Population monotonic allocation schemes for games with externalities

Abstract

This paper provides conditions for a game with externalities to have a population monotonic allocation scheme (PMAS). We observe that the notion of convexity defined by Hafalir [Games Econ Behav 61:242–258, 2007] does not guarantee the existence of a PMAS in the presence of externalities. We introduce a new notion of convexity and show that while our convexity is not a stronger condition than Hafalir’s [Games Econ Behav 61:242–258, 2007] , it is a sufficient condition for a game to have a PMAS. Moreover, we show that the Aumann-Drèze value, which is defined for games with coalition structures, explicitly constructs a PMAS. In addition, we offer two necessary and sufficient conditions to guarantee a PMAS in the presence of externalities.

Appendix

Through the appendix, for notational simplicity, we omit v and $*$, and use $f(S,{\mathcal {P}})$ to denote $f^*(v,S,{\mathcal {P}})$ given by (6). Similarly, we write g(S) instead of g(v, S) for any rule g given by (8).

1.1 Proof of Proposition 1

Proof

We show that $ASMC^\sigma (v)$ satisfies (i) coalitonal efficiency and (ii) population monotonicity.

(i) For any ${\mathcal {P}}\in \varPi (N)$ and $S\in {\mathcal {P}}$, let $\{j_1,\ldots ,j_l,\ldots ,j_{l'},\ldots ,j_{|S|}\}=S$ and $l<l'\iff \sigma (j_l)<\sigma (j_{l'})$. We have

$$\begin{aligned}&\sum _{j\in S}ASMC^{\sigma ,{\mathcal {P}}}_j(v) \overset{(5)}{=} \sum _{k=1}^{|S|} \left[ v(L^{\sigma ,{\mathcal {P}}}(j_k)\cup \{j_k\}, f(L^{\sigma ,{\mathcal {P}}}(j_k)\cup \{j_k\},{\mathcal {P}}))\right. \\&\qquad - \left. v(L^{\sigma ,{\mathcal {P}}}(j_k), f(L^{\sigma ,{\mathcal {P}}}(j_k),{\mathcal {P}}))\right] \\&\quad = v(\{j_1\}, f(\{j_1\},{\mathcal {P}})) - 0 \\&\qquad +\,v(\{j_1,j_2\}, f(\{j_1,j_2\},{\mathcal {P}})) - v(\{j_1\}, f(\{j_1\},{\mathcal {P}}))\\&\qquad +\cdots +\\&\qquad v(S, f(S,{\mathcal {P}})) - v(\{j_1,\ldots ,j_{|S|-1}\}, f(\{j_1,\ldots ,j_{|S|-1}\},{\mathcal {P}}))\\&\quad = v(S, f(S,{\mathcal {P}}))\\&\quad \overset{(6)}{=} v(S, {\mathcal {P}}). \end{aligned}$$

(ii) Let $i\in N$ and $S,T\subseteq N$ with $i\in S\subsetneq T$. We define and . It suffices to show

$$\begin{aligned} ASMC^{\sigma ,\mathcal {Q}_*}_{i} \ge ASMC^{\sigma ,{\mathcal {P}}^*}_{i}. \end{aligned}$$

We focus on $ASMC^{\sigma ,\mathcal {Q}_*}_{i}$.

$$\begin{aligned} ASMC^{\sigma ,\mathcal {Q}_*}_{i} = v(L^{\sigma ,\mathcal {Q}_*}(i)\cup \{i\}, f(L^{\sigma ,\mathcal {Q}_*}(i)\cup \{i\},\mathcal {Q}_*)) - v(L^{\sigma ,\mathcal {Q}_*}(i), f(L^{\sigma ,\mathcal {Q}_*}(i),\mathcal {Q}_*)). \end{aligned}$$

(13)

For simplicity, let $\tilde{L}:= L^{\sigma ,\mathcal {Q}_*}(i)$. Note that $\tilde{L}= L^{\sigma ,\mathcal {Q}_*}(i)= L^{\sigma }(i)\cap \mathcal {Q}_*(i)= L^{\sigma }(i)\cap T$. Let $T{\setminus } S=\{j_1,\ldots ,j_m\}$. We have

$$\begin{aligned}&(\text {}13) = v(\tilde{L}\cup \{i\}, f(\tilde{L}\cup \{i\},\mathcal {Q}_*)) - v(\tilde{L}, f(\tilde{L},\mathcal {Q}_*))\\&\quad \overset{(2)}{\ge } v(\tilde{L}\cup \{i\}, min) - v(\tilde{L}, max)\\&\quad \overset{(3)}{\ge } v((\tilde{L}{\setminus } \{j_1\})\cup \{i\}, max) - v(\tilde{L}{\setminus }\{j_1\}, max)\\&\quad \ge v((\tilde{L}{\setminus } \{j_1\})\cup \{i\}, min) - v(\tilde{L}{\setminus }\{j_1\}, max)\\&\quad \overset{(3)}{\ge } v((\tilde{L}{\setminus } \{j_1,j_2\})\cup \{i\}, max) - v(\tilde{L}{\setminus }\{j_1,j_2\}, max)\\&\quad \ge v((\tilde{L}{\setminus } \{j_1,j_2\})\cup \{i\}, min) - v(\tilde{L}{\setminus }\{j_1,j_2\}, max)\\&\quad \ge \cdots \\&\quad \overset{(3)}{\ge } v((\tilde{L}{\setminus } \{j_1,\ldots ,j_m\})\cup \{i\}, max) - v(\tilde{L}{\setminus }\{j_1,\ldots ,j_m\}, max)\\&\quad = v((L^{\sigma }(i)\cap S)\cup \{i\}, max) - v(L^{\sigma }(i)\cap S, max)\\&\quad \ge v((L^{\sigma }(i)\cap S)\cup \{i\}, f((L^{\sigma }(i)\cap S)\cup \{i\},\,P^*)) - v(L^{\sigma }(i)\cap S, max)\\&\quad \overset{(6)}{=} v((L^{\sigma }(i)\cap S)\cup \{i\}, f((L^{\sigma }(i)\cap S)\cup \{i\},{\mathcal {P}}^*)) - v(L^{\sigma }(i)\cap S, f(L^{\sigma }(i)\cap S,{\mathcal {P}}^*)), \end{aligned}$$

where the last equality holds because $S\in {\mathcal {P}}^*$ and $L^{\sigma }(i)\cap S \ne S$ imply $L^{\sigma }(i)\cap S \not \in {\mathcal {P}}^*$. Now, in view of $S={\mathcal {P}}^*(i)$, we have $L^{\sigma }(i)\cap S= L^{\sigma ,{\mathcal {P}}^*}(i)$. Hence, the final equality results in

$$\begin{aligned} v(L^{\sigma ,{\mathcal {P}}^*}(i)\cup \{i\}, f(L^{\sigma ,{\mathcal {P}}^*}(i)\cup \{i\},{\mathcal {P}}^*)) - v(L^{\sigma ,{\mathcal {P}}^*}(i), f(L^{\sigma ,{\mathcal {P}}^*}(i),{\mathcal {P}}^*))=ASMC^{\sigma ,{\mathcal {P}}^*}_{i}. \end{aligned}$$

This completes the proof. $\square$

1.2 Proof of Proposition 2

Proof

Fix $i \in N$ and ${\mathcal {P}}\in \varPi (N)$. We use $\tau$ to denote a permutation of ${\mathcal {P}}(i)$, namely, $\tau \in \varPsi ({\mathcal {P}}(i))$. For each $\tau$, we can find $(n-|{\mathcal {P}}(i)|)!\frac{n!}{|{\mathcal {P}}(i)|!\cdot (n-|{\mathcal {P}}(i)|)!}$ permutations $\sigma$ of N satisfying $\sigma (j)<\sigma (j')\iff \tau (j)<\tau (j')$ for $j,j'\in {\mathcal {P}}(i)$. Hence, we have

$$\begin{aligned}&\frac{1}{n!}\sum _{\sigma \in \varPsi (N)}ASMC^{\sigma ,{\mathcal {P}}}_i(v) \nonumber \\&\quad \overset{(5)}{=}\frac{1}{n!}\sum _{\sigma \in \varPsi (N)}v(L^{\sigma ,{\mathcal {P}}}(i)\cup \{i\}, f(L^{\sigma ,{\mathcal {P}}}(i)\cup \{i\},{\mathcal {P}}))-v(L^{\sigma ,{\mathcal {P}}}(i), f(L^{\sigma ,{\mathcal {P}}}(i),{\mathcal {P}})) \nonumber \\&\quad =\frac{1}{n!}\frac{(n-|{\mathcal {P}}(i)|)!\cdot n!}{|{\mathcal {P}}(i)|!\cdot (n-|{\mathcal {P}}(i)|)!} \sum _{\tau \in \varPsi ({\mathcal {P}}(i))}v(L^{\tau }(i)\cup \{i\}, f(L^{\tau }(i)\cup \{i\},{\mathcal {P}}))-v(L^{\tau }(i), f(L^{\tau }(i),{\mathcal {P}})) \nonumber \\&\quad =\frac{1}{|{\mathcal {P}}(i)|!} \sum _{\tau \in \varPsi ({\mathcal {P}}(i))}v(L^{\tau }(i)\cup \{i\}, f(L^{\tau }(i)\cup \{i\},{\mathcal {P}}))-v(L^{\tau }(i), f(L^{\tau }(i),{\mathcal {P}})) \end{aligned}$$

(14)

where $L^{\tau }(i)$ is the set of predecessors of i within ${\mathcal {P}}(i)$ with respect to $\tau$, $L^\tau (i)=\{j\in {\mathcal {P}}(i)| \tau (j)<\tau (i)\}$. In view of (11), we have

$$\begin{aligned} (14)= & {} \frac{1}{|{\mathcal {P}}(i)|!} \sum _{\tau \in \varPsi ({\mathcal {P}}(i))}v^*(L^{\tau }(i)\cup \{i\})-v^*(L^{\tau }(i)). \end{aligned}$$

(15)

Let $v^*|_{{\mathcal {P}}(i)}$ be the subgame of $v^*$ with the player set ${\mathcal {P}}(i)$. Since both $L^{\tau }(i)\cup \{i\}$ and $L^{\tau }(i)$ are subsets of ${\mathcal {P}}(i)$, we obtain

$$\begin{aligned}&(15) = Sh_i({\mathcal {P}}(i), v^*|_{{\mathcal {P}}(i)})\\&\quad \overset{(10)}{=}AD_i(v^*,{\mathcal {P}}). \end{aligned}$$

$\square$

1.3 Proof of Proposition 3

Proof

We show that if v satisfies IOC, then $Sh^g(v)\in Core^{opt}(v)$ for any g.

Let $S\subseteq N$ ($S\ne \emptyset$). We have

$$\begin{aligned}&\sum _{j\in S}Sh^g_j(v) \overset{(12)}{=}\sum _{j\in S}Sh_j(v^g) \nonumber \\&\quad =\frac{1}{n!}\sum _{j\in S}\sum _{\sigma \in \varPsi (N)}\left[ v^g(L^\sigma (j)\cup \{j\})-v^g(L^\sigma (j))\right] \nonumber \\&\quad =\frac{1}{n!}\sum _{\sigma \in \varPsi (N)}\sum _{j\in S}\left[ v^g(L^\sigma (j)\cup \{j\})-v^g(L^\sigma (j))\right] . \end{aligned}$$

(16)

For the term parenthesized with $[\cdot ]$, we have

$$\begin{aligned}&v^g(L^\sigma (j)\cup \{j\})-v^g(L^\sigma (j)) =v(L^\sigma (j)\cup \{j\}, g(L^\sigma (j)\cup \{j\}) )-v(L^\sigma (j), g(L^\sigma (j)))\\&\quad \ge v(L^\sigma (j)\cup \{j\}, min )-v(L^\sigma (j), max)\\&\quad \overset{(3)}{\ge }v((L^\sigma (j)\cap S)\cup \{j\}, max )-v(L^\sigma (j)\cap S, max). \end{aligned}$$

Hence, we have

$$\begin{aligned} (16)\ge & {} \frac{1}{n!}\sum _{\sigma \in \varPsi (N)}\sum _{j\in S}\left[ v((L^\sigma (j)\cap S)\cup \{j\},max)-v(L^\sigma (j)\cap S,max)\right] \\= & {} \frac{1}{n!}\sum _{\sigma \in \varPsi (N)}\sum _{k=1}^{|S|}\left[ v((L^\sigma (j_k)\cap S)\cup \{j_k\},max)-v(L^\sigma (j_k)\cap S,max)\right] \\= & {} \frac{1}{n!}\sum _{\sigma \in \varPsi (N)}(S,max)\\= & {} v(S,max). \end{aligned}$$

$\square$

1.4 Proof of Proposition 4

Proof

If Let $i\in N$ be a veto player and $v^i$ be a simple monotonic game with veto player i. For any ${\mathcal {P}}\in \varPi (N)$ and $j\in N$, define

$$\begin{aligned} x^{{\mathcal {P}}}_j= \left\{ \begin{array}{ll} v^{i}({\mathcal {P}}(j),{\mathcal {P}}) &{}\quad \text {if}\ \ j=i, \\ 0 &{} \quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

This constructs a PMAS. Moreover, for any two games v and $v'$, if x is a PMAS of v and y is a PMAS of $v'$, then $ax+by$ is a PMAS of $av+bv'$ for any nonnegative numbers a and b.

Only if Let v be a nonnegative game with a PMAS. Let x be the PMAS. Clearly, v is monotonic. For any player $j\in N$, we define $v^j$ as follows: for any $(S,{\mathcal {P}})\in EC(N)$,

$$\begin{aligned} v^j(S,{\mathcal {P}})= \left\{ \begin{array}{ll} x^{\mathcal {P}}_j &{}\quad \text {if}\ \ S={\mathcal {P}}(j) \ (\iff j\in S), \\ 0 &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

(17)

Since x is a PMAS, $v^j$ is monotonic. For any $(S,{\mathcal {P}})\in EC(N)$, we have

$$\begin{aligned} v(S,{\mathcal {P}}) \overset{\text {(i) of Def }1}{=} \sum _{j\in S}x_j^{\mathcal {P}}\overset{(17)}{=} \sum _{j\in S}v^j(S,{\mathcal {P}}) \overset{(17)}{=} \sum _{j\in N}v^j(S,{\mathcal {P}}), \end{aligned}$$

equivalently,

$$\begin{aligned} v=\sum _{j\in N}v^j. \end{aligned}$$

(18)

We fix a player $i\in N$ and define $\varPi ^i_{++}$ as the set of partitions in which the worth of player i’s coalition is positive, formally, $\varPi ^i_{++}=\{{\mathcal {P}}\in \varPi (N)| v^i({\mathcal {P}}(i),{\mathcal {P}}) >0\}$. For some ${\mathcal {P}},{\mathcal {P}}'\in \varPi ^i_{++}$, $v^i({\mathcal {P}}(i),{\mathcal {P}})$ may be equal to $v^i({\mathcal {P}}'(i),{\mathcal {P}}')$. We choose one from such partitions (if any) and define $\widehat{\varPi }^i_{++}\subseteq \varPi ^i_{++}$. Hence, for any ${\mathcal {P}}\in \widehat{\varPi }^i_{++}$, we have $v^i({\mathcal {P}}(i),{\mathcal {P}})>0$ and no ${\mathcal {P}}'\in \widehat{\varPi }^i_{++}$ satisfies $v^i({\mathcal {P}}(i),{\mathcal {P}})= v^i({\mathcal {P}}'(i),{\mathcal {P}}')$. We define an ordering of partitions in $\widehat{\varPi }^i_{++}$ as ${\mathcal {P}}_1,\ldots ,{\mathcal {P}}_m$, where $v^i({\mathcal {P}}_k(i),{\mathcal {P}}_k)< v^i({\mathcal {P}}_{k+1}(i),{\mathcal {P}}_{k+1})$ for every k ($1\le k\le K-1$), and $K=|\widehat{\varPi }^i_{++}|$. Note that $|{\mathcal {P}}_k(i)|\le |{\mathcal {P}}_{k+1}(i)|$ follows from the monotonicity of $v^i$.

Now, for every k ($1\le k\le K$), we define $\lambda ^{i,{\mathcal {P}}_k}$ as follows:

$$\begin{aligned} \lambda ^{i,{\mathcal {P}}_1}= & {} v^i({\mathcal {P}}_1(i),{\mathcal {P}}_1),\\ \lambda ^{i,{\mathcal {P}}_2}= & {} v^i({\mathcal {P}}_2(i),{\mathcal {P}}_2)-v^i({\mathcal {P}}_1(i),{\mathcal {P}}_1),\\&\cdots&\\ \lambda ^{i,{\mathcal {P}}_k}= & {} v^i({\mathcal {P}}_k(i),{\mathcal {P}}_k)-v^i({\mathcal {P}}_{k-1}(i),{\mathcal {P}}_{k-1}),\\&\cdots&\\ \lambda ^{i,{\mathcal {P}}_K}= & {} v^i({\mathcal {P}}_K(i),{\mathcal {P}}_K)-v^i({\mathcal {P}}_{K-1}(i),{\mathcal {P}}_{K-1}). \end{aligned}$$

Moreover, we define $v^{i,{\mathcal {P}}_k}$ as follows: setting $u^{i,{\mathcal {P}}_0}=v^{i}$,

$$\begin{aligned}&v^{i,{\mathcal {P}}_1}(S,{\mathcal {P}})= \left\{ \begin{array}{ll} 1 &{} \quad \text {if}\ \ u^{i,{\mathcal {P}}_0}(S,{\mathcal {P}})>0, \\ 0 &{} \quad \text {otherwise}, \end{array}\right. u^{i,{\mathcal {P}}_1}=u^{i,{\mathcal {P}}_0}-v^{i,{\mathcal {P}}_1};\\&\cdots&\\&v^{i,{\mathcal {P}}_k}(S,{\mathcal {P}})= \left\{ \begin{array}{ll} 1 &{}\quad \text {if}\ \ u^{i,{\mathcal {P}}_{k-1}}(S,{\mathcal {P}})>0, \\ 0 &{} \quad \text {otherwise}, \end{array}\right. u^{i,{\mathcal {P}}_k}=u^{i,{\mathcal {P}}_{k-1}}-v^{i,{\mathcal {P}}_k};\\&\cdots&\\&v^{i,{\mathcal {P}}_K}(S,{\mathcal {P}})= \left\{ \begin{array}{ll} 1 &{} \quad \text {if}\ \ u^{i,{\mathcal {P}}_{K-1}}(S,{\mathcal {P}})>0, \\ 0 &{} \quad \text {otherwise}, \end{array}\right. u^{i,{\mathcal {P}}_K}=u^{i,{\mathcal {P}}_{K-1}}-v^{i,{\mathcal {P}}_K}. \end{aligned}$$

We obtain $v^i=\sum _{k=1}^{K}\lambda ^{i,{\mathcal {P}}_k}v^{i,{\mathcal {P}}_k}$. By the construction of $\lambda ^{i,{\mathcal {P}}_k}$, each $\lambda ^{i,{\mathcal {P}}_k}$ is positive. Moreover, each $v^{i,{\mathcal {P}}_k}$ is a simple monotonic veto-controlled game: simplicity follows from the construction, monotonicity from $|{\mathcal {P}}_k(i)|\le |{\mathcal {P}}_{k+1}(i)|$ for any k ($1\le k\le m-1$), and i is a veto player because of (17). In view of (18), we have

$$\begin{aligned} v=\sum _{j\in N}\sum _{k=1}^{K}\lambda ^{j,{\mathcal {P}}_k}v^{j,{\mathcal {P}}_k}. \end{aligned}$$

This completes the proof. $\square$

1.5 Proof of Proposition 5

Proof

The proof consists of the following equivalent statements.

• 1. A nonnegative game v has a PMAS.

• 2. A nonnegative game v is a nonnegative linear combination of monotonic simple veto-controlled games $v_1,\ldots ,v_M$.

• 3. For any $\delta$ satisfying $\delta \cdot v_m\le 0$ for any $m=1,\ldots ,M$, we have $\delta \cdot v\le 0$.

• 4. For any $\delta$ satisfying $\delta \cdot v_m\le 0$ for any $m=1,\ldots ,M$ and $\delta _{(N,\{N\})}=-1$, we have $\delta \cdot v\le 0$.

• 5. For any extended vector of subbalanced weights $\delta$,

  $$\begin{aligned} \sum _{(S,{\mathcal {P}})\in EC^*(N)}\delta _{(S,{\mathcal {P}})}v(S,{\mathcal {P}})\le v(N,\{N\}). \end{aligned}$$

In view of Proposition 4, (1) $\Rightarrow$ (2) holds. For (1) $\Leftarrow$ (2), using $\varPi ^i_{++}$ instead of $\widehat{\varPi }^i_{++}$ in the only if part of the proof of Proposition 4 results in nonnegative coefficients $\lambda _1,\ldots ,\lambda _M$ for some natural number M.

The Minkowski-Farkas lemma implies (2) $\Leftrightarrow$ (3). Statement (2) is formally written as follows: there exist nonnegative numbers $\lambda _1,\ldots ,\lambda _M$ such that $v=\sum _{m=1}^M \lambda _m v_m$. From the Minkowski-Farkas lemma, it follows that no $\delta =(\delta _{(S,{\mathcal {P}})})_{(S,{\mathcal {P}})\in EC(N)}$ satisfies both $\delta \cdot v<0$ and $\delta \cdot v_m\ge 0$ for any $m=1,\ldots ,M$. Hence, for any $\delta$ satisfying $\delta \cdot v_m\ge 0$ for any $m=1,\ldots ,M$, we have $\delta \cdot v\ge 0$, which is equivalent to (3) by replacing $\ge$ with $\le$.

Clearly, (3) implies (4). However, the opposite direction is not necessarily obvious. The proof of Claim (3) $\Leftarrow$ (4) is provided below.

For the equivalence between (4) and (5), $\delta$ satisfies $\delta \cdot v_m\le 0$ for any $m=1,\ldots ,M$ and $\delta _{(N,\{N\})}=-1$ if and only if for any $i\in N$, $S\subsetneq N$ with $i\in S$, and ${\mathscr {P}}\subseteq \{{\mathcal {P}}\in \varPi (N){\setminus } \{N\}| S\in {\mathcal {P}}\}$,

$$\begin{aligned} \sum _{(T,\mathcal {Q})\in EC^*(N)}\delta _{(T,\mathcal {Q})}\chi ^{i,S,{\mathscr {P}}}(T,\mathcal {Q})\le 1. \end{aligned}$$

This completes the proof.

$\textit{Claim (3)} \Leftarrow \textit{(4)}$. We prove this by contraposition. We assume that there exists $\delta$ such that

$$\begin{aligned}&\delta \cdot v_m\le 0\text { for any }m=1,\ldots ,M \text {, and } \end{aligned}$$

(19)

$$\begin{aligned}&\delta \cdot v> 0. \end{aligned}$$

(20)

We construct $\delta '$ satisfying (i) $\delta '\cdot v_m\le 0$ for any $m=1,\ldots ,M$, (ii) $\delta '\cdot v> 0$, and (iii) $\delta '_{(N,\{N\})}=-1$.

Case 1: $\delta _{(N,\{N\})}\ne 0$. We define $\delta '_{(S,{\mathcal {P}})}=-\frac{1}{\delta _{(N,\{N\})}}\delta _{(S,{\mathcal {P}})}$ for every $(S,{\mathcal {P}})\in EC(N)$. Note that $\delta _{(N,\{N\})}< 0$ because, among the games $v_1,\ldots ,v_M$, we can find the game $v_{m^*}$ such that $v_{m^*}(S,{\mathcal {P}})=0$ for $(S,{\mathcal {P}})\ne (N,\{N\})$ and $v_{m^*}(N,\{N\})=1$, and (19) implies $\delta \cdot v_{m^*}=\delta _{(N,\{N\})}\le 0$. Since $\delta _{(N,\{N\})}\ne 0$, we have

$$\begin{aligned} \delta _{(N,\{N\})}< 0. \end{aligned}$$

(21)

Hence, $\delta '$ satisfies (i) because for any $m=1,\ldots ,M$,

$$\begin{aligned} \delta '\cdot v_m= -\frac{1}{\delta _{(N,\{N\})}}\delta \cdot v_m \overset{(19),(21)}{\le } 0, \end{aligned}$$

(ii) because

$$\begin{aligned} \delta '\cdot v= -\frac{1}{\delta _{(N,\{N\})}}\delta \cdot v \overset{(20),(21)}{>} 0, \end{aligned}$$

and (iii) because $\delta '_{(N,\{N\})}=-\frac{1}{\delta _{(N,\{N\})}}\delta _{(N,\{N\})}=-1$.

Case 2: $\delta _{(N,\{N\})}= 0$. The following construction is a slight variant of Sprumont (1990)’s. For simplicity, let $(\delta \cdot v)^*=\sum _{(T,\mathcal {Q})\in EC^*(N)}\delta _{(T,\mathcal {Q})}v(T,\mathcal {Q})$ for any $\delta$ and v. Note that

$$\begin{aligned}&(\delta \cdot v_m)^*= \delta \cdot v_m - \delta _{(N,\{N\})}v_m(N,\{N\})= \delta \cdot v_m \overset{19}{\le }0\text { for any } m=1,\ldots ,M, \end{aligned}$$

(22)

$$\begin{aligned}&(\delta \cdot v)^*= \delta \cdot v - \delta _{(N,\{N\})}v(N,\{N\})= \delta \cdot v \overset{(20)}{>}0. \end{aligned}$$

(23)

We define

$$\begin{aligned} \delta '_{(S,{\mathcal {P}})}=\left( \frac{|v(N,\{N\})|}{(\delta \cdot v)^*} +\epsilon \right) \delta _{(S,{\mathcal {P}})} \text { for every }(S,{\mathcal {P}})\in EC^*(N), \end{aligned}$$

(24)

where $\epsilon >0$, and define $\delta '_{(N,\{N\})}= -1$. Hence, $\delta '$ satisfies (i) because for any $m=1,\ldots ,M$, we have

$$\begin{aligned}&\delta '\cdot v_m= (\delta '\cdot v_m)^* + \delta '_{(N,\{N\})}v_m(N,\{N\}) \overset{24}{=} \left( \frac{|v(N,\{N\})|}{(\delta \cdot v)^*} +\epsilon \right) (\delta \cdot v_m)^* -1\\&\quad \overset{(22),(23)}{\le } 0-1, \end{aligned}$$

(ii) because

$$\begin{aligned}&\delta '\cdot v= (\delta '\cdot v)^* + \delta '_{(N,\{N\})}v(N,\{N\}) \overset{(24)}{=} \left( \frac{|v(N,\{N\})|}{(\delta \cdot v)^*} +\epsilon \right) (\delta \cdot v)^* -v(N,\{N\})\\&\quad = |v(N,\{N\})| + \epsilon (\delta \cdot v)^* -v(N,\{N\})\\&\quad \overset{(23)}{>} 0, \end{aligned}$$

and (iii) because of the construction. This completes the proof of Claim (3) $\Leftarrow$ (4). $\square$