## Abstract

We consider issue-externality games in which agents can cooperate on multiple issues and externalities are present both within and across issues, that is, the amount a coalition receives in one issue depends on how the players are organized on all the issues. Examples of such games are several firms competing in multiple markets, and countries negotiating both a trade agreement (through, e.g., WTO) and an environmental agreement (e.g., Kyoto Protocol). We propose a way to extend (Shapley) values for partition function games to issue-externality games. We characterize our proposal through axioms that extend the Shapley axioms to our more general environment. The solution concept that we propose can be applied to many interesting games, including inter-temporal situations where players meet sequentially.

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## Notes

Macho-Stadler et al. (2006) provided mechanisms that implement a family of extensions of the Shapley value for games in partition function form.

Nax (2014) considers a similar class of games that he calls multiple membership games. His approach is, however, different from ours since he focuses on extending the core allocation proposed by Bloch and de Clippel (2010) for combined games, to games with externalities across issues (multiple membership games).

The efficiency axiom on issue-externality games does not require that the formation of the grand coalition on a particular issue maximizes the total value in that issue. It may be the case that forming the grand coalition on an issue is efficient because it maximizes the joint value of all the issues although it does not maximize the value on that issue.

See Maskin (2003) and de Clippel and Serrano (2008) for a discussion of the possible consequences of including externalities on the efficiency of the outcome. Maskin (2003) suggests that in situations in which coalitions generate significant positive externalities, we should not expect that the grand coalition will form. This might be a reason why the Shapley value and the core have not been used in settings with externalities.

Beja and Gilboa (1990) propose a class of “two-stage games” and characterize all the semivalues in this class of games.

\( \left| \Omega \right| \) denotes the cardinality of any set \(\Omega \).

For notational simplicity, we use \(A\backslash a, P^{A\backslash a}\) and \( {\mathcal {P}}^{A\backslash a}\) instead of \(A\backslash \{a\}, P^{A\backslash \left\{ a\right\} }\) and \({\mathcal {P}}^{A\backslash \left\{ a\right\} }\), and similarly for other sets throughout this paper.

Consequently, all solution concepts, including values, for this game depend only on the information embedded in \((N,u),\) not on the identity of the issue under consideration.

In games without any type of externalities, additivity (part \(1.1\)), dummy and anonymity axioms imply the property on the multiplication for a scalar (part \(1.2\)). As shown in Macho-Stadler et al. (2007), in games with externalities within an issue there are values that are additive but not linear, that is, they satisfy part \(1.1\) (and the other basic axioms) but not part \(1.2\).

For games in characteristic form, symmetry and anonymity are synonymous. In our environment, we shall use anonymity to refer to properties for players and symmetry for issues.

In fact, this axiom can be replaced by a stronger version. Let \(A\) and \(B\) be two sets of issues such that \(\left| A\right| =\left| B\right| \) and let \(\mu _{AB}\) be a bijection from the set \(A\) to the set \(B.\) Then, the \(\mu _{AB}\)-

*renaming of issues*in game \((N,A,v)\) denoted by \((N,\mu _{AB}A,\mu _{AB}v)= (N,B,\mu _{AB}v)\) is defined by \( (\mu _{AB}v)(S;b;P^{B})=v(S;\mu _{AB}^{-1}(b);\mu _{AB}^{-1}P^{B})\) for all \( (S;b;P^{B})\in ECL(N,B),\) where \(\mu _{AB}^{-1}P^{B}\) applies the bijection \( \mu _{AB}^{-1}\) to the components of the vector of partitions \(P^{B}\). A value \(\Phi \) satisfies the (stronger version of) issue symmetry axiom if \( \Phi \left( N,\mu _{AB}A,\mu _{AB}v\right) =\Phi \left( N,A,v\right) \) for all \(\mu _{AB}\)-renaming of issues in \(\left( N,A,v\right) \).Among the values based on the “average approach” defined in Macho-Stadler et al. (2007), some satisfy the strong dummy player axiom while others do not. To illustrate this, note that all the values just mentioned but Myerson’s are in the family of the average approach. To show that there are some values that do not satisfy the axiom, let us define the “value alternate,” which consists of applying a value in the class of average values (for example, the value proposed by Macho-Stadler et al. 2007) to games with an odd number of players and another one (for example, the one by de Clippel and Serrano 2008) to games with an even number of players.

These two values are not in the family of values that satisfy the average approach.

For \(P\in {\mathcal {P}}\) and \(S\subseteq N, P\cap S\) is the partition on the set \(S\) obtained from \(P\ \)by removing the players in \(N\setminus S.\)

Note that for each \(c\ne a,\) if \(S\in P^{c},\) then \(S\in O^{c}\). However, it is possible that \(S\in P^{a}\) and \(S\notin O^{a}.\) In the original game, \( M\) exerts externalities through \(O^{a}\cap M\), while in the transformed game, \(M\) exerts externalities through \(P^{b}\cap M.\)

Note that issue-externality anonymity, issue-externality symmetry, and dummy issues are the key new axioms that are specific to our issue-externality games and they become superfluous in games with a single issue (corresponding to partition function games). As we shall see, these three axioms enable us to “transform” an issue-externality game to a partition function game, based on which our value concept is defined. To see, for example, that issue-externality symmetry axiom is independent from the rest of axioms, consider a simple example with three player (1, 2, and 3) and two issues (\(a\) and \(b\)) where \( \{23\}\) is the set of \(b\)-externality players: When \(2\) and \(3\) belong to the same coalition in \(P^{b}, v(S;a;P^{A})=1\) if \(1\in S\) and \(0\) if \( 1\notin S\); \(v(S;b;P^{A})=0\) for all \(P^{A}.\) Without issue-externality symmetry, the game cannot be reduced to a partition function game and there are multitude of values compatible with the rest of the axioms. With issue-externality symmetry, the value can be constructed from an auxiliary partition function game.

For example, denote \(\phi _{1}\) the value proposed by Macho-Stadler et al. (2007), and \(\phi _{2}\) the one proposed by de Clippel and Serrano (2008). Both satisfy efficiency, linearity, player anonymity, and strong dummy player. Consider the value \(\phi \) defined as follows:

$$\begin{aligned} \phi (N,v)= & {} \phi _{1}(N,v)\quad \text { if}\quad v(N)\le 5 \\ \phi (N,v)= & {} \phi _{2}(N,v)\quad \text { if}\quad v(N)>5. \end{aligned}$$It is immediate that the value \(\phi \) satisfies efficiency, player anonymity, and strong dummy player, but it does not satisfy linearity.

In Macho-Stadler et al. (2007), the “strong player anonymity” axiom was called the “strong symmetry” axiom.

In Macho-Stadler et al. (2007), it is proven that the “strong player anonymity” axiom, together with linearity and dummy player, leads to a natural method of constructing a solution, that is called the average approach: Each coalition is associated a worth that is some average of what the coalition can obtain in the different scenarios, and then it allocates to each player her Shapley value in this average game.

As previously done, we denote \(T(b)=T\cap N(b)\) for any coalition \(T\) of \( N(A)\) and \(Q(b)=\{T\cap N(b)\mid T\in Q\}\) for any partition \(Q\) of \(N(A),\) for any \(b\in A\).

Thus, \(R^{b}(a)=Q^{b}(b)\) for all \(b\in A\backslash a.\)

Recall that \(Q^{b}\) is a partition of \(N(A)\) on issue \(b\) and \(Q^{b}(b)\) is the partition of \(N(b)\) induced by \(Q^{b};\widetilde{Q^{b}}(b)\) is obtained from \(Q^{b}(b)\) by replacing each \(i(b)\in N(b)\) with \(i.\)

## References

Albizuri, M.J., Arin, J., Rubio, J.: An axiom system for a value for games in partition function form. Int. Game Theory Rev.

**7**, 63–73 (2005)Beja, A., Gilboa, I.: Values for two-stage games: another view of the Shapley axioms. Int. J. Game Theory

**19**, 17–31 (1990)Bloch, F., de Clippel, G.: Core of combined games. J. Econ. Theory

**145**, 2424–2434 (2010)Bolger, E.M.: A set of axioms for a value for partition function games. Int. J. Game Theory

**18**, 37–44 (1989)Bulow, J.I., Geanakoplos, J.D., Klemperer, P.D.: Multimarket oligopoly: strategic substitutes and complements. J. Polit. Econ.

**93**, 488–511 (1985)de Clippel, G., Serrano, R.: Marginal contributions and externalities in the value. Econometrica

**76**, 1413–1436 (2008)Dutta, B., Ehlers, L., Kar, A.: Externalities, potential, value and consistency. J. Econ. Theory

**134**, 2380–2411 (2010)Feldman, B.E.: Bargaining, coalition formation, and value. Ph.D. thesis, State University of New York at Stony Brook (1996)

Harsanyi, J.C.: A bargaining model for the cooperative \(n\)-person game. In: Tucker, A.W., Luce, R.D. (eds.) Contributions to the Theory of Games IV, pp. 325–355. Princeton University Press, Princeton (1959)

Hart, S., Mas-Colell, A.: Potential, value and consistency. Econometrica

**57**, 589–614 (1989)Macho-Stadler, I., Pérez-Castrillo, D., Wettstein, D.: Efficient bidding with externalities. Games Econ. Behav.

**57**, 304–320 (2006)Macho-Stadler, I., Pérez-Castrillo, D., Wettstein, D.: Sharing the surplus: an extension of the Shapley value for environments with externalities. J. Econ. Theory

**135**, 339–356 (2007)Maskin, E.: Bargaining, coalitions and externalities. In: Presidential Address to the Econometric Society. Institute for Advanced Study, Princeton (2003)

McQuillin, B.: The extended and generalized Shapley value: simultaneous consideration of coalitional externalities and coalitional structure. J. Econ. Theory

**144**, 696–721 (2009)Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)

Myerson, R.B.: Values of games in partition function form. Int. J. Game Theory

**6**, 23–31 (1977)Nax, H.H.: A note on the core of TU-cooperative games with multiple membership externalities. Games

**5**, 191–203 (2014)Pham Do, K.H., Norde, H.: The Shapley value for partition function form games. Int. Game Theory Rev.

**9**, 353–360 (2007)Shapley, L.S.: A value for \(n\)-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, pp. 307–317 (1953)

Thrall, R.M., Lucas, W.F.: \(n\)-Person games in partition function form. Nav. Res. Logist. Q.

**10**, 281–298 (1963)

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

We thank David Wettstein, an associate editor, and two reviewers for helpful comments. The financial support from ECO2009-7616, ECO2012-31962, 2014SGR-142, the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2011-0075), ICREA Academia, and FQRSC (Quebéc) is gratefully acknowledged. Inés Macho-Stadler and David Pérez-Castrillo are MOVE fellows.

## Appendix

### Appendix

###
*Proof of Proposition 1*

(i) Consider two games \((N,A,v)\) and \((N,A,v^{\prime })\). Since \(\phi ^{*}\) satisfies linearity, we have

Also, following (1), it is easy to check that \(\widehat{ v+v^{\prime }}=\hat{v}+\widehat{v^{\prime }}\). Hence,

for all \(i\in N,\) and \(\Phi ^{*}\) satisfies part \(1.1\) of the linearity axiom. Similarly, for the multiplication by a scalar \(\lambda \), it is the case that \(\lambda \phi _{k}^{*}(N(A),\hat{v})=\phi _{k}^{*}(N(A),\lambda \hat{v})\) for every \(k\in N(A)\) and \(\widehat{\lambda v} =\lambda \hat{v}.\) Hence,

for all \(i\in N,\) and \(\Phi ^{*}\) satisfies part \(1.2\) of the linearity axiom.

(ii) The player anonymity axiom of \(\phi ^{*}\) implies that \(\phi _{\sigma k}^{*}(N(A),\sigma \hat{v})=\phi _{k}^{*}(N(A),\hat{v})\) for any \(k\in N(A)\) and for any permutation \(\sigma \) on the set \(N(A).\) Take now a permutation \(\sigma _{N}\) on the set \(N\) and denote \(\sigma _{N(A)}\) the permutation on the set \(N(A)\) that associates player \(i(a)\) with \( (\sigma _{N}\left( i\right) )(a),\) for every \(i\in N, a\in A\). Consider the game \((N,A,v)\). Then,

Consequently,

for each \(i\in N\). Hence, \(\Phi ^{*}\) satisfies the player anonymity axiom.

(iii) We first prove that if \(j\in N\) is a dummy player in the game \( (N,A,v), \) then all the replicas \(j(a),\) for all \(a\in A,\) are dummy players in \((N(A),\hat{v})\). Consider any \((T,Q)\in ECL (N(A))\) and any \( (T^{\prime },Q^{\prime })\) obtained from \((T,Q)\) by changing the affiliation of player \(j(a).\) For any such \((T^{\prime },Q^{\prime }),\) it is always the case that \(Q^{\prime }(b)=Q(b)\) for any \(b\ne a,\) since we are changing the affiliation of a player that belongs to \(N(a).\) There are two possibilities:

a) It can be the case that \(Q^{\prime }(a)=Q(a)\). Then,

b) Or it can be the case that \(Q^{\prime }(a)\ne Q(a)\) when \(j(a)\) changes affiliation. In this case,

for any embedded coalition \((\widetilde{T}(b);b;(\widetilde{Q}(c))_{c\in A})\) and for all \(b\in A\) because \((\widetilde{T^{\prime }}(b);b;(\widetilde{ Q^{\prime }}(c))_{c\in A})\) can be deduced from \((\widetilde{T}(b);b;( \widetilde{Q}(c))_{c\in A})\) by changing the affiliation of the dummy player \(j\) within issue \(a\) in \((N,A,v).\) Hence again, \(\hat{v}(T^{\prime },Q^{\prime })=\hat{v}(T,Q).\)

This ends the proof that all the replicas \(j(a),\) for all \(a\in A,\) are dummy players in \((N(A),\hat{v}).\)

If \(\phi ^{*}\) satisfies the dummy player axiom, then \(\phi _{j(a)}^{*}(N(A),\hat{v})=0\) for all \(a\in A\) since \(j(a)\) is a dummy player in \((N(A),\hat{v}).\) Therefore,

and \(\Phi ^{*}\) satisfies the dummy player axiom.

(iv) Consider a dummy player \(j\in N\) in the game \((N,A,v)\) and a particular issue \(a\in A\). First, since \(\phi ^{*}\) satisfies the strong dummy player property and \(j(a)\) is a dummy player in \((N(A),\hat{v}),\)

for all \(k\in N(A)\backslash j(a)\). Second, player \(j(b)\), for \(b\ne a\), is also a dummy player in the game \((N(A)\backslash j(a),\hat{v}_{-j(a)})\). (If we have two dummy players in any *PFG*, the second dummy player is still dummy in the game where we have eliminated the first one). Applying this procedure sequentially to all the issues in \(A\), and denoting \( j(A)=\underset{a\in A}{\cup }j(a)\), we have that

for all \(k\in N(A)\backslash j(A)\). Therefore,

for all \(i\in N\backslash j\) and \(\Phi ^{*}\) satisfies the strong dummy player axiom.\(\square \)

###
*Proof of Proposition 2*

(i) This property is trivially satisfied.

(ii) If \(d\) is a dummy issue in the game \((N,A,v),\) then all the replicas \( i(d),\) for any \(i\in N,\) are dummy players in \((N(A),\hat{v}),\) because, by the definition of dummy issue, \(\hat{v}(T,Q)=\hat{v}(T^{\prime },Q^{\prime }) \) for all \((T^{\prime },Q^{\prime })\) obtained from \((T,Q)\) by changing the affiliation of player \(i(d),\) for any \(i\in N\).

Given that \(\phi ^{*}\) satisfies the strong dummy player axiom, then if the \(n\) dummy players \(i(d)\) are dropped off \(N(A)\)

which implies that for all \(i\in N\)

and \(\Phi ^{*}\) satisfies the dummy issue axiom.\(\square \)

###
*Proof of Proposition 3*

(i) Consider the game \(\left( N,A,v\right) \) and, for any \(a\in A\), define \( \left( N,A,v_{a}\right) \) as

It is immediate that \(v=\sum _{a\in A}v_{a}\). The linearity of \(\phi ^{*}\) implies the linearity of \(\Phi ^{*}\) (Proposition 1); hence,

Similarly, consider the game \(\left( N,A,v_{\sigma _{N}}\right) \), where \( \sigma _{N}\) is a permutation of the set of players \(N\). Remember that the function \(v_{\sigma _{N}}\) is defined as \(v_{\sigma _{N}}(S;a;P^{A})\equiv v\left( S;a;\left( P^{a},O^{A\backslash a}\right) \right) \) for all \( (S;a;P^{A})\in ECL (N,A),\) where \(O^{b}=\sigma _{N}P^{b}\) or \( O^{b}=P^{b}\) for all \(b\in A\backslash a\). Let \(B\subset A\backslash a\) be the subset of issues where \(\sigma _{N}\) applies, i.e., \(O^{b}=\sigma _{N}P^{b}\) for all \(b\in B\) and \(O^{b}=P^{b}\) for all \(b\in A\backslash B\backslash a\). For any particular \(a\in A\), we define \(\left( N,A,\left( v_{\sigma _{N}}\right) _{a}\right) \) as

Given that \(v_{\sigma _{N}}=\sum _{a\in A}\left( v_{\sigma _{N}}\right) _{a}\), the linearity of \(\Phi ^{*}\) implies

We now prove that \(\Phi _{i}^{*}(N,A,\left( v_{\sigma _{N}}\right) _{a})=\Phi _{i}^{*}(N,A,v_{a})\) for all \(i\in N\) for whom \(\sigma _{N}(i)=i\), which will prove part (i) of the proposition.

For any \(i\in N\),

where

for any \((T,Q)\in ECL (N(A))\), since the other terms in the sum are zero by construction of the function \(v_{a}\). Also,

where

for any \((T,Q)\in ECL (N(A))\). We notice that, by definition of \( v_{\sigma _{N}}, \left( v_{\sigma _{N}}\right) _{a}(S;a;P^{A})=v(S;a;\left( P^{a},O^{A\backslash a}\right) )\) for all \( (S;a;P^{A})\in ECL (N,A),\) where \(O^{b}=\sigma _{N}P^{b}\) or \( O^{b}=P^{b}\) for all \(b\in A\backslash a\) (and \(\left( v_{\sigma _{N}}\right) _{a}(S;b;P^{A})=0\) for all \(b\in A\backslash a, (S;b;P^{A})\in ECL (N,A)\)). Since \(\left( v_{\sigma _{N}}\right) _{a} \) only permutes the roles of the players involved in a subset of issues \(B\subset A\backslash a, \widehat{\left( v_{\sigma _{N}}\right) _{a}}\) only permutes the roles of the players in each \(N(b)\), for all \(b\in B\). In fact, \(\widehat{\left( v_{\sigma _{N}}\right) _{a}}=\sigma _{N(A)}\widehat{ v_{a}}\), where the permutation \(\sigma _{N(A)}\) is as follows:

Given that \(\phi ^{*}\) satisfies player anonymity,

for all \(i\in N\) and \(c\in A\backslash B\) and

for all \(i\in N\) and all \(b\in B\). In particular, \(\phi _{i(b)}^{*}(N(A), \widehat{\left( v_{\sigma _{N}}\right) _{a}})=\phi _{i(b)}^{*}(N(A), \widehat{v_{a}})\) for all \(i\in N\) for whom \(\sigma _{N}(i)=i\). This implies that \(\Phi _{i}^{*}(N,A,\left( v_{\sigma _{N}}\right) _{a})=\Phi _{i}^{*}(N,A,v_{a})\) for any \(i\in N\) for whom \(\sigma _{N}(i)=i\), and the result holds.

(ii) Consider the game \(\left( N,A,v\right) \), a set \(M\) of \(a\)-externality players, and \(b\ne a.\) We will show that if \(\phi ^{*}\) satisfies linearity and player anonymity in *PFG*, then \(\Phi _{i}^{*}\left( N,A,v_{M,ab}\right) =\Phi _{i}^{*}\left( N,A,v\right) \) for all \( i\in N\). Notice that \(\Phi _{i}^{*}(N,A,v)=\underset{c\in A}{\sum }\phi _{i(c)}^{*}(N(A),\widehat{v}),\) where \(\widehat{v}(T,Q)=\underset{c\in A}{\sum }v(\widetilde{T}(c);c;\left( \widetilde{Q}(d))_{d\in A}\right) \), and \( \Phi _{i}^{*}(N,A, v_{M,ab})=\underset{c\in A}{\sum }\phi _{i(c)}^{*}(N(A),\widehat{v_{M,ab}}),\) where \(\widehat{v_{M,ab}}(T,Q)=\underset{c\in A}{\sum }v_{M,ab}(\widetilde{T}(c);c;\left( \widetilde{Q}(d))_{d\in A}\right) \). We consider the following permutation \(\sigma _{N(A)}\) on the set \(N(A):\ \sigma _{N(A)}(i(a))=i(b)\) and \(\sigma _{N(A)}(i(b))=i(a)\) for all \(i\in M\) and \(\sigma _{N(A)}(k)=k\) otherwise. Applying the permutation \(\sigma _{N(A)} \) to the value function \(\widehat{v}\) has the same effect as going from \(v\) to \(v_{M,ab}\): It moves the roles of players in \(M\ \)from issue \(a\) to issue \(b\). Hence, \(\sigma _{N(A)}\widehat{v}=\widehat{v_{M,ab}}\).

Given that the value \(\phi ^{*}\) satisfies anonymity, it is the case that

Given that \(\sigma _{N(A)}\) only permutes replicas of the same players (those in \(M\)), it is the case that

(since \(\phi _{\sigma _{N(A)}(i(a))}^{*}(N(A),\widehat{v})+\phi _{\sigma _{N(A)}(i(b))}^{*}(N(A),\widehat{v})\!=\!\phi _{i(b)}^{*}(N(A),\widehat{v })+\phi _{i(a)}^{*}(N(A),\widehat{v})\) for \(i\in M\)). Therefore, \(\Phi _{i}^{*}\left( N,A,v_{M,ab}\right) =\Phi _{i}^{*}\left( N,A,v\right) \) as we wanted to prove.\(\square \)

###
*Proof of Theorem 1*

The sufficiency part of the theorem is a corollary of Propositions 1, 2, and 3. We prove the necessity part through a series of steps. Take any game \((N,A,v)\) in \({\mathcal {G}}\).

*Step* 1. For any \(a\in A\), we define the following game \( (N,A,v_{a}) \):

That is, the worth of a coalition on issue \(a\) in the game \(v_{a}\) is the same as that in \(v\); however, the worth of a coalition on any other issue is zero in game \(v_{a}\). Note that the organization of the players on issues other than \(a\) influences the worth of coalitions in issue \(a\) in the game \( v_{a}\) in the same way as it does in \(v\).

It is immediate that

Therefore, if \(\Phi \) satisfies the axiom of *linearity* then,

*Step* 2. For each \((N,A,v_{a}),\) we now define a related game \( (N(A),A,w_{a})\), which is similar to \((N,A,v_{a})\) except that we add \( (\left| A\right| -1)n\) dummy players. More precisely, for each \(b\in A\setminus a,\) let \(N(b)=\left\{ i(b)\mid i\in N\right\} \) be the \(b\)-replica of \(N\) and for convenience, let \(N(a)\equiv N\) (i.e., \(N(a)\) is the original set of players). Then, the set of players in the new game is \( N(A)=\cup _{b\in A}N(b)\) with \(N(A)\setminus N(a)\) being dummy players. Therefore, for every \(a\in A, (N(A),A,w_{a})\) is defined as follows:^{Footnote 20}

for all \((T;a;Q^{A})\in ECL (N(A),A)\) (i.e., for all vector \(Q^{A}\) of \(\left| A\right| \) partitions of \(N(A)\) and any \(T\in Q^{a}\)), and

for all \(b\in A\backslash a\) and all \((T;b;Q^{A})\in ECL (N(A),A)\).

Given that \(\Phi \) satisfies the axioms of *strong dummy player* and *player anonymity* (2’), we have

*Step* 3. Next, for each \(a\in A,\) we define another game \( (N(A),A,z_{a})\) that is related to \((N(A),A,w_{a})\) in the following sense. First, as in \((N(A),A,w_{a})\), a coalition of players obtains worth only on issue \(a\). Second, only players in \(N(a)\) create worth. Third, the inter-issue externalities in \((N(A),A,z_{a})\) are “similar” to those in \((N(A),A,w_{a})\); however, there is one important difference: In game \((N(A),A,z_{a}),\) the externalities originating from each issue \(b\in A\backslash a\) are exerted by players in \( N(b),\) rather than by players in \(N(a)\) as in game \((N(A),A,w_{a})\). That is, the game \((N(A),A,z_{a})\) is defined as follows:

for all \((T;a;Q^{A})\in ECL (N(A),A)\), where \(R^{A}\) is a vector of \( \left| A\right| \) partitions of \(N(A)\) such that \(R^{a}=Q^{a}\) and for every \(b\in A\backslash a, R^{b}\) is obtained from \(Q^{b}\) by exchanging the memberships of \(i(a)\) and \(i(b)\) for each \(i\in N,\)
^{Footnote 21} and

for all \(b\in A\backslash a\) and all \((T;b;Q^{A})\in ECL (N(A),A)\).

Note that \(z_{a}(T;a;Q^{A})=v_{a}\left( \widetilde{T}(a);a;\left( \widetilde{ Q^{b}}(b)\right) _{b\in A}\right) \) for all \((T;a;Q^{A})\in ECL (N(A),A)\).^{Footnote 22}

We claim that, by *issue-externality anonymity* axiom,

We prove this claim by decomposing the change from \((N(A),A,w_{a})\) to \( (N(A),A,z_{a})\) in \(\left| N\right| \left( \left| A\right| -1\right) \) stages. In each stage, we switch the membership of some \(i(a)\in N(a)\) with that of \(i(b)\in N(b)\) in the partition \(P^{b}\) on some issue \( b\in A\backslash a\). In doing so, \(i(b)\) takes the role of \(i(a)\) in generating externalities from issue \(b.\) Note that the identities of the players who create worth (always on issue \(a)\) remain the same. Then, by the issue-externality anonymity axiom, the value of every player different from \( i(a)\) and \(i(b)\) should not change; hence, the sum of the values for players \(i(a)\) and \(i(b)\) should not change either. Repeating this argument cross-issues implies that after \(\left| A\right| -1\) stages of switching the membership of \(i(a)\in N(a)\) with \(i(b)\in N(b)\) for every issue \(b\in A\backslash a,\) the sum of the values for all replicas of player \(i\) remains unchanged, while the value of each of the remaining players stays the same throughout these stages. By repeating the above stages for all \(i(a)\in N(a), \) we complete our transformation from \((N(A),A,w_{a})\) to \( (N(A),A,z_{a})\) and obtain Eq. (4).

*Step* 4. For each \((N(A),A,z_{a}),\) we now define a related game \( (N(A),A,r_{a})\) such that all externalities are generated from issue \(a.\) Recall that in \((N(A),A,z_{a}),\) for any \((T;a;Q^{A})\in ECL (N(A),A) \), the worth of \(T\) depends only on \(\left( Q^{b}(b\right) )_{b\in A}\); moreover, only a coalition of players in \(N(a)\) can create worth and it does so only on issue \(a.\) In fact, for each \(b\in A\backslash a, N(b)\) is a set of \(b\)-externality players in \((N(A),A,z_{a}).\) We define the game \( (N(A),A,r_{a})\) by encoding the externalities exerted by \(N(b)\) for all \( b\in A\backslash b\) in \(z_{a}\):

for all \((T;a;Q^{A})\in ECL (N(A),A)\), where \(R^{A}\) is a vector of \( \left| A\right| \) partitions of \(N(A)\) such that \(R^{a}=Q^{a}\) and for every \(b\in A\backslash a, R^{b}\) is such that \(R^{b}\cap N(b)=Q^{a}\cap N(b).\) Thus, \(r_{a}\) can be obtained from \(z^{a}\) from \( (\left| A\right| -1)\) steps of transformation, each involving moving the externalities induced by \(N(b)\), for a particular \(b\in A\backslash a\), from issue \(b\) to issue \(a\).

Note that \(r_{a}(T;a;Q^{A})=v_{a}\left( \widetilde{T}(a);a;\left( \widetilde{ Q^{a}}(b)\right) _{b\in A}\right) \) for all \((T;a;Q^{A})\in ECL (N(A),A)\).

By the *issue-externality symmetry* axiom,

We also note that all issues in \(A\backslash a\) are dummy issues in \( (N(A),A,r_{a})\).

*Step* 5. Finally, we define game \((N(A),a,s_{a})\) by eliminating the set of *dummy issues*
\(A\backslash a\) in \((N(A),A,r_{a}),\) that is,

for any \((T;a;Q)\in ECL (N(A),a)\) and any vector \(Q^{A}\) of \( \left| A\right| \) partitions of \(N(A)\) that satisfies \(Q^{a}=Q\). By the dummy issue axiom, we have

Note that \((N(A),a,s_{a})\) is a game with a single issue (\(a\) in this case). Therefore, we can consider \((N(A),a,s_{a})\) as a *PFG*, which we denote \((N(A),\widetilde{s_{a}})\). Moreover, when it is applied to games with only one issue, the issue symmetry axiom implies that the value \(\Phi \) depends only on the function that gives the worth of each embedded coalition, not on the identity of the issue itself. Thus, \(\Phi \) also defines a value for \( PFG .\) Let \(\phi \) be this value. Hence,

Therefore, steps 1–5 allow us to obtain the following series of equalities for every \(i\in N\):

We now prove that \(\hat{v}=\sum _{a\in A}\widetilde{s_{a}}.\) Consider any partition \(Q\) of \(N(A)\) and any coalition \(T\in Q\). By construction,

where \(Q^{A}\) is any vector of \(\left| A\right| \) partitions of \( N(A) \) that satisfies \(Q^{a}=Q\). Also,

Hence,

Finally, linearity of \(\Phi \) implies that the value \(\phi \) is also linear and \(\phi _{k}(N(A),\hat{v})=\sum _{a\in A}\phi _{k}(N(A),\widetilde{s_{a}})\) for all \(k\in N(A).\) Therefore,

which completes the proof of Theorem 1.\(\square \)

###
*Proof of Proposition 4*

Take two games \((N,A,v)\) and \((N,B,w),\) with \(A\cap B=\varnothing \), and consider a value \(\Phi \) that satisfies the dummy issue axiom. We add to the first game \(\left| B\right| \) dummy issues, obtaining the game \( (N,A\cup B,v^{\prime })\) where \(v^{\prime }\) is a characteristic function such that

By the dummy issue property, \(\Phi \) assigns the same payoff in both games to any player \(i\in N,\) i.e.,

Similarly, if we add to the game \((N,B,w)\) a set of \(\left| A\right| \) dummy issues, we obtain the game \((N,A\cup B,w^{\prime })\) where \( w^{\prime }\) is a characteristic function such that

Again, by the dummy issue axiom, we have

Since \(\Phi \) satisfies *linearity*,

Finally, we notice that the game \((N,A\cup B,v^{\prime }+w^{\prime })\) is equivalent to \((N,A\cup B,v^{\prime }\cup w^{\prime });\) hence,

and the independence axiom is satisfied.\(\square \)

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Diamantoudi, E., Macho-Stadler, I., Pérez-Castrillo, D. *et al.* Sharing the surplus in games with externalities within and across issues.
*Econ Theory* **60**, 315–343 (2015). https://doi.org/10.1007/s00199-015-0867-x

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DOI: https://doi.org/10.1007/s00199-015-0867-x