N}{n+1}\right) +\sum _{k\ne i}\left( u_k-\frac{u_N}{n+1}\right) +\Delta (u)\frac{n+1}{n}=u_i + \frac{\Delta (u)}{n}=\varphi _i[u], \end{aligned}$$

so $f=\varphi$. $\square$

As to the logical independence of the axiomatic system described above, it suffices to find four rules that satisfy all axioms but one. Only a problem that shows the failure is needed in each case (a counterexample). All counterexamples are (super)additive and all of them might be extended to any number n of agents by adding $n-2$ null agents.

• A rule that fails to satisfy (1) Group rationality.

  $$\begin{aligned} f_i[u]=u_i\quad \text{ for } \text{ all } u\in E_{n+1} \text{ and } \text{ all } i\in N. \end{aligned}$$

  It is easy to verify that f satisfies (2), (3) and (4). Instead, for $u=(1,1|3)$ we find that

  $$ f_1[u]+f_2[u] = 2 \neq 3 = u_N. $$

• A rule that fails to satisfy (2) Individual rationality.

  $$\begin{aligned} f_i[u]=\frac{u_N}{n}\quad \text{ for } \text{ all } u\in E_{n+1} \text{ and } \text{ all } i\in N. \end{aligned}$$

  It is easy to verify that f satisfies (1), (3) and (4). Instead, for $u=(1,3|4)$, where $\Delta (u)=0$, we find that

  $$\begin{aligned} f_2[u] = 2 \neq 3 = u_2. \end{aligned}$$

• A rule that fails to satisfy (3) Symmetry.

  $$\begin{aligned} f_i[u]= {\left\{ \begin{array}{ll} u_1+\Delta (u) & \quad{ \text {if }} \,\,i=1 \\ u_i & \quad {\text {otherwise}}\end{array}\right. } \end{aligned}$$

  for all $u\in E_{n+1}$ and all $i\in N$. It is easy to verify that f satisfies (1), (2) and (4). Instead, for $u=(1,1|3)$ we find that

  $$\begin{aligned} u_1=u_2\quad \text {but}\quad f_1[u] = 2 \ne 1 = f_2[u]. \end{aligned}$$

• A rule that fails to satisfy (4) Additivity. It combines proportional rule and Shapley rule.¹¹

  $$\begin{aligned} f_i[u]= {\left\{ \begin{array}{ll} \frac{u_i}{\sum u_j}u_N & \text {if}\, u_j>0 \quad \text{for all}\,\,j\in N \\ u_i+\frac{\Delta (u)}{n} & \text {otherwise}\end{array}\right. } \end{aligned}$$

  for all $u\in E_{n+1}$ and all $i\in N$. It is easy to verify that f satisfies (1), (2) and (3). Instead, for $u=(1,2|9)$ and $v=(0,1|3)$ we find $u+v=(1,3|12)$ and

  $$\begin{aligned} f[u+v]=(5,7) \ne (3,6) + (1,2) = f[u] + f[v]. \end{aligned}$$

1.2 The Domain of All CS-Problems

Let $N=\{1,2,\dots ,n\}$ be the set of agents, with $n\ge 1$. A CS-problem $[u,B,u^*]$ in N is defined by three objects:

• $u=(u_1,u_2,\dots ,u_n|u_N)$, which is a problem in N

• $B=\{B_1,B_2,\dots ,B_m\}$, which is a CS in N, with $M=\{1,2,\dots ,m\}$ and $m\ge 1$

• $u^*=(u_1^*,u_2^*,\dots ,u_m^*|u_M^*)$, which is a problem in M

with the following restrictions:

• (r1) If $B=B^n$ or $B=B^N$ then $u_M^*=u_N$

• (r2) if $n=1$ then $u_N=u_1$

• (r3) if $m=1$ then $u_M^*=u_1^*$

• (r4) if $b_k=1$ for some $k\in M$, i.e. if $B_k=\{i\}$ is a singleton, then $u_k^*=u_i$

$E_N$ will denote the set of all CS-problems defined in N, and we put $E=\displaystyle \bigcup\nolimits_{n=1}^\infty E_N$. These two sets do not have any structure. Instead, the set $E_N^B$, formed by all CS-problems defined in N with a fixed coalition structure B, becomes a vector space under the natural linear operations given by

• $[u,B,u^*]+[v,B,v^*]=[u+v,B,u^*+v^*]$

• $\lambda [u,B,u^*]=[\lambda u,B,\lambda u^*]$   for all $\lambda \in \mathbb {R}$

Thus, having B in mind, any vector of $E_N^B$ may be understood simply as a pair of problems u and $u^*$, a pair that we will denote as $u\odot u^*$ when working within $E_N^B$ for a given B.

Theorem 8.3

The dimension of $E_N^B$ is as follows:

• (a) If $n=1$  then  dim $E_N^B=1$.

• (b) If $n>1$ and $m=1$  then  dim $E_N^B=n+1$.

• (c) If $n>1$ and $m=n$  then  dim $E_N^B=n+1$.

• (d) If $n>1$ and $1<m<n$  then  dim $E_N^B=n+m-p+2$, where p is the number of unions in B that are singletons ($0\le p<m$).

Proof

At first glance, any vector $u\odot u^*$ requires $n+1+m+1$ linear parameters to be defined. However, restrictions (r1)–(r4) may reduce this freedom degree in some cases. And, indeed:

• (a) If $n=1$ then $m=1$ and $u_N=u_1=u_1^*=u_M^*$ is the only parameter.

• (b) In this case, $B=B^N$, the only union $B_1$ is not a singleton, and $u_1^*=u_M^*=u_N$, which reduces the freedom degree to $n+1$.

• (c) Here, $B=B^n$ and $u_M^*=u_N$, each union is a singleton with $u_i^*=u_i$ for all $i\in M$, and the freedom degree reduces again to $n+1$.

• (d) The number p of singletons satisfies $0\le p<m$, since $p=m$ would imply $m=n$. The only restrictions are $u_k^*=u_i$ for each singleton $B_k=\{i\}$, so the freedom degree reduces to $n+1+m+1-p$.

$\square$

In the sequel we will set  $d=\dim E_N^B$ .

1.3 Axiomatic Characterization of the Modified Shapley Rule

Let us consider the following properties for a coalitional sharing rule $g:E_N\longrightarrow \mathbb {R}^n$.

• 1. Group rationality: $\displaystyle \sum\nolimits_{i\in N}g_i[u,B,u^*]=u_M^*$ for every $[u,B,u^*]$ in N.

• 2. Individual rationality: if $\Delta (u^*)\ge 0$ and $\Delta (u^k)\ge 0$ for all $k\in M$ then $g_i[u,B,u^*]\ge u_i$ for all $i\in N$.

• 3. Coalitional rationality: if $\Delta (u^*)\ge 0$ then $\sum _{i\in B_k}g_i[u,B,u^*]\ge u_k^*$ for all $k\in M$.

• 4. Symmetry: if $u_i=u_j$ and $i,j\in B_k$ for some $k\in M$ then $g_i[u,B,u^*]=g_j[u,B,u^*]$.

• 5. Additivity: for all B and all $[u,B,u^*]$ and $[v,B,v^*]$ in $E_N^B$,

  $$\begin{aligned} g[u+v,B,u^*+v^*]=g[u,B,u^*]+g[v,B,v^*]. \end{aligned}$$

• 6. Singletons: if $B_k=\{i\}$ then $g_i[u,B,u^*]=\varphi _k[u^*]$.

We shall show that this set of properties characterizes the modified Shapley rule $\psi$, defined by Eq. (2). The statement is as follows.

Theorem 8.4

(Theorem 6.3) There is one and only one coalitional sharing rule on $E_N$ that satisfies properties (1)–(6). It is the modified Shapley rule $\psi$.

The proof of this result will be similar to that of Theorem 2.3. The existence part is straightforward using Eq. (2), whereas for the uniqueness part we will need to describe a suitable basis ${\mathcal{B}_0}=\{w_0^1,w_0^2,\dots ,w_0^d\}$ of $E_N^B$, where $d=\dim E_N^B$, for each nontrivial B. Each member of this basis will be of the form $w_0^h=u_0^h\odot (u_0^h)^*$, for $1\le h\le d$.

Example 8.5

• (a) Let us proceed to show this basis for a “large enough” numerical example, and to give the rules to construct it in general. In this example, $n=9$ and

  $$\begin{aligned} B=\{\{1\},\{2\},\{3,4\},\{5,6\},\{7,8,9\}\}, \end{aligned}$$

so $m=5$, $b_1=1$, $b_2=1$, $b_3=2$, $b_4=2$ and $b_5=3$, $p=2$, and  $\dim E_N^B=14$. The basis is described in Table 2. The local problems $(u_0^h)^1$ and $(u_0^h)^2$ corresponding to the unions that are singletons are trivial and do not appear, but we include the local problems corresponding to the remaining unions, not indispensable to build the basis, because they may well help to fully understand the procedure. All problems are (super)additive.

Table 2 The basis ${\mathcal{B}_0}$ of $E_N^B$ for $n=9$ and $B=\{\{1\},\{2\},\{3,4\},\{5,6\},\{7,8,9\}\}$

The rules for constructing such a basis for arbitrary n and nontrivial B are the following:

• First column $u_0^h$:

  • From 1 to n, the individual utilities form the unit matrix $n\times n$ and $\Delta =0$ (additive problems).

  • For $n+1$, all individual utilities equal 1 and $\Delta =1$ (superadditive problem).

  • From $n+2$ to d, the individual utilities vanish and $\Delta =0$ (additive problems).

• Second column $(u_0^h)^*$:

  • From 1 to $n+1$, the individual utility of each union is the sum of the individual utilities of its members in $u_0^h$ and $\Delta =0$ (additive problems).

  • From $n+2$ to d, and starting from the bottom ($w_0^d$), we prepare problems where all individual utilities initially vanish but are successively replaced by utility 1, one-by-one, from the left to the right, and $\Delta =1$ for $w_0^d$ (superadditive problem) but $\Delta =0$ for all its predecessors (additive problems).

• Next columns: all local problems, derived from the previous rules, are (super)additive

• (b) The interested reader is invited to try this procedure for different n and B. For example, if $n=4$ and $B=\{\{1,2\},\{3,4\}\}$ the basis ${\mathcal{B}_0}$ is given by Table 3.

Table 3 The basis ${\mathcal{B}_0}$ of $E_N^B$ for $n=4$ and $B=\{\{1,2\},\{3,4\}\}$

$$\begin{aligned} \left( \begin{array}{llllllllllllllll} 1&0&0&0&0&0&0&0&0&1 &0&0&0&1 \\ 0&1&0&0&0&0&0&0&0&1 &0&0&0&1 \\ 0&0&1&0&0&0&0&0&0&1 &1&0&0&1 \\ 0&0&0&1&0&0&0&0&0&1 &1&0&0&1 \\ 0&0&0&0&1&0&0&0&0&1 &0&1&0&1 \\ 0&0&0&0&0&1&0&0&0&1 &0&1&0&1 \\ 0&0&0&0&0&0&1&0&0&1 &0&0&1&1 \\ 0&0&0&0&0&0&0&1&0&1 &0&0&1&1 \\ 0&0&0&0&0&0&0&0&1&1 &0&0&1&1 \\ 1&1&1&1&1&1&1&1&1&10&2&2&3&9 \\ 0&0&0&0&0&0&0&0&0&0 &1&1&1&3 \\ 0&0&0&0&0&0&0&0&0&0 &0&1&1&2 \\ 0&0&0&0&0&0&0&0&0&0 &0&0&1&1 \\ 0&0&0&0&0&0&0&0&0&0 &0&0&0&1 \\ \end{array} \right) \end{aligned}$$

(4)

Matrix of coordinates of the ${\mathcal{B}_0}$ members for $n=9$

Lemma 8.6

Assume that a nontrivial coalition structure B has been fixed and ordered in such a way that the singletons (if any) occupy the first places. Then $n>1$,$1<m<n$ and  $d = \dim E_N^B=n+m-p+2$. A basis ${\mathcal{B}_0}=\{w_0^1,w_0^2,\dots ,w_0^d\}$ of $E_N^B$, where all problems involved for each member of ${\mathcal{B}_0}$ are (super)additive, can be obtained as indicated in Example 8(a).5.

Proof

In Example 8.5(a), since $(u_0^h)^*_1$ and $(u_0^h)^*_2$ are imposed by restriction (r4) and hence they cannot be chosen, the parameters that define each $w_0^h$ are only

$$\begin{aligned} (u_0^h)_1,\quad (u_0^h)_2,\quad \dots ,\quad (u_0^h)_9,\quad (u_0^h)_N, \quad (u_0^h)^*_3,\quad (u_0^h)^*_4,\quad (u_0^h)^*_5,\quad (u_0^h)^*_M. \end{aligned}$$

Eq. (4) provides the matrix defined by these parameters, the rows of which represent the coordinates of each $w_0^h$ in an obvious “canonical basis” of $E_N^B$—which is not suitable to our interests because most of its members are not (super)additive.

It is easy to see that this is a regular matrix, so $\{w_0^h\,\,h=1,\dots ,14\}$ is a linearly independent set and therefore a basis of $E_N^B$. Indeed, by subtracting rows 1–9 from row 10 we obtain an equivalent matrix (in rank terms) of the form

$$\begin{aligned} \left( \begin{array}{cc} \boxed {\,I_{10}\,} & \boxed {\,K\,} \\ & \\ \boxed {\,0\,} & \boxed {\,J_4\,} \\ \end{array} \right) \end{aligned}$$

where $I_{10}$ is the unit submatrix $10\times 10$, $J_4$ is a triangular regular submatrix $4\times 4$, 0 represents the null submatrix $4\times 10$, and K is a $10\times 4$ submatrix that does not matter to see that the rank of the entire matrix is 14. Of course, mutatis mutandis the argument also holds for Example 8.5(b) and remains clearly valid for any n and any nontrivial coalition structure B. $\square$

Remark 8.7

Moreover, there is a nice linear expression in terms of ${\mathcal{B}_0}$ of any CS-problem $w=u\odot u^*$, with B as coalition structure, but at this point our assertion is a well founded conjecture only. It can be checked in every particular case, using e.g. MATHEMATICA, but we have no general proof. Anyway, this is not necessary for the proof of the characterization theorem.

Thus, we have: in Example 8.5(a),

$$\begin{aligned} \begin{array}{ll} &w=u\odot u^*=[u_1-\Delta (u)]w_0^1+[u_2-\Delta (u)]w_0^2+[u_3-\Delta (u)]w_0^3+[u_4-\Delta (u)]w_0^4+\\ &[u_5-\Delta (u)]w_0^5+[u_6-\Delta (u)]w_0^6+[u_7-\Delta (u)]w_0^7+[u_8-\Delta (u)]w_0^8+[u_9-\Delta (u)]w_0^9+\\ &\Delta (u)w_0^{10}+\Delta (u^3)w_0^{11}+[\Delta (u^4)-\Delta (u^3)]w_0^{12}+[\Delta (u^5)-\Delta (u^4)]w_0^{13} +\Delta (u^*)w_0^{14}, \end{array} \end{aligned}$$

and, in Example 8.5(b),

$$\begin{aligned} \begin{array}{ll} &w=u\odot u^*=[u_1-\Delta (u)]w_0^1+[u_2-\Delta (u)]w_0^2+[u_3-\Delta (u)]w_0^3+[u_4-\Delta (u)]w_0^4+\\ &\Delta (u)w_0^5+\Delta (u^1)w_0^6+[\Delta (u^2)-\Delta (u^1)]w_0^7+\Delta (u^*)w_0^8. \end{array} \end{aligned}$$

The general rule for the coefficients of such a linear combination is as follows:

• For $h=1,\dots ,n$, the coefficient of $w_0^h$ is $u_h-\Delta (u)$.

• The coefficient of $w_0^{n+1}$ is $\Delta (u)$.

• For $h=n+2,\dots ,d$, if $B_k$ is the first union that is not a singleton, the coefficients are: $\Delta (u^k)$ for $w_0^{n+2}$, $\Delta (u^{k+1})-\Delta (u^k)$ for $w_0^{n+3}$, $\Delta (u^{k+2})-\Delta (u^{k+1})$ for $w_0^{n+4}$, and so on, but the last one, for $w_0^d$, is simply $\Delta (u^*)$.

The regularity of these expressions confirms the suitability of the basis ${\mathcal{B}_0}$.

Proof of Theorem 8.8

(Theorem 6.3) (Existence) It suffices to check that the modified Shapley rule $\psi$ satisfies (1)–(6), and this will follow at once from Eq. (2).

• 1. Group rationality. We have

  $$\begin{aligned} \displaystyle \sum _{i\in N}\psi _i[u,B,u^*]& = \displaystyle \sum _{i\in N}u_i+ \displaystyle \sum _{k\in M}\,\displaystyle \sum _{i\in B_k}\Big [\frac{\Delta (u^k)}{b_k}+ \frac{\Delta (u^*)}{mb_k}\Big ]\\ &= \displaystyle \sum _{k\in M}\, \Big [\displaystyle \sum _{i\in B_k}u_i+\Delta (u^k)\Big ]+\Delta (u^*)\\ &= \displaystyle \sum _{k\in M}u_k^*+\Delta (u^*)=u_M^*. \end{aligned}$$

• 2. Individual rationality. It is clear that if $\Delta (u^*)\ge 0$ and $\Delta (u^k)\ge 0$ for all $k\in M$ then $\psi _i[u,B,u^*]\ge u_i$ for all $i\in N$.

• 3. Coalitional rationality. If $\Delta (u^*)\ge 0$ then, for each $k\in M$,

  $$\begin{aligned} \displaystyle \sum _{i\in B_k}\psi _i[u,B,u^*]=\displaystyle \sum _{i\in B_k}u_i+\Delta (u^k)+ \frac{\Delta (u^*)}{m}=u_k^*+\frac{\Delta (u^*)}{m}\ge u_k^*. \end{aligned}$$

• 4. Symmetry. Since the second and third terms of Eq. (2) are constant in each $B_k$, if $u_i=u_j$ and $i,j\in B_k$ for some $k\in M$ then $\psi _i[u,B,u^*]=\psi _j[u,B,u^*]$.

• 5. Additivity. This property is clearly satisfied by the linearity of Eq. (2) with respect to u, $u^*$ and all $u^k$.

• 6. Singletons. This is a particular case of Proposition 6.2(b).

(Uniqueness) We shall see that, for every coalition structure B, if g satisfies (1)–(6) on $E_N^B$ then g is uniquely determined on $E_N^B$. Thus, we will have $g=\psi$ in each $E_N^B$ and hence in $E_N$.

• 1. If $B=B^n$ then only u matters since $M=N$ and $u^*=u$. Properties (1), (2), (4) and (5) become the properties of the Shapley rule $\varphi$ and, according to Theorem 2.3 and Proposition 6.2(a), it follows that $g[u,B^n,u^*]=\varphi [u]=\psi [u,B^n,u^*]$.

• 2. Similarly, if $B=B^N$ then only u matters since $B_1=N$, $u^*$ is trivial and $u^1=u$. Again, properties (1), (2), (4) and (5) become the properties of the Shapley rule $\varphi$ and, according to Theorem 2.3 and Proposition 6.2(a), it follows that $g[u,B^N,u^*]=\varphi [u]=\psi [u,B^N,u^*]$.

• 3. Therefore, in the sequel we can assume that B is not trivial. We will use the basis

  $$\begin{aligned} {\mathcal{B}_0}=\{w_0^1,w_0^2,\dots ,w_0^d\}\qquad (d=\dim E_N^B). \end{aligned}$$

  whose existence has been proven in Lemma 8.6. Since all problems involved in this basis are (super)additive by construction, it is straightforward to check that properties (1)–(4) and (6) univocally determine the action of g on each member of ${\mathcal{B}_0}$.

Now, using property (5), we will prove that g is univocally determined also on any CS-problem $w\in E_N^B$. First, we note that, for any $\lambda \ge 0$ and $h=1,2,\dots ,d$, g is also univocally determined on any vector of the form $\lambda w_0^h$, still (super)additive, and $g[\lambda w_0^h]=\lambda g[w_0^h]$. Let us assume that

$$\begin{aligned} w=\sum _{h=1}^d \lambda _h w_0^h. \end{aligned}$$

Even if w is (super)additive, some $\lambda <0$ might appear in this expressions. Then we write

$$\begin{aligned} w + \sum _{\lambda _h<0}(-\lambda _h)w_0^h = \sum _{\lambda _h>0}\lambda _h w_0^h \end{aligned}$$

where three vectors appear that are defined by (super)additive problems. Then we can apply property (5) to obtain

$$\begin{aligned} g[w] + \sum _{\lambda _h<0}(-\lambda _h)g[w_0^h] = \sum _{\lambda _h>0}\lambda _h g[w_0^h], \end{aligned}$$

that finally gives

$$\begin{aligned} g[w] = \sum _{h=1}^d \lambda _h g[w_0^h]. \end{aligned}$$

Summing up, g is univocally determined on $E_N^B$ and hence $g=\psi$ on this vector space. $\square$

As to the logical independence of the axiomatic system described above, first we solve some particular and quite trivial cases:

• If $n=1$ then group rationality (1) suffices to characterize $\psi$.

• If $n=2$ then $B^n$ and $B^N$ are the only coalition structures. Properties (1), (2), (4) and (5) suffice to characterize $\psi$ and are, therefore, logically independent. In fact, $\psi [u,B,u^*]$ and $\varphi [u]$ coincide (cf. Proposition 6.2 or the uniqueness part in the proof of Theorem 6.3) and these four properties reduce to the axioms of the Shapley rule used in Theorem 2.3.

• If $n=3$ then (1), (2), (4), (5) and (6) suffice to characterize $\psi$ and are logically independent.

Then, in the sequel we can assume $n\ge 4$. In this general case we show that the six properties are logically independent. To this end, it suffices to find six rules that satisfy all axioms but one. Only a CS-problem that shows the failure is needed in each case (a counterexample). All counterexamples given below concern $n=4$ agents and are based on (super)additive problems, and all of them might be extended to any number n of agents by adding $n-4$ null agents.

• A rule that fails to satisfy (1) Group rationality.

  $$\begin{aligned} g_i[u,B,u^*]= {\left\{ \begin{array}{ll} \varphi _k[u^*] & \text {if}\,B_k=\{i\} \,\text {is\;a\;singleton} \\ u_i+\frac{\Delta (u^k)}{b_k} & \text {otherwise}\end{array}\right. } \end{aligned}$$

  It is easy to verify that g satisfies (2), (3), (4), (5) and (6). Instead, for  $u=(1,1,2,3|8)$,  $B=\{\{1,2\},\{3,4\}\}$,  and  $u^*=(4,7|15)$  we find that

  $$\begin{aligned} \sum _{i\in N} g_i[u,B,u^*] = 11 \ne 15 = u_M^*. \end{aligned}$$

• A rule that fails to satisfy (2) Individual rationality.

  $$\begin{aligned} g_i[u,B,u^*]=\frac{\varphi _k[u^*]}{b_k}\quad \text {if}\, i\in B_k \end{aligned}$$

  It is easy to verify that g satisfies (1), (3), (4), (5) and (6). Instead, for  $u=(1,3,2,5|12)$,  $B=\{\{1,2\},\{3,4\}\}$,  and  $u^*=(4,8|12)$  we find that

  $$\begin{aligned} g_2[u,B,u^*] = 2 \ngeq 3 = u_2. \end{aligned}$$

• A rule that fails to satisfy (3) Coalitional rationality. We will say that B is wide if $m\ge 2$, and $b_1,b_2>1$.

  $$\begin{aligned} g_i[u,B,u^*]= {\left\{ \begin{array}{ll} u_i+\frac{\Delta (u^*)}{mb_1} & \text {if} \,B \text {is\;wide\;and\;} i\in B_1 \\ & \\ u_i+\frac{\Delta (u^1)}{b_2}+\frac{\Delta (u^2)}{b_2}+\frac{\Delta (u^*)}{mb_2} & \text {if} \,B \text {is\;wide\;and\;} i\in B_2 \\ & \\ \psi _i[u,B,u^*] & \text {otherwise,\;for\;this\;and\;any\;other\;} B\end{array}\right. } \end{aligned}$$

  It is easy to verify that g satisfies (1), (2), (4), (5) and (6). Instead, for  $u=(1,2,2,3|9)$,  $B=\{\{1,2\},\{3,4\}\}$,  and  $u^*=(9,7|20)$  we find that

  $$\begin{aligned} \sum _{i\in B_1} g_i[u,B,u^*] = 5 \ngeq 9 = u_1^*. \end{aligned}$$

• A rule that fails to satisfy (4) Symmetry. We assume that the natural ordering of N is preserved within each $B_k$ for any B. We will call the leader of $B_k$ to the first member of $B_k$ according to this induced ordering.

  $$\begin{aligned} g_i[u,B,u^*]= {\left\{ \begin{array}{ll} u_i+\varphi _k[u^*]-\displaystyle \sum _{j\in B_k} u_j & \text {if} \, \text {\,is\,the\, leader\,of\,} B_k \\ u_i & \text {otherwise}\end{array}\right. } \end{aligned}$$

  It is easy to verify that g satisfies (1), (2), (3), (5) and (6). Instead, for  $u=(1,1,2,2|7)$,  $B=\{\{1,2\},\{3,4\}\}$,  and  $u^*=(4,6|14)$  we find that

  $$\begin{aligned} u_1=u_2\quad {\text {but}}\quad g_1[u,B,u^*] = 5 \ne 1 = g_2[u,B,u^*]. \end{aligned}$$

• A rule that fails to satisfy (5) Additivity. It combines proportional rule and Shapley rule.

  $$\begin{aligned} g_i[u,B,u^*]= {\left\{ \begin{array}{ll} \frac{u_i}{\displaystyle \sum _{j\in B_k} u_j}\,\varphi _k[u^*] & \text {if}\, i\in B_k \,\,\text {such\;that\,} u_j>0 \,\,\text {for\,all} \,j\in B_k \\ & \\ \psi _i[u,B,u^*] & \text {otherwise,\;for\;this\;and\;any\;other\;} B\end{array}\right. } \end{aligned}$$

  It is easy to verify that g satisfies (1), (2), (3), (4) and (6). Instead, if

  $$\begin{aligned} \begin{array}{ll} &u=(1,1,0,3|6),\quad B=\{\{1,2\},\{3,4\}\},\quad u^*=(4,8|20) \\ &v=(4,0,1,1|7),\quad B=\{\{1,2\},\{3,4\}\},\quad v^*=(8,4|20) \end{array} \end{aligned}$$

  then

  $$\begin{aligned} u+v=(5,1,1,4|13),\quad B=\{\{1,2\},\{3,4\}\},\quad u^*+v^*=(12,12|40) \end{aligned}$$

  and

  $$\begin{aligned} \begin{array}{llllll}&\,&g[u+v,B,u^*+v^*]=(16.67,3.33,4,16) \ne (4,4,4,8)&\,&+ (8,4,4,4) =g[u,B,u^*] + g[v,B,v^*]. \end{array} \end{aligned}$$

• A rule that fails to satisfy (6) Singletons.

  $$\begin{aligned} g_i[u,B,u^*]=u_i+\frac{\Delta (u^k)}{b_k}+\frac{\Delta (u^*)}{n}\quad {\text {if}}\,i\in B_k \end{aligned}$$

  It is easy to verify that g satisfies (1), (2), (3), (4) and (5). Instead, for  $u=(1,1,2,3|8)$,  $B=\{\{1\},\{2,3,4\}\}$,  and  $u^*=(1,9|14)$  we find that $B_1$ is a singleton but

  $$\begin{aligned} g_1[u,B,u^*] = 2 \ne 3 = \varphi _1[u^*]. \end{aligned}$$

Theorem 8.8

(Theorem 6.10) There is one and only one coalitional sharing rule defined on the cone $(E_N^B)^*$ that satisfies properties (1)–(6) in this cone. It is (the restriction of) the modified Shapley rule $\psi$.

Proof

(Existence) The proof is exactly the same as in Theorem 6.3.

(Uniqueness) The proof is the same as in Theorem 6.3 provided that we slightly modify the basis ${\mathcal{B}_0}$ introduced in Example 8.5(a) and Lemma 8.6 and replace it by a new basis $\overline{\mathcal{B}_0}$.

For each fixed and nontrivial coalition structure B, all members of ${\mathcal{B}_0}$ were CS-problems of the form $w=u\odot u^*$ where u, $u^*$ and all $u^k$ were (super)additive. However, this does not ensure that all of them belong to the cone $(E_N^B)^*$. Hence a little modification is needed in ${\mathcal{B}_0}$.

The new basis $\overline{\mathcal{B}_0}$ derives from ${\mathcal{B}_0}$ by only replacing, just for $h=1,2,\dots ,n,n+1$, the total utility $(u_0^h)^*_M$ by $m(u_0^h)_N$. Then we obtain a new set $\overline{\mathcal{B}_0}=\{\overline{w}_0\,^1,\overline{w}_0\,^2,\dots ,\overline{w}_0\,^d\}$, the matrix of which is still regular as the matrix given in Eq. (4), so $\overline{\mathcal{B}_0}$ is also a basis of $(E_N^B)$.

However, now, for each member $\overline{w}=\overline{u}\odot \overline{u}\,^*$ of $\overline{\mathcal{B}_0}$, we have

• (1) $\Delta (\overline{u}\,^*)>0$

• (2) $\Delta (\overline{u}\,^k)\ge 0$ for all k, and

• (3) $\Delta (\overline{u}\,^*)=m\Delta (\overline{u})>\frac{mb_k}{n}\Delta (\overline{u})$ for all k,

which are the sufficient conditions stated in Example 6.6 for belonging to $(E_N^B)^*$. Hence, since all members of the new basis belong to the cone, the uniqueness proof follows the same pattern as in Theorem 6.3. $\square$