Sharing the cost of hazardous transportation networks and the Priority Shapley value for multi-choice games

Abstract

We consider the cost sharing issue resulting from the maintenance of a hazardous waste transportation network represented by a sink tree. The participating agents are located on the nodes of the network and must transport their waste to the sink through costly network portions. We introduce the Liability rule, which is inspired by the principles applied by the courts to settle cost-allocation disputes in the context of hazardous waste. We provide an axiomatic characterization of this rule. Furthermore, we show that the Liability rule coincides with the Priority Shapley value, a new value on an appropriate domain of multi-choice games arising from hazardous waste transportation problems. Finally, we also axiomatize the Priority Shapley value on the full domain of multi-choice games.

Appendix

1.1 Proof of Theorem 1

This proof is based on several intermediate results stated in the following lemmas.

Lemma 1

Let f be an allocation rule on $\mathfrak {P}$ satisfying Independence of other higher waste amounts, and Intra balanced contributions for highest contributors. Then, for each $(g, w, C) \in \mathfrak {P}$, each agent $i \in N$, the following holds:

$$\begin{aligned} \forall j \in \{1, \ldots , w_i -1\}, \quad f_{(i, j)} (g, w, C) = f_{(i, w_i)} (g, w, C) + f_{(i, j)} (g, w \wedge \overrightarrow{(w_i -1)}, C).\end{aligned}$$

Proof

Pick any $(g, w, C) \in \mathfrak {P}$ and any agent $i \in N$. Thanks to Remark 1,

$$\begin{aligned}\forall j \in \{1, \ldots , w_i\}, \quad f_{(i, j)} (g, w, C) = f_{(i, j)} (g, w \wedge \overrightarrow{w_{i}}, C).\end{aligned}$$

Starting from $(g, w \wedge \overrightarrow{w_i}, C) \in \mathfrak {P}$, Remark 2 leads to:

$$\begin{aligned}{} & {} \forall j \in \{1, \ldots , w_i -1\}, \\ {}{} & {} f_{(i, j)} (g, w \wedge \overrightarrow{w_i}, C) = f_{(i, w_i)} (g, w \wedge \overrightarrow{w_i}, C) + f_{(i, j)} (g, (w \wedge \overrightarrow{w_i}) - e^{i}, C), \end{aligned}$$

so that

$$\begin{aligned}{} & {} \forall j \in \{1, \ldots , w_i-1\}, \nonumber \\ {}{} & {} f_{(i, j)} (g, w, C) = f_{(i, w_i)} (g, w \wedge \overrightarrow{w_i}, C)+ f_{(i, j)} (g, (w \wedge \overrightarrow{w_i}) - e^{i}, C). \end{aligned}$$

(10)

Finally, consider the right-hand side of (10). Using Remark 1 again, one obtains that

$$\begin{aligned}{} & {} f_{(i,w_i)} (g, w, C)= f_{(i, j)} (g, w \wedge \overrightarrow{w_i}, C) \text{ and } \nonumber \\ {}{} & {} f_{(i, j)} (g, (w \wedge \overrightarrow{w_i}) - e^{i}, C) = f_{(i, j)} (g, (w \wedge \overrightarrow{w_i}) \wedge \overrightarrow{(w_i -1)}, C).\end{aligned}$$

By noting that

$$\begin{aligned}f_{(i, j)} (g, (w \wedge \overrightarrow{w_i}) \wedge \overrightarrow{(w_i -1)}, C) = f_{(i, j)} (g, w \wedge \overrightarrow{(w_i -1)}, C), \end{aligned}$$

the result follows. $\square$

Lemma 2

Consider any $(g, w, C) \in \mathfrak {P}$, and any two leaves $\ell$ and $\ell '$ of g with the same successor i in g such that $C_\ell = C_{\ell '}$ and $w_\ell = w_{\ell '}$. If an allocation rule f satisfies Anonymity, then

$$\begin{aligned} \forall j \in \{1, \ldots , w_\ell \}, \quad f_{(\ell , j)} (g, w, C) = f_{(\ell ', j)} (g, w, C). \end{aligned}$$

Proof

Pick any $(g, w, C) \in \mathfrak {P}$, and two leaves $\ell$ and $\ell '$ in g as hypothesized. Let $\pi$ be the permutation on N such that $\pi (\ell ) = \ell '$, $\pi (\ell ') = \ell$, and $\pi (\ell '') = \ell ''$ for each other $\ell '' \in N {\setminus } \{\ell , \ell ' \}$. From $(g, w, C) \in \mathfrak {P}$, construct the corresponding hazardous transportation waste problem $(g^{\pi }, w^{\pi }, C^{\pi }) \in \mathfrak {P}$ as defined before the statement of Anonymity. From the structure of g and the fact that $C_\ell = C_{\ell '}$ and $w_\ell = w_{\ell '}$, $(g^{\pi }, w^{\pi }, C^{\pi })$ coincides with (g, w, C). By Anonymity, one obtains that

$$\begin{aligned} \forall j \in \{1, \ldots , w_\ell \},\quad f_{(\ell , j)} (g, w, C) = f_{(\pi (\ell ), j)} ((g^{\pi }, w^{\pi }, C^{\pi }) = f_{(\ell ', j)} (g, w, C).\end{aligned}$$

$\square$

Lemma 3

Let f be an allocation rule on $\mathfrak {P}$ satisfying Anonymity and Invariance to a relocation on a null cost portion. For each $(g, w, C) \in \mathfrak {P}$, each pair $\{i, i'\} \subseteq N$ such that $(i, i') \in E$, $C_i = C^0$, and $w_i = w_{i'}$, it holds that

$$\begin{aligned} \forall j \in \{1, \ldots , w_i\},\quad f_{(i, j)} (g, w, C) = f_{(i', j)} (g, w, C). \end{aligned}$$

Proof

Consider $(g, w, C) \in \mathfrak {P}$ and i and $i'$ as hypothesized. Let $\pi$ be the permutation on N such that $\pi (i) = i'$, $\pi (i') = i$ and $\pi (\ell ) = \ell$ for each other $\ell \in N {\setminus } \{i, i'\}$. Because $w_i = w_{i'}$, the following equality holds:

$$\begin{aligned}(g^{i \leftrightarrow i'}, w^{i\leftrightarrow i'}, C^{i \leftrightarrow i'} ) = (g^{\pi }, w^{\pi }, C^{\pi }).\end{aligned}$$

By Invariance to a relocation on a null cost portion,

$$\begin{aligned} \forall j \in \{1, \ldots , w_i \}, \quad f_{(i, j)} (g, w, C) = f_{(i, j)} (g^{i \leftrightarrow i'}, w^{i\leftrightarrow i'}, C^{i \leftrightarrow i'} ).\end{aligned}$$

By Anonymity,

$$\begin{aligned} \forall j \in \{1, \ldots , w_i \}, \quad f_{(i, j)} (g, w, C) = f_{(i', j)} (g^{\pi }, w^{\pi }, C^{\pi })= f_{(i', j)} (g^{i \leftrightarrow i'}, w^{i\leftrightarrow i'}, C^{i \leftrightarrow i'} ). \end{aligned}$$

Thus,

$$\begin{aligned} \forall j \in \{1, \ldots , w_i \}, \quad f_{(i, j)} (g^{i \leftrightarrow i'}, w^{i\leftrightarrow i'}, C^{i \leftrightarrow i'} ) = f_{(i', j)} (g^{i \leftrightarrow i'}, w^{i\leftrightarrow i'}, C^{i \leftrightarrow i'} ). \end{aligned}$$

In particular, by setting $(g', w', C'):= (g^{i \leftrightarrow i'}, w^{i\leftrightarrow i'}, C^{i \leftrightarrow i'} )$, one obtains that

$$\begin{aligned} \forall j \in \{1, \ldots , w_i \}, \quad f_{(i', j)} ((g')^{i' \leftrightarrow i}, (w')^{i'\leftrightarrow i}, (C')^{i' \leftrightarrow i} ) = f_{(i, j)} ((g')^{i' \leftrightarrow i}, (w')^{i'\leftrightarrow i}, (C')^{i' \leftrightarrow i} ). \end{aligned}$$

Since $((g')^{i' \leftrightarrow i}, (w')^{i'\leftrightarrow i}, (C')^{i' \leftrightarrow i} ) = (g, w, C)$, then

$$\begin{aligned} \forall j \in \{1, \ldots , w_i \}, \quad f_{(i', j)} (g, w, C) = f_{(i, j)} (g, w, C), \end{aligned}$$

$\square$

Proof of Theorem 1

Uniqueness part: The combination of the seven axioms induces at most one allocation rule on $\mathfrak {P}$. So, let f be an allocation rule on ${{\mathfrak {P}}}$ satisfying these seven axioms. To show that f is uniquely determined. Pick any hazardous waste transportation problem $(g,w, C) \in {\mathfrak P}$. Set ${{\mathfrak {K}}} (g, w, C)$ as the set of portions in g with a non-null cost:

$$\begin{aligned}{{\mathfrak {K}}} (g, w, C) = \bigl \{i \in N: C_i \not = C^0 \bigr \}. \end{aligned}$$

The proof proceeds by induction on the number of elements in ${{\mathfrak {K}}} (g, w, C)$.

Induction basis: ${{\mathfrak {K}}} (g, w, C)$ is empty. By Efficiency,

$$\begin{aligned}\sum _{i \in N } \sum _{j =1}^{w_i} f_{(i,j)} (g, w, C) = 0. \end{aligned}$$

Consider any directed path of length two of the form $(i', i, d)$. Cut the directed link $(i', i) \in E$ and connect $i'$ to the treatment facility d by adding the directed link $(i', d)$, ceteris paribus. Repeat this operation until there is no directed path of length two leading to d. The resulting sink tree $g^{\star }$ is such that each $i \in N$ is directly connected to the treatment facility d. Apply repeatedly Invariance to a cut-and-connect operation on a null cost portion to obtain (with a slight abuse of notation),

$$\begin{aligned}\forall i \in N, \forall j \in \{1, \ldots , w_i\}, \quad f_{(i,j)} (g, w, C) = f_{(i,j)} (g^{\star }, w, C).\end{aligned}$$

To complete the induction basis, it suffices to prove that $f_{(i,j)} (g^{\star }, w, C) = 0$, for each agent $i \in N$, and each $j \in \{1, \ldots , w_i\}$. To that end, one again proceeds by induction on the value of $\max (w)$.

• Induction basis: If $\max (w) = 1$, then $Q^{w} (1) = N$. The Efficiency condition applied to $(g^{\star }, w, C)$ becomes

  $$\begin{aligned}\sum _{i \in N } f_{(i,1)} (g^{\star }, w, C) = 0.\end{aligned}$$

  From Lemma 2, all agents in N obtain the same payoff. Together with the Efficiency condition, this forces that $f_{(i, 1)} (g^{\star }, w, C) = 0$ for each $i \in N$.

• Induction hypothesis: Assume that the assertion is true whenever $\max (w) \le k$, for $k \ge 1.$

• Induction step: Consider the situation where $\max (w) = k+1$. If $i \in N {\setminus } Q^w(k +1)$, then Remark 1 and the induction hypothesis yield the following equalities:

  $$\begin{aligned} \forall j \in \{1, \ldots , w_{i} \}, \quad f_{(i,j)} (g^{\star }, w, C) = f_{(i,j)} (g^{\star }, w \wedge \overrightarrow{k}, C) = 0. \end{aligned}$$

  It thus remains to deal with the agents in $Q^w(k +1)$. By Lemma 2,

  $$\begin{aligned}\forall \{i, i'\} \subseteq Q^w(k +1), \quad f_{(i, k+1)} (g^{\star }, w, C) = f_{(i', k+1)} (g^{\star }, w, C). \end{aligned}$$

  By the induction hypothesis, for each agent $i \in Q^w(k +1)$, one has

  $$\begin{aligned} \forall j \in \{1, \ldots , k\}, \quad f_{(i, j)} (g^{\star }, w \wedge \overrightarrow{k}, C) = 0.\end{aligned}$$

  Using the above equality and Lemma 1, one obtains

  $$\begin{aligned} \forall i\in & {} Q^w(k +1), \forall j \in \{1, \ldots , k\}, \nonumber \\ f_{(i, j)} (g^{\star }, w, C)= & {} f_{(i, k+1)} (g^{\star }, w, C) + f_{(i, j)} (g^{\star }, w \wedge \overrightarrow{k}, C) \nonumber \\= & {} f_{(i, k+1)} (g^{\star }, w, C). \end{aligned}$$

  (11)

  It amounts to saying that, for any $i' \in Q^w(k +1)$,

  $$\begin{aligned} \sum _{i \in Q^w(k +1)} \sum _{j = 1}^{k+1} f_{(i, j)} (g^{\star }, w, C) = (k+1) \vert Q^w(k +1) \vert f_{(i', k+1)} (g^{\star }, w, C). \end{aligned}$$

  On the other hand, by Efficiency,

  $$\begin{aligned} \sum _{i \in Q^w(k +1)} \sum _{j = 1}^{k+1} f_{(i, j)} (g^{\star }, w, C)= \sum _{i \in N} \sum _{j=1}^{w_i} f_{(i, j)} (g^{\star }, w, C) = 0. \end{aligned}$$

  Therefore, one concludes that

  $$\begin{aligned}\forall i' \in Q^w(k +1), \quad f_{(i', k+1)} (g^{\star }, w, C) = 0,\end{aligned}$$

  and so, thanks to (11), for each $j \in \{ 1,\ldots , k+1\}$, $f_{(i', j)} (g^{\star }, w, C)= 0$, as desired.

Induction hypothesis: Suppose that f(g, w, C) is uniquely determined when ${{\mathfrak {K}}} (g, w, C)$ contains t elements, for $t \ge 0$.

Induction step: Suppose that ${{\mathfrak {K}}} (g, w, C)$ contains $t+1$ elements. Let

$$\begin{aligned}U_g = \bigcap _{i \in {\mathfrak K} (g, w, C)} {U}_g [i]. \end{aligned}$$

Case 1. If $U_g = \emptyset$, then, for each $i \in N$, there exists $i' \in {{\mathfrak {K}}} (g, w, C)$ such that $i \notin U_g [i']$, and so $D_g [i] \cap U_g [i'] = \emptyset$. Now define the cost profile $C - \lambda ^{C, i'}$ (see (2) for the definition of $\lambda ^{C, i'}$). Because $i' \notin D_g [i]$, by Path consistency, one obtains

$$\begin{aligned} \forall j \in \{1, \ldots , w_i\}, f_{(i, j)}(g, w, C ) = f_{(i, j)}(g, w, C - \lambda ^{C, i'}).\end{aligned}$$

By the induction hypothesis, for each

$$\begin{aligned} j \in \{1, \ldots , w_i\}, \quad f_{(i, j)}(g, w, C - \lambda ^{C, i'}) \text{ is } \text{ uniquely } \text{ determined } \end{aligned}$$

and so is $f_{(i, j)}(g, w, C )$, as desired.

Case 2. If $U_g \ne \emptyset$, then, since g is a sink tree, there exists $i' \in {{\mathfrak {K}}} (g, w, C)$ such that $U_g = U_g [i'].$ If $i \notin U_g$, then proceeding as in Case 1, one concludes that, for each $j \in \{1, \ldots , w_i\}$, $f_{(i, j)}(g, w, C )$ is uniquely determined. It remains to deal with the payoffs of agents belonging to $U_g$. By definition of $U_g$, for each $i \in U_g {\setminus } i'$, $C_i = C^0$. Consider any directed path of length two of the form $(\ell , \ell ', i')$. Cut the directed link $(\ell , \ell ') \in E$ and connect $\ell '$ to i by adding the directed link $(\ell , i')$, ceteris paribus. Repeat this operation until there is no directed path of length two leading to $i'$. The resulting sink tree $g^{\star }$ is such that each $i \in U_g\setminus i'$ is now directly connected to $i'$. Apply repeatedly Invariance to a cut-and-connect operation on a null cost portion to obtain

$$\begin{aligned} \forall i \in U_g\setminus i', \forall j \in \{1, \ldots , w_i\}, \quad f_{(i,j)} (g, w, C) = f_{(i,j)} (g^{\star }, w, C).\end{aligned}$$

To complete the proof, one again proceeds by induction on the value of $\max (w_{U_{g}})$ where $w_{U_{g}}$ stands for the restriction of w to the components of $U_g$.

• Induction basis: If $\max (w_{U_{g}}) = 1$, then, the Efficiency condition applied to (g, w, C) and the above equalities give

  $$\begin{aligned}\sum _{i \in U_g } f_{(i,1)} (g^{\star }, w, C) = \sum _{i \in N} C_i \biggl (\sum _{\ell \in U_g[i]} w_{\ell } \biggr ) - \sum _{i \in N \setminus U_g} \sum _{j = 1}^{w_i} f_{(i,j)} (g, w, C).\end{aligned}$$

  Note that the right-hand side of the above equality is uniquely determined. On the other hand, by Lemma 2, for each pair $\{\ell , \ell '\} \subseteq U_g \setminus i'$, $f_{(\ell ,1)} (g^{\star }, w, C) = f_{(\ell ',1)} (g^{\star }, w, C)$. Furthermore, by Lemma 3, $f_{(\ell ,1)} (g^{\star }, w, C) = f_{(i', 1)} (g^{\star }, w, C),$ meaning that each agent in $U_g$ obtains the same payoff. Together with the Efficiency condition, this forces that

  $$\begin{aligned} \forall \ell \in U_g, \quad f_{(\ell , 1)} (g^{\star }, w, C) = \frac{1}{ \vert U_g \vert } \biggl [ \sum _{i \in N} C_i \biggl (\sum _{\ell ' \in U_g[i]} w_{\ell '} \biggr ) - \sum _{i \in N \setminus U_g} \sum _{j =1}^ {w_i} f_{(i,j)} (g, w, C) \biggr ]. \end{aligned}$$

  Therefore, for each $\ell \in U$, $f_{(\ell , 1)} (g^{\star }, w, C)$ is uniquely determined, as desired.

• Induction hypothesis: Assume that the assertion is true whenever $\max (w_{U_g}) \le k$, for $k \ge 1.$

• Induction step: Consider the situation where $\max (w_{U_g})= k+1$. If $i \in U_g {\setminus } Q^w(k +1)$, then by Remark 1,

  $$\begin{aligned}\forall j \in \{1, \ldots , w_{i} \}, \quad f_{(i,j)} (g^{\star }, w, C) = f_{(i,j)} (g^{\star }, w \wedge \overrightarrow{k}, C),\end{aligned}$$

  which is uniquely determined by the induction hypothesis. So, it remains to deal with the agents in $U_g\cap Q^w(k +1)$. By Lemma 1,

  $$\begin{aligned} \forall i\in & {} U_g \cap Q^w(k +1), \forall j \in \{1, \ldots , k\},\nonumber \\ f_{(i, j)} (g^{\star }, w, C)= & {} f_{(i, k+1)} (g^{\star }, w, C) + f_{(i, j)} (g^{\star }, w \wedge \overrightarrow{k}, C). \end{aligned}$$

  (12)

  By the induction hypothesis, for each $j \in \{1, \ldots , k\}$, $f_{(i, j)}(g^{\star }, w \wedge \overrightarrow{k}, C)$ is uniquely determined. It remains to prove that $f_{(i, k+1)} (g^{\star }, w, C)$ is uniquely determined. By Lemma 2,

  $$\begin{aligned}\forall \{\ell , \ell '\} \subseteq (U_g \setminus i)\cap Q^w(k +1), \quad f_{(\ell , k+1)} (g^{\star }, w, C) = f_{(\ell ', k+1)} (g^{\star }, w, C). \end{aligned}$$

  Furthermore, if $i' \in Q^w(k +1)$, then, by Lemma 3, one obtains

  $$\begin{aligned} f_{(\ell , k+1)} (g^{\star }, w, C) = f_{(i', k+1)} (g^{\star }, w, C). \end{aligned}$$

  All in all, all agents of $U_g\cap Q^w(k +1)$ obtain the same payoff at level $k+1$. Let $\alpha _{k+1}$ denote this common value. Using the Efficiency condition and the fact that, for each pair (i, j) such that $i \in N {\setminus } (U_g\cap Q^w(k +1))$ and $1 \le j \le w_i$, the payoff $f_{(i, j)} (g^{\star }, w, C)$ is uniquely determined (see above), one concludes that the quantity

  $$\begin{aligned}{} & {} \sum _{i \in U_g\cap Q^w(k +1)} \sum _{ j = 1}^{k+1} f_{(i, j)} (g^{\star }, w, C) \\ {}{} & {} = \sum _{i \in N} \sum _{ j = 1}^{w_i} f_{(i, j)} (g^{\star }, w, C) - \sum _{i \in N \setminus (U_g\cap Q^w(k +1))} \sum _{j =1}^{w_i} f_{(i, j)} (g^{\star }, w, C)\end{aligned}$$

  is uniquely determined. Furthermore, by (12), one obtains

  $$\begin{aligned}{} & {} \sum _{i \in U_g\cap Q^w(k +1)} \sum _{j = 1}^{k+1} f_{(i, j)} (g^{\star }, w, C) \\ {}{} & {} \quad = \sum _{i \in U_g\cap Q^w(k +1)} \sum _{j=1}^{k} \biggl ( f_{(i, j)} (w \wedge \overrightarrow{k}, C) + \alpha _{k+1} \biggr )+ \sum _{i \in U_g\cap Q^w(k +1)} \alpha _{k+1}, \end{aligned}$$

  which leads to

  $$\begin{aligned}{} & {} \sum _{i \in U_g\cap Q^w(k +1)} \sum _{j = 1}^{k+1} f_{(i, j)} (g^{\star }, w, C) = \sum _{i \in U_g\cap Q^w(k +1)} \sum _{ j=1}^{k} f_{(i, j)} (g^{\star }, w \wedge \overrightarrow{k}, C) \\ {}{} & {} \quad + (k+1) \vert U_g \cap Q^w(k +1) \vert \alpha _{k+1}. \end{aligned}$$

  By the induction hypothesis,

  $$\begin{aligned}\sum _{i \in U_g\cap Q^w(k +1)} \sum _{j = 1}^{k} f_{(i, j)} (g^{\star }, w \wedge \overrightarrow{k}, C)\end{aligned}$$

  is uniquely determined, which allows to conclude that $\alpha _{k+1}$ is uniquely determined. The proof is complete.

Existence part: One verifies that $f^L$ satisfies the seven axioms. From the definition given in (4) of $f^L$, the cost increase at level k,

$$\begin{aligned}C_{i'} \biggl (\sum _{\ell \in U_g [i']} w_{\ell } \wedge k \biggr ) - C_{i'} \biggl (\sum _{\ell \in U_g [i']} w_{\ell } \wedge (k - 1) \biggr ),\end{aligned}$$

is paid only by the agents i located upstream of $i'$ provided that $w_i \ge k$, that is, this cost increase is paid only by the agents belonging to $U_g[i'] \cap Q^w(k)$. Therefore, $f^L$ trivially satisfies Path consistency and Independence of other higher waste amounts. Each $i \in U_g[i'] \cap Q^w(k)$ pays an equal share of this cost increase by charging it equally to each waste amount $k, k - 1, \ldots , 1$, that is, the Liability rule charges an amount

$$\begin{aligned}\frac{1}{k} \cdot \frac{C_{i'} (\sum _{\ell \in U_g [i']} w_{\ell } \wedge k) - C_{i'} (\sum _{\ell \in U_g [i']} w_{\ell } \wedge (k - 1))}{|U_g[i'] \cap Q^w(k)|},\end{aligned}$$

to each waste amount $j \in \{1, \ldots , k\}$. It follows that if agent i ships $w_i - 1$ units of waste instead of $w_i$, each unit $w_i - 1, \ldots , 1$, will reduce the amount to be paid by the same amount,

$$\begin{aligned} \frac{1}{w_i} \cdot \frac{C_{i'} (\sum _{\ell \in U_g [i']} w_{\ell } \wedge w_i) - C_{i'} (\sum _{\ell \in U_g [i']} w_{\ell } \wedge (w_i - 1))}{|U_g[i'] \cap Q^w(w_i)|}.\end{aligned}$$

This shows that $f^L$ satisfies Intra balanced contributions for highest contributors. It must also be clear that the cost charged to each waste amount j of i does not depend on the label i. Thus, $f^L$ satisfies Anonymity as well. From the above discussion, it follows that if the cost function $C_{i'} = C^0$ is the null cost function, the portion connecting each predecessor $i \in P_g(i')$ to $i'$ can be removed and replaced by a portion connecting each of these predecessors to the successor of $i'$, without modifying the cost share to be paid for i under $f^L$. This ensures that $f^L$ satisfies Invariance to a cut-and-connect operation on a null cost portion. Furthermore, i has trivially no cost to pay when it passes its waste on its portion, and each successor of i has nothing to pay either, since it does not use that portion. Then, under $f^L$, these agents will pay part of the cost increases of the portions from i’s successor to d. As a result, the positions of i and its successor can be exchanged without modifying their payoffs under $f^L$. This ensures that $f^L$ satisfies Invariance to a relocation on a null cost portion. It remains to prove that $f^L$ indeed satisfies Efficiency. First the total cost paid by $i \in N$ in (g, w, C) under $f^L$ is

$$\begin{aligned} \sum _{j = 1}^{w_i} f_{(i, j)}^L (g, w,C)= & {} \sum _{j = 1}^{w_i} \sum _{i' \in D_g [i] } \sum _{\begin{array}{c} k \ge j: \\ i \in Q^w(k) \end{array}} \frac{1}{k} \cdot \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] \cap Q^w(k) | } \\= & {} \sum _{i' \in D_g [i] } \sum _{k= 1}^{w_i} \sum _{j = 1}^{k} \frac{1}{k} \cdot \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] \cap Q^w(k) | } \\= & {} \sum _{i' \in D_g [i] } \sum _{k= 1}^{w_i} \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] \cap Q^w(k) | }. \end{aligned}$$

Then,

$$\begin{aligned} \sum _{i \in N} \sum _{j = 1}^{w_i} f_{(i, j)}^L (g, w,C)= & {} \sum _{i \in N} \sum _{i' \in D_g [i] } \sum _{k= 1}^{w_i} \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] \cap Q^w(k) | } \\= & {} \sum _{i' \in N} \sum _{i \in U_g [i'] } \sum _{k= 1}^{w_i} \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] \cap Q^w(k) | } \\= & {} \sum _{i' \in N} \sum _{k= 1}^{\max (w)} \sum _{i \in U_g [i'] \cap Q^w(k)} \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \biggl ) }{| U_g [i'] \cap Q^w(k) | } \\= & {} \sum _{i' \in N} \sum _{k= 1}^{\max (w)} C_{i'} \biggl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \biggr ) - C_{i'} \biggl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \biggl ) \\= & {} \sum _{i' \in N} C_{i'} \biggl (\sum _{\ell \in U_{g} [i']} w_{\ell } \biggr ), \end{aligned}$$

which proves that $f^L$ satisfies Efficiency. This completes the proof of Theorem 1. $\square$

Logical independence of the axioms. The axioms invoked in Theorem 1 are logically independent, as shown by the following alternative allocation rules.

• The allocation rule f on $\mathfrak {P}$ defined as:

  $$\begin{aligned}\forall (i, j) \in M^+, \quad f_{(i,j)} (g,w, C) = 0\end{aligned}$$

  satisfies all the axioms except Efficiency.

• The allocation rule f on $\mathfrak {P}$ defined as:

  $$\begin{aligned}{} & {} \forall (i, j) \in M^+, \quad f_{(i,j)} (g, w, C) = \\ {}{} & {} \sum _{i' \in D_g [i] } \sum _{k = j }^{\max (w)} \frac{1}{ w_i \wedge k} \cdot \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] | } \end{aligned}$$

  satisfies all the axioms except Independence of other higher waste amounts.

• The allocation rule f on $\mathfrak {P}$ defined as:

  $$\begin{aligned}{} & {} \forall (i, j) \in M^+, \quad f_{(i,j)} (g, w, C) \\ {}{} & {} = \sum _{i' \in N } \sum _{\begin{array}{c} k \ge j: \\ i \in Q^w(k) \end{array}} \frac{1}{ k}\cdot \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| Q^w (k) | }\end{aligned}$$

  satisfies all the axioms except Path consistency.

• The Responsibility rule $f^R$ on $\mathfrak {P}$ defined as:

  $$\begin{aligned}{} & {} \forall (i, j) \in M^+, \quad f^R_{(i,j)} (g, w, C) \\ {}{} & {} = \sum _{i' \in D_g [i]} \sum _{\begin{array}{c} k \ge j: \\ i \in Q^w(k) \end{array}} \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k \wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{| U_g [i'] \cap Q^w (k) | }\end{aligned}$$

  satisfies all the axioms except Intra balanced contributions for highest contributors.

• The allocation rule f on $\mathfrak {P}$ defined as:

  $$\begin{aligned}{} & {} \forall (i, j) \in M^+, \quad f_{(i,j)} (g, w, C) \\{} & {} = \sum _{i' \in D_g [i]} \sum _{\begin{array}{c} k \ge j: \\ i \in Q^w(k) \end{array}} i \cdot \frac{C_{i'} \bigl (\sum _{\ell \in U_{g} [i']} k\wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_g [i']} (k - 1) \wedge w_{\ell } \bigl ) }{\sum _{ \ell \in U_g [i'] \cap Q^w(k)} \ell }\end{aligned}$$

  satisfies all the axioms except Anonymity.

• To define the next allocation rule, recall that by (2), C can be decomposed as:

  $$\begin{aligned}C = \sum _{i' \in N} \lambda ^{C, i'}.\end{aligned}$$

  And by (3), one has the following decomposition of C:

  $$\begin{aligned} \lambda ^{C, i'} = \sum _{ k = 1}^{\max (w)} \lambda ^{C, i', k}\, \text{ and } \text{ so } \, C = \sum _{i' \in N} \sum _{ k = 1}^{\max (w)} \lambda ^{C, i', k}, \end{aligned}$$

  where,

  $$\begin{aligned}\forall s \in \prod _{i \in N} \{0, \ldots , w_i\}, \quad \lambda ^{C, i', k}(s) = \lambda ^{C, i'} \biggl ( \sum _{i \in U_{g}[i']} s_{i} \wedge k \biggr )- \lambda ^{C, i'} \biggl ( \sum _{i \in U_{g}[i']} s_{i} \wedge (k - 1) \biggr ). \end{aligned}$$

  From this, define the additive allocation rule f as follows: if $i'$ is a successor of a leaf of g and $i' \in Q^w(k)$, then

  $$\begin{aligned} \forall j \in \{1, \ldots , k \}, \quad f_{(i', j)} (g,w, \lambda ^{C,i', k} ) = \frac{1}{ k} \biggl (C_{i'} \bigl (\sum _{\ell \in U_{[i']}} k\wedge w_{\ell } \bigr ) - C_{i'} \bigl (\sum _{\ell \in U_{[i']}} (k - 1) \wedge w_{\ell } \bigr ) \biggr ), \end{aligned}$$

  and $f_{(i, j)} (g, w,\lambda ^{C,i', k} ) = 0$ if either $i \in N {\setminus } i'$ or $i = i'$ and $j \in \{k + 1, \ldots , w_i\}$.

  In any other case, that is, if $i'$ is neither a successor of a leaf of g nor $i' \in Q^w(k)$,

  $$\begin{aligned} \forall i \in N, \forall j \in \{1, \ldots , w_i\}, \quad f_{(i, j)}(g, w, \lambda ^{C,i', k} ) = f^L_{(i, j)}(g, w, \lambda ^{C,i', k} ). \end{aligned}$$

  By additivity of f,

  $$\begin{aligned}f(g,w, C ) = \sum _{i' \in N} \sum _{ k = 1}^{\max (w)} f_{}(g, w, \lambda ^{C,i', k} ). \end{aligned}$$

  This allocation rule f on $\mathfrak {P}$ satisfies all the axioms except Invariance to a relocation on a null cost portion.

• Define the allocation rule f as follows. Pick any $(g,w, C) \in \mathfrak {P}$. Consider the set $P_g(d)$ of predecessors of the sink d and define $S_g = \{i_1, i_2, \ldots , i_p\} \subseteq P_g (d)$ as the possibly empty subset of predecessors of the sink d that are also leaves of g and such that $w_{i_t} = 1$ for $t \in \{1, \ldots ,p\}$; and the complementary subset $P_g (d) {\setminus } S_g = \{i'_1, i'_2, \ldots , i'_q\}$. One distinguishes two cases.

  Case 1. If $P_g (d) \setminus S_g \ne \emptyset$ and $S_{{g}} \ne \emptyset$, define the cost function $C'$ as:

  $$\begin{aligned} C'_{i} = {\left\{ \begin{array}{ll} C_{i} + \frac{\displaystyle 1}{\displaystyle q} &{} \text{ if } i \in P_g (d) \setminus S_g, \\ C_i &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

  In this case,

  $$\begin{aligned} f_{(i,j)} (g,w, C) = {\left\{ \begin{array}{ll} f^L_{(i, 1)} (g, w, C ) - \frac{\displaystyle 1}{ \displaystyle p} &{} \text{ if } i \in S_g, \\ &{} \\ f_{(i, j)}^L(g, w, C' ) &{} \text{ if } i \in N \setminus S_g \text{ and } j \in \{1, \ldots , w_i\}. \end{array}\right. } \end{aligned}$$

  Case 2. Either $P_g (d) \setminus S_g = \emptyset$ or $S_g = \emptyset$. Then, set $f (g, w,C) = f^L (g, w, c)$.

  This allocation rule f on $\mathfrak {P}$ satisfies all the axioms except Invariance to a cut-and-connect operation on a null cost portion.

1.2 Proof of Theorem 2

Proof of Theorem 2

From Theorem 1, we know that the Liability rule $f^L$ satisfies the axioms on ${{\mathfrak {P}}}^*$. Hence, only the uniqueness part has to be shown. So, let f be an allocation rule on ${{\mathfrak {P}}}^*$ satisfying the axioms of the statement of Theorem 2. Pick any hazardous waste transportation problem $(g,w, C) \in {{\mathfrak {P}}}^*$. As in the proof of Theorem 1, set ${{\mathfrak {K}}} (g, w, C)$ as the set of portions in g with a non-null cost:

$$\begin{aligned}{{\mathfrak {K}}} (g, w, C) = \bigl \{i \in N: C_i \not = C^0 \bigr \}. \end{aligned}$$

The proof proceeds by induction on the number of elements in ${{\mathfrak {K}}} (g, w, C)$.

Induction basis: ${{\mathfrak {K}}} (g, w, C)$ is empty. By Efficiency,

$$\begin{aligned}\sum _{i \in N } f_{(i,1)} (g, w, C) = 0. \end{aligned}$$

By Lemma 3, all agents in N obtain the same payoff. By Efficiency, it follows that $f (g, w, C) = \overrightarrow{0}.$

Induction hypothesis: Suppose that f(g, w, C) is uniquely determined when ${{\mathfrak {K}}} (g, w, C)$ contains t elements, for $t \ge 0$.

Induction step: Suppose that ${{\mathfrak {K}}} (g, w, C)$ contains $t+1$ elements. Let $\ell$ be the most distant agent from d such that $C_k = C^0$. This means that each $i \in U_g [\ell ]$ is such that $C_i = C^0$. By Lemma 3, all agents in $U_g [\ell ]$ obtain the same payoff, say $c^{f} \in \mathbb {R}$. Next, consider any $i \in N {\setminus } U_g[k]$. By Path consistency,

$$\begin{aligned}f_{(i, 1)} (g, w,C) = f_{(i, 1)} (g, w, C - \lambda ^{\ell , C} ). \end{aligned}$$

By induction hypothesis, $f_{(i, 1)} (g, w, C - \lambda ^{\lambda , C} )$ is uniquely determined, so is $f_{(i, 1)} (g, w, C)$, for each $i \in N {\setminus } U_g[k]$. By Efficiency,

$$\begin{aligned}c^{f} = \frac{\sum _{i \in N} C_i (i) - \sum _{i \in N \setminus U_g[k]} f_{(i, 1)} (g, w, C)}{|U_g[k]|} \end{aligned}$$

is uniquely determined. This complete the proof of the induction step. $\square$

1.3 Proof of Proposition 1

Before starting with the proof of Proposition 1, we provide the following definition. A coalition $s \in \mathcal{C}$ is induced by an ordering $\sigma \in O^m$ if there is a pair $(i, j) \in M^+$ such that $s^{\sigma , (i, j) } = s$. Let $I^{\sigma }$ be the subset of coalitions in $\mathcal{C}$ induced by $\sigma$. Obviously, the grand coalition m and the null coalition $\overrightarrow{0}$ belong to $I^{\sigma }$ whatever $\sigma \in O^m$.

Proof of Proposition 1

Point 1. From the definition of the Priority Shapley value given in (9), one observes that $s^{\sigma , (i, j)}$ and $s^{\sigma , (i, j)} - e^i$ are in $\mathcal{C}$, and coalition $s^{\sigma , (i, j)}$ is such that $\max (s^{\sigma , (i, j)}) = j$. From this observation, one deduces that

$$\begin{aligned}\forall (i, j) \in M^+, \quad \phi _{(i, j)}(m,v)= \frac{1}{\prod _{r = 1}^{\max (m)} q^m_r !} \sum _{\begin{array}{c} s \in \mathcal{C}: \\ s_i \ge j \end{array}} \quad \sum _{\begin{array}{c} \sigma \in O^m: \\ I^{\sigma }\ni s \end{array}} \frac{v(s ) - v(s-e^i ) }{\max (s)}. \end{aligned}$$

Using Remark 3, for any coalition $s \in \mathcal{C}$, the number of orderings $\sigma \in O^m$ such that $I^{\sigma }\ni s$ is given by:

$$\begin{aligned} \biggl ( \prod _{r= 1}^{\max (s)-1} q^m_r ! \biggr ) \bigl (q^s_{\max (s)} -1\bigr )! \bigl (q^m_{\max (s)} - q^s_{\max (s)}\bigr ) ! \biggl ( \prod _{r = \max (s)+1}^{\max (m)} q^m_r ! \biggr ). \end{aligned}$$

From this, one directly gets the desired result:

$$\begin{aligned} \forall (i, j) \in M^+, \quad \phi _{(i, j)}(m,v)= \sum _{\begin{array}{c} s \in \mathcal{C}: \\ s_i \ge j \end{array}} \frac{ (q^m_{\max (s)} - q^s_{\max (s)}) ! (q^s_{\max (s)} -1)! }{q^m_{\max (s)}!} \biggl [\frac{v(s ) - v(s-e^i ) }{\max (s) } \biggr ].\end{aligned}$$

Point 2. The Priority Shapley value $\phi$ is obviously a linear function in v. By (7), it follows that

$$\begin{aligned} \forall (i, j) \in M^+, \quad \phi _{(i, j)} (m, v) = \sum _{s \le m} \Delta _v(s) \phi _{(i, j)} (m, u_s). \end{aligned}$$

(13)

Note that, for any ordering $\sigma \in O^m$, $\eta ^{\sigma }_{(i, j)}(m,u_s) = 1$ if and only if the following conditions hold:

• $i \in T(s)$;

• $j = \max (s)$;

• For any other agent $\ell \in T(s)$, $\sigma (i, \max (s)) > \sigma (\ell , \max (s))$.

In all other cases, $\eta ^{\sigma }_{(i, j)}(m,u_s) = 0$. By the above fact and definition (8) of $\lambda ^{\sigma }_{(i, j)} (m, u_s)$, it follows that

$$\begin{aligned}{} & {} \forall (i, j) \in M^+, \quad \lambda ^{\sigma }_{(i, j)} (m, u_s) \\ {}{} & {} \quad = \left\{ \begin{array}{ll} \frac{\displaystyle 1}{\displaystyle \max (s)} &{} \text{ if } i \in T(s), j \le \max (s), \forall \ell \in T(s)\setminus i, \sigma (i, \max (s)) > \sigma (\ell , \max (s)), \\ 0 &{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

By definition (9) of $\phi _{(i, j)} (m, u_s)$, it holds that

$$\begin{aligned}{} & {} \forall (i, j) \in M^+: i \in T(s), j \le \max (s), \\{} & {} \phi _{(i, j)}(m, u_s) = \frac{1}{|O^m | } \sum _{\sigma \in O^m } \lambda ^\sigma _{(i,j)}(m, u_s) \\{} & {} \quad = \frac{1}{ |O^m |} \sum _{{\mathop { \forall \ell \in T(s), \sigma (i, \max (s)) > \sigma (\ell , \max (s))}\limits ^{ \sigma \in O^m:}}} \frac{1}{\max (s)}. \end{aligned}$$

And, for each $i \in N {\setminus } T(s)$ and $j \in M_i$, $\phi _{(i, j)}(m, u_t) = 0$.

To complete the proof, note that, for $i, \ell \in T(s)$, the number of restricted orders $\sigma$ such that $\sigma (i, \max (s)) > \sigma (\ell , \max (s))$ is equal to the number of restricted orders $\sigma$ such that $\sigma (\ell , \max (s)) > \sigma (i, \max (s))$, and it is given by

$$\begin{aligned} \frac{\vert O^m \vert }{\vert T(s) \vert }.\end{aligned}$$

It follows that

$$\begin{aligned}{} & {} \forall (i, j) \in M^+: i \in T(s), j \le \max (s), \\{} & {} \phi _{(i, j)}(m, u_s) = \frac{1}{ |O^m |} \sum _{{\mathop { \forall \ell \in T(s), \sigma (i, \max (s)) > \sigma (\ell , \max (s))}\limits ^{ \sigma \in O^m:}}} \frac{1}{\max (s)} \\{} & {} \quad = \frac{1}{ |O^m |} \frac{\vert O^m \vert }{\vert T(s) \vert }\frac{1}{\max (s)} \\{} & {} \quad = \frac{1}{|T(s)| \max (s)}, \end{aligned}$$

and so

$$\begin{aligned} \forall (i, j) \in M^+, \quad \phi _{(i, j)} (m, u_s) = \left\{ \begin{array}{ll} \frac{\displaystyle 1}{\displaystyle |T(s)| \max (s)} &{} \text{ if } i \in T(s), j \le \max (s), \\ 0 &{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

The desired result follows by (13). $\square$

1.4 Proof of Proposition 2

Proof of Proposition 2

Consider any hazardous waste transportation problem $(g,w, C) \in {{\mathfrak {P}}}$ and its associated multi-choice game $(w, v_{g,C}) \in \mathfrak {G}$. By (2), C can be decomposed as:

$$\begin{aligned}C = \sum _{i' \in N} \lambda ^{C, i'}.\end{aligned}$$

By (3), one has the following decomposition of C:

$$\begin{aligned} \lambda ^{C, i'} = \sum _{ k = 1}^{\max (w)} \lambda ^{C, i', k}, \text{ and } \text{ so } \, C = \sum _{i' \in N} \sum _{ k = 1}^{\max (w)} \lambda ^{C, i', k}, \end{aligned}$$

where

$$\begin{aligned}\forall s \in \prod _{i \in N} \{0, \ldots , w_i\}, \quad \lambda ^{C, i', k}(s) = \lambda ^{C, i'} \biggl ( \sum _{i \in U_{g}[i']} s_{i} \wedge k \biggr )- \lambda ^{C, i'} \biggl ( \sum _{i \in U_{g}[i']} s_{i} \wedge (k - 1) \biggr ). \end{aligned}$$

Pick another hazardous waste transportation problem $(g, w,C') \in {{\mathfrak {P}}}$ with the same sink tree and waste profile as (g, w, C) and define the hazardous waste transportation problem $(g, w,C + C') \in {{\mathfrak {P}}}$. By definition of the coalition functions $v_{g, C}$ and $v_{g, C'}$, it holds that $v_{g, C+ C'} = v_{g, C} + v_{g, C'}$. Observe also that the Priority Shapley value $\phi$ given in (9) is additive in v and the Liability rule $f^L$ given in (4) is additive in C. Therefore, using (3), it suffices to prove that

$$\begin{aligned}\forall k \in \{1, \ldots , \max (w)\}, \quad \phi (w, v_{g, \lambda ^{C, i', k}}) = f^L (g,w, \lambda ^{C, i', k}).\end{aligned}$$

From the definition of the Liability rule $f^L$, one has that

$$\begin{aligned}{} & {} f_{(i, j)}^L (g,w, \lambda ^{C, i', k}) = \\ {}{} & {} {\left\{ \begin{array}{ll} \frac{\displaystyle 1}{\displaystyle k} \cdot \frac{\displaystyle C_{i'} \bigl ( \sum _{\ell \in U_{g}[i']} w_{\ell } \wedge k \bigr ) - C_{i'} \bigl ( \sum _{\ell \in U_{g}[i']} w_{\ell } \wedge (k - 1) \bigr )}{\displaystyle | U_{g}[i']\cap Q^{w} (k)|} &{} \text{ if } i \in U_{g}[i'] \cap Q^{w} (k), j \in \{1, \ldots , k\}, \\ &{} \\ 0 &{} \text{ otherwise. } \\ \end{array}\right. } \end{aligned}$$

On the other hand,

$$\begin{aligned} \forall s \in \mathcal {M}, \quad v_{g, \lambda ^{C, i', k}} (s)= & {} \lambda ^{C, i', k} (s) \\= & {} C_{i'} \biggl ( \sum _{\ell \in U_{g}[i']}s_{\ell } \wedge k \biggr ) - C_{i'} \biggl ( \sum _{\ell \in U_{g}[i']} s_{\ell } \wedge (k - 1) \biggr ) \\= & {} C_{i'} \biggl ( \sum _{\ell \in {U}_{g}[i']} s_{\ell } \wedge (k - 1) + |U_{g}[i'] \cap Q^{s} (k)|\biggr ) - C_{i'} \biggl ( \sum _{\ell \in U_{g}[i']} s_{\ell } \wedge (k - 1) \biggr ). \end{aligned}$$

Let $\sigma$ be any ordering in $O^m$. Using the definition of $v_{g, \lambda ^{C, i', k}}$, one has that

• for each $i \in N \setminus U_{g}[i']$ and each $j \in \{1, \ldots , w_i\}$,    $\eta ^{\sigma }_{(i, j)} (w, v_{g, \lambda ^{C, i', k}}) = 0;$

• for each $i \in N$ and each $j \in \{1, \ldots , w_i\} {\setminus } k$,    $\eta ^{\sigma }_{(i, j)} (w, v_{g, \lambda ^{C, i', k}}) = 0.$

By definition of the Priority Shapley value $\phi$ given in (9), one obtains the following payoffs:

$$\begin{aligned} \phi _{(i,j)} (w, v_{g, \lambda ^{C, i', k}}) = {\left\{ \begin{array}{ll} \frac{\displaystyle 1}{\displaystyle k} \cdot \frac{\displaystyle \sum _{\sigma \in O^w}\displaystyle \eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}})}{\displaystyle |O^w|} &{} \text{ if } i \in U_{g}[i'] \cap Q^{w} (k), j \in \{1, \ldots , k\}, \\ &{} \\ 0 &{} \text{ otherwise. } \\ \end{array}\right. } \end{aligned}$$

(14)

Furthermore, by the definition of a marginal vector,

$$\begin{aligned} \forall \sigma \in O^w, \sum _{(i, j) \in M^+} \eta ^{\sigma }_{(i, j)} (w, v_{g, \lambda ^{C, i', k}})= & {} \sum _{i \in U_{g}[i'] \cap Q^{w} (k)} \eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}}) \\= & {} v_{g, \lambda ^{C, i', k}} (w) \\= & {} C_{i'} \biggl (\sum _{\ell \in U_{g}[i']} w_{\ell } \wedge k \biggr ) - C_{i'} \biggl ( \sum _{\ell \in U_{g}[i']} w_{\ell } \wedge (k - 1) \biggr ), \end{aligned}$$

and so,

$$\begin{aligned} \sum _{\sigma \in O^w} \sum _{i \in U_{g}[i'] \cap Q^{w} (k)} \eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}})= & {} \sum _{i \in U_{g}[i'] \cap Q^{w} (k)} \sum _{\sigma \in O^w} \eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}}) \nonumber \\= & {} |O^{w} | \cdot \biggl (C_{i'} \bigl (\sum _{\ell \in U_{g}[i']} w_{\ell } \wedge k \bigr ) - C_{i'} \bigl ( \sum _{\ell \in U_{g}[i']} w_{\ell } \wedge (k - 1) \bigr ) \biggr ).\nonumber \\ \end{aligned}$$

(15)

Next, let $\ell$ and $\ell '$ be two agents in $U_{g}[i'] \cap Q^{w} (k)$ and let $inv^{\ell , \ell '} (\sigma ) \in O^w$ be defined as:

• $inv^{\ell , \ell '} (\sigma ) (\ell , k) = \sigma (\ell ', k)$;

• $inv^{\ell , \ell '} (\sigma ) (\ell ', k) = \sigma (\ell , k)$;

• $inv^{\ell , \ell '} (\sigma ) (i, j) = \sigma (i, j), \quad \forall (i, j) \in M^+ {\setminus } \{(\ell , k), (\ell ', k) \}$.

Because $C_{i'}$ only depends on the sum of the amount of waste shipped by the agents of $U_{g}[i']$, it holds that

$$\begin{aligned}\eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}})= \eta ^{inv^{\ell , \ell '} (\sigma )}_{(i, k)} (w, v_{g, \lambda ^{C, i', k}}). \end{aligned}$$

From this observation, one concludes that

$$\begin{aligned} \sum _{\sigma \in O^w} \eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}})= & {} \sum _{\sigma \in O^w} \eta ^{inv^{\ell , \ell '} (\sigma )}_{(i, k)} (w, v_{g, \lambda ^{C, i', k}}) \nonumber \\= & {} \sum _{\sigma ' \in O^w} \eta ^{\sigma '}_{(i, k)} (w, v_{g, \lambda ^{C, i', k}}). \end{aligned}$$

(16)

By (15) and (16), it holds that

$$\begin{aligned}\sum _{\sigma \in O^w} \eta ^{\sigma }_{(i, k)} (w, v_{g, \lambda ^{C, i', k}}) = \frac{|O^{w} | \cdot \biggl (C_{i'} \bigl (\sum _{\ell \in U_{g}[i']} w_{\ell } \wedge k \bigr ) - C_{i'} \bigl ( \sum _{\ell \in U_{g}[i']} w_{\ell } \wedge (k - 1) \bigr ) \biggr ) }{| U_{g}[i'] \cap Q^{w} (k)|}.\end{aligned}$$

Substituting the above equality in (14), one obtains that $\phi (w, v_{g, \lambda ^{C, i', k}}) = f^L (g,w, \lambda ^{C, i', k})$, as desired. $\square$

1.5 Proof of Theorem 5

Proof of Theorem 5

Uniqueness part: We prove that Efficiency, Independence of more active agents, Inter balanced contributions for top agents, and Intra balanced contributions for top agents determine $\phi$ in a unique way. Assume that a value f on $\mathfrak {G}$ satisfies the above four axioms. The proof is by induction on $\sum _{i \in N} m_i$.

Induction basis: If $\sum _{i \in N} m_i = 1$, then there is exactly one active agent in the game, say $i'$, with $m_{i'} = 1$. Therefore, $M^+ = \{(i', 1)\}$ and, by Efficiency,

$$\begin{aligned}f_{(i', 1)} (m, v) = v (0, \ldots , 1, \ldots , 0) = v(m),\end{aligned}$$

meaning that $f = \phi$.

Induction hypothesis: Assume that $f = \phi$ for all multi-choice games in $\mathfrak {G}$ such that $\sum _{i \in N} m_i \le p$, for some $p\ge 1$.

Induction step: Consider any multi-choice game $(m, v) \in \mathfrak {G}$ such that $\sum _{i \in N} m_i = p +1$. Two separate cases are distinguished.

Case 1. Consider any $i \in N {\setminus } T(m)$. Consider any activity level $j \in M_i^+$ and any agent $i' \in T(m)$. By Independence of more active agents and the induction hypothesis, one obtains

$$\begin{aligned} f_{(i, j) }(m, v ) = f_{(i, j)} (m - e^{i'}, v ) = \phi _{(i, j)} (m- e^{i'}, v ) = \phi _{(i, j)} (m, v),\end{aligned}$$

where the first equality follows from Independence of more active agents applied to f, the second equality is a consequence of the induction hypothesis, and the third equality follows from Independence of more active agents applied to $\phi$.

Case 2. Consider any $i \in T(m)$. First, Intra balanced contributions for top agents applied to f implies that

$$\begin{aligned} \forall i \in T(m), \forall j \in M_i^+, \quad f_{(i, j) }(m, v ) - f_{(i, j) }(m - e^i, v ) = f_{(i, \max (m)) }(m, v ). \end{aligned}$$

(17)

In a similar way, Intra balanced contributions for top agents applied to $\phi$ leads to

$$\begin{aligned} \forall i \in T(m), \forall j \in M_i^+, \quad \phi _{(i, j) }(m, v ) - \phi _{(i, j) }(m - e^i, v ) = \phi _{(i, \max (m)) }(m, v ). \end{aligned}$$

(18)

By induction hypothesis, $f_{(i, j) }(m - e^i, v ) = \phi _{(i, j) }(m - e^i, v )$. Thus, subtracting (18) from (17), one obtains that

$$\begin{aligned} \forall i \in T(m), \forall j \in M_i^+, \quad f_{(i, j) }(m, v ) - \phi _{(i, j) }(m, v ) = f_{(i, \max (m)) }(m, v ) - \phi _{(i, \max (m)) }(m, v ),\nonumber \\ \end{aligned}$$

(19)

which shows that the difference $f_{(i, j) }(m, v ) - \phi _{(i, j) }(m, v )$ does not depend on the activity level j. On the other hand, for any pair $\{i, i'\} \subseteq T(m)$, Inter balanced contributions for top agents applied to f implies that

$$\begin{aligned} f_{(i, \max (m))}(m, v ) - f_{(i, \max (m))} (m - e^{i'}, v) = f_{(i', \max (m))}(m, v ) - f_{(i', \max (m))} (m - e^i, v). \nonumber \\ \end{aligned}$$

(20)

In a similar way, Inter balanced contributions for top agents applied to $\phi$ leads to

$$\begin{aligned} \phi _{(i, \max (m)) }(m, v ) - \phi _{(i, \max (m))} (m- e^{i'}, v )= \phi _{(i', \max (m))} (m, v ) - \phi _{(i', \max (m))} (m - e^i, v).\nonumber \\ \end{aligned}$$

(21)

Subtracting (21) from (20) and using the induction hypothesis as above, one obtains that

$$\begin{aligned} f_{(i, \max (m)))}(m, v ) - \phi _{(i, \max (m)) }(m, v ) = f_{(i', \max (m))}(m, v ) - \phi _{(i', \max (m))} (m, v ),\end{aligned}$$

which proves that $f_{(i, \max (m))}(m, v ) - \phi _{(i, \max (m)) }(m, v )$ does not depend on the agent $i \in T(m)$. Taking into account (19), one finally obtains that

$$\begin{aligned} \exists c^{f, \phi } \in \mathbb {R}: \forall i \in T(m), \forall j \in \{1, \ldots , \max (m)\}, \quad f_{(i, j)}(m, v ) - \phi _{(i, j) }(m, v ) = c^{f, \phi }.\nonumber \\ \end{aligned}$$

(22)

By Efficiency,

$$\begin{aligned} v(m)= & {} \sum _{(i, j) \in M^+ } f_{(i, j) }(m, v) \nonumber \\= & {} \sum _{\begin{array}{c} (i, j) \in M^+: \\ i \in T(m) \end{array}} f_{(i, j) } (m, v) + \sum _{\begin{array}{c} (i, j) \in M^+: \\ i \notin T(m) \end{array}} f_{(i, j) } (m, v) \nonumber \\&{\mathop {=}\limits ^{(22), \text{ Case } \text{1 }}}&\sum _{\begin{array}{c} (i, j) \in M^+: \\ i \in T(m) \end{array}} ( \phi _{(i, j) } (m, v) + c^{f,\phi }) + \sum _{\begin{array}{c} (i, j) \in M^+: \\ i \notin T(m) \end{array}} \phi _{(i, j) } (m, v) \nonumber \\= & {} v(m) + \max (m) |T(m)| c^{f,\phi }, \end{aligned}$$

(23)

which implies that $c^{f,\phi } = 0$. Therefore,

$$\begin{aligned}\forall i \in T(m), \forall j \in M_i^+, \quad f_{(i, j) }(m, v) = \phi _{(i, j) }(m, v), \end{aligned}$$

which concludes the proof of the induction step.

Existence part: We prove that $\phi$ satisfies the four axioms. Consider any $(m, v) \in \mathfrak {G}$. For each ordering $\sigma \in O^m$,

$$\begin{aligned} \sum _{(i, j) \in M^+} \lambda ^\sigma _{(i,j)}(m, v)= & {} \sum _{i \in N} \sum _{j =1}^{m_i} \sum _{k =j}^{m_i} \frac{\eta ^{\sigma }_{(i,k)}(m,v) }{k} \\= & {} \sum _{i \in N} \sum _{k=1}^{m_i} \sum _{j= 1}^{k} \frac{\eta ^{\sigma }_{(i,k)}(m,v) }{k} \\= & {} \sum _{i \in N} \sum _{k =1}^{m_i} \eta ^{\sigma }_{(i,k)}(m,v) \\= & {} \sum _{(i, j) \in M^+} \eta ^{\sigma }_{(i,k)}(m,v). \end{aligned}$$

It follows that

$$\begin{aligned} \sum _{(i, j) \in M^+} \lambda ^\sigma _{(i,j)}(m, v)= & {} \sum _{(i, k) \in M^+} \bigl (v(s^{\sigma , (i, k) } - v(s^{\sigma , p^\sigma (i, k)} \bigr ) \nonumber \\= & {} \sum _{(i, k) \in M^+} v(s^{\sigma , (i,k) }) - \sum _{(i,k) \in M^+, \sigma (i, k) \ne 1 } v(s^{\sigma , p^\sigma (i, k)}), \nonumber \\= & {} \sum _{(i, k) \in M^+} v(s^{\sigma , (i, k) }) - \sum _{(i, k) \in M^+, \sigma (i, k) \ne \vert M^+ \vert } v(s^{\sigma , (i, k) }) \nonumber \\= & {} v(s^{\sigma , \sigma ^{-1}(\vert M^+ \vert ) }) \nonumber \\= & {} v(m). \end{aligned}$$

(24)

By (24) and the definition of $\phi$, it follows that

$$\begin{aligned} \sum _{(i, j) \in M^+}\phi _{(i,j)}(m,v)= & {} \frac{1}{ \vert O^m \vert } \sum _{\sigma \in O^m } \sum _{(i, j) \in M^+} \lambda ^\sigma _{(i,j)}(m, v) \\= & {} \frac{1}{ \vert O^m \vert } \sum _{\sigma \in O^m } v(m) \\= & {} v(m), \end{aligned}$$

which proves that $\phi$ satisfies Efficiency.

Pick any pair of agents $\{i, i'\} \subseteq N$ such that $m_i < m_{i'}$, and any $(i,j) \in M^+$. By definition of $O^m$, for each $\sigma \in O^m$, $\sigma (i, j) < \sigma (i', m_{i'})$. Therefore, for each $\sigma \in O^m$, $\eta ^{\sigma }_{ij}(m,v) = \eta ^{\sigma }_{ij}(m - e^{i'},v)$, and so, $\lambda ^{\sigma }_{(i, j)} (m, v) = \lambda ^{\sigma }_{(i, j)} (m - e^{i'}, v)$. Because the latter equality holds for each $\sigma \in O^m$, one gets $\phi _{(i, j)} (m, v) = \phi _{(i, j)} (m- e^{i'}, v)$. This shows that $\phi$ satisfies Independence of more active agents.

Now, pick any pair of top agents $\{i, i' \} \subseteq T(m)$. By Point 2. of Proposition 1,

$$\begin{aligned}{} & {} \phi _{(i, \max (m))} (m, v) -\phi _{(i, \max (m))} (m - e^{i'}, v) = \sum _{{\mathop { \max (t) = \max (m)}\limits ^{ t \le m: T(t)\ni i,}}}\frac{\Delta _v(t) }{\max (t) \vert T(t) \vert }\\ {}{} & {} \quad - \sum _{{\mathop { \max (t)= \max (m - e^{i'})}\limits ^{t \le m - e^{i'}, T(t) \ni i,}}}\frac{\Delta _v(t) }{\max (t) \vert T(t) \vert }. \end{aligned}$$

A coalition t appears in the first sum and not in the second one if and only if $t_{i'} =\max (m)$. Thus,

$$\begin{aligned} \phi _{(i, \max (m))} (m, v) -\phi _{(i, \max (m))} (m - e^{i'}, v)= & {} \sum _{{\mathop { \max (t) = \max (m)}\limits ^{t \le m, T(t) \supseteq \{i, i'\},}}}\frac{\Delta _v(t) }{\max (t) \vert T(t) \vert } \\= & {} \phi _{(i', \max (m))} (m, v) -\phi _{(i', \max (m))} (m - e^i, v), \end{aligned}$$

where the second equality follows from a symmetric argument. This ensures that $\phi$ satisfies Inter balanced contributions for top agents.

Finally, similarly as above, by choosing a top agent $i \in T(m)$ and any pair of distinct activity levels $\{j,j'\} \subseteq \{1, \ldots , \max (m)\}$, one obtains that

$$\begin{aligned} \phi _{(i, j)} (m, v) -\phi _{(i, j)} (m - e^i, v)= & {} \sum _{t \le m, T(t) \ni i, \max (t) = \max (m)}\frac{\Delta _v(t) }{\max (t) \vert T(t) \vert } \\= & {} \phi _{(i,j')} (m, v) -\phi _{(i,j')} (m - e^i, v), \end{aligned}$$

which entails that $\phi$ satisfies Intra balanced contributions for top agents. $\square$

Logical independence of the axioms. The axioms invoked in Theorem 5 are logically independent, as shown by the following alternative values. The second and third examples are taken from Table 2.

• The null value on $\mathfrak {G}$ that distributes a zero payoff to each pair $(i, j) \in M_+$ satisfies all the axioms except Efficiency.

• The Shapley-like value $\phi ^{DP}$ on $\mathfrak {G}$ defined as:

  $$\begin{aligned} \forall (i, j) \in M^+, \quad \phi ^{DP}_{(i, j)}(m,v) = \sum _{{\mathop { s_i\ge j }\limits ^{s \in \mathcal{M} \setminus \overrightarrow{0}:}}} \frac{\Delta _s(v)}{| B(s)| } \end{aligned}$$

  satisfies all the axioms except Independence of more active agents.

• The Shapley-like value $\phi ^{LT}$ on $\mathfrak {G}$ defined as:

  $$\begin{aligned}\forall (i, j) \in M_+, \quad \phi ^{LT}_{(i, j)} (m, v) = \sum _{{\mathop {T(s) \ni i, \max (s) = j}\limits ^{s \in \mathcal{M \setminus \overrightarrow{0}}:}}} \frac{\Delta _s(v)}{| T(s)|}\end{aligned}$$

  satisfies all the axioms except Intra balanced contributions for top agents.

• Associate with each $i \in N$ a positive real number $\beta _{i} \in \mathbb {R}_{++}$. The value $f^{\beta }$ on $\mathfrak {G}$ defined as:

  $$\begin{aligned}\forall (i, j) \in M_+, \quad f^{\beta }_{(i, j)} (m, v) = \sum _{{\mathop {T(s) \ni i, \max (s) = j}\limits ^{s \in \mathcal{M} \setminus \overrightarrow{0}:}}} \frac{\beta _{i}}{\sum _{i' \in T(s)} \beta _{i'}} \cdot \frac{\Delta _s(v)}{| \max (s)|}\end{aligned}$$

  satisfies all the axioms except Inter balanced contributions for top agents.

1.6 Proof of Theorem 6

Proof of Theorem 6

Uniqueness part: Consider any value f on $\mathfrak {G}$ satisfying Efficiency, Independence of more active agents, Intra balanced contributions for top agents, Additivity, Null max activity level out, and Equal treatment for top-veto agents. Pick any $(m,v)\in \mathfrak {G}$. By Additivity,

$$\begin{aligned}f(m, v) = \sum _{t \le m} f(m, \Delta _v(t) u_t). \end{aligned}$$

It must be shown that $\forall t\le m$, $f(m, \Delta _v(t) u_t) = \phi (m, \Delta _v(t) u_t)$. To that end, consider any minimum activity level game $(m, \Delta _v(t) u_t)$. Each pair $(i, m_i)$ where $m_i > t_i$, if it exists, is a null pair in $(m, \Delta _v(t) u_t)$. Applying successively Null max activity level out, one obtains that

$$\begin{aligned}\forall (i, j) \in M_+: j \le t_i, \quad f_{(i, j)}(m, \Delta _v(t) u_t) = f_{(i, j)}(t, \Delta _v(t) u_t). \end{aligned}$$

Null max activity level out and Efficiency leads to

$$\begin{aligned} \forall (i, j) \in M_+: j > t_i, \quad f_{(i, j)}(m, \Delta _v(t) u_t) = 0 = \phi _{(i, j)}(m, \Delta _v(t) u_t), \end{aligned}$$

as desired. Pick any $i ' \in T(t)$. By Independence of more active agents,

$$\begin{aligned}\forall i \not \in T(t), \forall j \in \{1,\ldots , t_i\}, \quad f_{(i, j)}(t, \Delta _v(t) u_t) = f_{(i, j)}(t - e^{i'}, \Delta _v(t) u_t).\end{aligned}$$

Note that $(t - e^{i'}, \Delta _v(t) u_t)$ is the null game $(t- e^{i'},\textbf{0})$. Additivity applied to a null game implies zero payoffs for each pair $(i, j) \in M_+$. It follows that

$$\begin{aligned} \forall i \in N \setminus T(t), \forall j \in \{1,\ldots , t_i\}, \quad f_{(i, j)}(t, \Delta _v(t) u_t)= & {} 0 \\= & {} \phi _{(i, j)}(t, \Delta _v(t) u_t). \end{aligned}$$

It remains to deal with the agents in T(t). Note that each $i \in T(t)$ is a top-veto agent in $(t, \Delta _v(t) u_t)$. By Equal treatment of top-veto agents applied to f and $\phi$,

$$\begin{aligned}{} & {} \forall i, i' \in T(t), \quad f_{(i, \max (t))}(t, \Delta _v(t) u_t) = f_{(i', \max (t))}(t, \Delta _v(t) u_t), \\{} & {} \forall i, i' \in T(t), \quad \phi _{(i, \max (t))}(t, \Delta _v(t) u_t) = \phi _{(i', \max (t))}(t, \Delta _v(t) u_t), \end{aligned}$$

and so

$$\begin{aligned}{} & {} f_{(i, \max (t))}(t, \Delta _v(t) u_t) - \phi _{(i, \max (t))}(t, \Delta _v(t) u_t) \nonumber \\ {}{} & {} = f_{(i', \max (t))}(t, \Delta _v(t) u_t) - \phi _{(i', \max (t))}(t, \Delta _v(t) u_t). \end{aligned}$$

(25)

Using Intra balanced contributions for top agents, Independence of more active agents, and proceeding as in Case 2 in the proof of Theorem 5, one obtains that

$$\begin{aligned}{} & {} \forall i \in T(t), \exists c_i^{f, \phi } \in \mathbb {R}, \forall j \in \{1, \ldots , \max (m)\}, \nonumber \\ {}{} & {} \quad f_{(i, j)} (t, \Delta _v(t) u_t) - \phi _{(i, j)}(t, \Delta _v(t) u_t) = c_i^{f, \phi }, \end{aligned}$$

(26)

which shows that the above difference does not depend on the activity level j. Combining (25) and (26), one deduces that

$$\begin{aligned}{} & {} \exists c^{f, \phi } \in \mathbb {R}: \forall i \in T(m), \forall j \in \{1, \ldots , \max (m)\}, \\ {}{} & {} \quad f_{(i, j)} (t, \Delta _v(t) u_t) - \phi _{(i, j)}(t, \Delta _v(t) u_t) = c^{f, \phi }, \end{aligned}$$

which ensures that the above difference does not depend upon either j or i. Using Efficiency as in the proof of Theorem 5, one easily obtains $c^{f, \phi } = 0$, and so $f_{(i, j)} (t, \Delta _v(t) u_t) = \phi _{(i, j)} (t, \Delta _v(t) u_t)$ for each $i \in T(t)$, as desired.

Existence part: One verifies that the Priority Shapley value $\phi$ satisfies the six axioms of the statement of Theorem 6. It was seen earlier that $\phi$ satisfies Efficiency, Independence of more active agents, Intra balanced contributions for top agents, and Additivity.

Let any $(m,v)\in \mathfrak {G}$, and let $(i', m_{i'})$ be a null maximal pair. For each $\sigma \in O^m$, let $p^{i'}( \sigma )$ be the ordering of $O^{m - e^{i'}}$ induced by $\sigma$. Obviously, by definition of a null maximal pair, $\eta ^{\sigma }_{(i',m_{i'})}(m,v) = 0$, and it holds that

$$\begin{aligned}\forall (i,j)\in M^{+}\setminus \{(i', m^{i'})\}, \quad \eta ^{\sigma }_{(i,j)}(m,v) = \eta ^{p^{i'}(\sigma )}_{(i,j)}(m- e^{i'},v), \end{aligned}$$

and so

$$\begin{aligned} \forall (i,j)\in M^{+}\setminus \{(i', m^{i'})\}, \quad \lambda ^{\sigma }_{(i, j)} (m, v) = \lambda ^{p^{i'}(\sigma )}_{(i, j)} (m-e^{i'}, v). \end{aligned}$$

(27)

Moreover, for each $\sigma ' \in O^{m - e^{i'}}$, there are $q^m_{m_{i'}}$ orderings $\sigma \in O^m$ such that $p^{i'}( \sigma ) = \sigma '$. Thus,

$$\begin{aligned} \forall (i,j)\in M^{+}\setminus \{(i', m^{i'})\}, \quad \phi _{(i, j)} (m, v)= & {} \frac{1}{ \vert O^m \vert } \sum _{\sigma \in O^m } \lambda ^{\sigma }_{(i, j)} (m, v) \\= & {} \frac{1}{ \vert O^m \vert } \sum _{\sigma ' \in O^{m- e^{i'}} } \sum _{\begin{array}{c} \sigma \in O^{m}: \\ p^{i'}(\sigma ) = \sigma ' \end{array}} \lambda ^{\sigma }_{(i, j)} (m, v) \\&{\mathop {=}\limits ^{(27)}}&\frac{1}{q^m_{m_{i'} } \vert O^{m- e^{i'} } \vert } \sum _{\sigma ' \in O^{m- e^{i'} } } q^m_{m_{i'}} \lambda ^{\sigma '}_{(i, j)} (m-e^{i'}, v) \\= & {} \phi _{(i, j)} (m -e^{i'} v), \end{aligned}$$

which ensures that $\phi$ satisfies Null max activity level out.

Now, consider any $(m,v)\in \mathfrak {G}$ that contains two distinct top-veto agents $i, i' \in T(m)$. For $\sigma \in O^m$, let $inv^{i, i' \max (m)} (\sigma ) \in O^m$ defined as:

• $inv^{i, i', \max (m)} (\sigma )(i, \max (m) ) = \sigma (i', \max (m) )$;

• $inv^{i, i', \max (m)} (\sigma )(i', \max (m) ) = \sigma (i, \max (m) )$;

• $inv^{i, i', \max (m)} (\ell , j ) = \sigma (\ell , j), \quad \forall (\ell , j) \in M^+ \setminus \{(i, \max (m)), (i', \max (m)) \}$.

Note that

$$\begin{aligned}s^{\sigma , (i', \max (m))} = s^{inv^{i, i', \max (m)} (\sigma ),(i, \max (m))}.\end{aligned}$$

From the above equality, one deduces that

$$\begin{aligned}\eta ^{\sigma }_{(i, \max (m))}(m,v) = \eta ^{ inv^{i, i' \max (m)} (\sigma )}_{(i', \max (m) )}(m,v). \end{aligned}$$

Therefore, one obtains that

$$\begin{aligned} \forall \sigma \in O^m, \quad \lambda ^{\sigma }_{(i, \max (m))} (m, v)= & {} \frac{\eta ^{\sigma }_{(i, \max (m))}(m,v)}{\max (m) } \nonumber \\= & {} \frac{\eta ^{inv^{i, i' \max (m)} (\sigma )}_{(i', \max (m))}(m,v)}{\max (m) } \nonumber \\= & {} \lambda ^{inv^{i, i' \max (m)} (\sigma )}_{(i', \max (m))} (m, v), \end{aligned}$$

(28)

and so

$$\begin{aligned} \phi _{(i, \max (m))} (m, v)= & {} \frac{1}{ \vert O^m \vert } \sum _{\sigma \in O^m } \lambda ^{\sigma }_{(i, \max (m))} (m, v) \\= & {} \frac{1}{ \vert O^m \vert } \sum _{inv^{i, i' \max (m)} (\sigma ) \in O^m } \lambda ^{inv^{i, i', \max (m)} (\sigma )}_{(i', \max (m))} (m, v) \\= & {} \phi _{(i', \max (m))} (m, v), \end{aligned}$$

which proves that $\phi$ satisfies Equal treatment for top-veto agents. $\square$

Logical independence of the axioms. The axioms invoked in Theorem 6 are logically independent, as shown by the following alternative values. The first four are those used to show the logical independence of the axioms invoked in Theorem 5.

• The null value on $\mathfrak {G}$ that distributes a zero payoff to each pair $(i, j) \in M_+$ satisfies all the axioms except Efficiency.

• The Shapley-like value $\phi ^{DP}$ on $\mathfrak {G}$ satisfies all the axioms except Independence of more active agents.

• The Shapley-like value $\phi ^{LT}$ on $\mathfrak {G}$ satisfies all the axioms except Intra balanced contributions for top agents.

• The value $f^{\beta }$ on $\mathfrak {G}$ satisfies all the axioms except Equal treatment of top-veto agents.

• The value f on $\mathfrak {G}$ defined as:

  $$\begin{aligned}{} & {} \forall (i, j) \in M_+, \quad f^{}_{(i, j)} (m, v) \\ {}{} & {} = \sum _{{\mathop {T(s) \ni i, \max (s) \ge j}\limits ^{s \in \mathcal{M \setminus \overrightarrow{0}:}}}} \frac{(v ((m \wedge \overrightarrow{(\max (s) - 1)}) +e^i))^2 + 1}{\sum _{i' \in T(s)} (v ((m \wedge \overrightarrow{(\max (s) - 1)}) +e^{i'}))^2 + 1} \cdot \frac{\Delta _s(v)}{| \max (s)|} \end{aligned}$$

  satisfies all the axioms except Additivity.

• The value f on $\mathfrak {G}$ defined as:

  $$\begin{aligned}\forall (i, j) \in M^+, \quad f_{(i, j)}(m,v) = \sum _{\begin{array}{c} k \ge j: \\ Q^m (k) \ni i \end{array}} \frac{1}{k} \cdot \frac{ v (m \wedge \overrightarrow{k}) - v (m \wedge \overrightarrow{(k-1)})}{|Q^w (k)| } \end{aligned}$$

  satisfies all the axioms except Null max activity level out.