Probabilistic opinion pooling generalized. Part two: the premise-based approach

How can several individuals’ probability functions on a given \(\sigma \)-algebra of events be aggregated into a collective probability function? Classic approaches to this problem usually require ‘event-wise independence’: the collective probability for each event should depend only on the individuals’ probabilities for that event. In practice, however, some events may be ‘basic’ and others ‘derivative’, so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize this idea by introducing a ‘premise-based’ approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the ‘premises’.

1. The correlation might be due to causal effects between, or common causes of, rain and heat.

2. One could construct basic events from non-basic events, using the operations of negation and disjunction. Formally, while the basic events typically form a generating system of the \(\sigma \)-algebra, there exist many alternative generating systems, and usually none of them is canonical in a technical sense. The task of determining the ‘basic’ events therefore involves some interpretation and may be context-dependent and open to disagreement. One might, however, employ a syntactic criterion which counts an event as ‘basic’ if, in a suitable language (perhaps one deemed ‘natural’), it can be expressed by an atomic sentence (one that is not a combination of other sentences using Boolean connectives). An event expressible by the sentence ‘it will rain or it will snow’ would then count as non-basic. This syntactic criterion relies on our choice of language, which, though not a purely technical matter, is arguably not ad hoc. An n-place connective (e.g., the two-place connective ‘or’) is called Boolean or truth-functional if the truth-value of every sentence constructed by applying this connective to n other sentences is determined by the truth values of the latter sentences. For instance, ‘or’ is Boolean since ‘p or q’ is true if and only if ‘p’ is true or ‘q’ is true. Many languages, especially ones that mimic natural language, contain non-Boolean connectives, for instance non-material conditionals for which the truth-value of ‘if p then q’ is not always determined by the truth-values of p and q. When the sentence ‘if p then q’ is not truth-functionally decomposable, an event represented by it would count as ‘basic’ under the present syntactic criterion. The sentence ‘\({\hbox {CO}_{2}}\) emissions cause global warming’ can be viewed as the non-material (specifically, causal) conditional ‘if p then q’, hence would describe a basic event. See Priest (2001) for an introduction to non-classical logic.

3. The terminology ‘premise’ is still justified, though not in the sense of ‘premise of individual probability assignment’ (since we no longer assume that premises are basic in the individuals’ formation of probabilistic beliefs), but in the sense of ‘premise of collective probability assignment’ (because the collective probabilities for these events are determined independently of the probabilities of other events and then constrain other collective probabilities).

4. Historically, linear pooling goes back at least to Stone (1961). Linear pooling is by no means the only plausible way to aggregate subjective probabilities. Other approaches include geometric and, more generally, externally Bayesian pooling (e.g., McConway 1978; Genest 1984b; Genest et al. 1986; Russell et al. 2015; Dietrich 2016), multiplicative pooling (Dietrich 2010; Dietrich and List 2016), supra-Bayesian pooling (e.g., Morris 1974), and pooling of ordinal probabilities (Weymark 1997). For literature reviews, see Genest and Zidek (1986), Clemen and Winkler (1999) and Dietrich and List (2016).

5. These auxiliary assumptions might be given exogenously; or they might be determined endogenously based on the experts’ probability functions (e.g., based on how dependent or independent the conjuncts are according to these probability functions).

6. Equivalently, one can demand the preservation of any unanimously assigned probability 0.

7. In Part I, we make the opposite move of extending consensus preservation to events outside the agenda, i.e., we extend it to events constructible from events in the agenda using conjunction (intersection), disjunction (union), or negation (complementation). In the present paper, there is no point in extending consensus preservation to other events, since there are no events outside the agenda constructible from events in it (as a \(\sigma \)-algebra, the agenda is closed under the relevant operations).

8. We are indebted to Richard Bradley for suggesting this formulation of the requirement.

9. For those special profiles of individual probability functions for which the collective probabilities for \((-\infty ,-1]\) and \((-\infty ,+1]\) coincide or one of them is zero or one, there is no such normal distribution. A different, non-normal extension must then be used.

10. The finiteness assumption in Theorems 1(a),  1(b),  2(a),  2(b),  3(a),  4(a),  4(b),  5(a), and  6(a) could be replaced by the assumption that the \(\sigma \)-algebra generated by X is \(\Sigma \) (rather than a sub-\(\sigma \)-algebra of \(\Sigma \)). It might be that some of these finiteness assumptions (or their substitutes)—especially in Theorems 1(b),  2(b), and  4(b)—could be dropped.

11. For example, for every event of the form \(A=(-\infty ,\omega ]\), we might use the decision criterion defined by \(D_{A}(t_{1},\ldots ,t_{n} )=(\frac{1}{n}t_{1}+\cdots +\frac{1}{n}t_{n})^{2}\), and for every event of the form \(A=(\omega ,\infty )\), we might use the decision criterion defined by \(D_{A}(t_{1},\ldots ,t_{n})=1-(\frac{1}{n}(1-t_{1})+\cdots +\frac{1}{n} (1-t_{n}))^{2}\).

12. To be precise, path-connectedness implies non-simplicity as long as \(X\ne \{\varnothing ,\Omega \}\).

13. In this proposition, we assume that the agenda \(\Sigma \) is not very small, i.e., contains more than \(2^{3}=8\) events (e.g., \(\Sigma =2^{\Omega }\) with \(\left| \Omega \right| >3\)). Note that, as \(\Sigma \) is a \(\sigma \)-algebra, it has the size \(2^{k}\) for some \(k\in \{1,2,3,\ldots \}\) or is infinite.

14. We require no restriction to a finite \(\Sigma \), as observed in footnote 10.

15. In this case, each opinion function in \({\mathcal {P}}_{X}\) is extendable not just to a probability function on \(\sigma (X)\), but also to one on \(\Sigma \). Probability theorists will be aware that the extendability of a probability function to a larger \(\sigma \)-algebra cannot be taken for granted.

16. In that proof it suffices to choose the \(Q_{A}\)s appropriately, since each \(Q_{A}\) equals \(P(\cdot |A)\), provided \(P(A)\ne 0\).

17. That proof took the four mentioned events to be singleton, but nothing depends on this.

Authors and Affiliations

1. Paris School of Economics and CNRS, Paris, France

   Franz Dietrich

2. London School of Economics, London, UK

   Christian List

Corresponding author

Correspondence to Christian List.

We are grateful to the referees and the editor for very helpful and detailed comments. Although we are jointly responsible for this work, Christian List wishes to note that Franz Dietrich should be considered the lead author, to whom the credit for the present mathematical proofs is due. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. Both papers build on our earlier, unpublished paper ‘Opinion pooling on general agendas’ (September 2007). Dietrich was supported by a Ludwig Lachmann Fellowship at the LSE and the French Agence Nationale de la Recherche (ANR-12-INEG-0006-01). List was supported by a Leverhulme Major Research Fellowship (MRF-2012-100) and a Harsanyi Fellowship at the Australian National University, Canberra.

Appendix A: Proofs

We now give all proofs. In Sect. A.1, we prove parts (a) of all our theorems by reducing them to results in Part I. In Sects. A.2, A.3, A.4, and A.5, we prove parts (b) of Theorems 1, 3, 4, and 5. Parts (b) of Theorems 2 and  6 require no separate proofs: Theorem 2(b) follows from Theorem 1(b) (since consensus preservation implies conditional consensus preservation on X by Proposition 1) and Theorem 6(b) follows from Theorem 3(b) (since non-neutrality on X implies non-linearity on X). In Sect. A.6, we prove Proposition 2.

1.1 A.1 Proof of part (a) of each theorem

We now prove Theorems 1(a) to  6(a). To do so, we first relate premise-based opinion pooling to opinion pooling on a general agenda as introduced in Part I. We begin by generalizing the present paper’s framework to agendas that need not be \(\sigma \)-algebras. In general, an agenda is a non-empty set X of events (each of which is of the form \(A\subseteq \Omega \)), where X is closed under complementation (i.e., \(A\in X\Leftrightarrow A^{c}\in X \)). It contains the events on which opinions are formed. Given an agenda X, an opinion function is a function \(P:X\rightarrow [0,1]\) which is coherent, i.e., extendable to a probability function on the \(\sigma \)-algebra \(\sigma (X)\) generated by X (i.e., the smallest \(\sigma \)-algebra which includes X, constructible by closing X under countable unions and complements). Let \({\mathcal {P}}_{X}\) be the set of all opinion functions for agenda X. If X is a \(\sigma \)-algebra, \({\mathcal {P}}_{X}\) consists of all probability functions on X, in line with the notation used above. An opinion pooling function for agenda X is a function \({\mathcal {P}} _{X}^{n}\rightarrow {\mathcal {P}}_{X}\) which assigns to each profile \((P_{1},\ldots ,P_{n})\) of individual opinion functions a collective opinion function, usually denoted \(P_{P_{1},\ldots ,P_{n}}\). We call the pooling function linear and neutral, respectively, if it is linear and neutral on X in line with the definition above.

Crucially, a pooling function for a \(\sigma \)-algebra \(\Sigma \) induces new pooling functions for any sub-agendas X on which it is independent. Formally, a pooling function \(F:{\mathcal {P}}_{\Sigma }^{n}\rightarrow {\mathcal {P}}_{\Sigma }\) for agenda \(\Sigma \) is said to induce the pooling function \(F^{\prime }:{\mathcal {P}}_{X}^{n}\rightarrow {\mathcal {P}}_{X}\) for (sub-)agenda X if F and \(F^{\prime }\) generate the same collective opinions within X, i.e.,

(and if, in addition, \({\mathcal {P}}_{X}=\{P|_{X}:P\in {\mathcal {P}}_{\Sigma }\}\), where this addition holds automatically whenever X is finite or \(\sigma (X)=\Sigma \)).^{Footnote 15} Our axiomatic requirements on a pooling function for agenda \(\Sigma \)—i.e., independence on a sub-agenda X and various consensus requirements—should be compared with the following requirements on a pooling function for the agenda X (introduced and discussed in Part I). The first two requirements are unrestricted versions of independence and consensus preservation:

Independence For each event \(A\in X\), there exists a function \(D_{A}:[0,1]^{n}\rightarrow [0,1]\) (the local pooling criterion for A) such that \(P_{P_{1},\ldots ,P_{n}}(A)=D_{A}(P_{1} (A),\ldots ,P_{n}(A))\) for any \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{X}\).

Consensus preservation For all \(A\in X\) and \(P_{1} ,\ldots ,P_{n}\in {\mathcal {P}}_{X}\), if \(P_{i}(A)=1\) for all individuals i, then \(P_{P_{1},\ldots ,P_{n}}(A)=1\).

Note the following criterion for the existence of induced pooling functions:

Lemma 2

(cf. Part I, Lemma 14) If a pooling function for a \(\sigma \)-algebra \(\Sigma \) is independent on a sub-agenda X (where X is finite or \(\sigma (X)=\Sigma \)), then it induces a pooling function for agenda X.

The next two axiomatic requirements are two different extensions of consensus preservation, namely to either implicitly revealed or unrevealed beliefs. An individual i’s explicitly revealed beliefs are given by the individual’s submitted opinion function \(P_{i}\). Her implicitly revealed beliefs are given by the probabilities of events in \(\sigma (X)\backslash X\) which are implied by her explicitly revealed beliefs, i.e., hold under every extension of \(P_{i}\) to a probability function on \(\sigma (X)\). If, for instance, \(P_{i}\) assigns probability 1 to \(A\in X\), then the agent implicitly reveals certainty of all events \(B\supseteq A\) in \(\sigma (X)\backslash X\). The following axiom extends consensus preservation to implicitly revealed beliefs:

Implicit consensus preservation       For all \(A\in \sigma (X)\) and all \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{X}\), if each \(P_{i}\) implies certainty of A (i.e., \(\overline{P}_{i}(A)=1\) for every extension \(\overline{P}_{i}\) of \(P_{i}\) to a probability function on \(\sigma (X)\)), then so does \(P_{P_{1} ,\ldots ,P_{n}}\).

By contrast, individual i’s unrevealed beliefs are any probabilistic beliefs which she privately holds relative to events in \(\sigma (X)\backslash X\) and which cannot be inferred from the submitted opinion function \(P_{i}\) because different extensions of \(P_{i}\) assign different probabilities to the events in question. The following axiom requires the collective opinion function to be compatible with any unanimously held certainty of an event—including any unrevealed certainty, which is not implied by the submitted opinion functions but is consistent with them. This ensures that no consensus (not even an unrevealed consensus) is ever overruled.

Consensus compatibility      For all \(A\in \sigma (X)\) and all \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{X}\), if each \(P_{i}\) is consistent with certainty of A (i.e., \(\overline{P}_{i}(A)=1\) for some extension \(\overline{P}_{i}\) of \(P_{i}\) to a probability function on \(\sigma (X)\)), then so is \(P_{P_{1},\ldots ,P_{n}}\).

A final requirement pertains to conditional beliefs. Note that, based on individual i’s opinion function \(P_{i}\), the conditional belief \(P_{i}(A|B)=P_{i}(A\cap B)/P_{i}(B)\) of one agenda event A given another B (where \(P_{i}(B)\ne 0\)) may be undefined, since we may have \(A\cap B\not \in X\) so that \(P_{i}(A\cap B)\) is undefined. Hence, if the agent happens to be privately certain of A given B, then this conditional certainty may be unrevealed. Our axiom of conditional consensus compatibility requires that any (possibly unrevealed) unanimous conditional certainty should not be overruled. In fact, we require something subtly stronger: any set of (possibly unrevealed) unanimous conditional certainties should not be overruled (see Part I for details).

Conditional consensus compatibility      For all \(P_{1} ,\ldots ,P_{n}\in {\mathcal {P}}_{X}\), and all finite sets S of pairs (A, B) of events in X, if every opinion function \(P_{i}\) is consistent with certainty of A given B for all (A, B) in S (i.e., some extension \(\overline{P}_{i}\) of \(P_{i}\) to a probability function on \(\sigma (X)\) satisfies \(\overline{P}_{i}(A|B)=1\) for all pairs \((A,B)\in S\) such that \(P_{i}(B)\ne 0\)), then so is the collective opinion function \(P_{P_{1},\ldots ,P_{n}}\).

The following lemma shows how properties of a pooling function for a \(\sigma \)-algebra translate into corresponding properties of an induced pooling function for a sub-agenda:

Lemma 3

(cf. Part I, Lemma 12) Suppose pooling function F for \(\sigma \)-algebra \(\Sigma \) induces pooling function \(F^{\prime }\) for sub-agenda X (where X is finite or \(\sigma (X)=\Sigma \)). Then:

• \(F^{\prime }\) is independent (respectively neutral, linear) if and only if F is independent (respectively neutral, linear) on X,

• \(F^{\prime }\) is consensus-preserving if and only if F is consensus-preserving on X,

• \(F^{\prime }\) is consensus-compatible if F is consensus-preserving,

• \(F^{\prime }\) is conditional-consensus-compatible if F is conditional-consensus-preserving on X.

This lemma follows from a more general result:

Lemma 4

(cf. Part I, Lemma 13) Consider a \(\sigma \)-algebra \(\Sigma \) and a sub-agenda X (where X is finite or \(\sigma (X)=\Sigma \)). Any pooling function for X is

1. (a)

   induced by some pooling function for agenda \(\Sigma \),

2. (b)

   independent (respectively neutral, linear) if and only if every inducing pooling function for agenda \(\Sigma \) is independent (respectively neutral, linear) on X, where ‘every’ can be replaced by ‘some’,

3. (c)

   consensus-preserving if and only if every inducing pooling function for agenda \(\Sigma \) is consensus-preserving on X, where ‘every’ can be replaced by ‘some’,

4. (d)

   consensus-compatible if and only if some inducing pooling function for agenda \(\Sigma \) is consensus-preserving,

5. (e)

   conditional-consensus-compatible if and only if some inducing pooling function for agenda \(\Sigma \) is conditional-consensus-preserving on X

(where in (d) and (e) the ‘only if’ claim assumes that X is finite).

Proof of parts (a) of Theorems 1–6

Using the above translation machinery, one can reduce Theorem 1(a) to Part I’s Theorem 1(a), Theorem 2(a) to Part I’s Theorem 2(a), and so on until Theorem 6(a). Since the reduction is analogous for each theorem, we only spell it out for Theorem 1. Let X be a non-nested finite sub-agenda of the \(\sigma \)-algebra agenda \(\Sigma \), and let \(F:{\mathcal {P}}_{\Sigma }^{n}\rightarrow {\mathcal {P}}_{\Sigma }\) be independent on X and (globally) consensus preserving. By Lemma 2, F induces a pooling function \(F^{\prime }\) for agenda X, which is independent and consensus-compatible by Lemma 3, hence neutral by Part I’s Theorem 1(a). So F is neutral on X by Lemma 3. \(\square \)

1.2 A.2 Proof of Theorem 1(b)

We now write \(\mathbf {1}\) and \(\mathbf {0}\) for the n-dimensional vectors \((1,\ldots ,1)\) and \((0,\ldots ,0)\), respectively. We draw on a measure-theoretic fact:

Lemma 5

(cf. Part I, Lemma 15) Every probability function on a finite sub-\(\sigma \)-algebra of \(\sigma \)-algebra \(\Sigma \) can be extended to a probability function on \(\Sigma \).

Proof of Theorem 1(b)

Consider a finite nested sub-agenda \(X\ne \{\varnothing ,\Omega \}\) of the \(\sigma \)-algebra agenda \(\Sigma \). (As will become clear, finiteness could be replaced by the assumption that \(\sigma (X)=\Sigma \). Under this alternative assumption, the ‘Claim’ below can be skipped, and the rest of the proof remains almost unaffected.) We construct a pooling function \((P_{1},\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n}}\) for agenda \(\Sigma \) with all relevant properties. Without loss of generality, let \(\varnothing ,\Omega \in X\).

Claim. If Theorem 1(b) holds in the case that \(\sigma (X)=\Sigma \), then it holds in general.

Let Theorem 1(b) hold in the special case. Let \(\Sigma ^{\prime }:=\sigma (X) \) (\(\subseteq \Sigma \)). By assumption, there is a pooling function \(F^{\prime }:{\mathcal {P}}_{\Sigma ^{\prime }}^{n}\rightarrow {\mathcal {P}}_{\Sigma ^{\prime }}\) with all relevant properties. Let \(\mathcal {A}\) be the set of atoms of the (finite) \(\sigma \)-algebra \(\Sigma ^{\prime }\). We define \(F:{\mathcal {P}}_{\Sigma }^{n}\rightarrow {\mathcal {P}}_{\Sigma }\) as follows. Consider \(P_{1},\ldots ,P_{n} \in {\mathcal {P}}_{\Sigma }\). Let \(P^{\prime }:=F^{\prime }(P_{1}|_{\Sigma ^{\prime } },\ldots ,P_{n}|_{\Sigma ^{\prime }})\). For all \(A\in \mathcal {A}\) such that \(P^{\prime }(A)\ne 0\), there is an individual \(i_{A}\) such that \(P_{i_{A} }(A)\ne 0\), since otherwise everyone assigns probability one to \(\Omega \backslash A\) while \(P^{\prime }(\Omega \backslash A)\ne 1\), violating consensus-preservation. By Lemma 5, \(P^{\prime }\) can be extended to a probability function P on \(\Sigma \). As is clear from that lemma’s proof (in Part I), we may assume without loss of generality that^{Footnote 16}

Now let \(F(P_{1},\ldots ,P_{n})\) be this P. It remains to show that the pooling function F just defined inherits all relevant properties from \(F^{\prime }\). This is clear for independence on X and non-neutrality on X. To show that F is (globally) consensus-preserving, consider \(B\in \Sigma \) and \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\) such that \(P_{1}(B)=\cdots =P_{n}(B)=1\). To show that \(P(B)=1\), where \(P:=F(P_{1},\ldots ,P_{n})\), note first that \(P(B)=\sum _{A\in {\mathcal {A}}:P(A)\ne 0}P(B|A)P(A)\). Here (in the notation above) each P(B|A) equals \(P_{i_{A}}(B|A)\), which equals 1 as \(P_{i_{A} }(B)=1\). So \(P(B)=\sum _{A\in {\mathcal {A}}:P(A)\ne 0}P(A)=1\). This proves the claim.

Now let \(\sigma (X)=\Sigma \), drawing on the above ‘Claim’. As X is nested, we may express it as \(X=\{A,A^{c}:A\in X_{+}\}\) for some subset \(X_{+}\subseteq X\) which is linearly ordered and contains both \(\varnothing \) and \(\Omega \).

As an ingredient of our construction, we consider any pooling function for agenda \(\Sigma \) which is neutral (at least) on X and consensus-preserving and whose pooling criterion on X, denoted \(D:[0,1]^{n}\rightarrow [0,1]\), is at least weakly increasing in each argument. (For instance, we might use dictatorship by individual 1, given by \((P_{1},\ldots ,P_{n})\mapsto P_{1}\), with pooling criterion given by \(D(t_{1},\ldots ,t_{n})=t_{1}\).) As \(X\ne \{\varnothing ,\Omega \}\), there is some \(A\in X\backslash \{\Omega ,\varnothing \}\). As \(A\ne \Omega ,\varnothing \), there are \(P_{1},\ldots ,P_{n} \in {\mathcal {P}}_{\Sigma }\) which all assign probability 1 / 2 to A (hence to \(A^{c}\)), so that the collective probabilities of A and of \(A^{c}\) are each given by \(D(1/2,\ldots ,1/2)\). As these probabilities sum to 1, it follows that

We now transform this pooling function, which is neutral on X, into a pooling function \((P_{1},\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n}}\) which is non-neutral on X, but still independent on X and consensus-preserving. To do so, we consider a function \(T:[0,1]\rightarrow [0,1]\) such that (i) \(T(1/2)\ne 1/2\), (ii) \(T(0)=0\) and \(T(1)=1\), (iii) T is at least weakly increasing, and (iv) T is Lipschitz continuous, i.e., there is a \(K>0\) such that \(\left| T(x)-T(y)\right| \le K\left| x-y\right| \) for all \(x,y\in [0,1]\). (T could be defined by \(T(x)=\min \{2x,1\}\).) Now consider any \( P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\). We have to define \(P_{P_{1},\ldots ,P_{n}}\). We write P for the result of applying the neutral pooling function to \((P_{1},\ldots ,P_{n})\). To anticipate, our definition will imply that

As a first step towards our definition, we define \(P_{P_{1},\ldots ,P_{n}}\) on the subdomain

The restriction of \(P_{P_{1},\ldots ,P_{n}}\) to \({\widetilde{X}}\), to be denoted g, is defined as follows. Each \(C\in {\widetilde{X}}\) is uniquely representable as \(C=B\backslash A\) with \(A,B\in X_{+}\) and \(A\subseteq B\) (and \(A=B=\varnothing \) if \(C=\varnothing \)), and we let

It follows that

because, firstly, each \(C\in X_{+}\) can be written as \(C\backslash \varnothing \) where \(C,\varnothing \in X_{+}\), and, secondly, each \(C\in X\backslash X_{+}\) can be written as \(\Omega \backslash C^{c}\) where \(\Omega ,C^{c}\in X_{+}\) and where \(T(P(\Omega ))=T(1)=1\).

Note that \({{\widetilde{X}}}\) is a semi-ring on \(\Omega \), since (i) \(\varnothing \in {\widetilde{X}}\), (ii) \(C,C^{\prime }\in {\widetilde{X}}\Rightarrow C\cap C^{\prime }\in {\widetilde{X}}\), and (iii) for all \(C,C^{\prime } \in {\widetilde{X}}\), the difference \(C\backslash C^{\prime }\) is a union of finitely many—in fact, at most two—events in \({\widetilde{X}}\). We next show that the function g on this semi-ring is \(\sigma \)-additive. First, g is finitely additive, i.e., for all disjoint \(C_{1},C_{2}\in {\widetilde{X}} \), if \(C_{1}\cup C_{2}\in {\widetilde{X}}\), then \(g(C_{1}\cup C_{2} )=g(C_{1})+g(C_{2})\), by definition of g and additivity of P. To show \(\sigma \)-additivity, consider pairwise disjoint \(C_{1},C_{2},\ldots \in {\widetilde{X}}\) such that \(\cup _{m=1}^{\infty }C_{m}\in {\widetilde{X}}\). We have to show that

For all \(M\in \{1,2,\ldots \}\), note that the difference \(\left( \cup _{m=1}^{\infty }C_{m}\right) \backslash \left( \cup _{m=1}^{M}C_{m}\right) =\cup _{m=M+1}^{\infty }C_{m}\) need not belong to \({\widetilde{X}}\), but can be partitioned into a finite set \(\mathcal {C}^{M}\) of events in \({\widetilde{X}}\) (as \(\cup _{m=1}^{\infty }C_{m}\) belongs the the semi-ring \({\widetilde{X}}\)). So, \(\mathcal {C}^{M}\cup \{C_{1},\ldots ,C_{M}\}\) partitions \(\cup _{m=1}^{\infty }C_{m} \). Careful inspection of g’s definition reveals that \(\delta _{M}=\sum _{C\in \mathcal {C}^{M}}g(C)\). So, as \(g(C)\le KP(C)\) for each \(C\in {\widetilde{X}}\) (by definition of g and property (iv) of T), \(\delta _{M}\le K\sum _{C\in \mathcal {C}^{M}}P(C)=KP(\cup _{m=M+1}^{\infty }C_{m})\). As \(M\rightarrow \infty \) we have \(P(\cup _{m=M+1}^{\infty }C_{m})\rightarrow 0\) (by \(\sigma \)-additivity of P), and so \(\delta _{M}\rightarrow 0\), as required.

As g is non-negative, \(\sigma \)-additive, and also \(\sigma \)-finite (i.e., \(\Omega \) is a union of countably many events in \({\widetilde{X}}\) of finite g-measure, which trivially holds as \(\Omega \in {\widetilde{X}}\)), Caratheodory’s Extension Theorem tells us that g extends uniquely to a measure on \(\sigma ({\widetilde{X}})=\sigma (X)=\Sigma \). Let \(P_{P_{1},\ldots ,P_{n}}\) be this extension. \(P_{P_{1},\ldots ,P_{n}}\) is indeed a probability function since \(P_{P_{1},\ldots ,P_{n}}(\Omega )=1\) as \(\Omega \in {\widetilde{X}}\) and \(g(\Omega )=T(1)=1\).

Finally, we must prove that the pooling function \((P_{1},\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n}}\), as just defined, is independent on X, (globally) consensus-preserving, and non-neutral on X.

Independence on X This holds because, for all \(P_{1},\ldots ,P_{n} \in {\mathcal {P}}_{\Sigma }\), the function \(P_{P_{1},\ldots ,P_{n}}\) extends g, which satisfies (2). Note that the pooling criterion \(D_{C}\) for \(C\in X_{+}\) is defined as \(T\circ D\), while the pooling criterion \(D_{C}\) for \(C\in C\backslash X_{+}\) is defined by \(\mathbf {t}\mapsto 1-T\circ D(\mathbf {1} -\mathbf {t})\).

Non-neutrality on X Here it suffices to show that, for some \(C\in X\backslash \{\Omega ,\varnothing \}\), the pooling criteria \(D_{C}\) and \(D_{C^{c}}\) differ. This follows from the following argument. First, \(X\backslash \{\Omega ,\varnothing \}\ne \varnothing \) as \(X\ne \{\varnothing ,\Omega \}\). So we may pick \(C,C^{c}\in X\backslash \{\Omega ,\varnothing \}\); say, assume \(C\in X_{+}\) and \(C^{c}\in X\backslash X_{+}\). So, as just shown, \(D_{C}=T\circ D\) and \(D_{C^{c}}=1-T\circ D(\mathbf {1}-\cdot )\). Hence \(D_{C}\ne D_{C^{c}}\), since \(D_{C}(1/2,\ldots ,1/2)\ne D_{C^{c}}(1/2,\ldots ,1/2)\), as is clear from the fact that \(T(1/2)\ne 1/2\) and that

Consensus preservation Let \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\) and \(A\in \Sigma \) such that \(P_{1}(A)=\cdots =P_{n}(A)=1\). We show that \(P_{P_{1},\ldots ,P_{n}}(A)=1\). Let P be the result of pooling \(P_{1},\ldots ,P_{n}\) using the (at least on X) neutral pooling function defined above. As that pooling function is consensus-preserving, \(P(A)=1\). It suffices to show that \(P_{P_{1},\ldots ,P_{n}}\le KP\), as this implies that \(P_{P_{1},\ldots ,P_{n}} (A^{c})\le KP(A^{c})=K(1-P(A))=K(1-1)=0\), so that \(P_{P_{1},\ldots ,P_{n}}(A)=1\). Now, to show that \(P_{P_{1},\ldots ,P_{n}}\le KP\), note first that, by property (iv) of T, \(g\le KP|_{{\widetilde{X}}}\), and so \(KP|_{{\widetilde{X}}}-g\ge 0\). Since g and \(KP|_{{\widetilde{X}}}\), and hence also \(KP|_{{\widetilde{X}}}-g\), are \(\sigma \)-additive, \(\sigma \)-finite and non-negative functions on the semi-ring \({\widetilde{X}}\), each of them extends uniquely to a measure on \(\sigma ({\widetilde{X}})=\Sigma \) by Caratheodory’s Extension Theorem. The first two extensions are \(P_{P_{1},\ldots ,P_{n}}\) and KP, respectively. So the third one must be \(KP-P_{P_{1},\ldots ,P_{n}}\). Hence \(KP-P_{P_{1},\ldots ,P_{n}}\ge 0\), and thus \(P_{P_{1},\ldots ,P_{n}}\le KP\). \(\square \)

1.3 A.3 Proof of Theorem 3(b)

Let X be a non-path-connected and finite sub-agenda of the \(\sigma \)-algebra \(\Sigma \). As in the proof of Theorem 1(b), we begin by proving that we may assume without loss of generality that \(\sigma (X)=\Sigma \).

Claim 1 If Theorem 3(b) holds when \(\sigma (X)=\Sigma \), then it holds in general.

Assume Theorem 3(b) holds if \(\sigma (X)=\Sigma \) and let \(\Sigma ^{\prime }:=\sigma (X)\) (\(\subseteq \Sigma \)). By assumption, there exists \(F^{\prime }:{\mathcal {P}}_{\Sigma ^{\prime }}^{n}\rightarrow {\mathcal {P}}_{\Sigma ^{\prime }}\) which, on X, is independent, consensus-preserving, and non-neutral. Consider some \(F:{\mathcal {P}}_{\Sigma }^{n}\rightarrow {\mathcal {P}}_{\Sigma }\) which, for any \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\), generates a probability function in \({\mathcal {P}}_{\Sigma }\) extending \(F^{\prime }(P_{1}|_{\Sigma ^{\prime } },\ldots ,P_{n}|_{\Sigma ^{\prime }})\) (where such an extension exists by Lemma 5 and finiteness of \(\Sigma ^{\prime }\)). The so-defined F inherits all relevant properties from \(F^{\prime }\): it is, on X, independent, consensus preserving, and non-neutral. This proves the claim.

Now let \(\sigma (X)=\Sigma \). Notationally, for any sub-\(\sigma \)-algebra \(\bar{\Sigma }\subseteq \Sigma \), let \(\mathcal {A}(\bar{\Sigma })\) be its set of atoms (i.e., minimal elements of \(\tilde{\Sigma }\backslash \{\varnothing \}\)). We now define a pooling function for agenda \(\Sigma \) and show that it has the desired properties. As an ingredient to the definition, let \(D^{\prime }:[0,1]^{n}\rightarrow [0,1]\) and \(D^{\prime \prime }:[0,1]^{n} \rightarrow [0,1]\) be the local pooling criteria of two distinct linear pooling functions; and let \(\bar{A}\in X\backslash \{\varnothing ,\Omega \}\) be a (by assumption existing) event such that not for all \(A\in X\backslash \{\varnothing ,\Omega \}\) there is \(\bar{A}\vdash \vdash ^{*}A\), where \(\vdash \vdash ^{*}\) is the transitive closure of \(\vdash ^{*}\). Consider any \((P_{1},\ldots ,P_{n})\in {\mathcal {P}}_{\Sigma }^{n}\). To define \(P_{P_{1} ,\ldots ,P_{n}}\in {\mathcal {P}}_{\Sigma }\), we start by defining probability functions on two sub-\(\sigma \)-algebras of \(\Sigma \), denoted \(\Sigma ^{\prime }\) and \(\Sigma ^{\prime \prime }\) and defined as the \(\sigma \)-algebras generated by the sets

respectively. \((X^{\prime }\) and \(X^{\prime \prime }\) might be empty, in which case \(\Sigma ^{\prime }\) and \(\Sigma ^{\prime \prime }\),  respectively, are \(\{\varnothing ,\Omega \}\).) Let \(P_{P_{1},\ldots ,P_{n}}^{\prime }\in {\mathcal {P}} _{\Sigma ^{\prime }}\) and \(P_{P_{1},\ldots ,P_{n}}^{\prime \prime }\in {\mathcal {P}} _{\Sigma ^{\prime \prime }}\) be defined by

These two functions are indeed probability functions (on \(\Sigma ^{\prime }\) and \(\Sigma ^{\prime \prime }\), respectively), as they are linear averages of of probability functions.

Claim 2 The \(\sigma \)-algebras \(\Sigma ^{\prime }\) and \(\Sigma ^{\prime \prime }\) are logically independent, that is: if \(A^{\prime }\in \Sigma ^{\prime }\) and \(A^{\prime \prime }\in \Sigma ^{\prime \prime }\) are non-empty, so is \(A^{\prime }\cap A^{\prime \prime }\).

Suppose the contrary. Then, as each non-empty element of \(\Sigma ^{\prime }\) includes an atom of \(\Sigma ^{\prime }\) and hence a non-empty intersection of events in \(X^{\prime }\), and similarly for \(\Sigma ^{\prime \prime }\), there are consistent sets \(Y^{\prime }\subseteq X^{\prime }\) and \(Y^{\prime \prime }\subseteq X^{\prime \prime }\) such that \(Y^{\prime }\cup Y^{\prime \prime }\) is inconsistent. Let Y be a minimal inconsistent subset of \(Y^{\prime }\cup Y^{\prime \prime }\). Then Y is not a subset of \(Y^{\prime } \) or \(Y^{\prime \prime }\), as \(Y^{\prime }\) and \(Y^{\prime \prime }\) are consistent. So there are \(A\in Y\cap X^{\prime }\) and \(B\in Y\cap X^{\prime \prime }\). Note that \(A\vdash ^{*}B^{c}\), a contradiction since \(A\in X^{\prime }\) and \(B^{c}\in X^{\prime \prime }\). This proves Claim 2.

We now extend \(P_{P_{1},\ldots ,P_{n}}^{\prime }\) and \(P_{P_{1},\ldots ,P_{n}} ^{\prime \prime }\) to a probability function on the \(\sigma \)-algebra \(\tilde{\Sigma }:=\sigma (\Sigma ^{\prime }\cup \Sigma ^{\prime \prime } )=\sigma (X^{\prime }\cup X^{\prime \prime })\), in such a way that the events in \(\Sigma ^{\prime }\) are probabilistically independent of those in \(\Sigma ^{\prime \prime }\). By Claim 2, the atoms of \(\tilde{\Sigma }\) are precisely the intersections of an atom of \(\Sigma ^{\prime }\) and one of \(\Sigma ^{\prime \prime }\): \(\mathcal {A}(\tilde{\Sigma })=\{A^{\prime }\cap A^{\prime \prime }:A^{\prime }\in \mathcal {A}(\Sigma ^{\prime }),A^{\prime \prime }\in \mathcal {A} (\Sigma ^{\prime \prime })\}\). Let \(\tilde{P}_{P_{1},\ldots ,P_{n}}\) be the unique measure on \(\tilde{\Sigma }\) that behaves as follows on the atoms:

for all \(A^{\prime }\in \mathcal {A}(\Sigma ^{\prime }),\) \(A^{\prime \prime } \in \mathcal {A}(\Sigma ^{\prime \prime })\). Now \(\tilde{P}_{P_{1},\ldots ,P_{n}} \) is a probability function as

Check that restricting \(\tilde{P}_{P_{1},\ldots ,P_{n}}\) to \(\Sigma ^{\prime }\) and \(\Sigma ^{\prime \prime }\) yields \(P_{P_{1},\ldots ,P_{n}}^{\prime }\) and \(P_{P_{1},\ldots ,P_{n}}^{\prime \prime }\), respectively. So

Before we can extend \(\tilde{P}_{P_{1},\ldots ,P_{n}}\) to the full \(\sigma \)-algebra \(\Sigma \), we prove another claim. For all \(A\in X\) such that \(\bar{A}\vdash \vdash ^{*}A\) but not \(\bar{A}\vdash \vdash ^{*}A^{c}\), define

Claim 3 For all atoms C of \(\tilde{\Sigma }\) (\(=\sigma (X^{\prime }\cup X^{\prime \prime })\)) with \(\tilde{P}_{P_{1},\ldots ,P_{n}}(C)>0\), the event \(C\cap \left( \cap _{A\in X:\bar{A}\vdash \vdash ^{*}A\text { and not }\bar{A}\vdash \vdash ^{*}A^{c}}A_{P_{1},\ldots ,P_{n}}\right) \) is an atom of \(\Sigma \).

Let C be as specified, and write \(C_{P_{1},\ldots ,P_{n}}\) for the event in question. As noted above, \(C=A^{\prime }\cap A^{\prime \prime }\) with \(A^{\prime }\in \mathcal {A}(\Sigma ^{\prime })\) and \(A^{\prime \prime }\in \mathcal {A} (\Sigma ^{\prime \prime })\). By \({\tilde{P}}_{P_{1},\ldots ,P_{n}}(C)>0\) and (3), we have \({\tilde{P}}_{P_{1},\ldots ,P_{n}}^{\prime }(A^{\prime })>0\) and \({\tilde{P}}_{P_{1},\ldots ,P_{n}}^{\prime \prime }(A^{\prime \prime })>0 \). Since \(A^{\prime }\in \mathcal {A}(\Sigma ^{\prime })\), we may write \(A^{\prime } =\cap _{A\in Y^{\prime }}A\) for some set \(Y^{\prime }\subseteq X^{\prime }\) containing exactly one member of each pair \(A,A^{c}\in X^{\prime }\). Similarly, \(A^{\prime \prime }=\cap _{A\in Y^{\prime \prime }}A \) for some set \(Y^{\prime \prime }\subseteq X^{\prime \prime }\) containing exactly one member of each pair \(A,A^{c}\in X^{\prime \prime }\). Note also that \(\cap _{A\in X:\bar{A} \vdash \vdash ^{*}A\text { and not }\bar{A}\vdash \vdash ^{*}A^{c}} A_{P_{1},\ldots ,P_{n}}\) can be written as \(\cap _{A\in Y_{P_{1},\ldots ,P_{n}}}A\), where the set

consists of exactly one member of each pair \(A,A^{c}\in X\backslash (X^{\prime }\cup X^{\prime \prime })\). So \(C_{P_{1},\ldots ,P_{n}}=\cap _{A\in Y^{\prime }\cup Y^{\prime \prime }\cup Y_{P_{1},\ldots ,P_{n}}}A\), where the set \(Y^{\prime }\cup Y^{\prime \prime }\cup Y_{P_{1},\ldots ,P_{n}}\) consists of exactly one member of each pair \(A,A^{c}\in X\). So, since \(\Sigma =\sigma (X) \), \(C_{P_{1},\ldots ,P_{n}}\) is an atom or is empty. Hence it suffices to show that \(C_{P_{1},\ldots ,P_{n} }\ne \varnothing \). Suppose the contrary. Then \(Y^{\prime }\cup Y^{\prime \prime }\cup Y_{P_{1},\ldots ,P_{n}}\) is inconsistent, hence has a minimal inconsistent subset Y. We distinguish two cases and derive a contradiction in each.

Case 1 There is some \(B\in Y\cap Y_{P_{1},\ldots ,P_{n}}\) with \(\bar{A}\vdash \vdash ^{*}B\). Consider some \(B^{\prime }\in Y\backslash \{B\}\). We have (i) not \(\bar{A}\vdash \vdash ^{*}B^{\prime }\) (otherwise by \(B^{\prime }\vdash ^{*}B^{c}\) we would have \(\bar{A}\vdash \vdash ^{*}B^{c}\), hence \(B\in X^{\prime }\), a contradiction as \(B\in Y_{P_{1},\ldots ,P_{n}}\)). Further, (ii) \(\bar{A}\vdash \vdash ^{*}(B^{\prime })^{c}\) (as \(\bar{A}\vdash \vdash ^{*}B\) and \(B\vdash ^{*}(B^{\prime })^{c}\)). By (i) and (ii), letting \(A:=(B^{\prime })^{c}\), the event \(A_{P_{1},\ldots ,P_{n}}\) (\(\in \{A,A^{c}\}\)) is well-defined. Since \(Y_{P_{1},\ldots ,P_{n}}\) contains \(A_{P_{1},\ldots ,P_{n}}\) (\(\in \{A,A^{c}\}\)) and contains \(B^{\prime }=A^{c}\) but not \((B^{\prime })^{c}=A\), we must have \(A_{P_{1},\ldots ,P_{n}}=A^{c}\). So, for all i, \(P_{i}(A)=0\) and hence \(P_{i}(B^{\prime })=1\). Note that this holds for all \(B^{\prime }\in Y\backslash \{B\}\). So \(P_{i}(\cap _{B^{\prime }\in Y}B^{\prime })=P_{i}(B)\) for all i. Hence, as Y is inconsistent, \(P_{i}(B)=0\) for all i. Thus \(B_{P_{1},\ldots ,P_{n}}=B^{c}\). So \(B^{c}\in Y_{P_{1},\ldots ,P_{n}}\), a contradiction as \(B\in Y_{P_{1},\ldots ,P_{n}}\).

Case 2 There is no \(B\in Y\cap Y_{P_{1},\ldots ,P_{n}}\) with \(\bar{A}\vdash \vdash ^{*}B\). Then all \(B\in Y\cap Y_{P_{1},\ldots ,P_{n}}\) take the form \(A_{P_{1},\ldots ,P_{n}}=A^{c}\), so that \(P_{i}(A)=0\) for all i, i.e., \(P_{i}(B)=1\) for all i. So, (*) \(P_{i}(\cap _{B\in Y}B)=P_{i}(\cap _{B\in Y\backslash Y_{P_{1},\ldots ,P_{n}}}B)\) for all i. Now, either (i) \(Y\subseteq Y_{P_{1},\ldots ,P_{n}}\cup Y^{\prime }\), or (ii) \(Y\subseteq Y_{P_{1},\ldots ,P_{n} }\cup Y^{\prime \prime }\), because otherwise there are \(A^{\prime }\in Y^{\prime }\) and \(A^{\prime \prime }\in Y^{\prime \prime }\), and \(A^{\prime }\vdash ^{*}(A^{\prime \prime })^{c}\), whence \(\bar{A}\vdash \vdash ^{*}(A^{\prime \prime })^{c}\), a contradiction as \((A^{\prime \prime })^{c}\in X^{\prime \prime }\). First suppose case (i) holds. Then \(Y\backslash Y_{P_{1},\ldots ,P_{n}}\subseteq Y^{\prime }\), and so (*) implies that (**) \(P_{i}(\cap _{B\in Y}B)\ge P_{i}(\cap _{B\in Y^{\prime }}B)=P_{i}(A^{\prime })\) for all i. Since by assumption \(\tilde{P}_{P_{1},\ldots ,P_{n}}(A^{\prime })>0\), there is (by (4)) at least one i with \(P_{i}(A^{\prime })>0\), hence by (**) with \(P_{i}(\cap _{B\in Y}B)>0\). So \(\cap _{B\in Y}B\ne \varnothing \), i.e., Y is consistent, a contradiction. Similarly, in case (ii), one can show that Y is consistent, a contradiction. This completes the proof of Claim 3.

Let \(P_{P_{1},\ldots ,P_{n}}\) be the unique measure on \(\Sigma \) behaving as follows on any atom C of \(\Sigma \). If C takes the form as in Claim 3, i.e., \(B=C\cap \left( \cap _{A\in X:\bar{A}\vdash \vdash ^{*}A\text { and not }\bar{A}\vdash \vdash ^{*}A^{c}}A_{P_{1},\ldots ,P_{n}}\right) \) where \(C\in \mathcal {A}(\tilde{\Sigma })\) and \(\tilde{P}_{P_{1},\ldots ,P_{n}}(C)>0\), then let \(P_{P_{1},\ldots ,P_{n}}(B):=\tilde{P}_{P_{1},\ldots ,P_{n}}(C)\). Otherwise let \(P_{P_{1},\ldots ,P_{n}}(B):=0\).

Claim 4 \(P_{P_{1},\ldots ,P_{n}}\) extends \(\tilde{P}_{P_{1},\ldots ,P_{n}}\) (in particular, is a probability function).

It suffices to show that \(P_{P_{1},\ldots ,P_{n}}\) coincides with \(\tilde{P}_{P_{1},\ldots ,P_{n}}\) on \(\mathcal {A}(\tilde{\Sigma })\). Consider any \(C\in \mathcal {A}(\tilde{\Sigma })\). As \(\Sigma \) is a refinement of \(\tilde{\Sigma } \),

There are two cases.

Case 1 \(\tilde{P}_{P_{1},\ldots ,P_{n}}(C)=0\). Then, for all \(B\in \mathcal {A}(\Sigma )\) with \(B\subseteq C\), we have \(P_{P_{1},\ldots ,P_{n} }(B)=0\) (by definition of \(P_{P_{1},\ldots ,P_{n}}\)), and so by (5) we have \(P_{P_{1},\ldots ,P_{n}}(C)=0=\tilde{P}_{P_{1},\ldots ,P_{n}}(C)\), as desired.

Case 2 \(\tilde{P}_{P_{1},\ldots ,P_{n}}(C)>0\). Then, among all atoms \(B\in \mathcal {A}(\Sigma )\) with \(B\subseteq C\), there is by definition of \(P_{P_{1},\ldots ,P_{n}}\) exactly one such that \(P_{P_{1},\ldots ,P_{n}}(B)>0\) (namely \(B=C\cap (\cap _{A\in X:\bar{A}\vdash \vdash ^{*}A\text { and not }\bar{A} \vdash \vdash ^{*}A^{c}}A_{P_{1},\ldots ,P_{n}})\)), and \(P_{P_{1},\ldots ,P_{n} }(B)=\tilde{P}_{P_{1},\ldots ,P_{n}}(C)\). So by (5) \(P_{P_{1},\ldots ,P_{n} }(C)=\tilde{P}_{P_{1},\ldots ,P_{n}}(C)\). This completes the proof of Claim 4.

Claim 5 For all \(A\in X\) such that \(\bar{A}\vdash \vdash ^{*}A\) and not \(\bar{A}\vdash \vdash ^{*}A^{c}\), \(P_{P_{1},\ldots ,P_{n}}(A)\) is 1 if, for some individual i, \(P_{i}(A)>0\), and 0 otherwise.

By definition of \(P_{P_{1},\ldots ,P_{n}}\), all atoms of \(\Sigma \) with positive probability are subsets of the event \(\cap _{A\in X:\bar{A}\vdash \vdash ^{*}A\text { and not }\bar{A}\vdash \vdash ^{*}A^{c}}A_{P_{1},\ldots ,P_{n}}\). So this event has probability 1. Hence, for all \(A\in X\) such that \(\bar{A} \vdash \vdash ^{*}A\) and not \(\bar{A}\vdash \vdash ^{*}A^{c}\), we have \(P_{P_{1},\ldots ,P_{n}}(A_{P_{1},\ldots ,P_{n}})=1\), and so

This proves Claim 5.

By Claim 4, we have constructed a well-defined pooling function \((P_{1} ,\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n}}\) for agenda \(\Sigma \). By (4) and Claims 4 and 5, we know its behaviour on the entire sub-agenda X: the pooling function is independent on X and the local pooling criteria \(D_{A}\) of events \(A\in X\) are given by

1. (i)

   the linear criterion \(D^{\prime }\) if \(A\in X^{\prime }\),

2. (ii)

   the different linear criterion \(D^{\prime \prime }\) if \(A\in X^{\prime \prime }\),

3. (iii)

   a non-linear criterion \({\hat{D}}\) (taking the value 0 at \(\mathbf {0}\) and the value 1 everywhere else) if \(\bar{A}\vdash \vdash ^{*}A\) but not \(\bar{A}\vdash \vdash ^{*}A^{c}\),

4. (iv)

   the different non-linear criterion \(1-\hat{D}(\mathbf {1} -\mathbf {\cdot )}\) if not \(\bar{A}\vdash \vdash ^{*}A\) but \(\bar{A} \vdash \vdash ^{*}A^{c}\).

These pooling criteria also ensure unanimity preservation on X. To check non-neutrality, it suffices to show that at least two of the four different types of events (i)–(iv) do indeed occur. This is so because \(\bar{A}\) is of type (i) or (iii) and because by assumption there exists an \(A\in X \) such that not \(\bar{A}\vdash \vdash ^{*}A\), i.e., such that A has type (ii) or (iv). \(\square \)

1.4 A.4 Proof of Theorem  4(b)

Consider any finite sub-agenda \(X\ne \{\varnothing ,\Omega \}\) (of the \(\sigma \)-algebra agenda \(\Sigma \)) which is nested or satisfies \(\left| X\backslash \{\varnothing ,\Omega \}\right| \le 4\). If X is nested, the claim follows from Theorem 1(b), as non-neutrality on X implies non-linearity on X. Now assume \(\left| X\backslash \{\varnothing ,\Omega \}\right| \le 4\). We reduce the claim to Part I’s Theorem 4(b). By that result, there is a pooling function \(F^{\prime }\) for agenda X which is independent, consensus compatible, and not linear. By Lemma 4, \(F^{\prime }\) is induced by a pooling function for agenda \(\Sigma \) which is independent on X, (globally) consensus-preserving, and not linear on X. \(\square \)

1.5 A.5 Proof of Theorem 5(b)

Consider a simple sub-agenda X of \(\sigma \)-algebra \(\Sigma \), where X is finite and not \(\{\varnothing ,\Omega \}\). We construct a pooling function which, on X, is independent (in fact, neutral), conditional-consensus-preserving, and non-linear. We may assume without loss of generality that \(\sigma (X)=\Sigma \), because the ‘Claim’ in the proof of Theorem 1(b) holds analogously here as well.

As an ingredient of the construction, we use an arbitrary pooling function \((P_{1},\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n}}^{\mathrm {lin}}\) which, at least on X, is linear and conditional-consensus-preserving. The function could be simply given by \((P_{1},\ldots ,P_{n})\mapsto P_{1}\), which is even globally linear and conditional-consensus-preserving. Let \(D^{\mathrm {lin}}\) be its pooling criterion for all events in X. To anticipate, the pooling function \((P_{1},\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n} }\) to be constructed will have the pooling criterion \(D:[0,1]^{n} \rightarrow [0,1]\) for each event in X, where

Consider any \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\). We must define \(P_{P_{1},\ldots ,P_{n}}\). We use the following notation (which suppresses the parameters \(P_{1},\ldots ,P_{n}\)):

Notice that for all \(A\in X\) we have \(A\in X_{>1/2}\Rightarrow A^{c} \not \in X_{>1/2}\) and \(A\in X_{=1/2}\Leftrightarrow A^{c}\in X_{=1/2}\). We now prove two claims (which use X’s simplicity).

Claim 1 \(X_{=1/2}\) can be partitioned into two (possibly empty) sets \(X_{=1/2}^{1}\) and \(X_{=1/2}^{2}\) such that (i) each \(X_{=1/2}^{j}\) satisfies \(p(A\cap B)>0\) for all \(A,B\in X_{=1/2}^{j}\) and (ii) each \(X_{=1/2}^{j}\cup X_{>1/2}\) is consistent (whence \(X_{=1/2}^{j}\) contains exactly one member of each pair \(A,A^{c}\in X_{=1/2}\)).

To show this, note first that \(X_{=1/2}\) has a subset Y such that \(p(A\cap B)>0\) for all \(A,B\in Y\) (e.g., \(Y=\varnothing \)). Among all such subsets \(Y\subseteq X_{=1/2}\), let \(X_{=1/2}^{1}\) a maximal one, and let \(X_{=1/2}^{2}:=X_{=1/2}\backslash X_{=1/2}^{1}\). By definition, \(X_{=1/2}^{1}\) and \(X_{=1/2}^{2}\) form a partition of \(X_{=1/2}\). We show that (i) and (ii) hold.

1. (i)

   Property (i) holds for \(X_{=1/2}^{1}\) by definition, and for \(X_{=1/2}^{2}\) by the following argument. Let \(A,B\in X_{=1/2}^{2}\) and for a contradiction let \(p(A\cap B)=0\). By the maximality property of \(X_{=1/2}^{1} \), there are \(A^{\prime },B^{\prime }\in X_{=1/2}^{1}\) such that \(p(A\cap A^{\prime })=0\) and \(p(B\cap B^{\prime })=0\). Thus, \(p(A\cap C)=p(B\cap C)=0\) where \(C:=A^{\prime }\cap B^{\prime }\). Since the intersection of any two of the sets A, B, C has zero p-probability, we must have \(p(A)+p(B)+p(C)=p(A\cup B\cup C)\le 1\), a contradiction because \(p(A)=p(B)=1/2\) and \(p(C)=p(A^{\prime }\cap B^{\prime })>0\) (the latter because \(X_{=1/2}^{1}\) satisfies (i)).

2. (ii)

   For a contradiction, let some \(X_{=1/2}^{j}\cup X_{>1/2}\) be inconsistent. Then (since X and hence \(X_{=1/2}^{j}\cup X_{>1/2}\) are finite) there is a minimal inconsistent subset \(Y\subseteq X_{=1/2}^{j}\cup X_{>1/2}\). Since X is simple, we have \(\left| Y\right| \le 2\), say \(Y=\{A,B\}\). Since \(A\cap B=\varnothing \), we have \(p(A)+p(B)=p(A\cup B)\le 1\). And since \(p(A),p(B)\ge 1/2\), it follows that \(p(A)=p(B)=1/2\), i.e., \(A,B\in X_{=1/2}^{j}\). Hence, by (i), we have \(p(A\cap B)>0\), a contradiction as \(A\cap B=\varnothing \).

Claim 2 \(\cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}}C\) and \(\cap _{C\in X_{=1/2}^{2}\cup X_{>1/2}}C\) are atoms of the \(\sigma \)-algebra \(\Sigma \), i.e., (\(\subseteq \)-)minimal elements of \(\Sigma \backslash \{\varnothing \}\) (they are the same atoms if and only if \(X_{=1/2}=\varnothing \), i.e., if and only if \(X_{=1/2}^{1}=X_{=1/2}^{2}=\varnothing \)).

To show this, first write X as \(\{C_{j}^{0},C_{j}^{1}:j=1,\ldots ,J\}\), where \(J=\left| X\right| /2\) and each pair \(C_{j}^{0},C_{j}^{1}\) consists of an event and its complement. We may write \(\Sigma \) as

Recall that \(\Sigma \) is the \(\sigma \)-algebra generated by X. The inclusion ‘\(\supseteq \)’ in (7) is obvious, and the inclusion ‘\(\subseteq \)’ holds because the right side of (7) includes X (since any \(C_{j}^{k}\in X\) can be written as the union of all \(C_{1}^{k_{1}}\cap \cdots \cap C_{J}^{k_{J}}\) for which \(k_{j}=k\)) and is a \(\sigma \)-algebra (check closedness under taking unions and complements).

From (7) and the pairwise disjointness of the intersections of the form \(C_{1}^{k_{1}}\cap \cdots \cap C_{J}^{k_{J}}\), it is clear that every consistent such intersection is an atom of \(\Sigma \). Now \(\cap _{C\in X_{=1/2}^{j}\cup X_{>1/2}}C\) is (for \(j\in \{0,1\}\)) precisely such a consistent intersection. Indeed, \(\cap _{C\in X_{=1/2}^{j}\cup X_{>1/2}}C\) is consistent by Claim 1, and contains a member of each pair \(A,A^{c}\) in X. The latter holds by Claim 1 if \(p(A)=p(A^{c})\) (\(=1/2\)), and otherwise because there is a \(B\in \{A,A^{c}\}\) with \(p(B)>1/2\), i.e., with \(B\in X_{>1/2} \subseteq X_{=1/2}^{j}\cup X_{>1/2}\). This proves Claim 2.

We can now define \(P_{P_{1},\ldots ,P_{n}}\). By Claim 1, we may pick \(\omega ^{1}\in \cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}}C\) and \(\omega ^{2}\in \cap _{C\in X_{=1/2}^{2}\cup X_{>1/2}}C\), where we assume that \(\omega ^{1}=\omega ^{2}\) if \(X_{=1/2}=\varnothing \), i.e., if \(\cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}} C=\cap _{C\in X_{=1/2}^{2}\cup X_{>1/2}}C=\cap _{C\in X_{>1/2}}C\). Let \(\delta _{\omega ^{1}}\) and \(\delta _{\omega ^{2}}\) be, respectively, the Dirac measures on \(\Sigma \) at \(\omega ^{1}\) and \(\omega ^{2}\), given for all \(A\in \Sigma \) by \(\delta _{\omega ^{j}}(A)=1\) if \(\omega ^{j}\in A\) and \(\delta _{\omega ^{j}}(A)=0\) if \(\omega ^{j}\notin A\). Let

where \(\omega ^{1}\) and \(\omega ^{2}\) depend on \(P_{1},\ldots ,P_{n}\) via \(X_{=1/2}^{1},X_{=1/2}^{2},X_{>1/2}\). So \(P_{P_{1},\ldots ,P_{n}}(A)\) is 1 or 1/2 or 0 depending on whether A (\(\in \Sigma \)) contains both, exactly one, or none of \(\omega ^{1}\) and \(\omega ^{2}\); and \(P_{P_{1},\ldots ,P_{n}}=\delta _{\omega }\) if \(\omega ^{1}=\omega ^{2}=\omega \), i.e., if \(X_{=1/2}=\varnothing \). We finally show that the so-defined pooling function \((P_{1},\ldots ,P_{n})\mapsto P_{P_{1},\ldots ,P_{n}}\) has all desired properties.

Independence on X We in fact show something stronger, i.e., neutrality on X with pooling criterion D given in (6). Let \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\), \(A\in X\) and \((t_{1},\ldots ,t_{n} ):=(P_{1}(A),\ldots ,P_{n}(A))\). We prove that \(P_{P_{1},\ldots ,P_{n}}(A)=D(t_{1} ,\ldots ,t_{n})\) by considering three cases and using the above notation p,  \(X_{>1/2},\) \(X_{=1/2}^{1},X_{=1/2}^{2},\omega ^{1},\omega ^{2}\).

Case 1 \(p(A)=D^{\mathrm {lin}}(t_{1},\ldots ,t_{n})<1/2\). Here \(D(t_{1},\ldots ,t_{n})=0\). So we must prove that \(P_{P_{1},\ldots ,P_{n}}(A)=0\), i.e., that \(\omega ^{1},\omega ^{2}\not \in A\). Assume for a contradiction that \(\omega ^{1}\in A\) (the proof is analogous if we instead assume \(\omega ^{2}\in A\)). Then A includes \(\cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}}C\), as this set contains \(\omega ^{1}\) and is by Claim 2 an atom of \(\Sigma \). So \(A^{c} \cap [\cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}}C]=\varnothing \). Hence the set \(\{A^{c}\}\cup X_{=1/2}^{1}\cup X_{>1/2}\) is inconsistent, so has a minimal inconsistent subset Y. As X is simple, \(\left| Y\right| \le 2\). Now \(\varnothing \not \in Y\) as \(A^{c}\ne \varnothing \) (by \(p(A^{c} )=1-p(A)>1/2\)) and as all \(B\in X_{=1/2}^{1}\cup X_{>1/2}\) are non-empty (by \(p(B)\ge 1/2\)). So \(\left| Y\right| =2\). Further, Y is not a subset of \(X_{=1/2}^{1}\cup X_{>1/2}\), as this set is consistent by Claim 1. So \(Y=\{A^{c},B\}\) for some \(B\in X_{=1/2}^{1}\cup X_{>1/2}\). As \(A^{c}\cap B=\varnothing \) and as \(p(A^{c})=1-p(A)>1/2\) and \(p(B)\ge 1/2\), we have \(p(A^{c}\cup B)=p(A^{c})+p(B)>1/2+1/2=1\), a contradiction.

Case 2 \(p(A)=D^{\mathrm {lin}}(t_{1},\ldots ,t_{n})>1/2\). Then \(D(t_{1},\ldots ,t_{n})=1\). Hence we must prove that \(P_{P_{1},\ldots ,P_{n}}(A)=1\), i.e., that \(P_{P_{1},\ldots ,P_{n}}(A^{c})=0\). The latter follows from case 1 as applied to \(A^{c}\), since \(p(A^{c})=1-p(A)<1/2\).

Case 3 \(p(A)=D^{\mathrm {lin}}(t_{1},\ldots ,t_{n})=1/2\). Then \(D(t_{1},\ldots ,t_{n})=1/2\). So we must prove that \(P_{P_{1},\ldots ,P_{n}}(A)=1/2\), i.e., that A contains exactly one of \(\omega ^{1}\) and \(\omega ^{2}\). As \(p(A)=1/2\), exactly one of \(X_{=1/2}^{1}\) and \(X_{=1/2}^{2}\) contains A and the other one contains \(A^{c}\), by Claim 1. Say \(A\in X_{=1/2}^{1}\) and \(A^{c}\in X_{=1/2}^{2}\) (the proof is analogous if instead \(A\in X_{=1/2}^{2}\) and \(A^{c}\in X_{=1/2}^{1}\)). So \(A\supseteq \cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}}C\), whence \(\omega ^{1}\in A\). Further, \(\omega ^{2}\notin A\) because A is disjoint from \(A^{c}\), hence from its subset \(\cap _{C\in X_{=1/2} ^{2}\cup X_{>1/2}}C\) which contains \(\omega ^{2}\).

Non-linearity on X Pooling cannot be linear, since otherwise for any fixed \(A\in X\backslash \{\Omega ,\varnothing \}\) (\(\ne \varnothing \)) the collective probabilities \(P_{P_{1},\ldots ,P_{n}}(A)\) could take any given values \(t\in [0,1]\) (for instance by letting \(P_{1}(A)=\cdots =P_{n}(A)=t\)), a contradiction, since by definition \(P_{P_{1},\ldots ,P_{n}}(A)\in \{0,1/2,1\}\).

Conditional-consensus-preservation on X Let \(A,B\in X\) and \(P_{1},\ldots ,P_{n}\in {\mathcal {P}}_{\Sigma }\) such that \(P_{i}(A\cup B)=1\) for all i. We show that \(P_{P_{1},\ldots ,P_{n}}(A\cup B)=1\), which establishes conditional-consensus-preservation on X by Proposition 1(a). For all i, \(P_{i}(A)+P_{i}(B)\ge P_{i}(A\cup B)=1\), and hence \(P_{i}(A)\ge 1-P_{i}(B)=P_{i}(B^{c})\). So, as \(D^{\mathrm {lin}}:[0,1]^{n}\rightarrow [0,1]\) takes a linear form with non-negative coefficients and hence is weakly increasing in every component,

So, with p as defined earlier, \(p(A)\ge 1-p(B)\), i.e., \(p(A)+p(B)\ge 1\). We distinguish between three cases.

Case 1 \(p(A)>1/2\). Then, by the above proof of independence on X, \(P_{P_{1},\ldots ,P_{n}}(A)=1\). So \(P_{P_{1},\ldots ,P_{n}}(A\cup B)=1\), as desired.

Case 2 \(p(B)>1/2\). Then, again by the above proof of independence on X, \(P_{P_{1},\ldots ,P_{n}}(B)=1\). Hence, \(P_{P_{1},\ldots ,P_{n}}(A\cup B)=1\), as desired.

Case 3 \(p(A),p(B)\le 1/2\). Then, as \(p(A)+p(B)\ge 1\), we have \(p(A)=p(B)=1/2\). Let \(X_{>1/2},X_{=1/2}^{1},X_{=1/2}^{2},\omega ^{1},\omega ^{2}\) be as defined above. Note that \(A,B\in X_{=1/2}^{1}\cup X_{=1/2}^{2}\). It cannot be that A and B are both in \(X_{=1/2}^{1}\): otherwise \(A^{c}\) and \(B^{c}\) are both in \(X_{=1/2}^{2}\) by Claim 1, whence \(p(A^{c}\cap B^{c})>0\) (again by Claim 1), a contradiction since

(where \(p(A\cup B)=1\) because \(p(A\cup B)=P_{P_{1},\ldots ,P_{n}}^{\mathrm {lin} }(A\cup B)\) and \(P_{i}(A\cup B)=1\) for all i). Analogously, it cannot be that A and B are both in \(X_{=1/2}^{2}\). So one of A and B is in \(X_{=1/2}^{1}\) and the other one in \(X_{=1/2}^{2}\); say \(A\in X_{=1/2}^{1}\) and \(B\in X_{=1/2}^{2}\) (the proof is analogous otherwise). So \(A\supseteq \cap _{C\in X_{=1/2}^{1}\cup X_{>1/2}}C\) and \(B\supseteq \cap _{C\in X_{=1/2} ^{2}\cup X_{>1/2}}C\), and hence \(\omega ^{1}\in A\) and \(\omega ^{2}\in B\). Thus \(\omega ^{1},\omega ^{2}\in A\cup B\), whence \(P_{P_{1},\ldots ,P_{n}}(A\cup B)=1\). \(\square \)

1.6 A.6 Proof of Proposition 2

Consider the \(\sigma \)-algebra agenda \(\Sigma \), and let \(\left| \Sigma \right| >2^{3}=8\), i.e., \(\left| \Sigma \right| \ge 2^{4} =16\). Then \(\Sigma \) includes a partition of \(\Omega \) into four non-empty events. Let X be the sub-agenda consisting of any union of two of these four events. In the proof of Part I’s Proposition 2 we construct a pooling function for this agenda X which is neutral, consensus-preserving, and non-linear.^{Footnote 17} By Lemma 4, this pooling function is induced by a pooling function for agenda \(\Sigma \) which, on X, is neutral, consensus-preserving, and non-linear. \(\square \)

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