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Information within coalitions: risk and ambiguity

Abstract

We address economies with asymmetric information where agents are not perfectly aware of the informational structure for coalitions. Thus, when joining a coalition, each consumer considers the informational risk and may be uncertain about the prior relevant to her decision. In this context, we introduce cooperative solutions that we refer to as risky core, ambiguous core, and meu-core. We provide existence results and a variety of properties of these concepts, including their coalitional incentive compatibility. We also formalize the intuition that the blocking power of coalitions is increasing with their information but decreasing with the degree of risk or ambiguity aversion faced by their members.

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Notes

  1. 1.

    Given a partition P of \(\varOmega \), \(x=(x(w))_{w \in \varOmega } \in ({\mathbb {R}}_{+}^{\ell })^{k}\) is said to be P-measurable when it is constant on the elements of the partition P. That is, \(x(\omega )=x(\omega ')\) for all states \(\omega \) and \(\omega '\) belonging to the same element of P.

  2. 2.

    To unify notations, members of a coalition consider probability distributions on the set of informational profiles. However, the veto power of a coalition S does not depend on the information that agents in \(N{\setminus } S\) have.

  3. 3.

    Following the related literature, we consider the strong veto condition requiring that every member in a blocking coalition becomes better off. Even with continuous and monotone utility functions, in differential information economies the weak and strong veto are not equivalent. To show this, consider an economy with three consumers, three states and one commodity. Private information structures are \(P_{1}= \{ \{a,b\}, \{c\} \}, \ P_{2}= \{ \{a,c\}, \{b\} \} \) and \(P_{3}= \{ \{a\}, \{b, c\} \}\). Endowments are \(e_{1}=(1,1,0) \ e_{2}=(1,0,1),\) and \(e_{3}=(0,0,0)\). The expected utility functions are \(U_{1} (x_{a}, x_{b} ,x_{c})= (x_{a}+x_{b})/4 + x_{c}/2, \ U_{2} (x_{a}, x_{b}, x_{c})= (x_{a}+ x_{c})/4 + x_{b}/2\) and \(U_{3} (x_{a}, x_{b}, x_{c})= (x_{b}+ x_{c})/4 + x_{a}/2\). The endowment allocation is blocked in the weak sense by the big coalition via the allocation that assigns (0, 0, 1) to agent 1, (0, 1, 0) to agent 2 and (1, 0, 0) to agent 3. However, there is no coalition that blocks the endowments in the strong sense.

  4. 4.

    That is, \(A \in \bigvee _{h \in H} P_{h}\) if and only if \(A= \bigcap _{h \in H} A_{h},\) with \( A_{h} \in P_{h}\) for every h. In addition, \(A \in \bigwedge _{h \in H} P_{h}\) if and only if for \(A= \bigcup _{h \in H} A_{h},\) with \( A_{h} \in P_{h}\) for every h.

  5. 5.

    Given \(P,P' \in {{{\mathbb {P}}}}_{j}, \ P\) is either finer or coarser than \(P'\).

  6. 6.

    Assumption A(b) implies that the support of possible individual beliefs can be identified with a subset \({\mathbb {R}}^{n}\). Thus, we can apply the properties of multivariate first-order stochastic dominance in Levhari et al. (1975).

  7. 7.

    Note that implicitly \(e_{i}(\omega ) + x^{{\mathcal {P}}}_{i}(\omega ') - e_{i}(\omega ')\) is required to be in \({\mathbb {R}}^{\ell }_{+} \) for every \(i \in S\).

  8. 8.

    The requirement of Theorem 4 implies that any efficient interior allocation can be obtained by a net trade that satisfies the physical feasibility restriction as an equality. This condition can be dropped in a model without free disposal.

  9. 9.

    To simplify, the set \(\varTheta \) is assumed to be finite. However, the results we obtain can be recast without this assumption by adding the adequate topological properties on the set of signals.

  10. 10.

    Recall that \(\lim \nolimits _{n \rightarrow \infty } A_{n} = \bigcup \nolimits _{n \in {\mathbb {N}}} A_{n}\) for any increasing sequence of sets \(\{A_{n}\}_{n \in {\mathbb {N}}}\).

  11. 11.

    Since \({\mathcal {C}}^\vee ({\mathcal {E}}_{|\varPi })=\emptyset \), we have that \(\varPhi (x,1)>0,\forall x \in {\mathcal {F}}\). Thus, (ii) is a direct consequence of the compactness of \({\mathcal {F}}\) and the continuity of \(\varPhi \).

  12. 12.

    For any \(x \in {\mathcal {F}}\), \(1\in \varTheta (x)\). Also, the continuity of \(\varPhi \) guarantees that \(\varTheta (x)\) is a closed subset of [0, 1]. Therefore, \(\varTheta \) has non-empty and compact values. \(\varTheta \) is upper hemicontinuous, because has closed graph and compact codomain.

    Since \(x\twoheadrightarrow \dot{\varTheta }(x):=\{\theta \in [0,1]: \varPhi (x,\theta )> \bar{a}\}\) has open graph, its closure is lower-hemicontinuous. In addition, given \(x \in {\mathcal {F}}\) and \(\theta \in \varTheta (x)\cap [0,1)\), it follows from (i) that for any \(n \in {\mathbb {N}}\) we have that \(\varPhi (x,n^{-1} + (1-n^{-1})\theta )>\varPhi (x,\theta )\ge \overline{a}\). Hence \(\{n^{-1} +(1-n^{-1}) \theta \}_{n \in {\mathbb {N}}}\subseteq \dot{\varTheta }(x)\) and converges to \(\theta \). This ensures that \(\varTheta (x)\subseteq \overline{\dot{\varTheta }}(x)\). We conclude that \(\varTheta =\overline{\dot{\varTheta }}\), which implies in the lower hemicontinuity of \(\varTheta \).

  13. 13.

    Note that \({\mathscr {C}}_\tau ({\mathcal {E}})= \{ x \in {\mathcal {F}}: \varUpsilon (x)\le 0 \},\) where \(\varUpsilon (x)= \max \nolimits _{S\subseteq N} \max \nolimits _{y\in {\mathcal {F}}(S) } \min \nolimits _{i \in S} \left( \min \nolimits _{\theta \in \varTheta } \sum \nolimits _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) - U_{i} ( x_i)\right) \) is a continuous function in \({\mathcal {F}}\).

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Corresponding author

Correspondence to Juan Pablo Torres-Martínez.

Additional information

We are especially grateful to two anonymous referees that have helped to improve this work. Moreno-García acknowledges support by the Research Grant ECO2016-75712-P (Ministerio de Economía y Competitividad). Torres-Martínez acknowledges the financial support of Conicyt-Chile through Fondecyt project 1150207.

Appendix

Appendix

Proof of Theorem 1

Consider the NTU game (NV) where the correspondence \(V:2^{N}{\setminus } \{\emptyset \} \twoheadrightarrow {\mathbb {R}}^n\) associates to each coalition \(S\ne N\) the set

$$\begin{aligned} \begin{array}{ll} V(S)= \{ a \in {\mathbb {R}}^n \ | &{} \hbox {there exists a net trade } (z^{\mathcal {P}}_i)_{i\in S,{\mathcal {P}}\in {\mathbb {P}}} \text{ attainable } \text{ for } S \\ &{} \text{ such } \text{ that, } \,(e_i+z^{{\mathcal {P}}}_i)_{{\mathcal {P}} \in {\mathbb {P}}} \ge 0 \\ &{} \hbox {and } a_i \le \sum _{{\mathcal {P}}\in {\mathbb {P}}} \rho ^{\mathcal {P}}_i(S) u_i(e_i+ z^{\mathcal {P}}_i), \hbox { for every } i \in S\}, \end{array} \end{aligned}$$

and associates to the grand coalition the set

$$\begin{aligned} \begin{array}{ll}V(N)= \{ a \in {\mathbb {R}}^n \ | &{} \hbox {there exists a feasible net trade } (z^{\mathcal {P}}_i)_{i\in N,{\mathcal {P}}\in {\mathbb {P}}} \hbox { such that, } \\ &{} (e_i+z^{{\mathcal {P}}}_i )_{{\mathcal {P}}\in {\mathbb {P}}}\ge 0 \\ &{} \hbox {and } a_i \le \sum _{{\mathcal {P}}\in {\mathbb {P}}} r^{\mathcal {P}}_i u_i(e_i+ z^{\mathcal {P}}_i), \hbox { for every } i \in N\}. \end{array} \end{aligned}$$

By Scarf (1967), to prove that (NV) has a non-empty core, i.e., \(V(N){\setminus } \bigcup _{S\subseteq N \begin{array}{c} S \end{array}\ne \emptyset } \text{ int }(V(S)) \ne \emptyset ,\) it suffices to show that (NV) is balanced, that is, \(\bigcap _{S\in {\mathcal {S}}} V(S) \subseteq V(N)\) for any balanced set of coalitions \({\mathcal {S}}\).

Let \({\mathcal {S}}\) be a balanced set of coalitions and let \((a_i)_{i \in N}\in \bigcap _{S\in {\mathcal {S}}} V(S)\). It follows that, for every coalition \(S\in {\mathcal {S}}\) there exists a net trade \((z^{\mathcal {P}}_i(S))_{i\in S,{\mathcal {P}}\in {\mathbb {P}}}\) attainable for S such that \((e_i+z^{\mathcal {P}}_i(S))_{{\mathcal {P}}\in {\mathbb {P}}}\ge 0\) and \(a_i \le \sum _{{\mathcal {P}} \in {\mathbb {P}}} \rho ^{\mathcal {P}}_i(S) u_i(e_i + z^{ {\mathcal {P}} }_i(S)),\forall i \in S\). Moreover, for any \(S\ne N\), the net trade can be chosen verifying \(u_i(e_i+ z^{\mathcal {P}}_i(S))\le u_i(e_i+ z^{{\mathcal {P}}'}_{i}(S))\) whenever \({\mathcal {P}} < {{\mathcal {{P}}}}'\) and \({\mathcal {P}}, {\mathcal {P}}' \in \sigma (r_{i}) \cup \sigma _{S}\). To see this, note that if \(\mathcal {P}\in \sigma _{S},\) then Assumptions A(a) and A(c) guarantee that there is no \({\mathcal {P}}' \in \sigma _{S}\) such that either \({\mathcal {P}}'< \mathcal {P}\) or \(\mathcal {P}< {\mathcal {P}}'\). Therefore, to ensure the monotonicity of the mapping \({\mathcal {P}}\rightarrow u_i(e_i+z^{\mathcal {P}}_i(S))\) on \(\sigma (r_i) \cup \sigma _{S},\) we can assume that for every \({\mathcal {P}}\in \sigma ({r_i}) {\setminus } \sigma _{S}\) the net trades verify \(z^{{\mathcal {P}}}_j(S)=z^{\bar{{\mathcal {P}}}}_j(S)\), being \(\bar{{\mathcal {P}}}\in \sigma _{S}\) the unique informational profile satisfying \(\bar{{\mathcal {P}}} \le \mathcal {P}\) whose existence is guaranteed by Assumption A(c).

Without loss of generality, define \(z^{\mathcal {P}}_i(S)=0\) for every \(i \notin S\). Hence, \(\sum _{i \in N } z^{\mathcal {P}}_i(S)= \sum _{i \in S } z^{\mathcal {P}}_i(S) \le 0\). Since \({\mathcal {S}}\) is balanced, there exists a function \(\alpha :{\mathcal {S}}\rightarrow (0,1]\) such that \(\sum _{S\in {\mathcal {S}}: i \in S} \alpha (S)=1\) for every \(i \in N\). Given \(i \in N\) and \({\mathcal {P}}=(P_{i})_{i \in N} \in {\mathbb {P}}\), let \(z^{\mathcal {P}}_i:=\sum _{S \in {\mathcal {S}}: i\in S}\alpha (S) z^{{\mathcal {P}}}_i(S)= \sum _{S \in {\mathcal {S}}}\alpha (S) z^{{\mathcal {P}}}_i(S)\). Note that \(z^{\mathcal {P}}_i\) is \(P_i\)-measurable, \(e_i + z^{\mathcal {P}}_i\ge 0\), and \(\sum _{i \in N} z^{\mathcal {P}}_i \le 0\). Furthermore, the concavity of utility functions implies that, for every \(i \in N\),

$$\begin{aligned} \sum _{{\mathcal {P}} \in {\mathbb {P}}} r^{\mathcal {P}}_i u_i(e_i + z^{\mathcal {P}}_i)= & {} \sum _{{\mathcal {P}} \in {\mathbb {P}} } r^{\mathcal {P}}_i u_i\left( \sum _{S\in {\mathcal {S}}, i \in S}\, \alpha (S)(e_i + z^{\mathcal {P}}_i(S)) \right) \\\ge & {} \sum _{S\in {\mathcal {S}}, i \in S}\, \alpha (S)\,\sum _{{\mathcal {P}} \in {\mathbb {P}} } r^{\mathcal {P}}_i u_i(e_i + z^{\mathcal {P}}_i(S))\\\ge & {} \min _{S\in {\mathcal {S}}, i \in S}\, \left\{ \sum _{{\mathcal {P}} \in {\mathbb {P}} } r^{\mathcal {P}}_i u_i(e_i + z^{\mathcal {P}}_i(S))\right\} . \end{aligned}$$

For any coalition S,  by conditions (a) and (b) in Assumption A and the monotonicity of \({\mathcal {P}}\rightarrow u_i(e_i+z^{\mathcal {P}}_i(S))\) one obtains that \(\sum _{{\mathcal {P}} \in {\mathbb {P}} } r^{\mathcal {P}}_i u_i(e_i + z^{\mathcal {P}}_i(S))\ge \sum _{{\mathcal {P}} \in {\mathbb {P}} } \rho ^{\mathcal {P}}_i (S) u_i(e_i + z^{\mathcal {P}}_i(S))\ge a_i\) (see Levhari et al. 1975). Therefore, \(\sum _{{\mathcal {P}} \in {\mathbb {P}}} r^{\mathcal {P}}_i u_i (e_i + z^{\mathcal {P}}_i)\ge a_i\), which implies that \((a_i)_{i \in N}\in V(N)\). We conclude that the core of (NV) is non-empty.

Consider a vector a in the core of (NV). Since \(a \in V(N){\setminus } \bigcup _{S\subseteq N \begin{array}{c} S \end{array}\ne \emptyset } \text{ int }(V(S)),\) there is a feasible allocation \(x=(x_i)_{i \in N}\) such that

$$\begin{aligned} a\le (U_i(x_i))_{i \in N}, \left( \{(U_i(x_i))_{i \in N}\}+{\mathbb {R}}^n_{++}\right) \cap V(N) =\emptyset , \end{aligned}$$

and x is not blocked by any coalition \(S\ne N\). To prove that x belongs to the risky core it remains to ensure that it cannot be blocked by the grand coalition. By contradiction, suppose that there is a feasible net trade \((\tilde{z}^{\mathcal {P}}_i)_{i\in N,{\mathcal {P}}\in {\mathbb {P}}}\) such that \((e_i+\tilde{z}^{{\mathcal {P}}}_i)_{{\mathcal {P}} \in {\mathbb {P}}} \ge 0\) and \(\sum _{{\mathcal {P}}\in {\mathbb {P}}} \rho ^{\mathcal {P}}_i(N) u_i(e_i+ \tilde{z}^{\mathcal {P}}_i)> U_i(x_i)\) for every \(i\in N\). As before, \((\tilde{z}^{\mathcal {P}}_i)_{i \in N, {\mathcal {P}} \in {\mathbb {P}}}\) can be chosen verifying \(u_i(e_i+ \tilde{z}^{\mathcal {P}}_i)\le u_i(e_i+ \tilde{z}^{\mathcal {P}'}_{i})\) whenever \({\mathcal {P}} < {\mathcal {P}}'\) and \({\mathcal {P}}, {{\mathcal {{P}}}}' \in \sigma (r_i) \cup \sigma _{N}\). Hence, (a) and (b) in Assumption A guarantee that,

$$\begin{aligned} \sum _{{\mathcal {P}}\in {\mathbb {P}}} r^{\mathcal {P}}_i u_i(e_i+ \tilde{z}^{\mathcal {P}}_i)\ge \sum _{{\mathcal {P}}\in {\mathbb {P}}} \rho ^{\mathcal {P}}_i(N) u_i(e_i+ \tilde{z}^{\mathcal {P}}_i)> U_i(x_i),\quad \forall i \in N, \end{aligned}$$

which is a contradiction with the fact that \(\left( \{(U_i(x_i))_{i \in N}\}+{\mathbb {R}}^n_{++} \right) \cap V(N) =\emptyset \). \(\square \)

Proof of Theorem 2

Under continuity, concavity, and locally non-satiability of utility functions, the set of Walrasian expectations equilibrium allocations is non-empty and it is included in the private core \({\mathcal {C}}^\circ ({\mathcal {E}}_{|\varPi })\). Let \(x\in {\mathcal {C}}^\circ ({\mathcal {E}}_{|\varPi })\) such that \(x\notin {\mathcal {C}}_{\rho }({\mathcal {E}})\). Then, we can find a coalition S and \(y \in {\mathcal {F}}(S)\) such that \(\sum _{ {\mathcal {P}} \in {\mathbb {P}}} \rho ^{{\mathcal {P}} }_{S}\, \ u_{i} ( y^{{\mathcal {P}}}_{i})> u_i(x_i^{\varPi })\) for every \( i \in S\). Since, by condition (b), agents are unable to get more information than the given by \(\varPi \) we have that \(y^{{\mathcal {P}}}_{i}\) is \(\varPi _{i}\)-measurable. The concavity of the functions \(u_{i}\) implies that \(u_{i} \left( \sum _{ {\mathcal {P}} \in {\mathbb {P}}} \rho ^{{\mathcal {P}} }_{S}\, y^{{\mathcal {P}}}_{i}\right) >u_i(x^{\varPi }_i)\) for every \( i \in S\). Condition (a) allows us to obtain that the allocation that assigns to each \(i \in S\) the bundle \(\sum _{ {\mathcal {P}} \in {\mathbb {P}}} \rho ^{{\mathcal {P}} }_{S}\, y^{{\mathcal {P}}}_{i} \) is physically attainable for S. This is in contradiction with the fact that \(x \in {\mathcal {C}}^\circ ({\mathcal {E}}_{|\varPi })\). \(\square \)

Proof of Theorem 3

Our economy can be recast as a coalitional exchange economy (see Del Mercato 2006). In this framework, neither utility functions nor consumption sets depend on coalitions. Therefore, the non-emptiness of the risky core is a consequence of Theorem 5 in Del Mercato (2006). \(\square \)

Proof of Proposition 1

The continuity of utility functions, the compactness of sets \({\mathcal {F}}\) and \(\{{\mathcal {F}}(S)\}_{S\subseteq N}\), and the emptiness of the fine core \({\mathcal {C}}^\vee ({\mathcal {E}}_{|\varPi })\) ensure that the continuous mapping

$$\begin{aligned} (x,\theta )\in {\mathcal {F}}\times [0,1]\,\longrightarrow \,\varPhi (x,\theta ):=\max \limits _{S\subseteq N}\,\max \limits _{y \in {\mathcal {F}}(S)}\,\min \limits _{i \in S} \left( \theta u_i (y^{\varPi ^{\vee }_S}_i)-u_i (x^{\varPi }_i)\right) \end{aligned}$$

satisfies the following properties: (i) for any \(\theta ',\theta \in [0,1]\), if \(\theta '>\theta \) and \(\varPhi (x,\theta )>0\), then \(\varPhi (x,\theta ')>\varPhi (x,\theta )\); and (ii) there exists \(a>0\) such that \(\varPhi (x,1)\ge a,\forall x \in {\mathcal {F}}\).Footnote 11 Given \(x \in {\mathcal {F}}\), let \(\varTheta (x)=\{\theta \in [0,1]: \varPhi (x,\theta )\ge \bar{a}\},\) where \(\bar{a} \in (0,a)\). It is not difficult to verify that \(\varTheta \) is a continuous correspondence with non-empty and compact values.Footnote 12 Therefore, the Berge’s Maximum Theorem guarantees that \(x\in {\mathcal {F}} \longrightarrow \min \{\theta : \theta \in \varTheta (x)\}\) is a continuous function and it has values strictly lower than one. We conclude that there exists \(\kappa \in (0,1)\) such that \(\varPhi (x,\theta )>0,\forall x \in {\mathcal {F}},\forall \theta \in [\kappa ,1]\).

Notice that \({\mathcal {C}}_\rho ({\mathcal {E}}_{|\varPi })=\{x \in {\mathcal {F}}: \varOmega (\rho , x)\le 0\}\) where

$$\begin{aligned} \varOmega (\rho , x):=\max _{S\subseteq N}\,\,\max _{y\in {\mathcal {F}}(S) }\,\,\min _{i \in S} \left( \sum _{ {\mathcal {P}} \in {\mathbb {P}}} \rho _{i}^{{\mathcal {P}} } (S) \ u_{i} ( y^{{\mathcal {P}}}_{i}) - u_{i} ( x^{\varPi }_{i})\right) . \end{aligned}$$

Therefore, \(x \notin {\mathcal {C}}_\rho ({\mathcal {E}}_{|\varPi })\) if and only if \(\varOmega (\rho , x)>0\).

Since \(u_i\) takes non-negative values, \(\varOmega (\rho , x) \ge \varPhi \left( x,\min \nolimits _{S\subseteq N}\,\min \nolimits _{i\in S} \rho ^{\varPi ^\vee _S}_{i}(S)\right) \). Therefore, if \(\min \nolimits _{S\subseteq N}\,\min \nolimits _{i\in S} \rho ^{\varPi ^\vee _S}_{i}(S)\ge \kappa \), then \( {\mathcal {C}}_\rho ({\mathcal {E}}_{|\varPi })=\emptyset \). \(\square \)

Proof of Theorem 4

Let \(x \gg 0\) be an efficient allocation. Thus, there is a feasible net trade z given by vectors \(z^{{\mathcal {P}}}_{i} = (z^{{\mathcal {P}}}_{i}(\omega ))_{\omega \in \varOmega }\) such that \(x^{{\mathcal {P}}}_{i}(\omega ) = e_{i}(\omega ) + z^{{\mathcal {P}}}_{i}(\omega ),\) for each consumer i and every \( {\mathcal {P}} \in {\mathbb {P}}\). If x is not coalitionally incentive compatible, then one can find a coalition S,  a profile \({\mathcal {P}} =(P_{i})_{i \in N},\) and states of nature \(\omega , \omega '\) for which conditions (a)–(d) in Definition 2 hold.

Since \(\omega ' \in \bigcap _{i \notin S} P_{i}(\omega ),\) one has \(\sum _{i \notin S} z_{i}^{{\mathcal {P}}} (\omega )= \sum _{i \notin S} z_{i}^{{\mathcal {P}}} (\omega ')\). By the assumptions in the statement of the theorem and monotonicity of preferences, we can take z such that \(\sum _{i \in N} z_{i}^{{\mathcal {P}}} (\omega )= 0\) and \(\sum _{i \in N} z_{i}^{{\mathcal {P}}} (\omega ')=0\). We can deduce that \(\sum _{i \in S} z_{i}^{{\mathcal {P}}} (\omega )= \sum _{i \in S} z_{i}^{{\mathcal {P}}} (\omega ')\). For each \(i \in S\) and \(\kappa \in \varOmega ,\) consider the net trade given by

$$\begin{aligned} \bar{z}^{{\mathcal {P}}} _{i}(\kappa )= \left\{ \begin{array}{l} z_{i}^{{\mathcal {P}}} (\kappa ) \text{ if } \kappa \notin P_{i}(\omega ),\\ z_{i}^{{\mathcal {P}}} (\omega ') \text{ if } \kappa \in P_{i}(\omega ). \end{array} \right. \end{aligned}$$

By construction, \(\bar{z}_{i}^{{\mathcal {P}}}\) is \(P_{i}\)-measurable for every \(i \in S\) and, by condition (c) in Definition 2, one has \(\sum _{i \in S} \bar{z}_{i}^{{\mathcal {P}}}(\kappa ) + \sum _{i \notin S} z_{i}^{{\mathcal {P}}}(\kappa )=0,\) for every \(\kappa \in \varOmega \). Consider the feasible allocation \(\bar{x}\) defined as

$$\begin{aligned} \bar{x}_{i}^{{\mathcal {P}}'}= \left\{ \begin{array}{l} x_{i}^{{\mathcal {P}}'} \hbox { if } {\mathcal {P}}' \ne {\mathcal {P}} \hbox { or } i \notin S,\\ e_{i} + \bar{z}_{i}^{{\mathcal {P}}}, \hbox { otherwise.} \end{array} \right. \end{aligned}$$

Note that \(\bar{x}_{i}^{{\mathcal {P}}}(\omega )= e_{i}(\omega )+ z_{i}^{{\mathcal {P}}}(\omega ')\) for every \(i \in S\). From Definition 2(d), it follows that \(v_{i}(\bar{x}_{i}^{{\mathcal {P}}}(\omega )) > v_{i}(x_{i}^{{\mathcal {P}}}(\omega ))\) for every \(i \in S\). Definition 2(a) implies that \(U_{i}(\bar{x}_{i}) > U_{i}(x_{i}), \) for every \(i \in S,\) and \(U_{i}(\bar{x}_{i}) = U_{i}(x_{i}),\) otherwise.

Fix an agent \(h \in S\). By continuity of \(v_{h}\) there is a non-null vector \(\varepsilon \in {\mathbb {R}}_{+}^{\ell }\) such that \(v_{h}(\bar{x}_{h}^{{\mathcal {P}}}(\omega ) - \varepsilon ) > v_{h}(x_{h}^{{\mathcal {P}}}(\omega ))\). By construction of \(\bar{x},\) since x is an interior allocation we can take \(\varepsilon \) such that \(U_{h}(\hat{x}_{h}) > U_{h}(x_{h}),\) where \(\hat{x}_{h}\) is given by \(\hat{x}_{h}^{{\mathcal {P}}}(\kappa )= \bar{x}_{h}^{{\mathcal {P}}}(\kappa ) - \varepsilon \) for every \(\kappa \in \varOmega \). Let m be the number of members in \(S^{c}\) and consider the allocation \(\tilde{x}\) given by \(\tilde{x}_{h}= \hat{x}_{h}, \ \tilde{x}_{i}= \bar{x}_{i}\) for every \(i \in S {\setminus } \{h\},\) and for each \(i \notin S\) and \(\kappa \in \varOmega :\)

$$\begin{aligned} \tilde{x}^{{\mathcal {P}}}_{i}(\kappa )= \left\{ \begin{array}{l} x_{i}^{{\mathcal {P}}'} (\kappa ) \hbox { if } {\mathcal {P}}' \ne {\mathcal {P}},\\ x_{i}^{{\mathcal {P}}} (\kappa ) + \frac{\varepsilon }{m}, \hbox { otherwise.} \end{array} \right. \end{aligned}$$

By strict monotonicity of the functions \(v_{i}\) we have \(U_{i}(\tilde{x}_{i}) > U_{i}(x_{i})\) for every \(i \in N\). A contradiction with the efficiency of x. \(\square \)

The following auxiliary result will be used to prove Proposition 2.

Lemma

Suppose that the probability distribution \(\hat{\nu }\) given by the vector of probabilities \((\hat{\nu }^1,\ldots , \hat{\nu }^m)\) over the ordered set \(\{1, \ldots , m\}\) first-order stochastically dominates \(\nu =(\nu ^1, \ldots , \nu ^m)\). Then, for each \( k \in \{1,\ldots , m\}\) and \(h \in \{1,\ldots , k\}\) there exists \(a_{k,h}\ge 0\) verifying

$$\begin{aligned} \sum _{h=k}^{m} a_{h,k} = \nu ^{k},\quad \sum _{h=1}^{k} a_{k,h}=\hat{\nu }^{k},\quad \forall k \in \{1,\ldots ,m \}. \end{aligned}$$

Proof

We show it by induction. When \(m=1\) there is no uncertainty and the result trivially holds. Assume that the result is true for \(m=t\) and let us prove that it is also true for \(m=t+1\). Notice that \(\hat{\nu }_{*}= (\hat{\nu }^1,\ldots , \hat{\nu }^{t-1}, \hat{\nu }^{t}+\hat{\nu }^{t+1})\) first-order stochastically dominates \( \nu _{*} = (\nu ^1,\ldots , \nu ^{t-1}, \nu ^{t}+\nu ^{t+1})\). Therefore, it follows from the induction hypothesis that, for each \( k \in \{1,\ldots , t\}\) and \(h \in \{1,\ldots , k\}\) there exists \(a^*_{k,h}\ge 0\) verifying \(\sum _{h=k}^{t} a^*_{h,k} = \nu ^{k}_*\) and \(\sum _{h=1}^{k} a^*_{k,h}=\hat{\nu }^{k}_*\).

Given \(k\in \{1,\ldots , t+1\}\) and \(h \in \{1,\ldots , k\}\), define

$$\begin{aligned} a_{k,h}=\left\{ \begin{array}{ll} a^*_{k,h}, &{} \quad h\le k< t; \\ a^*_{k,h} -\alpha _h, &{} \quad h< k= t;\\ a^*_{k,h}-\nu ^{t+1}-\alpha _h, &{} \quad h= k= t;\\ \alpha _h, &{} \quad h< k= t+1;\\ \nu ^{t+1}, &{} \quad h=k=t+1, \end{array} \right. \end{aligned}$$

where \((\alpha _h)_{1\le h\le t}\ge 0\) satisfies \(\sum _{h=1}^t \alpha _h = \hat{\nu }^{t+1}-\nu ^{t+1}\).

It follows that \(\sum _{h=k}^{t+1}a_{h,k}=\nu ^k\) and \(\sum _{h=1}^{k} a_{k,h}=\hat{\nu }^{k}\), for all \(k \in \{1,\ldots , t+1\}\). \(\square \)

Proof of Proposition 2

Let S be a coalition that blocks \(x\notin {\mathcal {C}}_{\rho }({\mathcal {E}}) \). That is, there exists \(y \in {\mathcal {F}}(S)\) such that \( \sum _{ {\mathcal {P}} \in {{\mathbb {P}}}} \rho _{i}^{{\mathcal {P}} } (S)\, \ u_{i} ( y^{{\mathcal {P}}}_{i})> U_i(x_i),\forall i \in S\).

Since (a) holds, the supports of \(\bar{\rho }_{i}(S)\) and \(\rho _{i}(S)\) are contained in a finite totally ordered set \(\psi _{i}(S)\subset {{\mathbb {P}}}\). Denoting by m(Si) the cardinality of \(\psi _{i}(S), \) we write \(\psi _{i}(S)= \{{\mathcal {P}}^{1}_{i}, \ldots , {\mathcal {P}}^{m(S,i)}_{i} \} \subset {{\mathbb {P}}},\) where \({\mathcal {P}}^{h}_{i}= (P_{i,j}^{h}, j \in S )\) and \(P_{i,j}^{h} \le P_{i,j}^{\tilde{h}},\) for every \(h \le \tilde{h}\) and every \(j \in S\). Furthermore, as \(\bar{\rho }\) first-order stochastically dominates \(\rho \), it follows from the previous Lemma that, for each \(k\in \{1,\ldots , m(S,i)\}\) and \(h\in \{1,\ldots , k\}\) there exists \(a^{i}_{k,h}\ge 0\) such that,

$$\begin{aligned} \sum _{h=k}^{m(S,i)}a^{i}_{h,k}= \rho _{i}^{\mathcal {P}^k_{i}}(S),\quad \quad \quad \sum _{h=1}^{k} a^{i}_{k,h}=\bar{\rho }_{i}^{\mathcal {P}^k_{i}}(S),\quad \quad \forall k \in \{1,\ldots , m(S,i)\}. \end{aligned}$$

Therefore, the fact that \(y \in {\mathcal {F}}(S)\) ensures that the allocation \(\hat{y}\) characterized by

$$\begin{aligned} \hat{y}_i^{\mathcal {P}^k_{i}}=\sum _{h=1}^k \frac{a^{i}_{k,h}}{\bar{\rho }_{i}^{\mathcal {P}^k_{i}}(S)} y_i^{\mathcal {P}^h_{i}},\quad \quad \forall i \in S,\,\forall k\in \{1,\ldots , m(S,i)\}, \end{aligned}$$

is attainable for S. Furthermore, the concavity of the functions \(u_{i}\) implies that

$$\begin{aligned} \sum _{k=1}^{m(S,i)} \bar{\rho }_{i}^{{\mathcal {P}}^k_{i} } (S)\, \ u_{i} ( \hat{y}^{{\mathcal {P}}^k_{i}}_{i})\ge & {} \sum _{h=1}^{m(S,i)} \left( \sum _{k=h}^{m(S,i)} a^{i}_{k,h}\right) u_{i} ( y^{{\mathcal {P}}^h_{i}}_{i}) \\= & {} \sum _{h=1}^{m(S,i)} \rho _{i}^{{\mathcal {P}}^h_{i}} (S)\, \ u_{i} ( y^{{\mathcal {P}}^h_{i}}_{i})\,>\,U_i(x_i),\quad \forall i \in S. \end{aligned}$$

We conclude that \(x\notin {\mathcal {C}}_{\bar{\rho }}({\mathcal {E}})\).

If (b) holds, as in the proof of Theorem 1, the blocking allocation y at the beginning of this proof can be chosen verifying \(u_i(y_i^{\mathcal {P}}(S))\le u_i(y^{{\mathcal {P}}'}_{i}(S))\) whenever \({\mathcal {P}} < {{\mathcal {{P}}}}'\) for every \(i \in S\). Indeed, given \(\bar{\mathcal {P}}\in {\mathbb {P}}\) such that \(\rho ^{\bar{\mathcal {P}}}_i(S)>0\) for some \(i \in S\), by Assumption A(c) one has \((\rho ^{{\mathcal {P}}}_j(S))_{j \in N}=0\) for any \({\mathcal {P}}\in {\mathbb {P}}\) satisfying \({\mathcal {P}}< \bar{\mathcal {P}}\) or \(\bar{\mathcal {P}}< {\mathcal {P}}\). Then, to ensure the monotonicity of the mapping \({\mathcal {P}}\rightarrow u_i(y_i^{\mathcal {P}}(S))\) on the set \(\{{\mathcal {P}}\in {\mathbb {P}}: {\mathcal {P}} \le \bar{\mathcal {P}}\,\vee \, \bar{\mathcal {P}} \le {\mathcal {P}}\}\), it suffices to define \(y^{{\mathcal {P}}}_j(S):=0\) when \({\mathcal {P}}< \bar{\mathcal {P}}\) and \(y^{{\mathcal {P}}}_j(S):=y^{\bar{\mathcal {P}}}_j(S)\) when \(\bar{\mathcal {P}}< {\mathcal {P}}\).

By Assumption A(b), the support of \(\bar{\rho }_{i}(S)\) (resp., \({\rho }_{i}(S)\)) is contained in a cartesian product \(\bar{{\mathbb {H}}}^{i}= \prod _{j \in N} \bar{{\mathbb {H}}}^{i}_{j}\) (resp., \({{\mathbb {H}}}^{i}= \prod _{j \in N} {{\mathbb {H}}}^{i}_{j}\)) of completely ordered sets of partitions of \(\varOmega \). Hence, for every \(i \in S,\) the supports of \(\bar{\rho }_{i}(S)\) and \(\rho _{i}(S)\) are contained in \(\prod _{j \in N} {{\mathbb {P}}}_{j},\) where \({{\mathbb {P}}}_{j}\) is the completely ordered set of partitions of \(\varOmega \) given by the j-projection of \(\bar{{\mathbb {H}}}^{i} \cup {{\mathbb {H}}}^{i}\). It follows from Levhari et al. (1975) that the first-order stochastic dominance of \(\bar{\rho }\) over \(\rho \) joint with the monotonicity of the mappings \({\mathcal {P}}\rightarrow u_i(y_i^{\mathcal {P}}(S)),\) imply that

$$\begin{aligned} \sum _{{\mathcal {P}} \in {{\mathbb {P}}}} \bar{\rho }_{i}^{{\mathcal {P}}} (S)\, \ u_{i} ( y^{{\mathcal {P}}}_{i}) \ge \sum _{{\mathcal {P}} \in {{\mathbb {P}}}} \rho _{i}^{{\mathcal {P}}} (S)\, \ u_{i} ( y^{{\mathcal {P}}}_{i}), \quad \forall i \in S. \end{aligned}$$

As in the previous case, we conclude that \(x\notin {\mathcal {C}}_{\bar{\rho }}({\mathcal {E}})\). \(\square \)

Proof of Proposition 3

Let S be a coalition that blocks \(x\notin {\mathcal {C}}_{\rho }((\overline{u}_i)_{i\in N})\). That is, there exists \(y \in {\mathcal {F}}(S)\) such that \(\sum _{ {\mathcal {P}}\in {\mathbb {P}}} \rho _{i}^{{\mathcal {P}} } (S) \ \overline{u}_{i} ( y^{{\mathcal {P}}}_{i}) > \sum _{ {\mathcal {P}}\in {\mathbb {P}}} r_{i}^{{\mathcal {P}} } \ \overline{u}_{i} ( x^{{\mathcal {P}}}_{i})=\overline{u}_i(x^{\varPi }_{i}) , \forall i \in S\). Since \((u_i)_{i\in N} \preceq (\overline{u}_i)_{i\in N}\), for each agent i there is a concave and strictly increasing function \(f_i:{\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) and \(\overline{u}_i=f_i\circ u_i\). The concavity of functions \(f_i\) implies that

$$\begin{aligned} f_i \left( \sum _{ {\mathcal {P}}\in {\mathbb {P}}} \rho _{i}^{{\mathcal {P}} } (S) u_{i}( y^{{\mathcal {P}}}_{i})\right)\ge & {} \sum _{ {\mathcal {P}}\in {\mathbb {P}}} \rho _{i}^{{\mathcal {P}} } (S) \ f_i\circ u_{i}( y^{{\mathcal {P}}}_{i})\\= & {} \sum _{ {\mathcal {P}}\in {\mathbb {P}}} \rho _{i}^{{\mathcal {P}} } (S) \ \overline{u}_{i} ( y^{{\mathcal {P}}}_{i}) > \overline{u}_i(x^{\varPi }_{i})= f_i(u_i(x^{\varPi }_i)). \end{aligned}$$

The strict monotonicity of \(f_i\) guarantee that \(f^{-1}_i\) is well-defined and strictly increasing. Thus, we conclude that \(x\notin {\mathcal {C}}_{\rho }((u_i)_{i\in N})\). \(\square \)

Proof of Theorem 5

Let \(x\notin {\mathscr {C}}_{\tau }^{\pi } ({\mathcal {E}},\bar{\varPhi })\). Then, there exist a coalition S and \(y\in {\mathcal {F}}(S)\) such that \( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \bar{\varPhi } \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) \right) > \bar{\varPhi }(U_{i}(x_{i}))\) for every \( i \in S\). Since \(\varPhi :{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) is a continuous, concave, and strictly increasing function such that \(\varPhi \preceq \overline{\varPhi }\), then

$$\begin{aligned} \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ f\circ \varPhi \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) \right) > f\circ \varPhi (U_{i}(x_{i})),\quad \text{ for } \text{ every }\, i \in S, \end{aligned}$$

where \(f:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) is a concave and strictly increasing mapping satisfying \(\bar{\varPhi }=f \circ \varPhi \). The concavity and invertibility of f imply that

$$\begin{aligned} \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) \right) > \varPhi (U_{i}(x_{i})) \end{aligned}$$

for all \( i \in S\). Therefore, \(x\notin {\mathscr {C}}_{\tau }^{\pi } ({\mathcal {E}},\varPhi )\) and the concavity and monotonicity of \(\varPhi \) imply that

$$\begin{aligned} \sum _{{\mathcal {P}\in {\mathbb {P}}}} \left( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta )\tau ^{{\mathcal {P}}}_{i}(S, \theta ) \right) u_{i} (y^{\mathcal {P}}_{i}) > U_{i}(x_{i}), \ \ \hbox { for every } i\in S. \end{aligned}$$

We conclude that x does not belong to the risky core \({\mathcal {C}}_{\rho _{\alpha }}({\mathcal {E}})\). \(\square \)

Proof of Theorem 6

If \(x\notin {\mathscr {C}}_{\tau } ({\mathcal {E}})\), then there is a coalition \(S\subseteq N\) and \(y\in {\mathcal {F}}(S)\) such that \( \min _{\theta \in \varTheta } \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) > U_{i}(x_{i})\) for all \( i \in S\). Therefore, for any \(\theta \in \varTheta \) and \(i \in S\) we have that \( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) > U_{i}(x_{i})\), which implies that \(x\notin {\mathcal {C}}_{\tau _{\theta }}({\mathcal {E}})\). Furthermore, for each \(\varPhi :{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) strictly increasing we have that

$$\begin{aligned} \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) \right)\ge & {} \varPhi \left( \min _{\theta \in \varTheta } \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) \right) \\> & {} \varPhi (U_{i}(x_{i})),\,\,\text{ for } \text{ every }\, i \in S. \end{aligned}$$

Since \(\varPhi \) is continuous, strictly increasing and \(\varPhi (0)=0\) we have that \(\varPhi ^{-1}\) is well-defined in \({\mathbb {R}}_{+}\) and strictly increasing. Then, we deduce that \(x\notin {\mathscr {C}}_{\tau }^{\pi } ({\mathcal {E}}, \varPhi )\). \(\square \)

Proof of Proposition 4

Assumption (a) implies that

$$\begin{aligned} \overline{\bigcup \limits _{n \in {\mathbb {N}}} {\mathscr {C}}_\tau ^\pi ({\mathcal {E}},\varPhi _n)}= \bigcap \limits _{\varepsilon >0} \,\bigcup \limits _{n\in {\mathbb {N}}}\, {\mathscr {C}}_\tau ^\pi [\varPhi _n,\epsilon ], \end{aligned}$$

where \({\mathscr {C}}_\tau ^\pi [\varPhi _n,\varepsilon ]\) is the set of allocations \(x\in {\mathcal {F}}\) such that

$$\begin{aligned} \max _{S\subseteq N}\max _{y\in {\mathcal {F}}(S) }\min _{i \in S} \left( \varPhi ^{-1}_n \left( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi _n \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{i}) \right) \right) - U_{i} ( x_i)\right) \le \varepsilon . \end{aligned}$$

Given \(\varepsilon >0\), if \(x\notin \bigcup _{n\in {\mathbb {N}}}\, {\mathscr {C}}_\tau ^\pi [\varPhi _n,\varepsilon ]\), then for each \(n\in {\mathbb {N}}\) there exist a coalition \(S_n\subseteq N\) and a feasible allocation \(y_n\in {\mathcal {F}}(S_n)\) such that

$$\begin{aligned} \varPhi ^{-1}_n \left( \sum _{\theta \in \varTheta } \pi _{i}(S_n, \theta ) \ \varPhi _n \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S_n, \theta ) u_{i} (y^{\mathcal {P}}_{n,i}) \right) \right) - U_{i}(x_{i})> \varepsilon , \quad \forall i \in S_n. \end{aligned}$$

Since there are finitely many coalitions and sets of attainable allocations are compact, there exist \(S\subseteq N\) and \(\{y_{n_k}\}_{k\in {\mathbb {N}}}\subseteq \{y_{n}\}_{n\in {\mathbb {N}}}\) belonging to \({\mathcal {F}}(S)\) and converging to some \(\overline{y}\) such that,

$$\begin{aligned} \varPhi ^{-1}_{n_k} \left( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi _{n_k} \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{n_k,i}) \right) \right) - U_{i}(x_{i})> \varepsilon , \quad \forall (i,k) \in S \times {\mathbb {N}}. \end{aligned}$$

Given \(k\in {\mathbb {N}}\), it follows from assumption (b) that for any \(k'>k\) we have that \(\varPhi _{n_{k'}}=f\circ \varPhi _{n_k}\) for some strictly increasing and concave function \(f:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) (see Proposition 6.C.2 in Mas-Colell et al. 1995). Hence, for every \(i\in S\) and \(k'>k\) the following property holds

$$\begin{aligned}&\varPhi ^{-1}_{n_k} \left( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi _{n_k} \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{n_{k'},i}) \right) \right) -U_i(x_i)\,\\&\quad \ge \, \varPhi ^{-1}_{n_{k'}} \left( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi _{n_{k'}} \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (y^{\mathcal {P}}_{n_{k'},i}) \right) \right) -U_i(x_i)\,>\,\varepsilon . \end{aligned}$$

Taking the limit as \(k'\) goes to infinity we conclude that

$$\begin{aligned} \varPhi ^{-1}_{n_k} \left( \sum _{\theta \in \varTheta } \pi _{i}(S, \theta ) \ \varPhi _{n_k} \! \left( \sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (\overline{y}_i^{\mathcal {P}}) \right) \right) - U_{i}(x_{i})\ge \varepsilon , \,\, \forall (i,k) \in S\times {\mathbb {N}}. \end{aligned}$$

Assumptions (a)–(c) and Lemma 8 in Klibanoff et al. (2005) imply that, by taking the limit as k goes to infinity, we have that \( \min _{\theta \in \varTheta }\,\sum _{{\mathcal {P}\in {\mathbb {P}}}} \tau ^{{\mathcal {P}}}_{i}(S, \theta ) u_{i} (\overline{y}_i^{\mathcal {P}}) - U_{i}(x_{i})\ge \varepsilon \) for every \(i \in S\). Hence \(x\notin {\mathscr {C}}_\tau ({\mathcal {E}})\) and by Proposition 3 and Theorem 6 we have

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } {\mathscr {C}}_\tau ^\pi ({\mathcal {E}},\varPhi _n)= \bigcup \limits _{n \in {\mathbb {N}}} {\mathscr {C}}_\tau ^\pi ({\mathcal {E}},\varPhi _n)\subseteq {\mathscr {C}}_\tau ({\mathcal {E}})\,\subseteq \, \bigcap \limits _{\varepsilon >0} \,\bigcup \limits _{n\in {\mathbb {N}}}\, {\mathscr {C}}_\tau ^\pi [\varPhi _n,\varepsilon ]. \end{aligned}$$

Since \({\mathscr {C}}_\tau ({\mathcal {E}})\) is closed,Footnote 13 to conclude the proof it remains to take the closure of the above sets. \(\square \)

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Moreno-García, E., Torres-Martínez, J.P. Information within coalitions: risk and ambiguity. Econ Theory 69, 125–147 (2020). https://doi.org/10.1007/s00199-018-1159-z

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Keywords

  • Differential information
  • Risky core
  • Ambiguous core
  • meu-core

JEL Classification

  • D71
  • D81
  • D82