Social preference under twofold uncertainty

Abstract

We investigate the conflict between the ex ante and ex post criteria of social welfare in a new framework of individual and social decisions, which distinguishes between two sources of uncertainty, here interpreted as being objective and subjective, respectively. This framework makes it possible to endow the individuals and society not only with ex ante and ex post preferences, as is usually done, but also with interim preferences of two kinds, and correspondingly, to introduce interim forms of the Pareto principle. After characterizing the two social welfare criteria, we present two compromises between them, one based on the ex ante criterion and absorbing as much as possible of the ex post criterion (Theorem 1), the other based on the ex post criterion and absorbing as much as possible of the ex ante criterion (Theorem 2). Both solutions translate the assumed Pareto conditions into weighted additive utility representations, as in Harsanyi’s Aggregation Theorem, and both attribute to the individuals common probability values on the objective source of uncertainty, and different probability values on the subjective source. We discuss these solutions in terms of the by now classic spurious unanimity argument and a novel informational argument labeled complementary ignorance. The paper complies with the standard economic methodology of basing probability and utility representations on preference axioms.

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Notes

  1. 1.

    Note the difference between a social welfare criterion and the corresponding Pareto principle. There is more to the ex ante (ex post) social welfare criterion than just the ex ante (ex post) Pareto principle, because a criterion also decides where rationality assumptions apply (to the individuals or society).

  2. 2.

    The utilitarian interpretation of Harsanyi remains controversial; see Sen’s (1986) objections and the discussion in Weymark (1991). Fleurbaey and Mongin (2016) propose a new defense of Harsanyi’s position.

  3. 3.

    The earlier literature on the linear pooling rule is vast; see among others (Genest and Zidek 1986; Clemen and Winkler 2007).

  4. 4.

    The requirement that separability conditions hold across the uncertainty type means that they are equivalent to dominance conditions for the given type.

  5. 5.

    Briefly: despite finiteness assumptions that resemble Anscombe and Aumann’s, our theorems follow Savage by relying on a pure uncertainty framework, and eschewing any probabilistic primitives.

  6. 6.

    Implicitly, we define all forms of the Pareto principle except for the ex ante one in terms of binary relations rather than bona fide preference orderings. This makes these Pareto conditions logically independent of the decision-theoretic conditions discussed above.

  7. 7.

    In the standard framework, with a single source of uncertainty, Mongin (1998), Chambers and Hayashi (2006) and Keeney and Nau (2011) have explored this solution. It is not merely formal, but normatively defensible, as it seems desirable that society could change the individuals’ weights in the utility sum according to which state of the world is realized. In the standard framework, this solution might leave society without any probabilistic beliefs at all. But Theorem 1 does away with social probability only for the subjective part of uncertainty.

  8. 8.

    This troubling argument indirectly connects with worries that Savage (1972) once expressed concerning “small worlds” and the problem they raise for his SEU theory.

  9. 9.

    By substituting (9) into (8), we can express \(W_{\mathrm {xa}}\) as a weighted sum of \(u^i\) values. But since society uses its own probability \({\mathbf {p}}\), this expression is not a sum of the individual SEU representations from (7).

  10. 10.

    We may assume that the bridge is financially feasible, whichever the main direction of the flow is.

  11. 11.

    In the Savage framework, the remaining part of the ex post criterion, i.e., the ex post Pareto principle, follows from either the ex ante principle or its GSS restricted form.

  12. 12.

    We may also note that this two-step process of revising and averaging does not give the same result as the direct averaging of posteriors. In technical jargon, the linear pooling rule is not “externally Bayesian,” an old observation of the literature (see, e.g., Genest and Zidek 1986; Clemen and Winkler 2007).

  13. 13.

    To check formally that there is no connection, suppose that, for any choice of \(u^i\), the \({{\mathbf {p}}}^i\) are the same and differ from \({\mathbf {p}}\). The assumptions of the theorem can be satisfied under this supposition.

  14. 14.

    Chambers and Hayashi further show that the ex ante social welfare function is a weighted sum of individual expected utilities, whereas Nehring assumes this.

  15. 15.

    Here, we focus on Alon and Gayer’s (2014) preprint rather than their final (2016) version because it better connects with the other work reviewed here. See Billot and Qu (2017) for more on the same line.

  16. 16.

    Indeed, applying the results of Qu or Danan et al. to this example yields SEU social preferences with social beliefs satisfying (ii). Applying Alon and Gayer’s result yields MEU social preferences in which all priors agree that \(P(E_1\cup E_3)=0.96\) and \(P(E_2)=0.04\), so the social ranking of f and g is still incorrect.

  17. 17.

    In the bridge example, the risk faced by the Islanders is not independent of the risk faced by the Mainlanders, and in the duel example, the risks faced by the duelists are not independent either.

  18. 18.

    For these definitions and basic facts, see Fishburn (1970), Keeney and Raiffa (1976) and Wakker (1989). What is called separable here is sometimes called weakly separable elsewhere.

  19. 19.

    Here, we make the obvious identifications of \(\left\{ s\right\} \times {\mathcal {O}}\) with \({\mathcal {O}}\), \({\mathcal {S}}\times \left\{ o\right\} \) with \({\mathcal {S}}\), etc. Similar routine identifications will occur below without being mentioned.

  20. 20.

    Thus, our statement “\(\succsim ^i\) induces conditional preferences \(\succsim _s^i\),” which definitionally says that the \(\succsim _s^i\) are orderings, equivalently says that \(\succsim ^i\) satisfies dominance with respect to the \(\succsim _s^i\).

  21. 21.

    Proposition A2(c) is stated for \({\mathbb {R}}^{{\mathcal {J}}\times {\mathcal {K}}}\), but it carries through to subsets \(Y^{{\mathcal {J}}\times {\mathcal {K}}}\subseteq {\mathbb {R}}^{{\mathcal {J}}\times {\mathcal {K}}}\), when these are open and take the form of a product of intervals.

  22. 22.

    To avoid burdening notation, we refer to the original and translated functions by the same symbol. This convention is applied throughout the proofs.

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Correspondence to Philippe Mongin.

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The authors thank an anonymous referee and the editors for their detailed advice on this paper. The authors benefited from comments from Takashi Hayashi, Michele Lombardi, Hervé Moulin and Clemens Puppe during seminar presentations. The first author acknowledges support from the Wissenschaftskolleg zu Berlin and the Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047), and the second author from NSERC Grant #262620-2008, Labex MME-DII (ANR11-LBX-0023-01) and CHOp (ANR-17-CE26-0003).

Appendices

Appendices

Technical background

We begin by restating the definition of a conditional relation in terms of its source relation, and the separability property that turns a conditional relation into an ordering.

Suppose that a weak preference ordering \({\mathsf {R}}\) is defined on a product set \(\mathcal {X=}\prod _{\ell \in {\mathcal {L}}}{\mathcal {X}}_{\ell }\), where \({\mathcal {L}}\) is a finite set of indexes. Take a subset of indexes \({\mathcal {J}}\subseteq {\mathcal {L}}\) and its complement \({\mathcal {K}}:={\mathcal {L}}\setminus {\mathcal {J}}\). Denote the subproduct sets \(\prod _{\ell \in {\mathcal {J}}}{\mathcal {X}}_{\ell }\) and \(\prod _{\ell \in {\mathcal {K}}}{\mathcal {X}}_{\ell }\) by \({\mathcal {X}}_{{\mathcal {J}}}\) and \({\mathcal {X}}_{{\mathcal {K}}}\), respectively. By definition, the conditional induced by \({\mathsf {R}}\)on \({\mathcal {J}}\) is the relation \({\mathsf {R}}_{{\mathcal {J}}} \) on \({\mathcal {X}}_{{\mathcal {J}}}\) thus defined: for all \(\xi _{{\mathcal {J}}},\xi _{{\mathcal {J}}}'\in {\mathcal {X}}_{{\mathcal {J}}}\),

$$\begin{aligned} \xi _{{\mathcal {J}}}\ {\mathsf {R}}_{{\mathcal {J}}}\ \xi _{{\mathcal {J}}}'\text { if and only if for some }\xi _{{\mathcal {K}}}\in {\mathcal {X}}_{{\mathcal {K}}},(\xi _{{\mathcal {J}}},\xi _{{\mathcal {K}}})\ {\mathsf {R}}\ (\xi _{{\mathcal {J}}}',\xi _{{\mathcal {K}}}). \end{aligned}$$

By a well-known fact, the conditional \({\mathsf {R}}_{{\mathcal {J}}}\) is an ordering if and only if \({\mathsf {R}}\) is separable in \({\mathcal {J}}\), that is: for all \(\xi _{{\mathcal {J}}},\xi _{{\mathcal {J}}}'\in {\mathcal {X}}_{{\mathcal {J}}}\) and \(\xi _{{\mathcal {K}}},\xi _{{\mathcal {K}}}'\in {\mathcal {X}}_{{\mathcal {K}}}\),

$$\begin{aligned} (\xi _{{\mathcal {J}}},\xi _{{\mathcal {K}}})\ {\mathsf {R}}\ (\xi _{{\mathcal {J}}}',\xi _{{\mathcal {K}}})\text { if and only if }(\xi _{{\mathcal {J}}},\xi _{{\mathcal {K}}}')\ {\mathsf {R}}\ (\xi _{{\mathcal {J}}}^{\prime },\xi _{{\mathcal {K}}}'). \end{aligned}$$

In this case, we may also say that \({\mathcal {J}}\) is a \({\mathsf {R}}\)-separable. Clearly, separability in \({\mathcal {J}}\) entails that \({\mathsf {R}}\) is increasing with \({\mathsf {R}}_{{\mathcal {J}}}\), i.e., that for all \(\xi _{{\mathcal {J}}},\xi _{{\mathcal {J}}}'\in {\mathcal {X}}_{{\mathcal {J}}}\) and \(\xi _{{\mathcal {K}}}\in {\mathcal {X}}_{{\mathcal {K}}}\),

$$\begin{aligned} \text {if }\xi _{{\mathcal {J}}}\ {\mathsf {R}}_{{\mathcal {J}}}\ \xi _{{\mathcal {J}}}'\text {, then }(\xi _{{\mathcal {J}}},\xi _{{\mathcal {K}}})\ {\mathsf {R}}\ (\xi _{{\mathcal {J}}}',\xi _{{\mathcal {K}}}), \end{aligned}$$

and if the \({\mathsf {R}}_{{\mathcal {J}}}\)-comparison is in fact strict, so is the resulting \({\mathsf {R}}\)-comparison. Conversely, if \({\mathsf {R}}_{{\mathcal {J}}}\) is some ordering on \({\mathcal {X}}_{{\mathcal {J}}}\), the property that \({\mathsf {R}}\) on \({\mathcal {X}}\) is increasing with \({\mathsf {R}}_{{\mathcal {J}}}\) entails that \({\mathsf {R}}\) is weakly separable in \({\mathcal {J}}\).Footnote 18

Section 2 considers several cases of this construction; in each of them, \(\mathcal { X}_{\ell }={\mathbb {R}}\) for all \(\ell \in {\mathcal {L}}\). If \({\mathcal {L}}={\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\) and \({\mathsf {R}}\) is the social preference \(\succeq \), then we can let \({\mathcal {J}}={\mathcal {I}}\times \left\{ s\right\} \times {\mathcal {O}}\) to obtain s-conditional social preference relations, let \({\mathcal {J}}={\mathcal {I}}\times {\mathcal {S}}\times \left\{ o\right\} \) to obtain o-conditional social preference relations, and let \({\mathcal {J}}=\{i\}\times {\mathcal {S}}\times {\mathcal {O}}\) for the ex ante Pareto principle. If \({\mathcal {L}}={\mathcal {S}}\times {\mathcal {O}}\) and \({\mathsf {R}}\) is an individual preference \(\succeq ^i\), then we let \({\mathcal {J}}=\left\{ s\right\} \times {\mathcal {O}}\) to obtain s-conditional individual preference relations, and let \({\mathcal {J}}={\mathcal {S}}\times \left\{ o\right\} \) to obtain o-conditional individual preference relations. If \({\mathcal {L}}={\mathcal {I}}\times {\mathcal {O}}\) and \({\mathsf {R}}\) is the s-conditional social preference \(\succeq _s\), then we let \({\mathcal {J}}=\{i\}\times {\mathcal {O}}\) to obtain the subjective interim Pareto principle. Other cases are similar.Footnote 19 These relations may or may not be orderings, i.e., from the first equivalence stated above, their source relation may or may be separable in the associated index set. As we assume that conditional preference orderings exist across the uncertainty type or not at all, it follows from the second equivalence that the ordering assumptions could be restated in terms of dominance.Footnote 20

Let us now take \({\mathcal {X}}\) to be an open box in \({{\mathbb {R}}}^{{\mathcal {L}}}\), i.e., \({\mathcal {X}}=\prod _{\ell \in {\mathcal {L}}}{\mathcal {X}}_{\ell }\), where the \({\mathcal {X}}_{\ell }\) are open intervals. An ordering \(\succeq \) on \({{\mathcal {X}}}\) has an additive representation if it is represented by a function \(U:{{\mathcal {X}}}{\longrightarrow }{{\mathbb {R}}}\) of the form

$$\begin{aligned} U({\mathbf {x}})\quad :=\quad \sum _{\ell \in {\mathcal {L}}}u_{\ell }(x_{\ell }), \end{aligned}$$
(A1)

where \(u_{\ell }:{\ {\mathcal {X}}}_{\ell }{\longrightarrow }{{\mathbb {R}}}\), \(\ell \in {\mathcal {L}}\). We now state a proposition that will be useful later. A proof can be devised from the formal arguments in Mongin and Pivato (2015).

Proposition A1

Take \({\mathcal {L}}={\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\), with \(\left| {\mathcal {I}}\right| ,\left| {\mathcal {S}}\right| ,\left| {\mathcal {O}}\right| \ge 2\), and view elements \({\mathbb {X}}\in {\mathbb {R}}^{{\mathcal {L}}}\) as arrays \([x_{so}^i]_{s\in {\mathcal {S}},o\in {\mathcal {O}}}^{i\in {\mathcal {I}}}\). If a continuous order \(\succeq \) on \({\mathbb {R}}^{{\mathcal {L}}}\) is increasing in every coordinate and is separable in each \(i\in {\mathcal {I}}\), each \(s\in {\mathcal {S}}\) and each \(o\in {\mathcal {O}}\), then it has an additive utility representation \(U:{{\mathbb {X}}}{\longrightarrow }{{\mathbb {R}}}\) of the form \(\displaystyle U({\mathbb {X}}):= \sum \nolimits _{i\in {\mathcal {I}}}\sum \nolimits _{s\in {\mathcal {S}}}\sum \nolimits _{o\in {\mathcal {O}}}u_{so}^i(x_{so}^i)\), where each \(u_{so}^i:{\mathbb {R}} \longrightarrow {\mathbb {R}}\) is a continuous, increasing function. Furthermore, the functions \(\{u_{so}^i\}_{s\in {\mathcal {S}},o\in {\mathcal {O}}}^{i\in {\mathcal {I}}}\) are unique up to positive affine transformation (PAT) with a common multiplier.

We now specialize the basic sets still differently. Take \({\mathcal {L}}={\mathcal {J}}\times {\mathcal {K}}\) with \(|{\mathcal {J}}|,|{\mathcal {K}}|\ge 2\), and \({\mathcal {X}}={{\mathbb {R}}}^{{\mathcal {L}}}\), viewing elements \({\mathbf {X}}\in {\mathcal {X}}\) as matrices \([x_k^j]_{k\in {\mathcal {K}}}^{j\in {\mathcal {J}}}\), with \(j\in {\mathcal {J}}\) indexing the rows and \(k\in {\mathcal {K}}\) indexing the columns. Alternatively, think of \({\mathbf {X}}\) as a \(\mathcal {\ J}\)-indexed array of row vectors \({\mathbf {x}}^j:=[x_k^j]_{k\in \mathcal {\ K}}\in {{\mathbb {R}}}^{{\mathcal {K}}}\), or as a \({\mathcal {K}}\)-indexed array of columns vectors \({\mathbf {x}}_k:=[x_k^j]^{j\in {\mathcal {J}}}\in {\mathbb {R}}^{{\mathcal {J}}}\). Now consider a continuous ordering \(\succeq \) on \({\mathcal {X}}\). Here are three axioms that \(\succeq \) might satisfy.

  • Coordinate Monotonicity: For all \({\mathbf {X}},{\mathbf {Y}}\in {\mathcal {X}}\), if \(x_k^j\ge y_k^j\) for all \((j,k)\in {\mathcal {J}} \times {\mathcal {K}}\), then \({\mathbf {X}}\succeq {\mathbf {Y}}\). If, in addition, \(x_k^j>y_k^j\) for some \((j,k)\in {\mathcal {J}}\times {\mathcal {K}}\), then \({\mathbf {X}}\succ {\mathbf {Y}}\).

  • Row Preferences: For each column \(j\in {\mathcal {J}}\), \(\succeq \) is separable in \(\{j\}\times {\mathcal {K}}\).

  • Column Preferences: For all rows \(k\in {\mathcal {K}}\), \(\succeq \) is separable in \({\mathcal {J}}\times \{k\}\).

Define \(\succeq ^j\) and \(\succeq _k\) to be the conditional relations of \(\succeq \) on j and k, respectively. It follows from Row Preferences that the \(\succeq ^j\) are orderings on \({{\mathbb {R}}}^{{\mathcal {K}}}\), and from Column Preferences that the \(\succeq _k\) are orderings on \({{\mathbb {R}}}^{{\mathcal {J}}}\). Moreover, \(\succeq \) is increasing with respect to each of these conditional relations. The next two axioms force the conditional orders to be invariant.

  • Invariant Row Preferences: Row Preferences holds, and there is an ordering \(\succeq ^{{\mathcal {J}}}\) on \({\mathcal {Y}}^{{\mathcal {K}}}\) such that \(\succeq ^j= \succeq ^{{\mathcal {J}}}\) for all \(j\in {\mathcal {J}}\).

  • Invariant Column Preferences: Column Preferences holds, and there is an ordering \(\succeq _{{\mathcal {K}}}\) on \({\mathcal {Y}}^{ {\mathcal {J}}}\) such that \(\succeq _k=\)\(\succeq _{{\mathcal {K}}}\) for all \(k\in {\mathcal {K}}\).

These five axioms draw their use from the following proposition, which the proofs in “Appendix B” will repeatedly use. (Each of these proofs will involve two of the sets \({\mathcal {I}}\), \({\mathcal {S}}\), \({\mathcal {O}}\) taking the place of the abstract indexing sets \({\mathcal {J}}\) and \({\mathcal {K}}\).)

Proposition A2

  1. (a)

    Suppose a continuous preference order \(\succeq \) on \({\mathcal {X}}= \ {{\mathbb {R}}}^{{\mathcal {L}}}\) satisfies Coordinate Monotonicity, Row Preferences and Column Preferences. Then for all \(j\in {\mathcal {J}}\) and \(k\in {\mathcal {K}}\), there is an increasing, continuous function \(v_k^j:{{\mathbb {R}}\longrightarrow }{{\mathbb {R}}}\), such that \(\succeq \) is represented by the function \(W:{\mathcal {X}}{\ \longrightarrow }{{\mathbb {R}}}\) defined by: \(\displaystyle W({\mathbf {X}}) := \sum \nolimits _{k\in {\mathcal {K}}}\sum \nolimits _{j\in {\mathcal {J}} } v^j_k (x_k^j)\). In this representation, the functions \(v^j_k\) are unique up to PAT with a common multiplier.

  2. (b)

    Assume Invariant Column Preferences instead of Column Preferences, holding the other conditions the same as in part (a). Then, there is a strictly positive probability vector \({\mathbf {p}}\in \Delta _{ {\mathcal {K}}}\), and for all \(j\in {\mathcal {J}}\), there is an increasing, continuous function \(u^j:{{\mathbb {R}}\longrightarrow }{{\mathbb {R}}}\), such that \(\succeq \) is represented by the function \(W:{\mathcal {X}}{\ \longrightarrow }{{\mathbb {R}}}\) defined by: \(\displaystyle W({\mathbf {X}}) := \sum \nolimits _{k\in {\mathcal {K}}}\sum \nolimits _{j\in {\mathcal {J}} }p_k\,u^j(x_k^j) \). In this representation, the probability vector \({\mathbf {p}}\) is unique, and the functions \(u^j\) are unique up to PAT with a common multiplier.

  3. (c)

    Assume Invariant Row Preferences instead of Row Preferences, holding the other conditions the same as in part (b). Then, there is an increasing, continuous function \(u:{\mathbb {R}}\longrightarrow {\mathbb {R}}\) and strictly positive probability vectors \({\mathbf {q}}\in \Delta _{{\mathcal {J}}}\) and \({\mathbf {p}}\in \Delta _{{\mathcal {K}}} \) such that \(\succeq \) is represented by the function \(W:{\mathcal {X}}\longrightarrow {\mathbb {R}}\) defined by \(\displaystyle W({\mathbf {X}}) := \sum \nolimits _{k\in {\mathcal {K}}}\sum \nolimits _{j\in {\mathcal {J}} }q^j\,p_k\,u(x_k^j)\). In this representation, \({\mathbf {q}}\) and\({\ }\mathbf {\ p}\) are unique, and u is unique up to a PAT.

Proof

See Mongin and Pivato (2015). Part (a) follows from Proposition 1(b). Part (b) follows from Theorem 1(c,d), and part (c) from Corollary 1(c,d). The axioms of that paper are stated differently, because the domains considered there are not necessarily Cartesian products. \(\square \)

Proofs of the results of the paper

Our framework may seem to raise the possibility that conditional orderings depend on how they are induced; e.g., that \(\succsim _{so}\), as directly induced by \(\succsim \), differs from \(\succsim _{so}\), as induced by the ordering \(\succsim _s\) induced by \(\succsim \), or from \(\succsim _{so}\), as induced by the ordering \(\succsim _o\). But such a discrepancy cannot occur, as the different forms of conditionalization commute with one another (we skip the proof). In what follows, we use this property without saying.

Lemma B1

Let \(\succeq \) be a continuous order on \({{\mathbb {R}}}^{{\mathcal {I}}\times {\mathcal {S}} \times {\mathcal {O}}}\).

  1. (a)

    If \(\succeq \) induces interim preferences \(\succeq _s\) and \(\succeq _o\), then it induces ex post preferences \(\succeq _{so} \).

  2. (b)

    If, moreover, the interim preferences \(\succeq _o\) are invariant, then for any given s, \(\succeq _s\) induces invariant ex post preferences \(\succeq _{so}\).

  3. (c)

    If, moreover, the interim preferences \(\succeq _o\) and \(\succeq _s\) are both invariant, then the ex post preferences \(\succeq _{so} \) are invariant.

Proof

Let \((s,o)\in {\mathcal {S}}\times {\mathcal {O}}\). For all \(o\in {\mathcal {O}}\), let \({\mathcal {J}}_o:=\{(i',s',o)\); \(i'\in {\mathcal {I}}\) and \(s'\in {\mathcal {S}}\}\). Then, \({\mathcal {J}}_o\) is a \(\succeq \)-separable subset of \({\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\), because, by hypothesis, \(\succeq \) induces interim preferences \(\succeq _o\). Similarly, for all \(s\in {\mathcal {S}}\), let \({\mathcal {K}}_s:=\{(i',s,o')\); \(i'\in {\mathcal {I}}\) and \(o'\in {\mathcal {O}}\}\); this is a \(\succeq \)-separable subset of \({\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\), because \(\succeq \) induces interim preferences \(\succeq _s\). The nonempty intersection \({\mathcal {I}}_{so}:={\mathcal {J}}_o\cap {\mathcal {K}}_s\) is \(\succeq \)-separable by a classic result of Gorman (1968). Thus, \(\succeq \) induces ex post preferences \(\succeq _{so}\).

Adding the assumption that the interim preferences \(\succeq _o\) induced by \(\succeq \) are invariant, we fix s and consider any pair \(o\ne o^{\prime }\). By commutativity of conditionalization, we can regard the ex post preferences \(\succeq _{so}\) and \(\succeq _{so'}\) as being induced by \(\succeq _o\) and \(\succeq _{o'}\), respectively. But \(\succeq _o= \succeq _{o'}\), so that \(\succeq _{so}= \succeq _{so^{\prime }} \), and now regarding these ex post preferences as being induced by \(\succeq _s\), we conclude that this ordering induces invariant ex post preferences.

Now we add the assumption that the interim preferences \(\succeq _s\) induced by \(\succeq \) are invariant, fix o and consider any pair \(s\ne s'\). By symmetric reasoning, we conclude that \(\succeq _{so}= \succeq _{s'o}\). The two paragraphs together prove that, for all \(o,o'\in {\mathcal {O}}\) and \(s,s'\in {\mathcal {S}}\), \(\succeq _{so}= \succeq _{s'o'}\), meaning that \(\succeq \) induces invariant ex post preferences. \(\square \)

Proof of Proposition 1

Let \({\mathcal {J}}:={\mathcal {S}}\) and \({\mathcal {K}}:={\mathcal {O}}\). We will check which of the axioms of “Appendix A” apply to the ordering \(\succeq ^i\), for any \(i\in {\mathcal {I}}\). Coordinate Monotonicity holds because \(\succeq ^i\) induces preference orderings \(\succeq _{so}^i\) that coincide with the natural ordering of real numbers, by statement (1). As the \(\succeq _s^i\) (resp. the \(\succeq _o^i\)) are invariant, Invariant Row Preferences (resp. Invariant Column Preferences) holds. Thus, Proposition A2(c) yields the expected utility representation (2) for \(\succeq _i\). Since \(\succeq \) has a numerical representation that is increasing with the \(\succeq ^i\) by the ex ante Pareto principle, the social representation (3) follows. The uniqueness condition for F is obvious, and the other uniqueness statements follow from Proposition A2(c). \(\Box \)

Proof of Proposition 2

By Lemma B1(c), the assumption that \(\succeq \) induces invariant interim preferences of both kinds guarantees that \(\succeq \) also induces invariant ex post preferences \(\succeq _{\mathrm {xp}}\) on \({\mathbb {R}}^{{\mathcal {I}}}\). These preferences inherit the continuity of \(\succeq \) and the ex post Pareto principle makes them increasing in every coordinate. Thus, each of them is represented by a continuous and increasing function \(v:{\mathbb {R}}^{{\mathcal {I}}}\longrightarrow {\mathbb {R}}\).

To any \({\mathbb {X}}\in {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}}\), we associate the element \(\widetilde{{\mathbf {X}}} \in {\mathbb {R}}^{{\mathcal {S}}\times {\mathcal {O}}}\) whose (so) component is \({\widetilde{x}}_{so}:=v({\mathbf {x}}_{so})\). The function \(V: {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}}\rightarrow {\mathbb {R}}^{{\mathcal {S}}\times {\mathcal {O}}}\) defined by \(V({\mathbb {X}}):=\widetilde{{\mathbf {X}}}\) is continuous and increasing in each component. By these two properties, the image set \(\widetilde{{\mathcal {X}}}:=V({\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}})\) is a set of the form \({\mathcal {Y}}^{{\mathcal {S}}\times {\mathcal {O}}}\), where \({\mathcal {Y}}:=v({\mathbb {R}}^{{\mathcal {I}}})\) is an open interval.

Define an ordering \({\widetilde{\succeq }}\) on \(\widetilde{{\mathcal {X}}}\) by the condition that for all \(\widetilde{{\mathbf {X}}},\widetilde{{\mathbf {Y}}}\in \widetilde{{\mathcal {X}}}\), if \(\widetilde{{\mathbf {X}}}=V({\mathbb {X}})\) and \(\widetilde{{\mathbf {Y}}}=V({\mathbb {Y}})\), then

$$\begin{aligned} \widetilde{{\mathbf {X}}}\ {\widetilde{\succeq }}\ \widetilde{{\mathbf {Y}}}\ \text { if and only if }{\mathbb {X}}\succeq {\mathbb {Y}}. \end{aligned}$$
(B1)

(To see that \({\widetilde{\succeq }}\) is mathematically well defined by (B1), suppose \(V({\mathbb {X}})=\widetilde{{\mathbf {X}}}=V({\mathbb {X}}')\) for some \({\mathbb {X}},{\mathbb {X}}'\in {\mathcal {X}}\). Then for all \((s,o)\in {\mathcal {S}}\times {\mathcal {O}}\), we have \(v({\mathbf {x}}_{so})=v({\mathbf {x}}_{so}')\), and hence \({\mathbf {x}}_{so}\approx _{\mathrm {xp}}\ {\mathbf {x}}_{so}'\). Thus, \({\mathbb {X}}\approx {\mathbb {X}}'\), because \(\succeq \) is increasing relative to \(\succeq _{\mathrm {xp}}\).) In terms of “Appendix A,” putting \({\mathcal {J}}:={\mathcal {S}}\) and \({\mathcal {K}}:={\mathcal {O}}\), we conclude that \({\widetilde{\succeq }}\) is continuous and satisfies Invariant Row Preferences and Invariant Column Preferences, and Coordinate Monotonicity, by using the respective properties that \({\succeq }\) is continuous, induces invariant interim orderings \(\succeq _s\), and induces invariant interim orderings \(\succeq _o\), and induces invariant ex post orderings \(\succeq _{\mathrm {xp}}\). Thus, Proposition A2(c) yields strictly positive probability vectors \({\mathbf {p}}\in \Delta _{{\mathcal {S}}}\) and \({\mathbf {q}}\in \Delta _{{\mathcal {O}}}\), and a continuous increasing function \(u:{\mathbb {R}}\longrightarrow {\mathbb {R}}\), such that \({\widetilde{\succeq }}\) is represented by the function \({\widetilde{W}}:\widetilde{{\mathcal {X}}}\longrightarrow {\mathbb {R}}\) defined by \(\displaystyle {\widetilde{W}}(\widetilde{{\mathbf {X}}}):=\sum \nolimits _{s\in {\mathcal {S}}}\sum \nolimits _{o\in {\mathcal {O}}}\,q_o\,p_s\,u({\widetilde{x}}_{so}) \). Now set \(W_{\mathrm {xa}}({\mathbb {X}}):={\widetilde{W}}\circ V({\mathbf {X}})\) for all \({\mathbf {X}}\in {{\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}}}\), and \(W_{\mathbf {\ xp}}({\mathbf {x}}):=u\circ v({\mathbf {x}})\) for all \({\mathbf {x}}\in {\mathbb {R}}^{{\mathcal {I}}}\) to get the desired representations. The uniqueness properties are those of Proposition A2(c).Footnote 21\(\square \)

Proof of Proposition 3

First, we show that \(\succeq \) is increasing in every coordinate. Let \((i,s,o)\in {\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\). Statement (1) implies that \(\succeq ^i\) is increasing with respect to the coordinate \(x_{s,o}^i\). By the ex ante Pareto principle, \(\succeq \) is also increasing with respect to \(x_{s,o}^i\).

The result now follows from Theorem A2(b), by setting \({\mathcal {J}}:={\mathcal {I}}\) and \({\mathcal {K}}:={\mathcal {S}}\times {\mathcal {O}}\). Ex ante Pareto then becomes Row Preferences, while the existence of invariant ex post preferences yields Invariant Column Preferences. Meanwhile, \(\succeq \) satisfies Coordinate Monotonicity by the previous paragraph. \(\square \)

Proof of Theorem 1

First note that \(\succeq \) is increasing in every coordinate, by exactly the same argument as the first paragraph in the proof of Proposition 3. Next, since the \(\succeq ^i\) relations are orderings and the ex ante Pareto principle makes \(\succeq \) increasing with them, \(\succeq \) is separable in each \(i\in {\mathcal {I}}\). As \(\succeq \) induces interim preferences of both types, \(\succeq \) is also separable in each \(s\in {\mathcal {S}}\) and \(o\in {\mathcal {O}}\). It then follows from Proposition A1 that, for all \((i,s,o)\in {\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\), there exist continuous and increasing functions \(u_{so}^i:{\mathbb {R}}\longrightarrow {\mathbb {R}}\) such that \(\succeq \) is represented by the function \(W_{\mathrm {xa}} :{\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} W_{\mathrm {xa}}\ ({\mathbb {X}})\quad :=\quad \sum _{i\in {\mathcal {I}}}\sum _{s\in {\mathcal {S}}}\sum _{o\in {\mathcal {O}}}u_{so}^i(x_{so}^i). \end{aligned}$$
(B2)

Furthermore, the \(u_{so}^i\) are unique up to positive affine transformations (PAT) with a common multiplier. We can fix any \({\mathbb {Y}} \in {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}}\) and add constants to these functions so as to ensure that \(u_{so}^i(y_{so}^i)=0\) for all \((i,s,o)\in {\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\).Footnote 22 For convenience, fix some \({\overline{y}}\in {\mathbb {R}}\), and suppose that \(y_{so}^i={\overline{y}}\) for all \((i,s,o)\in {\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}}\).

For all \(i\in {\mathcal {I}}\), Eq. (B2) implies that the preference ordering \(\succeq ^i\) can be represented by the function \(U^i:{\mathbb {R}}^{{\mathcal {S}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} U^i({\mathbf {X}})\ \ :=\ \ \sum _{s\in {\mathcal {S}}}\sum _{o\in {\mathcal {O}}}u_{so}^i(x_{so}). \end{aligned}$$
(B3)

From Proposition 1, there are continuous increasing utility functions \({\tilde{u}}^i :{\mathbb {R}}\longrightarrow {\mathbb {R}}\), and two strictly positive probability vectors \({\mathbf {p}}^i\in \Delta _{{\mathcal {S}}}\) and \({\mathbf {q}}^i\in \Delta _{{\mathcal {O}}}\), such that \(\succeq ^i\) is represented by the function \(U^i: {\mathbb {R}}^{{\mathcal {S}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} U^i({\mathbf {X}})\ \ :=\ \ \sum _{s\in {\mathcal {S}}}\sum _{o\in {\mathcal {O}}}\,q_o^i\,p_s^i\,{\tilde{u}}^i(x_{so}). \end{aligned}$$
(B4)

Furthermore, in this representation, \({\mathbf {p}}^i\) and \({\mathbf {q}}^i\) are unique, and \({\tilde{u}}^i\) is unique up to PAT. By adding a constant, we ensure that \({\tilde{u}}^i({\overline{y}})=0\).

From the uniqueness property applied to (B3) and (B4), there exist constants \(\alpha ^i>0\) and \(\beta ^i\in {\mathbb {R}}\) such that:

$$\begin{aligned} u_{so}^i(x)=\alpha ^i\,q_o^i\,p_s^i\,{\tilde{u}}^i(x)+\beta ^i,\ \text {for all } (s,o)\in {\mathcal {S}}\times {\mathcal {O}}. \end{aligned}$$
(B5)

Substituting \(x={\overline{y}}\) into (B5) leads to \(\beta ^i=0\). Then substituting (B5) (for all \(i\in {\mathcal {I}}\)) into representation (B2) yields:

$$\begin{aligned} W_{\mathrm {xa}}\ ({\mathbb {X}})\ \ =\ \ \sum _{i\in {\mathcal {I}}}\sum _{s\in {\mathcal {S}}}\sum _{o\in {\mathcal {O}}}\alpha ^i\,q_o^i\,p_s^i\,{\tilde{u}}^i(x_{so}^i). \end{aligned}$$
(B6)

For given \(s\in {\mathcal {S}}\) in this representation, we obtain a representation \(V_s: {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) of the interim preference \(\succeq _s\) on \({\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\):

$$\begin{aligned} V_s({\mathbf {X}})\ \,=\ \ \sum _{i\in {\mathcal {I}}}\sum _{o\in {\mathcal {O}}}\alpha ^i\,q_o^i\,p_s^i\,{\tilde{u}}^i(x_o^i). \end{aligned}$$
(B7)

Let \({\mathbf {Y}}_s:=({\overline{y}},\ldots ,{\overline{y}})\in {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\); then \(V_s({\mathbf {Y}}_s)\ =0\).

Let us now put \({\mathcal {J}}:={\mathcal {I}}\) and \({\mathcal {K}}:={\mathcal {O}}\), and check which axioms in “Appendix A” the interim preference \(\succeq _s\) satisfies. This is a continuous ordering by the continuity of \(\succeq \). By representation (B7), \(\succeq _s\) is separable in each \(\left\{ i\right\} \times {\mathcal {O}}\) and each \({\mathcal {I}}\times \left\{ o\right\} \), and increasing in every coordinate, and thus satisfies Row Preferences, Column Preferences and Coordinate Monotonicity. As \(\succeq \) induces invariant \(\succeq _o\), Lemma B1(b) entails that the induced preferences \(\succeq _{so}\) are invariant, meaning that the stronger axiom of Invariant Column Preferences holds. Hence, Proposition A2(b) yields a strictly positive probability vector \({\mathbf {r}}_s\in \Delta _{{\mathcal {O}}} \), and for all \(i\in {\mathcal {I}}\), continuous, increasing utility functions \({\widehat{u}}_s^i:{\mathbb {R}}\longrightarrow {\mathbb {R}}\) such that \(\succeq _s\) is represented by the function \({{\widehat{V}}}_s:{\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} {\widehat{V}}_s({\mathbf {X}})\quad :=\quad \sum _{i\in {\mathcal {I}}}\sum _{o\in {\mathcal {O}}}\,r_{so}\, {\widehat{u}}_s^i(x_o^i). \end{aligned}$$
(B8)

In this representation, \({\mathbf {r}}_s\) is unique and \(\{ {\widehat{u}}_s^i\}_{i\in {\mathcal {I}}}\) are unique up to PAT with a common multiplier. Add constants to ensure that \({\widehat{u}}_s^i({\overline{y}}) =0\) for all \(i\in {\mathcal {I}}\). It follows that \({\widehat{V}}_s({\mathbf {Y}}_s) =0\).

From the uniqueness property applied to (B7) and (B8), there exist \(\gamma _s>0\) and \(\delta _s\in {\mathbb {R}}\) such that \({\widehat{V}}_s=\gamma _s\,V_s+\delta _s\). Substituting \({\mathbf {Y}}_s\) leads to \(\delta _s=0\). Since this holds for all \(s\in {\mathcal {S}}\), we can conclude that

$$\begin{aligned} \gamma _s\,r_{so}\,{\widehat{u}}_s^i\quad = \quad \alpha ^i\,q_o^i\,p_s^i\,{\tilde{u}}^i,\ \text {for all }(i,s,o)\in {\mathcal {I}}\times {\mathcal {S}}\times {\mathcal {O}} \end{aligned}$$
(B9)

Let us now fix i and s in these equations. All the coefficients are positive, and the increasing functions \({\widehat{u}}_s^i\) and \({\tilde{u}}^i\) are nonzero for some \(y^{*}\in {\mathbb {R}}\). Thus, we can derive the relations:

$$\begin{aligned} \frac{r_{so}}{q_o^i}\quad =\quad \frac{\alpha ^i\,p_s^i\,{\tilde{u}}^i(y^{*})}{\gamma _s\,{\widehat{u}}_s^i(y^{*})},\quad \text {for all} \ o\in {\mathcal {O}}. \end{aligned}$$
(B10)

The right-hand side of (B10) does not depend on o. Thus, the left-hand side must also be independent of o, which means that the vectors \({\mathbf {q}}^i\) and \({\mathbf {r}}_s\) are scalar multiples of one another. Thus, since they are probability vectors, we have \({\mathbf {q}}^i={\mathbf {r}}_s\). Since this holds for all i and s, we can drop the indexes. Denote the common probability vector by \({\mathbf {q}}\). Substituting \({\mathbf {q}}\) into (B6) and defining \(u^i:=\alpha ^i\,{\tilde{u}}^i\), we get formula (5) of the theorem. The other parts readily follow. \(\square \)

Proof of Theorem 2

For each \(i\in {\mathcal {I}}\), \(\succeq ^i\) satisfies the assumptions of Proposition 1. Thus, by the argument used to prove this proposition, we conclude that there exist a continuous increasing utility function \(u^i:{\mathbb {R}}\longrightarrow {\mathbb {R}}\), and strictly positive probability vectors \({\mathbf {p}}^i\in \Delta _{{\mathcal {S}}}\) and \({\mathbf {q}}^i\in \Delta _{{\mathcal {O}}}\), such that \(\succeq ^i\) is represented by the function \(U^i:{\mathbb {R}}^{{\mathcal {S}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) defined by \(\displaystyle U^i({\mathbf {X}}) := \sum \nolimits _{s\in {\mathcal {S}}}\sum \nolimits _{o\in {\mathcal {O}}}\,q_o^i\,p_s^i\,u^i(x_{so}) \), and the \(u^i\) are unique up to PAT. This establishes the SEU representation (7). Fix \({\overline{x}}\in {\mathbb {R}}\). By adding constants, we ensure that \(u^i( {\overline{x}})=0\) for all \(i\in {\mathcal {I}}\).

Meanwhile, Proposition 2 yields strictly positive probability vectors \(\mathbf {\ p}\in \Delta _{{\mathcal {S}}}\) and \({\mathbf {q}}\in \Delta _{{\mathcal {O}}}\), and a continuous increasing function \(W_{\mathrm {xp}}:{\mathbb {R}}^{{\mathcal {I}}}\longrightarrow {\mathbb {R}} \), such that \(\succeq \) is represented by the function \(W_{\mathrm {xa}}:{\mathcal {X}}\longrightarrow {\mathbb {R}}\) defined by \(\displaystyle W_{\mathrm {xa}}({\mathbb {X}}) := \sum \nolimits _{s\in {\mathcal {S}}}\sum \nolimits _{o\in {\mathcal {O}}}p_s\,q_o\,W_{\mathrm {xp}}({\mathbf {x}}_{so}) \), where \({\mathbf {p}}\) and \({\mathbf {q}}\) are unique, and \(W_{\mathrm {xp}}\) is unique up to PAT. This establishes the SEU representation (8). Let \(\overline{{\mathbf {x}}}:=({\overline{x}},\ldots , {\overline{x}})\). By adding a constant, we ensure that \(W_{\mathrm {xp}}(\overline{{\mathbf {x}}})=0\).

Now let \({\mathcal {J}}={\mathcal {I}}\) and \({\mathcal {K}}={\mathcal {O}}\) and consider how the axioms of “Appendix A” apply to \(\succeq _s\) for any given \(s\in {\mathcal {S}}\), recalling that these interim social preferences are well defined and invariant (i.e., independent of s). The objective interim Pareto principle makes \(\succeq _s\) separable in each \(i\in {\mathcal {I}}\), so that Row Preferences holds. By Proposition 2, the ex post social preferences \(\succeq _{so}\) are well defined and invariant, so that Invariant Column Preferences holds. Then, by Proposition A2(b), there exists a probability vector \(\widetilde{{\mathbf {q}}} \in \Delta _{{\mathcal {O}}}\), and for all \(i\in {\mathcal {I}}\), continuous increasing functions \(v^i\) such that \(\succeq _s\) are represented by the function \(W:{\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\longrightarrow {\mathbb {R}}\) defined by \(\displaystyle W({\mathbf {X}}) := \sum \nolimits _{ i\in {\mathcal {I}} }\sum \nolimits _{o\in {\mathcal {O}}}{\widetilde{q}}_o\, v^i(x_{so}^i) \), where \(\widetilde{{\mathbf {q}}}\) is unique and the \(v^i\) are unique up to PAT with a common multiplier. The same representation holds for all \(s\in {\mathcal {S}}\). Adding a constant, we ensure that \(v^i({\overline{x}})=0\) for all \(i\in {\mathcal {I}}\).

We now show that \({\mathbf {q}}=\widetilde{{\mathbf {q}}}\). By fixing \(s\in {\mathcal {S}}\) and applying the representation \(W_{\mathrm {xa}}\) to elements \({\mathbb {X}}\) whose components for \(s'\ne s\) are fixed at some values, we obtain a new representation for \(\succeq _s\) and reduce it to the representation just obtained in terms of W by the standard uniqueness property. That is, there exist constants \(\alpha >0\) and \(\beta \) such that \(\displaystyle \sum _{o\in {\mathcal {O}}}q_o\,W_{\mathrm {xp}}({\mathbf {x}}_o) = \alpha \sum _{ i\in {\mathcal {I}} }\sum _{o\in {\mathcal {O}}}{\widetilde{q}}_o\,v^i(x_o^i)+\beta \), for all \({\mathbf {X}}\in {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\). Substituting \(x_o^i={\overline{x}}\) for all \(i\in {\mathcal {I}}\) and \(o\in {\mathcal {O}}\) leads to \(\beta =0\). Now fixing o and putting \(x_{o^{\prime }}^i = {\overline{x}}\) for all \(o'\ne o\) leads to the equation: \(\displaystyle W_{\mathrm {xp}}({\mathbf {x}}_o) = \frac{{\widetilde{q}}_o}{q_o} \sum _{ i\in {\mathcal {I}} }\alpha \, v^i(x_{so}^i)\), for all \({\mathbf {x}}_o\in {\mathbb {R}}^{{\mathcal {I}}}\). Since this holds for all \(o\in {\mathcal {O}}\), the two probability vectors \({\mathbf {q}}\) and \(\widetilde{{\mathbf {q}}}\) are proportional, hence equal. Hence,

$$\begin{aligned} W_{\mathrm {xp}}({\mathbf {x}}_o)\quad =\quad \sum _{ i\in {\mathcal {I}} }\alpha \, v^i(x_{so}^i),\quad \text { for all }{\mathbf {x}}_o\in {\mathbb {R}}^{{\mathcal {I}}}. \end{aligned}$$
(B11)

and the invariant conditional preference \(\succeq _s\) is represented by the function \({\widetilde{W}}: {\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}} \longrightarrow {\mathbb {R}}\) defined by \(\displaystyle {\widetilde{W}}({\mathbf {X}}) := \sum _{ i\in {\mathcal {I}} }\sum _{o\in {\mathcal {O}}} q_o\,\alpha \, v^i(x_{so}^i)\). We now use a similar argument to show that \({\mathbf {q}}={\mathbf {q}}^i\) for all \(i\in {\mathcal {I}}\). Fixing \(i\in {\mathcal {I}}\) and \(s\in {\mathcal {S}}\), we can obtain a representation for the invariant interim preferences \(\succeq _s^i\) in two ways: first, from \({\widetilde{W}}\) by applying this representation to elements of \({\mathbb {R}}^{{\mathcal {I}}\times {\mathcal {O}}}\) whose components for \(i'\ne i\) are fixed at some values (because \(\succeq _s\) satisfies the objective interim Pareto principle), and second, from \(U^i\) by applying this representation to elements of \({\mathbb {R}}^{{\mathcal {S}}\times {\mathcal {O}}}\) whose components for \(s'\ne s\) are fixed at some values. By the standard uniqueness property, there exist \(\gamma _s^i>0\) and \(\delta _s^i\) such that

$$\begin{aligned} \sum _{o\in {\mathcal {O}}}q_o\alpha v^i(x_o)\quad =\quad \gamma _s^i\sum _{o\in {\mathcal {O}}}\,q_o^i\,p_s^i\,u^i(x_o)+\delta _s^i,\quad \text {for all } {\mathbf {x}}\in {\mathbb {R}}^{{\mathcal {O}}}. \end{aligned}$$
(B12)

Substituting \(x_o={\overline{x}}\) into (B12) leads to \(\delta _s^i=0\). Fix \(o\in {\mathcal {O}}\). Put \(x_{o'} = {\overline{x}}\) for all \(o'\ne o\) yields:

$$\begin{aligned} \frac{q_o}{q_o^i}\alpha v^i(x)\quad =\quad \gamma _s^i\,p_s^i\,u^i(x)\text { for all }x\in {\mathbb {R}}. \end{aligned}$$
(B13)

The right-hand side of (B13) is independent of o. Thus, the probability vectors \({\mathbf {q}}\) and \({\mathbf {q}}^i\) are proportional, hence equal, and thus

$$\begin{aligned} \alpha v^i(x)\quad =\quad \gamma _s^i\,p_s^i\,u^i(x)\text { for all }x\in {\mathbb {R}}. \end{aligned}$$
(B14)

Equation (B14) holds for all \(s\in {\mathcal {S}}\). Hence, for all \(i\in {\mathcal {I}}\), the product \(r^i:=\gamma _s^i\,p_s^i\) is independent of s; note that \(r^i>0\). Equation (B14) now says \(\alpha v^i=r^i\,u^i\). Substituting this into representation (B11) yields representation (9) for \(\succeq _{\mathrm {xp}}\). \(\square \)

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Mongin, P., Pivato, M. Social preference under twofold uncertainty. Econ Theory 70, 633–663 (2020). https://doi.org/10.1007/s00199-019-01237-0

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Keywords

  • Ex ante social welfare
  • Ex post social welfare
  • Objective versus subjective uncertainty
  • Objective versus subjective probability
  • Pareto principle
  • separability
  • Harsanyi Social Aggregation theorem
  • Spurious unanimity
  • Complementary ignorance

JEL Classification

  • D70
  • D81