Abstract
This paper explores two nonstandard supermajority rules in the context of judgment aggregation over multiple logically connected issues. These rules set the supermajority threshold in a local, context sensitive way—partly as a function of the input profile of opinions. To motivate the interest of these rules, I prove two results. First, I characterize each rule in terms of a condition I call ‘Block Preservation’. Block preservation says that if a majority of group members accept a judgment set, then so should the group. Second, I show that one of these rules is, in a precise sense, a judgment aggregation analogue of a rule for connecting qualitative and quantitative belief that has been recently defended by Hannes Leitgeb. The structural analogy is due to the fact that Leitgeb sets thresholds for qualitative beliefs in a local, context sensitive way—partly as a function of the given credence function.
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Notes
This work was spearheaded by List and Pettit (2002) but the literature now encompasses a variety of methods and applications. The key results that set up the present paper are found in Dietrich and List (2007a, b) and Nehring and Puppe (2007). See Grossi and Pigozzi (2014) for an updated introduction with an ample reference list: §6.3 of this book—which discusses Lang et al. (2011, 2012) and Nehring and Pivato (2011)—is especially relevant to the present discussion.
For the sources of this framework see the references in fn. 1.
Notational Convention: In presenting aggregation conditions, F occurs as a free variable. A more explicit way of giving the same definition would be to say that the set of Consistency Preserving rules is: \({\{{\mathbf{F}}\mathbin {  } \forall \vec {J} \in {\mathbf{RP}}; [{\mathbf{F}}(\vec {J})\,\text { is\,consistent }].\}}\)
Two more reasons to broaden the focus beyond distancebased rules: first, distancebased rules are typically very difficult to compute (see Endriss 2015, §17.3.2); second, they are typically language dependent (see Cariani et al. 2008). Having said that, issues of computational complexity and language dependence for the rules I present in the next section are beyond the scope of this paper.
In general, there is no guarantee that there will be exactly one maximally inclusive threshold: for any threshold t that is maximally inclusive, there is a sufficiently small \(\epsilon \) such that either \(t\epsilon \) or \(t+\epsilon \) accept the same sentences. There always is, however, a set X of maximally inclusive thresholds.
Independence is the requirement that the collective judgment on \(\varphi \) depend only on the individual judgments on \(\varphi \).
There might even be cases of collective decisionmaking in which the rule cannot easily be exploited. Imagine a society whose members submit their judgments to a social planner (in ignorance of the judgments of others). The social planner is then be tasked with choosing an appropriate threshold, and derive some range of collective choices. Under such a system, there is relatively little threat that individuals would vote insincerely.
The principle is related to the principle of supermajoritarian efficiency in Nehring and Pivato (2011), which they describe informally as the requirement to overrule “if necessary, small supermajorities in favor of an equal or larger number of supermajorities of equal or greater size” (p. 9).
To see this, note (a) that both are quantified conditionals with one open variable \({\mathbf{F}}\), (b) that the antecedent of the Weak property is strictly stronger than the antecedent of the Strong one and, finally, (c) that the consequent of the former is strictly weaker than the consequent of the latter.
A side remark: it does not follow from Theorem 1 that there is no distancebased rule that is strongly block preserving, because distancebased rules are generally not uniform relaxed quota rules.
Some remarks about how the apparatus is lifted to the propositional setting. Say that a set of propositions P is: (i) consistent iff there is a world w such that \(w \in \bigcap P\); (ii) complete iff there is w such that \(\bigcap P\subseteq \{w\}\); (iii) closed relative to agenda A iff for every proposition \(q \in {\mathbf{A}}\) such that \(\bigcap P \subseteq q, q \in P\). The other definitions are also easily lifted: an agenda is a set of propositions with appropriate closure properties (we will, however, change the closure properties: see the next paragraph in the main text); a judgment set is a subset of the agenda; \({\texttt{support}}\) can be redefined by replacing the variables for sentences with variables for propositions. Similarly for \({\texttt{closed}}\) (the concept used in defining \({\mathbf{LS}}^{+}\)).
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Acknowledgments
Thanks to Branden Fitelson, Davide Grossi, Hannes Leitgeb, Gabriella Pigozzi and Patricia Rich for comments on earlier versions. Thanks to two reviewers for Erkenntnis for detailed and insightful reports. Special thanks to Branden Fitelson’s epistemology seminar at Rutgers where this material was presented in the Fall of 2014 and to the audience at the 2015 Formal Epistemology Workshop. Thanks also to Christian List for suggesting that I slice out one idea out of a much more baroque project and turn it into a paper in its own right (this one). The more baroque paper was delivered at DEON 2012 whose audience I also thank for their feedback.
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Appendix: Proofs of theorems
Appendix: Proofs of theorems
Proof of theorem 1
Suppose \({\mathbf{A}}\) and \(\vec {J}\) are respectively an arbitrary agenda and an arbitrary profile. First, one can easily check that \({\mathbf{LS}}^{}\) satisfies the three characterizing properties. Second, let F be an arbitrary rule that satisfies the three characterizing properties. We prove that \(\forall \varphi \in {\mathbf{A}}, \varphi \in {\mathbf{LS}}^{}(\vec {J})\Leftrightarrow \varphi \in {\mathbf{F}}(\vec {J})\).
\([\Rightarrow ]\): suppose \(\varphi \in {\mathbf{LS}}^{}(\vec {J})\). Then there is a threshold \(t > 0.5\) with \({\texttt{cons}}(\vec {J}, t)={\mathbf{LS}}^{}(\vec {J})\) and so \(\varphi \in {\texttt{cons}}(\vec {J},t)\). We show that \({\texttt{cons}}(\vec {J},t)\) is a set of sentences that meets the conditions in the antecedent of strong block preservation, instantiating the quantifiers with profile \(\vec {J}\), agenda A and threshold t. Because F satisfies strong block preservation, establishing this implies \({\texttt{cons}}(\vec {J},t)\subseteq {\mathbf{F}}(\vec {J})\) and so \(\varphi \in {\mathbf{F}}(\vec {J})\).
Obviously: \({\texttt{cons}}(\vec {J},t) \subseteq {\mathbf{A}}\). For conditions (i) and (ii), it is also obvious that \({\texttt{cons}}(\vec {J},t)\) is consistent (\({\texttt{cons}}\)sets must be consistent) and \({\texttt{support}}(\psi , \vec {J})\ge t\) for \(\psi \in {\texttt{cons}}(\vec {J},t)\). Finally, to establish (iii), there cannot be any set T inconsistent with \({\texttt{cons}}(\vec {J},t)\) such that for all \(\psi \in T, {\texttt{support}}(\psi , \vec {J})\ge t\). If there was, \(T \subseteq {\texttt{cons}}(\vec {J}, t)\), which is impossible because \({\texttt{cons}}(\vec {J}, t)\) is consistent. Since the antecedent of strong block preservation is met, \({\texttt{cons}}(\vec {J},t) \subseteq {\mathbf{F}}(\vec {J})\), so \(\varphi \in {\mathbf{F}}(\vec {J})\).
\([\Leftarrow ]\): suppose \(\varphi \in {\mathbf{F}}(\vec {J})\). Because F is a uniform relaxed quota rule, given \(\vec {J}\), there is a threshold \(t_{\vec {J}}\) such that
Fix some such \(t_{\vec {J}}\). Note that by (#) and the fact that \({\mathbf{F}}(\vec {J})\) is consistent, \({\mathbf{F}}(\vec {J})={\texttt{cons}}(\vec {J},t_{\vec {J}})\). But we know \({\texttt{cons}}(\vec {J},t_{\vec {J}}) \subseteq {\mathbf{LS}}^{}(\vec {J})\), so \({\mathbf{F}}(\vec {J}) \subseteq {\mathbf{LS}}^{}(\vec {J})\), and hence \(\varphi \in {\mathbf{LS}}^{}(\vec {J})\).
Proof of theorem 2
First, we note that \({\mathbf{LS}}^{+}\) satisfies all of the characterizing conditions. As before, we prove that for all \(\varphi \in {\mathbf{A}}, \varphi \in {\mathbf{LS}}^{+}(\vec {J}) \Leftrightarrow \varphi \in {\mathbf{F}}(\vec {J})\), where \({\mathbf{F}}\) is an arbitrary aggregation function satisfying the characterizing conditions.
\([\Rightarrow ]\): suppose \(\varphi \in {\mathbf{LS}}^{+}(\vec {J})\). Then there is a threshold t such that \({\texttt{closed}}(\vec {J}, t)={\mathbf{LS}}^{+}(\vec {J})\) and so \(\varphi \in {\texttt{closed}}(\vec {J}, t)\). To prove \(\varphi \in {\mathbf{F}}(\vec {J})\), we check that \({\texttt{closed}}(\vec {J}, t)\) satisfies the conditions in the antecedent of intermediate block preservation. First, \({\texttt{closed}}(\vec {J}, t)\) must be a closed and consistent subset of \({\mathbf{A}}\), thus satisfying condition (i). Condition (ii) is satisfied by construction of \({\texttt{closed}}(\vec {J}, t)\). To check condition (iii), suppose there was a \(\psi \) such that \({\texttt{closed}}(\vec {J}, t) \nvDash \psi \) and \({\texttt{support}}(\psi , \vec {J})\ge t\). This would imply \(\psi \in {\texttt{closed}}(\vec {J}, t)\), which contradicts the claim that \(\psi \) is not entailed by \({\texttt{closed}}(\vec {J}, t)\). By intermediate block preservation \(\varphi \in {\mathbf{F}}(\vec {J})\).
\([\Leftarrow ]\): suppose \(\varphi \in {\mathbf{F}}(\vec {J})\). We know that \({\mathbf{F}}(\vec {J})\) is consistent and closed. Because F is a uniform relaxed quota rule, given \(\vec {J}\), there is a threshold \(t_{\vec {J}}\,\) such that
Fix some such \(t_{\vec {J}}\). Because \({\mathbf{F}}(\vec {J})\) is consistent and closed, \({\mathbf{F}}(\vec {J})={\texttt{closed}}(\vec {J}, t_{\vec {J}}))\). Since, by definition, \({\texttt{closed}}(\vec {J}, t_{\vec {J}})) \subseteq {\mathbf{LS}}^{+}(\vec {J}), \varphi \in {\mathbf{LS}}^{+}(\vec {J})\).
Proof of Fact 1
We assume that that A is a \(\sigma \)algebra and \(\vec {J} \in {\mathbf{RP}}\). We check that \(P^{\vec {J}}(\cdot )\), that is \({\texttt{support}}(\cdot , \vec {J})\), satisfies the Kolmogorov axioms.
Nonnegativity: For \(p \in {\mathbf{A}}, P^{\vec {J}}(p)\ge 0\) follows from the definition of \({\texttt{support}}\).
Tautology: For any tautology, \(\top , P^{\vec {J}}(\top )=1\), because \(\vec {J}\) is a rational profile, which implies consistent and complete relative to \({\mathbf{A}}\) (\(\top \in {\mathbf{A}}\) because \({\mathbf{A}}\) is a \(\sigma \)algebra).
Additivity: Suppose p and q are mutually exclusive. Let \({\texttt{support}}(p, \vec {J})=x\) and \({\texttt{support}}(q, \vec {J})=y\). It follows that \({\texttt{support}}(p \cup q, \vec {J})=x+y\), because the fact that \(\vec {J}\) is a rational profile entails each of the following: (i) every group member who either accepts p or accepts q must also accept \(p \cup q\) (note that \(p \cup q\) must be in \({\mathbf{A}}\) because of \({\mathbf{A}}\)’s closure properties); (ii) no group member can believe both p and q (as they are mutually exclusive); (iii) no group member other than those who either accept p or q can accept \(p \cup q\).
Proof of theorem 3
\([\subseteq ]\): suppose \(p \in {\mathbf{LS}}^{+}(\vec {J})\). We want to show \(p \in {\mathbf{ST}}(P^{\vec {J}})\). Let \(q=\bigcap [{\mathbf{LS}}^{+}(\vec {J})]\). Since A is a finite \(\sigma \)algebra, \(q \in {\mathbf{A}}\). Because \({\mathbf{LS}}^{+}\)’s output is closed (relative to A), \(q \in {\mathbf{LS}}^{+}(\vec {J})\). We show that q is the strongest \(P^{\vec {J}}\)stable proposition (in \({\mathbf{A}}\)).

1.
\(q\;{\hbox{is}}\;P^{\vec {J}}{\hbox{stable:}}\) Suppose it is not, then \(P^{\vec {J}}(q)<1\) and there is \(w \in q\), s.t. \(P^{\vec {J}}(\{w\})\le P^{\vec {J}}(Wq)\). Let \(r:=(q\{w\}) \cup (Wq)\). Given \(P^{\vec {J}}(\{w\})\le P^{\vec {J}}(Wq)\), we must have: \(P^{\vec {J}}(r)\ge P^{\vec {J}}(q)\). This is equivalent to:
$$ {\texttt{support}}(r, \vec {J})\ge {\texttt{support}}(q, \vec {J}) $$Given this, the fact that \({\mathbf{LS}}^{+}\) is a uniform relaxed quota rule and \(q\in {\mathbf{LS}}^{+}(\vec {J}), r\in {\mathbf{LS}}^{+}(\vec {J})\). However, this is incompatible with \(q=\bigcap [{\mathbf{LS}}^{+}(\vec {J})]\). Because \(w \in q\), and \(q=\bigcap [{\mathbf{LS}}^{+}(\vec {J})], w \in r\), but that contradicts r’s definition.

2.
\(q\,{\hbox{is\,strongest\,among\,the}}\,P^{\vec {J}}{\hbox{stable\,propositions:}}\) suppose there is an r that is \(P^{\vec {J}}\)stable and \(r\,\subsetneq\,q\). Let w be the world that witnesses \(r\,\subsetneq\,q\) (so \(w \in q, w\notin r\)). Because r is \(P^{\vec {J}}\)stable, \({\{s\mathbin {  } P^{\vec {J}}(s)\ge P^{\vec {J}}(r)\}}\) must be nonempty (it includes r), consistent and closed. But then \({\texttt{closed}}(\vec {J}, P^{\vec {J}}(r))\) is nonempty, consistent and closed. Moreover, because \(r\,\subsetneq\,q, P^{\vec {J}}(r)\le P^{\vec {J}}(q)\). This, together with the fact that \({\texttt{closed}}(\vec {J}, P^{\vec {J}}(r))\) is nonempty, consistent and closed, implies:
$$ {\texttt{closed}}(\vec {J}, P^{\vec {J}}(q)) \subseteq {\texttt{closed}}(\vec {J}, P^{\vec {J}}(r)) $$Because of the definition of \({\mathbf{LS}}^{+}\), this implies:

(i)
\({\texttt{closed}}(\vec {J}, P^{\vec {J}}(r)) \subseteq {\mathbf{LS}}^{+}(\vec {J})\)
In addition, we must have all of the following.

(ii)
\(r \in {\texttt{closed}}(\vec {J}, P^{\vec {J}}(r)) \);

(iii)
\(q = \bigcap [{\mathbf{LS}}^{+}(\vec {J})]\);

(iv)
\(w \in q, w \notin r\)
(ii) is immediate because \(r \in {\{s\mathbin {  } P^{\vec {J}}(s)\ge P^{\vec {J}}(r)\}}\); (iii) and (iv) repeat how we fixed q and w respectively at the beginning of this proof.
But these four claims are contradictory. By (i) and (ii), \(r \in {\mathbf{LS}}^{+}(\vec {J})\). By (iii) and (iv), w is in every member of \({\mathbf{LS}}^{+}(\vec {J})\). But it is not in r, so \(r \notin {\mathbf{LS}}^{+}(\vec {J})\).

(i)
Putting together (1) and (2) implies that q is the strongest Pstable proposition.
\([\supseteq ]\): suppose \(p \in {\mathbf{ST}}(P^{\vec {J}})\). Let \(q=\bigcap {\mathbf{ST}}(P^{\vec {J}})\) and \(t'=P^{\vec {J}}(q)\). Note that it immediately follows that \(P^{\vec {J}}(p)\ge t'\). Furthermore, it follows that if \({\texttt{closed}}({\vec {J}},t')\) is nonempty, \(p \in {\texttt{closed}}({\vec {J}},t')\) and so \(p \in {\mathbf{LS}}^{+}(\vec {J})\).
All we have to show is that \({\texttt{closed}}(\vec {J}, t')\) is nonempty, which it is iff \({\{p\mathbin {  } {\texttt{support}}(p, \vec {J})\ge t'\}}\) (i.e. \({\{p\mathbin {  } P^{\vec {J}}(p)\ge t'\}}\)) is closed and consistent. This follows by Leitgeb’s theorem that for every probability function P, including \(P^{\vec {J}}, {\{p\mathbin {  } P(p)\ge P(q)\}}\) is consistent and closed (relative to \({\mathbf{A}}\)) when q is Pstable.
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Cariani, F. Local Supermajorities. Erkenn 81, 391–406 (2016). https://doi.org/10.1007/s106700159746x
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DOI: https://doi.org/10.1007/s106700159746x
Keywords
 Aggregation Rule
 Strategic Vote
 Judgment Aggregation
 Acceptance Rule
 Collective Judgment