Appendix
Proof of Theorem 1
Let the premises hold. It is fairly easy to check that if the social preference has the functional form given in (1), then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \) and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Thus, for each individual i, the individual i’s ex ante welfare ordering \(\succsim _{i}^{p}\) over \(\mathbb {R}^{\left| \mathcal {S} \right| }\), given in Definition 1, inherits all properties satisfied by the social preference \(\succsim _{0}^{p}\). By Gilboa and Schmeidler (1989), there exists a closed and convex set of strictly positive probability vectors on \(\mathcal {S}\), \(S_{i}\left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {S}\right) \), such that the individual i’s ex ante welfare ordering is represented in the form
$$\begin{aligned} u_{i}\succsim _{i}^{p}v_{i}\iff \min _{\sigma _{i}\in S_{i}\left( p\right) }\sum _{s\in \mathcal {S}}\sigma _{is}u_{is}\ge \min _{\sigma _{i}\in S_{i}\left( p\right) }\sum _{s\in \mathcal {S}}\sigma _{is}v_{is},\quad \text {for all }i\in \mathcal {I}\text {.} \end{aligned}$$
Next, given that by Lemma 1 there exists a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {I}\times \mathcal {S}\), \(\Lambda \left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {I\times S}\right) \), let us define the set of welfare weights I(p) by
$$\begin{aligned} I(p)=\left\{ \left( \sum _{s\in \mathcal {S}}\lambda _{is}\right) _{i\in \mathcal {I}}:\lambda \in \Lambda (p)\right\} \text {.} \end{aligned}$$
It follows from its definition that I(p) is a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {I}\).
It suffices to show
$$\begin{aligned} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S} }\lambda _{is}u_{is}=\min _{\alpha \in I(p)}\sum _{i\in \mathcal {I}}\alpha _{i}\left( \min _{\sigma _{i}\in S_{i}(p)}\sum _{s\in \mathcal {S}}\sigma _{is}u_{is}\right) ,\quad \text {for all }u\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\text {.} \end{aligned}$$
(5)
To this end, fix any prospect u. For each individual \(i\in \mathcal {I}\) and each state \(s\in \mathcal {S}\), define \(u_{is}^{*}\) by
$$\begin{aligned} u_{is}^{*}=\min _{\mu _{i}\in S_{i}(p)}\sum _{t\in \mathcal {S}}\mu _{it}u_{it}\equiv w_{i}^{*}\text {.} \end{aligned}$$
Thus, the matrix \(u^{*}=\left[ u_{is}^{*}\right] _{i\in \mathcal {I},s\in \mathcal {S}}\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\) is a prospect.
By this definition, we have
$$\begin{aligned} u_{i}^{*}\sim _{i}^{p}u_{i},\quad \text {for all }i\in \mathcal {I}\text {,} \end{aligned}$$
where \(u_{i}^{*}=\left[ u_{is}^{*}\right] _{s\in \mathcal {S}}\in \mathbb {R}^{\left| \mathcal {S}\right| }\) is the ith row vector of the prospect \(u^{*}\). Since by remark 1 the social preference satisfies the ex ante welfare ordering indifference condition, it follows that
$$\begin{aligned} u^{*}\sim _{0}^{p}u\text {,} \end{aligned}$$
thus
$$\begin{aligned} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S} }\lambda _{is}u_{is}^{*}=\min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S}}\lambda _{is}u_{is}\text {.} \end{aligned}$$
By the definition of the prospect \(u^{*}\), we have
$$\begin{aligned} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S} }\lambda _{is}u_{is}^{*}= & {} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}w_{i}^{*}\sum _{s\in \mathcal {S}}\lambda _{is} \\= & {} \min _{\alpha \in I(p)}\sum _{i\in \mathcal {I}}\alpha _{i}w_{i}^{*} \\= & {} \min _{\alpha \in I(p)}\sum _{i\in \mathcal {I}}\alpha _{i}\left( \min _{\sigma _{i}\in S_{i}(p)}\sum _{s\in \mathcal {S}}\sigma _{is}u_{is}\right) \text {.} \end{aligned}$$
Therefore, (5) holds. \(\square \)
Proof of Theorem 2
Though the proof of this representation theorem is similar to that of Theorem 1, we report it for the sake of completeness. Thus, let the premises hold. It is fairly easy to check that if the social preference has the functional form given in (2), then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \), and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Thus, for each state s, the ex post welfare ordering \(\succsim _{0}^{p,s}\) over \(\mathbb {R}^{\left| \mathcal {I}\right| }\), given in Definition 2, inherits all properties satisfied by the social preference \(\succsim _{0}^{p}\). By Gilboa and Schmeidler (1989), there exists a closed and convex set of strictly positive welfare weights on \(\mathcal {I}\), \(I_{s}\left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {I}\right) \), such that the ex post welfare ordering at the state s is represented in the form
$$\begin{aligned} u_{s}\succsim _{0}^{p,s}v_{s}\iff \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I}}\alpha _{si}u_{is}\ge \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I}}\alpha _{si}v_{is}\text {.} \end{aligned}$$
Next, given that by Lemma 1 there exists a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {I}\times \mathcal {S}\), \(\Lambda \left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {I\times S}\right) \), let us define the set of probability distributions S(p) by
$$\begin{aligned} S(p)=\left\{ \left( \sum _{i\in \mathcal {I}}\lambda _{is}\right) _{s\in \mathcal {S}}:\lambda \in \Lambda (p)\right\} \text {.} \end{aligned}$$
It follows from its definition that S(p) is a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {S}\).
It suffices to show
$$\begin{aligned} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S} }\lambda _{is}u_{is}=\min _{\sigma \in S(p)}\sum _{s\in \mathcal {S}}\sigma _{s}\left( \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I} }\alpha _{si}u_{is}\right) ,\quad \text {for all }u\in \mathbb {R} ^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\text {.} \end{aligned}$$
(6)
Fix any prospect u. For each individual \(i\in \mathcal {I}\) and each state \(s\in \mathcal {S}\), define \(u_{is}^{*}\) by
$$\begin{aligned} u_{is}^{*}=\min _{\alpha _{s}\in I_{s}(p)}\sum _{j\in \mathcal {I}}\alpha _{sj}u_{js}\equiv w_{s}^{*}\text {.} \end{aligned}$$
Thus, the matrix \(u^{*}=\left[ u_{is}^{*}\right] _{i\in \mathcal {I} ,s\in \mathcal {S}}\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\) is a prospect.
By this definition, we have
$$\begin{aligned} u_{s}^{*}\sim _{0}^{p,s}u_{s},\quad \text {for all }s\in \mathcal {S}\text {,} \end{aligned}$$
where \(u_{s}^{*}=\left[ u_{is}^{*}\right] _{i\in \mathcal {I}}\in \mathbb {R}^{\left| \mathcal {I}\right| }\) is the sth column vector of the prospect \(u^{*}\). Since by remark 3 the social preference satisfies the ex post welfare ordering indifference condition, it follows that
$$\begin{aligned} u^{*}\sim _{0}^{p}u\text {,} \end{aligned}$$
thus
$$\begin{aligned} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S} }\lambda _{is}u_{is}^{*}=\min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S}}\lambda _{is}u_{is}\text {.} \end{aligned}$$
By the definition of the prospect \(u^{*}\), we have
$$\begin{aligned} \min _{\lambda \in \Lambda (p)}\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S} }\lambda _{is}u_{is}^{*}= & {} \min _{\lambda \in \Lambda (p)}\sum _{s\in \mathcal {S}}w_{s}^{*}\sum _{i\in \mathcal {I}}\lambda _{is} \\= & {} \min _{\sigma \in S(p)}\sum _{s\in \mathcal {S}}\sigma _{s}w_{s}^{*} \\= & {} \min _{\sigma \in S(p)}\sum _{s\in \mathcal {S}}\sigma _{s}\left( \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I}}\alpha _{si}u_{is}\right) \text {.} \end{aligned}$$
Therefore, (6) holds. \(\square \)
Proof of Lemma 2
Fix any profile p of beliefs and any two social prospects u and v. Suppose that the social preference \(\succsim _{0}^{p}\) satisfies ex ante Pareto and ex ante strict Pareto.
Suppose that \(u\ge v\). Then, \(u_{i}\ge v_{i}\) for all \(i\in \mathcal {I}\). It immediately follows that \(p_{i}u_{i}\ge p_{i}v_{i}\) for all \(i\in \mathcal {I}\). Then, \(u\succsim _{0}^{p}v\) by ex ante Pareto. Hence, the social preference satisfies Monotonicity.
To show that the social preference satisfies \(\mathcal {I}\)-separability, fix any individual i and suppose that \(\left( u_{i},u_{-i}\right) \succsim _{0}^{p}\left( v_{i},u_{-i}\right) \). To prove that \(\left( u_{i},v_{-i}\right) \succsim _{0}^{p}\left( v_{i},v_{-i}\right) \), observe that according to Definition 1 it holds that \(u_{i}\succsim _{i}^{p}v_{i}\). Suppose that \(p_{i}v_{i}>p_{i}u_{i}\). Then, ex ante strict Pareto implies that \(\left( v_{i},u_{-i}\right) \succ _{0}^{p}\left( u_{i},u_{-i}\right) \), producing a contradiction. Therefore, we have that \(p_{i}u_{i}\ge p_{i}v_{i}\). Since \(p_{i}u_{i}\ge p_{i}v_{i}\) and \(p_{j}v_{j}=p_{j}v_{j}\) for every individual \(j\ne i\), Ex ante Pareto implies \(\left( u_{i},v_{-i}\right) \succsim _{0}^{p}\left( v_{i},v_{-i}\right) \), as we sought. Since the choice of \(u_{-i}\) and \(v_{-i}\) was arbitrary, we conclude that the social preference satisfies \(\mathcal {I}\)-separability. \(\square \)
Proof of Proposition 3
Let the premises hold. It is fairly easy to check that if the social preference has the functional form given in (1), then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \), and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Lemma 2 implies that the social preference \(\succsim _{0}^{p}\) satisfies Monotonicity and \(\mathcal {I}\)-separability. Theorem 1 implies that \(\succsim _{0}^{p}\) has the ex ante functional form given in (1). What is left is to show that (3) holds.
Fix any individual i. First, we show that \(p_{i}\in S_{i}(p)\). Assume, to the contrary, that \(p_{i}\notin S_{i}(p)\). Thus, the separation theorem implies that there is a nonzero vector \(u_{i}\in \mathbb {R}^{\left| \mathcal {S}\right| }\) such that
$$\begin{aligned} \beta \equiv \min _{\sigma _{i}\in S_{i}(p)}\sigma _{i}u_{i}>p_{i}u_{i}\text {.} \end{aligned}$$
Fix any real number \(\gamma \), and let \(\gamma _{-i}\) denote the \(\left| \mathcal {I}\backslash \left\{ i\right\} \right| \times \left| \mathcal {S}\right| \) matrix with all entries equal to \(\gamma \). Moreover, with abuse of notation, we also use \(\beta \) to denote the vector in \(\mathbb {R}^{\left| \mathcal {S}\right| }\) with all entries equal to real number \(\beta \). Thus, \((u_{i},\gamma _{-i})\) is a prospect such that i obtains \(u_{i}\) and everybody else obtains \(\gamma \) with certainty. Also, \((\beta ,\gamma _{-i})\) is a prospect such that i obtains \(\beta \) with certainty and everybody else obtains \(\gamma \) with surety.
Since
$$\begin{aligned} \min _{\alpha \in I(p)}\left( \alpha _{i}\min _{\sigma _{i}\in S_{i}(p)}\sigma _{i}u_{i}+\gamma \sum _{j\ne i}\alpha _{j}\right) =\min _{\alpha \in I(p)}\left( \alpha _{i}\beta +\gamma \sum _{j\ne i}\alpha _{j}\right) \text {,} \end{aligned}$$
it follows from (1) that
$$\begin{aligned} (u_{i},\gamma _{-i})\succsim _{0}^{p}(\beta ,\gamma _{-i})\text {.} \end{aligned}$$
Furthermore, since \(\succsim _{0}^{p}\) satisfies ex ante strict Pareto, it holds that \(p_{i}u_{i}\ge p_{i}\beta \). Since \(p_{i}\beta =\beta \), we have that \(p_{i}u_{i}\ge \beta \), which is a contradiction. Thus, we have that \(p_{i}\in S_{i}(p)\).
Next, we show that \(S_{i}(p)\subseteq \{p_{i}\}\). Assume, to the contrary, that \(\{p_{i}\}\) is a proper subset of \(S_{i}(p)\). Thus, the separation theorem implies that there is a nonzero vector \(u_{i}\in \mathbb {R}^{\left| \mathcal {S}\right| }\) such that
$$\begin{aligned} p_{i}u_{i}>\min _{\sigma _{i}\in S_{i}(p)}\sigma _{i}u_{i}\equiv \beta \text {.} \end{aligned}$$
Again, fix any real number \(\gamma \), and let \(\gamma _{-i}\) denote the \(\left| \mathcal {I}\backslash \left\{ i\right\} \right| \times \left| \mathcal {S}\right| \) matrix with all entries equal to \(\gamma \). As above, we also use \(\beta \) to denote the vector in \(\mathbb {R}^{\left| \mathcal {S}\right| }\) with all entries equal to real number \(\beta \). Thus, since
$$\begin{aligned} \min _{\alpha \in I(p)}\left( \alpha _{i}\beta +\gamma \sum _{j\ne i}\alpha _{j}\right) =\min _{\alpha \in I(p)}\left( \alpha _{i}\min _{\mu \in S_{i}(p)}\sigma _{i}u_{i}+\gamma \sum _{j\ne i}\alpha _{j}\right) \text {,} \end{aligned}$$
we have that
$$\begin{aligned} (\beta ,\gamma _{-i})\succsim _{0}^{p}(u_{i},\gamma _{-i})\text {.} \end{aligned}$$
Since \(\succsim _{0}^{p}\) satisfies ex ante strict Pareto, it holds that \(p_{i}\beta \ge p_{i}u_{i}\). Again, since \(p_{i}\beta =\beta \), we have that \(\beta \ge p_{i}u_{i}\), which is a contradiction. Thus, we conclude that \(S_{i}(p)=\left\{ p_{i}\right\} \). \(\square \)
Proof of Proposition 4
Let the premises hold. It is fairly easy to check that if the social preference has the functional form given in part (a) or in part (b) of the statement, then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \), and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Since the social preference \(\succsim _{0}^{p}\) satisfies ex ante Pareto and ex ante strict Pareto, Lemma 2 implies that it satisfies Monotonicity and \(\mathcal {I}\)-separability. Thus, the social preference \(\succsim _{0}^{p}\) is represented in the additive form by
$$\begin{aligned} \sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S}}\lambda _{is}(p)u_{is}\quad \text {for all }u\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\text {,} \end{aligned}$$
where \(\lambda _{is}(p)>0\) for each individual i and each state s and where \(\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S}}\lambda _{is}(p)=1\).
Since by ex ante Pareto the profile \(\left( \lambda _{is}(p)\right) _{s\in \mathcal {S}}\) must be proportional to individual i’s beliefs, \(p_{i} \), \(\lambda _{is}(p)\) has the form
$$\begin{aligned} \lambda _{is}(p)=\beta _{i}\left( p\right) p_{is}\quad \text {for all } s\in \mathcal {S}\text {.} \end{aligned}$$
Observe that \(\beta _{i}\left( p\right) >0\). Since, moreover,
$$\begin{aligned} 1=\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S}}\lambda _{is}(p)=\sum _{i\in \mathcal {I}}\sum _{s\in \mathcal {S}}\beta _{i}(p)p_{is}=\sum _{i\in \mathcal {I} }\beta _{i}(p)\sum _{s\in \mathcal {S}}p_{is}=\sum _{i\in \mathcal {I}}\beta _{i}(p)\text {,} \end{aligned}$$
we obtain \(\beta (p)\equiv \left( \beta _{i}\left( p\right) \right) _{i\in \mathcal {I}}\in \Delta ^{\circ }(\mathcal {I})\).
From this, it follows that the social preference \(\succsim _{0}^{p}\) is represented in the additive form by
$$\begin{aligned} \sum _{i\in \mathcal {I}}\beta _{i}(p)\sum _{s\in \mathcal {S}}p_{is}u_{is} \quad \text {for all }u\in \mathbb {R}^{\left| \mathcal {I} \right| \times \left| \mathcal {S}\right| }\text {.} \end{aligned}$$
One can easily see that this corresponds to the ex ante representation form of Theorem 1 with the specification that \(I\left( p\right) =\left\{ \beta \left( p\right) \right\} \) and \(S_{i}(p)=\{p_{i}\}\) for all \(i\in \mathcal {I}\). This completes part (a).
To prove part (b), note that
$$\begin{aligned} \sum _{i\in \mathcal {I}}\beta _{i}(p)\sum _{s\in \mathcal {S}}p_{is}u_{is}= & {} \sum _{s\in \mathcal {S}}\sum _{i\in \mathcal {I}}\beta _{i}(p)p_{is}u_{is} \\= & {} \sum _{s\in \mathcal {S}}\left( \sum _{j\in \mathcal {I}}\beta _{j}(p)p_{js}\right) \left( \sum _{i\in \mathcal {I}}\left( \frac{\beta _{i}(p)p_{is}}{\sum _{j\in \mathcal {I}}\beta _{j}(p)p_{js}}\right) u_{is}\right) \text {.} \end{aligned}$$
One can easily check that \(\left( \sum _{j\in \mathcal {I}}\beta _{j}(p)p_{js}\right) _{_{s\in \mathcal {S}}}\in \Delta ^{\circ }(\mathcal {S})\) and that \(\left( \frac{\beta _{i}\left( p\right) p_{is}}{\sum _{j\in \mathcal {I}}\beta _{j}\left( p\right) p_{js}}\right) _{i\in \mathcal {I}}\in \Delta ^{\circ }(\mathcal {I})\) for all \(s\in \mathcal {S}\). Therefore, part (b) corresponds to the ex post representation form of Theorem 2 where
$$\begin{aligned} S\left( p\right)= & {} \left\{ \left( \sum _{j\in \mathcal {I}}\beta _{j}\left( p\right) p_{js}\right) _{s\in \mathcal {S}}\right\} \text {, and } \\ I_{s}\left( p\right)= & {} \left\{ \left( \frac{\beta _{i}\left( p\right) p_{is}}{\sum _{j\in \mathcal {I}}\beta _{j}\left( p\right) p_{js}}\right) _{i\in \mathcal {I}}\right\} \quad \text {for all }s\in \mathcal {S} \text {,} \end{aligned}$$
as we sought. \(\square \)
Proof of Lemma 3
Let the premises hold. It is fairly easy to check that if the social preference has the functional form given in (4), then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \), and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Let us define the ranking \(\succsim _{0}^{*p}\) induced over \(\mathbb {R} ^{\left| \mathcal {I}\right| \times \left| \mathcal {I} (p)\right| }\) by the social preference \(\succsim _{0}^{p}\) as follows:
$$\begin{aligned} u\succsim _{0}^{p}v\iff x\succsim _{0}^{*p}y\text {,}\quad \text { for all }u,v\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\text {,} \end{aligned}$$
where \(x\equiv (p_{J}u_{i})_{i\in \mathcal {I},J\in \mathcal {I}(p)}\) and \(y\equiv (p_{J}v_{i})_{i\in \mathcal {I},J\in \mathcal {I}(p)}\). One can easily check that this ranking is complete, transitive, continuous, and convex and, moreover, it satisfies Homogeneity (A.4), UI-Aversion (A.6 ), Collective ex ante Pareto (A.9) and Collective ex ante strict Pareto (A.10).
To show that the ranking also satisfies CE-Independence (A.5), it suffices to show that for all \(u,v\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\) it holds
$$\begin{aligned} x\succsim _{0}^{*p}y\implies x+\alpha \mathbf {1}_{\mathcal {I}\times \mathcal {I}(p)}\succsim _{0}^{*p}y+\alpha \mathbf {1}_{\mathcal {I}\times \mathcal {I}(p)}\text {,} \end{aligned}$$
where \(x\equiv (p_{J}u_{i})_{i\in \mathcal {I},J\in \mathcal {I}(p)}\) and \( y\equiv (p_{J}v_{i})_{i\in \mathcal {I},J\in \mathcal {I}(p)}\), where \(\mathbf { 1}_{\mathcal {I}\times \mathcal {I}(p)}\in \mathbb {R}^{\left| \mathcal {I} \right| \times \left| \mathcal {I}(p)\right| }\) denotes the \( |I|\times |\mathcal {I}(p)|\) matrix whose entries are all 1 and \(\alpha \) is a scalar. This follows from the fact that \(\succsim _{0}^{p}\) satisfies CE-Independence (A.5) and from the fact that we can take \( (p_{J}(u_{i}+\alpha \mathbf {1}_{\mathcal {S}}))_{i\in \mathcal {I},J\in \mathcal {I}(p)}=x+\alpha \mathbf {1}_{\mathcal {I}\times \mathcal {I}(p)}\) and \( (p_{J}(u_{i}+\alpha \mathbf {1}_{\mathcal {S}}))_{i\in \mathcal {I},J\in \mathcal {I}(p)}=y+\alpha \mathbf {1}_{\mathcal {I}\times \mathcal {I}(p)}\).
Since \(\succsim _{0}^{*p}\) satisfies Collective ex ante Pareto (A.9), one can also see that the ranking satisfies Monotonicity (A.3). By Gilboa and Schmeidler (1989), there exists a closed convex set \(\Gamma (p)\subseteq \Delta ^{\circ }(\mathcal {I}\times \mathcal {I}(p))\) such that, for all \(u,v\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\),
$$\begin{aligned} x\succsim _{0}^{*p}y\Longleftrightarrow \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}x_{iJ}\ge \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I} (p)}\gamma _{iJ}y_{iJ}\text {,} \end{aligned}$$
where \(x\equiv (p_{J}u_{i})_{i\in \mathcal {I},J\in \mathcal {I}(p)}\) and \(y\equiv (p_{J}v_{i})_{i\in \mathcal {I},J\in \mathcal {I}(p)}\). Thus, by definition of the ranking \(\succsim _{0}^{*p}\), it follows that for all \(u,v\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\) it holds that
$$\begin{aligned} u\succsim _{0}^{p}v\Longleftrightarrow \min _{\gamma \in \Gamma (p)}\sum _{i\in I}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S} }p_{Js}u_{is}\ge \min _{\gamma \in \Gamma (p)}\sum _{i\in I}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}\text {,} \end{aligned}$$
as we sought. \(\square \)
Proof of Theorem 3
Let the premises hold. It is fairly easy to check that if the social preference has the functional form specified in the statement of Theorem 3, then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \), and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Thus, for each individual i, the individual i’s ex ante welfare ordering \(\succsim _{i}^{p}\) over \(\mathbb {R}^{\left| \mathcal {S}\right| }\), given in Definition 1, inherits all properties satisfied by the social preference \(\succsim _{0}^{p}\). By Gilboa and Schmeidler (1989), there exists a closed and convex set of strictly positive probability vectors on \(\mathcal {S}\), \(S_{i}\left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {S}\right) \), such that the individual i’s ex ante welfare ordering is represented in the form
$$\begin{aligned} u_{i}\succsim _{i}^{p}v_{i}\iff \min _{\sigma _{i}\in S_{i}\left( p\right) }\sum _{s\in \mathcal {S}}\sigma _{is}u_{is}\ge \min _{\sigma _{i}\in S_{i}\left( p\right) }\sum _{s\in \mathcal {S}}\sigma _{is}v_{is}\text {, }\quad \text {for all }i\in \mathcal {I}\text {.} \end{aligned}$$
Next, given that by Lemma 3 there exists a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {I}\times \mathcal {I}\left( p\right) \), \(\Gamma \left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {I\times I}\left( p\right) \right) \), let us define the set of welfare weights I(p) by
$$\begin{aligned} I(p)=\left\{ \left( \sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\right) _{i\in \mathcal {I}}:\gamma \in \Gamma (p)\right\} \text {.} \end{aligned}$$
It follows from its definition that I(p) is a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {I}\).
To complete the proof, it suffices to show, for all \(u\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\), it holds
$$\begin{aligned} \min _{\gamma \in \Gamma (p)}\left( \sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}\right) =\min _{\alpha \in I(p)}\sum _{i\in \mathcal {I}}\alpha _{i}\left( \min _{\sigma _{i}\in S_{i}(p)}\sum _{s\in \mathcal {S}}\sigma _{is}u_{is}\right) \text {.} \end{aligned}$$
(7)
To this end, fix any prospect u. For each individual \(i\in \mathcal {I}\) and each state \(s\in \mathcal {S}\), define \(u_{is}^{*}\) by
$$\begin{aligned} u_{is}^{*}=\min _{\mu _{i}\in S_{i}(p)}\sum _{t\in \mathcal {S}}\mu _{it}u_{it}\equiv w_{i}^{*}\text {.} \end{aligned}$$
Thus, the matrix \(u^{*}=\left[ u_{is}^{*}\right] _{i\in \mathcal {I} ,s\in \mathcal {S}}\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\) is a prospect.
By this definition, we have
$$\begin{aligned} u_{i}^{*}\sim _{i}^{p}u_{i},\quad \text {for all }i\in \mathcal {I} \text {,} \end{aligned}$$
where \(u_{i}^{*}=\left[ u_{is}^{*}\right] _{s\in \mathcal {S}}\in \mathbb {R}^{\left| \mathcal {S}\right| }\) is the ith row vector of the prospect \(u^{*}\). Since by remark 1 the social preference satisfies the ex ante welfare ordering indifference condition, it follows that
$$\begin{aligned} u^{*}\sim _{0}^{p}u\text {,} \end{aligned}$$
thus
$$\begin{aligned} \min _{\gamma \in \Gamma (p)}\left( \sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}^{*}\right) =\min _{\gamma \in \Gamma (p)}\left( \sum _{i\in \mathcal {I} }\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S} }p_{Js}u_{is}\right) \text {.} \end{aligned}$$
By the definition of the prospect \(u^{*}\), we have
$$\begin{aligned} \min _{\gamma \in \Gamma (p)}\left( \sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}^{*}\right)= & {} \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}w_{i}^{*}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js} \\= & {} \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}w_{i}^{*}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ} \\= & {} \min _{\alpha \in I(p)}\sum _{i\in \mathcal {I}}\alpha _{i}w_{i}^{*} \\= & {} \min _{\alpha \in I(p)}\sum _{i\in \mathcal {I}}\alpha _{i}\left( \min _{\sigma _{i}\in S_{i}(p)}\sum _{s\in \mathcal {S}}\sigma _{is}u_{is}\right) \text {.} \end{aligned}$$
Thus, (7) holds. This means that
$$\begin{aligned} S_{i}(p)=\left\{ \left( \sum _{J\in \mathcal {I}(p)}\gamma _{iJ}p_{Js}\right) _{s\in \mathcal {S}}:\gamma \in \Gamma (p)\right\} \quad \text {for all }i\in \mathcal {I}\text {,} \end{aligned}$$
as we sought. \(\square \)
Proof of Theorem 4
Let the premises hold. It is fairly easy to check that if the social preference has the functional form specified in the statement of Theorem 4, then it satisfies the axioms. Thus, fix any profile of beliefs \(p=\left( p_{i}\right) _{i\in \mathcal {I}}\) such that \(p_{i}\in \Delta ^{\circ }\left( \mathcal {S}\right) \), and suppose that the continuous weak order \(\succsim _{0}^{p}\) satisfies the axioms.
Thus, for each state s, the ex post welfare ordering \(\succsim _{0}^{p,s}\) over \(\mathbb {R}^{\left| \mathcal {I}\right| }\), given in Definition 2, inherits all properties satisfied by the social preference \(\succsim _{0}^{p}\). By Gilboa and Schmeidler (1989), there exists a closed and convex set of strictly positive welfare weights on \(\mathcal {I}\), \(I_{s}\left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {I}\right) \), such that the ex post welfare ordering at the state s is represented in the form
$$\begin{aligned} u_{s}\succsim _{0}^{p,s}v_{s}\iff \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I}}\alpha _{si}u_{is}\ge \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I}}\alpha _{si}v_{is}\text {, } \quad \text {for all }s\in \mathcal {S}\text {.} \end{aligned}$$
Next, given that by Lemma 3 there exists a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {I}\times \mathcal {I}\left( p\right) \), \(\Gamma \left( p\right) \subseteq \Delta ^{\circ }\left( \mathcal {I\times I}\left( p\right) \right) \), let us define the set of social beliefs S(p) by
$$\begin{aligned} S(p)=\left\{ \left( \sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}p_{Js}\right) _{s\in \mathcal {S}}:\gamma \in \Gamma (p)\right\} \text {.} \end{aligned}$$
It follows from its definition that S(p) is a unique, closed and convex set of strictly positive probability vectors on \(\mathcal {S}\).
To complete the proof, it suffices to show, for all \(u\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\), it holds that
$$\begin{aligned} \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I} (p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}=\min _{\mu \in S(p)}\sum _{s\in \mathcal {S}}\mu _{s}\min _{\alpha _{s}\in I_{s}(p)}\sum _{i\in \mathcal {I}}\alpha _{is}u_{is}\text {.} \end{aligned}$$
(8)
Fix any prospect u. For each individual \(i\in \mathcal {I}\) and each state \(s\in \mathcal {S}\), define \(u_{is}^{*}\) by
$$\begin{aligned} u_{is}^{*}=\min _{\alpha _{s}\in I_{s}(p)}\sum _{j\in \mathcal {I}}\alpha _{sj}u_{js}\equiv w_{s}^{*}\text {.} \end{aligned}$$
Thus, the matrix \(u^{*}=\left[ u_{is}^{*}\right] _{i\in \mathcal {I},s\in \mathcal {S}}\in \mathbb {R}^{\left| \mathcal {I}\right| \times \left| \mathcal {S}\right| }\) is a prospect.
By this definition, we have
$$\begin{aligned} u_{s}^{*}\sim _{0}^{p,s}u_{s},\quad \text {for all }s\in \mathcal {S}\text {,} \end{aligned}$$
where \(u_{s}^{*}=\left[ u_{is}^{*}\right] _{i\in \mathcal {I}}\in \mathbb {R}^{\left| \mathcal {I}\right| }\) is the sth column vector of the prospect \(u^{*}\). Since by remark 3 the social preference satisfies the ex post welfare ordering indifference condition, it follows that
$$\begin{aligned} u^{*}\sim _{0}^{p}u\text {,} \end{aligned}$$
thus
$$\begin{aligned} \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I} (p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}^{*}=\min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}\text {.} \end{aligned}$$
By the definition of the prospect \(u^{*}\), we also have
$$\begin{aligned} \min _{\gamma \in \Gamma (p)}\sum _{i\in \mathcal {I}}\sum _{J\in \mathcal {I} (p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}u_{is}^{*}= & {} \min _{\gamma \in \Gamma (p)}\sum _{s\in \mathcal {S}}\left( \sum _{J\in \mathcal {I} (p)}\gamma _{iJ}\sum _{s\in \mathcal {S}}p_{Js}\right) w_{s}^{*} \\= & {} \min _{\sigma \in S(p)}\sum _{s\in \mathcal {S}}\sigma _{s}w_{s}^{*} \\= & {} \min _{\sigma \in S(p)}\sum _{s\in \mathcal {S}}\sigma _{s}\left( \min _{\alpha _{s}\in I_{s}\left( p\right) }\sum _{i\in \mathcal {I}}\alpha _{si}u_{is}\right) \text {.} \end{aligned}$$
Thus, (8) holds. This means that
$$\begin{aligned} I_{s}(p)=\left\{ \left( \frac{\sum _{J\in \mathcal {I}(p)}\gamma _{iJ}p_{Js}}{\sum _{j\in I}\sum _{J\in \mathcal {I}(p)}\gamma _{jJ}p_{Js}}\right) _{i\in \mathcal {I}}:\gamma \in \Gamma (p)\right\} \quad \text {for all }s\in \mathcal {S}\text {,} \end{aligned}$$
as we sought. \(\square \)