Scoring rules for judgment aggregation
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Abstract
This paper introduces a new class of judgment aggregation rules, to be called ‘scoring rules’ after their famous counterparts in preference aggregation theory. A scoring rule generates the collective judgment set which reaches the highest total ‘score’ across the individuals, subject to the judgment set having to be rational. Depending on how we define ‘scores’, we obtain several (old and new) solutions to the judgment aggregation problem, such as distancebased aggregation, premise and conclusionbased aggregation, truthtracking rules, and a generalization of the Borda rule to judgment aggregation theory. Scoring rules are shown to generalize the classical scoring rules of preference aggregation theory.
1 Introduction
The classical example of logically inconsistent majority judgments
Premise \(p\)  Premise \(q\)  Conclusion \(r (\Leftrightarrow p\wedge q\))  

Individual 1  Yes  Yes  Yes 
Individual 2  Yes  No  No 
Individual 3  No  Yes  No 
Majority  Yes  Yes  No 
It is fair to say that judgment aggregation theory has until recently been dominated by ‘impossibility’ findings, as is evident from the Symposium on Judgment Aggregation in Journal of Economic Theory (List and Polak 2010). The recent conference ‘Judgment aggregation and voting’ in Freudenstadt in 2011 however marks a visible shift of attention towards constructing concrete aggregation rules and finding ‘second best’ solutions in the face of impossibility results. The new proposals range from a first Bordatype aggregation rule (Zwicker 2011) to, among others, new distancebased rules (Duddy and Piggins 2012) and rules which approximate the majority judgments when these are inconsistent (Nehring et al. 2011). The more traditional proposals include premise and conclusionbased rules (e.g., Kornhauser and Sager 1986; Pettit 2001; List and Pettit 2002; Dietrich 2006a; Dietrich and Mongin 2010), sequential rules (e.g., List 2004; Dietrich and List 2007b), distancebased rules (e.g., Konieczny and PinoPerez 2002; Pigozzi 2006; Miller and Osherson 2008; Eckert and Klamler 2009; Hartmann et al. 2010; Lang et al. 2011), and quota rules with wellcalibrated acceptance thresholds and various degrees of collective rationality (e.g., Dietrich and List 2007b; see also Nehring and Puppe 2010a).
The present paper contributes to the theory’s current ‘constructive’ effort by investigating a class of aggregation rules to be called scoring rules. The inspiration comes from classical scoring rules in preference aggregation theory. These rules generate collective preferences which rank each alternative according to the sumtotal ‘score’ it receives from the group members, where the ‘score’ could be defined in different ways, leading to different classical scoring rules such as the Borda rule (see Smith 1973; Young 1975; Zwicker 1991, and for abstract generalizations Myerson 1995; Zwicker 2008; Conitzer et al. 2009; Pivato 2011b). In a general judgment aggregation framework, however, there are no ‘alternatives’; so our scoring rules are based on assigning scores to propositions, not alternatives. Nonetheless, our scoring rules are related to classical scoring rules, and generalize them, as will be shown.
The paradigm underlying our scoring rules—i.e., the maximization of total score of collective judgments—differs from standard paradigms in judgment aggregation, such as the premise, conclusion or distancebased paradigms. Nonetheless, it will turn out that several existing rules can be remodelled as scoring rules, and can thus be ‘rationalized’ in terms of the maximization of total scores. Of course, the way scores are being assigned to propositions—the ‘scoring’—differs strongly across rules; for instance, the Kemeny rule and the premisebased rule can each be viewed as a scoring rule, but with respect to two very different scorings. This paper explores various plausible scorings. It uncovers the scorings which implicitly underlie several wellknown aggregation rules, and suggests other scorings which generate novel aggregation rules. For instance, a particularly natural scoring, to be called reversal scoring, will lead to a new generalization of the Borda rule from preference aggregation to judgment aggregation. The problem of how to generalize the Borda rule has been a longlasting open question in judgment aggregation theory. Recently, an interesting (so far incomplete) proposal was made by Zwicker (2011). Surprisingly, his and the present Borda generalizations are somewhat different, as detailed below.^{1}
Though large, the class of scoring rules is far from universal: some important aggregation rules fall outside this class (notably the mentioned rule approximating the majority judgments, proposed by Nehring et al. 2011). I will also investigate a natural generalization of scoring rules, to be called set scoring rules, which are based on assigning scores to entire judgment sets rather than single propositions (judgments). Such rules are examples of Zwicker’s (2008) ‘generalized scoring rules’ for an abstract aggregation problem whose inputs and outputs are arbitrary objects rather than judgment sets (see also Myerson 1995; Pivato 2011a).^{2}
After this introduction, Sect. 2 defines the general framework, Sect. 3 analyses various scoring rules, Sect. 4 goes on to analyse several set scoring rules, and Sect. 5 draws some conclusions about where we stand in terms of concrete aggregation procedures.
2 The framework, examples and interpretations
I now introduce the framework, following List and Pettit (2002) and Dietrich (2007).^{3} We consider a set of n (\(\ge \)2) individuals, denoted \(N=\{1,\ldots ,n\}\). They need to decide which of certain interconnected propositions to ‘believe’ or ‘accept’.
The agenda The set \(X\) of propositions under consideration is called the agenda. It is subdivided into issues, i.e., pairs of a proposition and its negation, such as ‘it will rain’ and ‘it won’t rain’. Rationally, an agent accepts a proposition from each issue (‘completeness’), while respecting any logical interconnections between propositions (‘consistency’). We write ‘\(\lnot p\)’ for the negation of a proposition ‘\(p\)’, so that the agenda takes the form \(X=\{p,\lnot p,q,\lnot q,\ldots \}\), with issues \(\{p,\lnot p\}, \{q,\lnot q\}\), etc. It is worth defining the present notion of an agenda formally:
Definition 1
 (a)
binary ‘issues’ \(\{p,p^{\prime }\}\) which partition \(X\) (where the members \(p\) and \(p'\) of an issue are the ‘negations’ of each other, written \(p=\lnot p^{\prime } \) and \(p^{\prime }=\lnot p\)),
 (b)
interconnections, i.e., a notion of which subsets of \(X\) are consistent, or formally, a system \(\mathcal C \equiv \mathcal C _{X}\) of (‘consistent’) subsets.^{4}
Given an agenda \(X\), an individual’s judgment set is the set \(J\subseteq X\) of propositions he accepts. It is complete if it contains a member of each issue \(\{p,\lnot p\}\), and (fully) rational if it is complete and consistent. The set of all rational judgment sets is denoted by \(\mathcal J \equiv \mathcal J _{\mathrm{X}}\).
 (b)
interconnections, i.e., a notion of which judgment sets are rational, or formally, a system \(\mathcal J \equiv \mathcal J _{\mathrm{X}} \ne \varnothing \) of (‘rational’) judgment sets \(J\subset X\), each containing exactly one member from any issue.^{5}
 C1:
no set \(\{p,\lnot p \}\) is consistent (selfentailment);
 C2:
subsets of consistent sets are consistent (monotonicity);
 C3:\(\varnothing \) is consistent and each consistent set can be extended to a complete and consistent set (completability). Wellbehavedness can be expressed equivalently by a single condition:

\(\mathcal{C } = \{C \subseteq J : J \in \mathcal{J }\} \ne \varnothing \), i.e., the consistent sets are the subsets of the fully rational sets.

Example 1
Example 2
Some thoughts about the model As an excursion for interested readers, let me add some considerations about the present model and its flexibility. Firstly, I mention three salient ways of specifying an agenda in practice. All three approaches could qualify broadly as ‘logical’:

Under the syntactic approach, the propositions are sentences of formal logic. The structure of the agenda—i.e., the issues \(\{p,\lnot p\}\) and the interconnections—need not be specified explicitly, as it is directly inherited from the logic: it is given by the logic’s negation symbol and consistency notion.^{7} The logic needs to have the right expressive power to adequately render the group’s decision problem: one might use standard propositional logic, or standard predicate logic, or various modal or conditional logics (see Dietrich 2007). Many reallife agendas draw on nonstandard logics by involving for instance modal operators or nonmaterial conditionals. Fortunately, most relevant logics are wellbehaved, i.e., satisfy the conditions C1–C3 (now read as conditions on sentences of the logic), so that the agenda is automatically wellbehaved.

Under the semantic approach, the propositions are subsets of some fixed underlying set \(\Omega \) of possibilities or worlds. The structure of the agenda (the issues \(\{p,\lnot p\}\) and the interconnections) need again not be specified explicitly, as it is inherited from set theory: the negation of a proposition \(p\) is the complement \(\lnot p:=\Omega \backslash p\), and a set of propositions is consistent just in case its intersection is nonempty.^{8} Such a semantic agenda is automatically wellbehaved.^{9}

Under an algebraic (or abstract semantic) approach, the agenda is a subset of an underlying Boolean algebra.^{10} The structure of the agenda (the issues \(\{p,\lnot p\}\) and the interconnections) is once again automatically inherited, this time from the Boolean algebra.^{11} The agenda is automatically wellbehaved.
Secondly, I mention an interpretational point (orthogonal to the question of whether one works with a syntactic, semantic, algebraic or other agenda). Under the standard interpretation, we are aggregating ‘judgments’, i.e., belieftype attitudes towards propositions. But one may reinterpret the nature of the attitude, so that judgment sets become desire sets, or hope sets, or normative approval sets, or intention sets, and so on—which leads to desire aggregation, or hope aggregation, and so on. In this case we still aggregate propositional attitudes, albeit not judgments. In a more radical departure, we may consider the aggregation of arbitrary individual properties (characteristics). Here the agenda contains not propositions which someone may or may not believe (or desire, or hope etc.), but properties which someone may or may not have or satisfy. For instance, the agenda might contain the properties of liking peace, being successful, and so on. Each individual \(i\) has some set of properties \(J_{i}\), and the goal is to derive a collective property set. This is the general property aggregation problem, distinct from the problem of aggregating propositional attitudes such as judgments.^{12}
The mathematics built on the model does not ‘know’ which kind of interpretation or application one has in mind. Nonetheless, interpretation and context matter, since the question of how to best aggregate and which axioms to require depends on it.
3 Scoring rules
3.1 Simple scoring and the Kemeny rule
Simple scoring (2) for the doctrinal paradox agenda and profile
Score of  

\(p\)  \(\lnot p\)  \(q\)  \(\lnot q\)  \(r\)  \(\lnot r\)  \(pqr\)  \(p\lnot q\lnot r\)  \(\lnot pq\lnot r\)  \(\lnot p\lnot q\lnot r\)  
Indiv. 1 \((pqr)\)  1  0  1  0  1  0  3  1  1  0 
Indiv. 2 \((p\lnot q\lnot r)\)  1  0  0  1  0  1  1  3  1  2 
Indiv. 3 \((\lnot pq\lnot r)\)  0  1  1  0  0  1  1  1  3  2 
Group  2  1  2  1  1  2  5*  5*  5*  4 
Proposition 1
The simple scoring rule is the Kemeny rule.
3.2 Classical scoring rules for preference aggregation
3.3 Reversal scoring and a Borda rule for judgment aggregation
Reversal scoring (5) for the doctrinal paradox agenda and profile
Score of  

\(p\)  \(\lnot p\)  \(q\)  \(\lnot q\)  \(r\)  \(\lnot r\)  \(pqr\)  \(p\lnot q\lnot r\)  \(\lnot pq\lnot r\)  \(\lnot p\lnot q\lnot r\)  
Indiv. 1 \((pqr)\)  2  0  2  0  2  0  6  2  2  0 
Indiv. 2 \((p\lnot q\lnot r)\)  1  0  0  2  0  2  1  5  2  4 
Indiv. 3 \((\lnot pq\lnot r)\)  0  2  1  0  0  2  1  2  5  4 
Group  3  2  3  2  2  4  8  9*  9*  8 
The remarkable feature of reversal scoring rule is that it generalizes the Borda rule from preference to judgment aggregation. The Borda rule is initially only defined for the preference agenda \(X\) (for a given finite set of alternatives), namely as the classical scoring rule w.r.t. Borda scoring; see the last subsection. The key observation is that reversal scoring is intimately linked to Borda scoring:
Remark 1
In the case of the preference agenda (for any finite set of alternatives), reversal scoring \(s\) is given by (4) with \(\textit{SCO}\) defined as classical Borda scoring.
Remark 1 and Proposition 2 imply that reversal scoring allows us to extend the Borda rule to arbitrary judgment aggregation problems:
Proposition 3
The reversal scoring rule generalizes the Borda rule, i.e., matches it in the case of the preference agenda (for any finite set of alternatives).
I note that one could use a perfectly equivalent variant of reversal scoring \(s\) which, in the case of the preference agenda, is related to classical Borda scoring \({ SCO}\) via (3) instead of (4):
Remark 2
For comparison, I now sketch Zwicker (2011) interesting approach to extending the Borda rule to judgment aggregation—let me call such an extension a ‘BordaZwicker’ rule. The motivation derives from a geometric characterization of Borda preference aggregation obtained by Zwicker (1991). Let me write the agenda as \(X=\{p_{1},\lnot p_{1},p_{2},\lnot p_{2} ,\ldots ,p_{m},\lnot p_{m}\}\), where \(m\) is the number of ‘issues’. Each profile gives rise to a vector \(\mathbf{v}\equiv (v_{1},\ldots ,v_{m})\) in \(\mathbb R ^{m}\) whose jth entry \(v_{j}\) is the net support for \(p_{j}\), i.e., the number of individuals accepting \(p_{j}\) minus the same number for \(\lnot p_{j}\). Now if \(X\) is the preference agenda for any finite set of alternatives \(A\), then each \(p_{j}\) takes the form \(xPy\) for certain alternatives \(x,y\in A\). Each preference cycle can be mapped to a vector in \(\mathbb R ^{m}\); for instance, if \(p_{1}=xPy, p_{2}=yPz\) and \(p_{3}=xPz\), then the cycle \(x\succ y\succ z\succ x\) becomes the vector \((1,1,1,0,\ldots ,0)\in \mathbb R ^{m}\). The linear span of all vectors corresponding to preference cycles defines the socalled ‘cycle space’ \(\mathbf{V}_{cycle}\subseteq \mathbb R ^{m}\), and its orthogonal complement defines the ‘cocycle space’ \(\mathbf{V}_{cocycle}\subseteq \mathbb R ^{m}\). Let \(\mathbf{v}_{cocycle}\) be the orthogonal projection of \(\mathbf{v}\) on the cocycle space \(\mathbf{V}_{cocycle}\). Intuitively, \(\mathbf{v}_{cocycle}\) contains the ‘consistent’ or ‘acyclic’ part of \(\mathbf{v}\). The upshot is that the Borda outcome can be read off from \(\mathbf{v}_{cocycle}\): for each \(p_{j}=xPy\), the Borda group preference ranks \(x\) above (below) \(y\) if the jth entry of \(\mathbf{v}_{cocycle}\) is positive (negative). Zwicker’s strategy for extending the Borda rule to judgment aggregation is to define a subspace \(\mathbf{V}_{cycle}\) analogously for agendas other than the preference agenda; one can then again project \(\mathbf{v}\) on the orthogonal complement of \(\mathbf{V}_{cycle}\) and determine collective ‘Borda’ judgments according to the signs of the entries of this projection. This approach has proved successful for simple agendas, in which there is a natural way to define \(\mathbf{V}_{cycle}\). Whether the approach is viable for general agendas (i.e., whether \(\mathbf{V}_{cycle}\) has a useful general definition) seems to be open so far.^{16}
A BordaZwicker rule is not just constructed differently from a scoring rule in our sense, but, as I conjecture, it also cannot generally be remodelled as a scoring rule, since most interesting scoring rules use information that goes beyond the information contained in the profile’s ‘net support vector’ \(\mathbf{v}\in \mathbb R ^{m}\). (Even more does the required information go beyond the projection of \(\mathbf{v}\) on the orthogonal complement of \(\mathbf{V}_{cycle}\).)
In summary, there seem to exist two quite different approaches to generalizing Borda aggregation. One approach, taken by Zwicker, seeks to filter out the profile’s ‘inconsistent component’ along the lines of the justdescribed geometric technique. The other approach, taken here, seeks to retain the principle of scoremaximization inherent in Borda aggregation (with scoring now defined at the level of propositions, not alternatives, as these do not exist outside the world of preferences). One might view the normative core of the scoring approach as being the use of the strength of someone’s judgments, i.e., the strength of acceptance of propositions, as measured by the score—just as Borda preference aggregation is sometimes interpreted as a way to incorporate the strength of someone’s preference between alternatives \(x\) and \(y\), as measured by the score of \(xPy\), i.e., the difference between \(x\)’s and \(y\)’s score.^{17}
3.4 A generalization of reversal scoring
For instance, we could adopt Duddy and Piggins’ 2012 distance function, i.e., we could define \(d(J,J^{\prime })\) as the number of minimal consistent modifications needed to transform \(J\) into \(J^{\prime } \).^{18} So, while Duddy and Piggins had introduced their distance in the context of distancebased aggregation to develop an alternative to the Kemeny rule, their distance can also be used in our context of (reversal) scoring rules to develop an alternative to Borda aggregation.
3.5 Scoring based on logical entrenchment
We now consider scoring rules which explicitly exploit the logical structure of the agenda. Let us think of the score of a proposition p (\(\in \)X) given the judgment set J (\(\in \mathcal J \)) as the degree to which \(p\) is logically entrenched in the belief system \(J\), i.e., as the ‘strength’ with which \(J\) entails \(p\). We measure this strength by the number of ways in which \(p\) is entailed by \(J\), where each ‘way’ is given by a particular judgment subset \(S\subseteq J\) which entails \(p\), i.e., for which \(S\cup \{\lnot p\}\) is inconsistent. If \(J\) does not contain \(p\), then no judgment subset—not even the full set \(J\)—can entail \(p\); so the strength of entailment (score) of \(p\) is zero. If \(J\) contains \(p\), then \(p\) is entailed by the judgment subset \(\{p\}\), and perhaps also by very different judgment subsets; so the strength of entailment (score) of \(p\) is positive and more or less high.
There are different ways to formalise this idea, depending on precisely which of the judgment subsets that entail \(p\) are deemed relevant. I now propose four formalizations. Two of them will once again allow us to generalize the Borda rule from preference to judgment aggregation. These generalizations differ from that based on reversal scoring in Sect. 3.3.
Disjointentailment scoring (8) for the doctrinal paradox agenda and profile
Score of  

\(p\)  \(\lnot p\)  \(q\)  \(\lnot q\)  \(r\)  \(\lnot r\)  \(pqr\)  \(p\lnot q\lnot r\)  \(\lnot pq\lnot r\)  \(\lnot p\lnot q\lnot r\)  
Indiv. 1 \((pqr)\)  2  0  2  0  2  0  6  2  2  0 
Indiv. 2 \((p\lnot q\lnot r)\)  1  0  0  2  0  2  1  5  2  4 
Indiv. 3 \((\lnot pq\lnot r)\)  0  2  1  0  0  2  1  2  5  4 
Group  3  2  3  2  2  4  8  9*  9*  8 
Disjointentailment scoring turns out to match reversal scoring for our doctrinal paradox agenda (check that Tables 3, 4 coincide), as well as for the preference agenda (as shown later). Is this pure coincidence? The general relationship is that the disjointentailment score of a proposition \(p\) is always at most the reversal score, as one may show.^{19}
Proposition 4
Disjointentailment scoring (8) and irreducibleentailment scoring (10) match reversal scoring (5) in the case of the preference agenda (for any finite set of alternatives).
Propositions 3 and 4 jointly have an immediate corollary.
3.6 Propositionwise scoring and a way to repair quota rules with nonrational outputs
We now consider a special class of scorings: propositionwise scorings. This will allow us to relate scoring rules to the wellknown judgment aggregation rules called quota rules (Dietrich and List 2007b)—in fact, to ‘repair’ these rules by rendering their outcomes rational across all profiles.
A nontrivial question is how the acceptance thresholds would have to be set to ensure that the collective judgment set satisfies some given degree of rationality, such as to be (i) consistent, or (ii) deductively closed, or (iii) consistent and deductively closed, or even (iv) fully rational, i.e., in \(\mathcal J \). These questions have been settled [see Nehring and Puppe 2010a for (iv), and, subsequently, Dietrich and List 2007b for (i)–(iv)]. Unfortunately, for many agendas the thresholds would have to be set at ‘extreme’ and normatively unattractive levels. Worse, often no thresholds achieve (iv) (see Nehring and Puppe 2010a). For our doctrinal paradox agenda \(X=\{p,q,r\}^{\pm }\) only the extreme thresholds \(m_{p} =m_{q}=m_{r}=n\) and \(m_{\lnot p}=m_{\lnot q}=m_{\lnot r}=1\) achieve (iv), and for the preference agenda (with more than two alternatives) no thresholds achieve (iv).
Proposition 5
For every threshold family \((m_{p})_{p\in X}\), the scoring rule w.r.t. scoring (12) matches the quota rule \(F_{(m_{p})_{p\in X}}\) at all profiles where the quota rule generates rational outputs (and still generates rational outputs at all other profiles).
Scoring (12) for the doctrinal paradox agenda and profile
Score of  

\(p\)  \(\lnot p\)  \(q\)  \(\lnot q\)  \(r\)  \(\lnot r\)  \(pqr\)  \(p\lnot q\lnot r\)  \(\lnot pq\lnot r\)  \(\lnot p\lnot q\lnot r\)  
Indiv. 1 \((pqr)\)  2  2  2  2  2  3  6  3  3  7 
Indiv. 2 \((p\lnot q\lnot r)\)  2  2  2  2  2  1  \(2\)  5  3  1 
Indiv. 3 \((\lnot pq\lnot r)\)  2  2  2  2  2  1  \(2\)  3  5  1 
Group  2  2  2  2  2  1  2*  1  1  5 
How does our scoring rule ‘repair’ those special quota rules which use a uniform threshold \(m\equiv m_{p}\ (p\in X)\), such as majority rule?
Remark 3
For a uniform threshold \(m\equiv m_{p}\), the scoring rule w.r.t. scoring (12) is the Kemeny rule, or equivalently, the simple scoring rule.
This remark follows from Proposition 1 and the fact that, for a uniform threshold \(m\equiv m_{p}\), scoring (12) is equivalent to simple scoring by footnote 13.
3.7 Premise and conclusionbased aggregation
I have just mentioned the possibility of a differential treatment of propositions when ‘repairing’ a quota rule. This possibility is particularly salient in the popular context of premise or conclusionbased aggregation.^{24} One may indeed view the classical premise and conclusionbased rules as two (rival) ways of repairing the simplest of all quota rules—majority rule—by privileging certain propositions over others, namely premise propositions or conclusion propositions, respectively.

the premisebased rule is the aggregation rule which for each profile in \(\mathcal J ^{n}\) delivers the (unique) judgment set in \(\mathcal J \) containing each premise proposition accepted by a majority;

the conclusionbased rule is the aggregation rule which for each profile in \(\mathcal J ^{n}\) delivers the judgment set (or sets) in \(\mathcal J \) containing the conclusion proposition accepted by a majority.^{26}
Remark 4

the premisebased rule if and only if \(\alpha _{\mathrm{premise}}>(n2)\alpha _{\mathrm{conclusion}}\),

the conclusionbased rule if and only if \(\alpha _{\mathrm{conclusion} }>\alpha _{\mathrm{premise}}=0\).
This result lets premise and conclusionbased aggregation appear in a rather extreme light: each rule is based on somewhat unequal importance coefficients \(\alpha _{\mathrm{premise}}\) and \(\alpha _{\mathrm{conclusion}}\), deeming one type of proposition to be overwhelmingly more important than the other. It might therefore be interesting to consider more equilibrated values of the importance coefficients, so as to achieve a compromise between democracy at the premise level and democracy at the conclusion level.
4 Set scoring rules: assigning scores to entire judgment sets
4.1 Naive set scoring and plurality voting
Remark 5
The naive set scoring rule is the plurality rule.
4.2 Distancebased set scoring
Remark 6
For every given distance function over \(\mathcal J \), the distancebased set scoring rule is the distancebased rule.
4.3 Averaging rules
Given an ordinary scoring \(s\), we have so far aimed for collective judgments with a high total score. But this is not the only plausible aim or approach. We now turn to an altogether different approach. Rather than using \(s\) to assign scores only from each individual’s perspective, we now care about how propositions score under the collective judgment set. Instead of wanting the collective judgments to achieve the highest total score from individuals, we now want them to resemble the ‘average individual judgments’ in the sense that the collective judgments should lead (approximately) to the same scores of propositions as the individual judgments do on average. In short, any proposition \(p\)’s collective score should be (approximately) \(p\)’s average individual score. This approach has its own, rather different intuitive appeal. But is it really totally different? As will turn out, aggregation rules which follow this approach—I call them ‘averaging rules’ as opposed to ‘scoring rules’—can be viewed as a particular kind of set scoring rules. This result is essentially a special case of a powerful precursor result by Zwicker (2008), as Marcus Pivato kindly pointed out to me. A comparison with Zwicker (2008) is given at the end of this subsection.
Given an ordinary scoring \(s\), we can represent judgment sets in \(\mathcal J \) as vectors in \(\mathbb R ^{X}\), by identifying each judgment set \(J\) in \(\mathcal J \) with its score vector, i.e., the vector in \(\mathbb R ^{X}\) whose \(p^{\mathrm{th}}\) component is the score of \(p, s_{J} (p)\).^{30} The score vector corresponding to \(J\in \mathcal J \) is denoted \(J^{s}\equiv (s_{J}(p))_{p\in X}\in \mathbb R ^{X}\). Having represented judgment sets as vectors of numbers, we can apply standard algebraic and geometric operations, such as adding judgment sets, taking their average, or measuring their distance—where, of course, sums or averages of (score vectors of) judgment sets in \(\mathcal J \) may be ‘infeasible’, i.e., not correspond to any judgment set in \(\mathcal J \).
The averaging rule (w.r.t. reversal scoring) for the doctrinal paradox agenda and profile
\(p\)  \(\lnot p\)  \(q\)  \(\lnot q\)  \(r\)  \(\lnot r\)  Distance to group’s average  

\(pqr\) (indiv. 1)  2  0  2  0  2  0  \(\sqrt{58}/3\approx 2.54\) 
\(p\lnot q\lnot r\) (indiv. 2)  1  0  0  2  0  2  \(\sqrt{37}/3\approx 2.03\) 
\(\lnot pq\lnot r\) (indiv. 3)  0  2  1  0  0  2  \(\sqrt{37}/3\approx 2.03\) 
\(\lnot p\lnot q\lnot r\) (no indiv.)  0  1  0  1  0  3  \(7/3\approx 2.33\) 
Group’s average  1  \(\frac{2}{3}\)  1  \(\frac{2}{3}\)  \(\frac{2}{3}\)  \(\frac{4}{3}\)  0 
Proposition 6
For any scoring \(s\), the averaging rule w.r.t. \(s\) is the set scoring rule w.r.t. set scoring (17).
Corollary 2
The Kemeny rule is both the scoring rule and the averaging rule w.r.t. simple scoring.
Our averaging rules are special cases of Zwicker’s (2008) mean proximity rules. Like an averaging rule, a mean proximity rule represents each possible vote as well as each possible output as a vector of characteristics (real numbers), and determines the winning output(s) by minimizing the distance—in that vector representation—to the average vote. In our case, this vector is \(\left X\right \)dimensional and contains the support (score) given to each proposition in \(X\). In Zwicker’s more general case, the vector could have any dimensionality and could contain any sorts of characteristics. Moreover, the votes need not be judgment sets, but could for instance be political candidates, characterized, say, by a threedimensional vector containing age, number of preelection speeches, and political orientation on a leftright spectrum. One may therefore view our averaging rules as a concrete way to implement Zwicker’s general approach inside judgment aggregation theory: we offer a concrete method of choosing the vector representation of votes, which is left open in Zwicker’s approach.^{33} Alternative vector representations are of course also imaginable.
4.4 Probabilitybased set scoring
I close the analysis by taking a brief (skippable) excursion into an important, but different approach to judgment aggregation: the epistemic or truthtracking approach. In this approach, each proposition \(p\in X\) is taken to have an objective, but unknown truth value (‘true’ or ‘false’), and the goal of aggregation is to track the truth, i.e., to generate true collective judgments.^{34} The truthtracking perspective has a long history elsewhere in social choice theory (e.g., Condorcet 1785; Grofman et al. 1983; AustenSmith and Banks 1996; Dietrich 2006b; Pivato 2011a); but within judgment aggregation theory specifically, rather little work has been done on the epistemic side (e.g., Bovens and Rabinowicz 2006; List 2005; Bozbay et al. 2011).
The epistemic approach warrants the use of particular set scoring rules. To show this, I import standard statistical estimation techniques (such as maximumlikelihood estimation), following the path taken by other authors in the context of preference aggregation (e.g., Young 1995) and other aggregation problems (e.g., Dietrich 2006b; Pivato 2011a). My goal is to give no more than a brief introduction to what could be done. The results given below are essentially variants of existing results; see in particular Pivato (2011a).^{35}
For each combination \((J_{1},\ldots ,J_{n},T)\in \mathcal J ^{n}\times \mathcal J \) of \(n+1\) judgment sets, let \(\Pr (J_{1},\ldots ,J_{n},T) >0\) measure the probability that people submit the profile \((J_{1},\ldots ,J_{n})\) and the set of true propositions is \(T\), where of course \(\sum _{(J_{1},\ldots ,J_{n} ,T)\in \mathcal J ^{n}\times \mathcal J } \Pr (J_{1},\ldots ,J_{n},T)=1\). From this joint probability function we can, as usual, derive various marginal and conditional probabilities, such as the probability that the truth is \(T\in \mathcal J , \Pr (T)=\sum _{(J_{1},\ldots ,J_{n})\in \mathcal J ^{n}}\Pr (J_{1},\ldots , J_{n},T)\), the probability that the profile is \((J_{1} ,\ldots ,J_{n}), \Pr (J_{1},\ldots ,J_{n})=\sum _{T\in \mathcal J }\Pr (J_{1} ,\ldots ,J_{n}, T)\), the conditional probability \(\Pr (TJ_{1},\ldots ,J_{n} )=\frac{\Pr (J_{1},\ldots ,J_{n},T)}{\Pr (J_{1},\ldots ,J_{n})}\) (called the posterior probability of \(T\) given the ‘data’ \(J_{1},\ldots ,J_{n}\)), and the conditional probability \(\Pr (J_{1},\ldots ,J_{n}T)=\frac{\Pr (J_{1} ,\ldots ,J_{n},T)}{\Pr (T)}\) (called the likelihood of the ‘data’ \(J_{1},\ldots ,J_{n}\) given \(T\)).
 (IND)
the individual judgment sets are independent conditional on \(T\) being the true judgment set, i.e., \(\Pr (J_{1},\ldots ,J_{n}T)=\Pr (J_{1} T)\cdots \Pr (J_{n}T)\) for all \(J_{1},\ldots ,J_{n}\in \mathcal J \) (‘independence’)
 (COM)
for each \(J\in \mathcal J \), each individual has the same probability, denoted \(\Pr (JT)\), of submitting the judgment set \(J\) conditional on \(T\) being the true judgment set (‘equal competence’).
5 Concluding remarks
While the positions of most rules in Fig. 1 have been established above or follow easily, a few positions are of the order of conjectures. This is so for the placement of our Borda generalization outside the class of distancebased rules.^{37}
Though several old and new aggregation rules are scoring rules (or at least set scoring rules), there are important counterexamples. One counterexample is the mentioned rule introduced by Nehring et al. (2011) (the socalled Condorcetadmissibility rule, which generates rational judgment set(s) that ‘approximate’ the majority judgment set). Other counterexamples are nonanonymous rules (such as rules prioritizing experts), and rules that return boundedly rational collective judgments (such as rules returning incomplete but still consistent and deductively closed judgments). The last two kinds of counterexamples suggest two generalizations of the notion of a scoring rule. Firstly, scoring might be allowed to depend on the individual; this leads to ‘nonanonymous scoring rules’. Secondly, the search for a collective judgment set with maximal total score might be done within a larger set than the set \(\mathcal J \) of fully rational judgment sets (such as the set of consistent but possibly incomplete judgment sets); this leads to ‘boundedly rational scoring rules’. The same generalizations could of course be made for set scoring rules.
Many challenges lie ahead of us. For instance, it would be interesting to characterize scoring rules and set scoring rules by some salient axiomatic properties.^{38} And, in the epistemic context of searching for objectively ‘true’ or ‘correct’ judgments, one might give a maximumlikelihood rationalization of (set) scoring rules, by proving that they are precisely the aggregation rules which generate maximumlikelihood estimators of the truth relative to some probability model from a suitable class. Clearly, the paper raises more questions than it answers.
Footnotes
 1.
Conal Duddy and Ashley Piggins also have independent work in progress on ‘generalizing Borda’, and I learned from Klaus Nehring that he had ideas similar to those in the present paper.
 2.
If the inputs and outputs of that aggregation problem are taken to be judgment sets, then Zwicker’s ‘generalized scoring rules’ become our set scoring rules (suitably translated into Zwicker’s setting of an anonymous variable population).
 3.
To be precise, I use a slimmer variant of their models, since the logic in which propositions are formed is not explicitly part of the model.
 4.
Algebraically, the agenda is thus the structure \((X,\mathcal I ,\mathcal C )\) (where \(\mathcal I \) is the partition into issues), or alternatively the structure \((X,\lnot ,\mathcal C )\). To see why the issues and the negation operator are interdefinable, note that we could alternatively start with an operator \(\lnot \) on \(X\) satisfying \(\lnot p\ne p=\lnot \lnot p\), and then define the issues as the pairs \(\{p,\lnot p\}\).
 5.
So, algebraically speaking, the agenda is the structure \((X,\mathcal I ,\mathcal J )\) instead of \((X,\mathcal I ,\mathcal C )\) (where \(\mathcal I \) is the partition of \(X\) into issues, replaceable by the negation operator \(\lnot \) on \(X\)).
 6.
More generally, dropping the requirement of collective rationality, we have a correspondence \(\mathcal J ^{n}\rightrightarrows 2^{X}\), where \(2^{X}\) is the set of all judgment sets, rational or not. As usual, I write ‘\(\rightrightarrows \)’ instead of ‘\(\rightarrow \)’ to indicate a multifunction.
 7.
Formally, the agenda \(X\) is any set of logical sentences which is partitionable into pairs containing a sentence and its negation. These pairs define the issues. (We have presupposed that the logic includes the negation symbol, as all interesting logics do.)
 8.
Formally, the agenda \(X\) is any set of subsets of \(\Omega \) such that \(p\in X\Rightarrow \Omega \backslash p\in X\).
 9.
Nehring and Puppe’s (2010a) property spaces are essentially semantically defined agendas.
 10.
A Boolean algebra is a distributive lattice \(\mathcal L \) (with its operations of join and meet) in which there exists a top element \(\intercal \) (tautology) and a bottom element \(\perp \) (contradiction) and in which every element \(p\) has a negation/complement (i.e., an element whose join with \(p\) is \(\intercal \) and whose meet with \(p\) is \(\perp \)). An important example is a concrete Boolean algebra \(\mathcal L \subseteq 2^{\Omega }\) (for an underlying set of worlds \(\Omega \)), in which the join is given by the union, the meet by the intersection, the top by \(\Omega \), the bottom by \(\varnothing \), and the negation by the settheoretic complement. In this case, the algebraic approach reduces to the standard semantic approach. Another example is the Boolean algebra generated from a logic, i.e., the set of sentences modulo logical equivalence (where the logic includes classical negation and conjunction, which induce the algebra’s join, meet and complement operations).
 11.
Formally, the agenda \(X\) is any subset of the Boolean algebra which is closed under Booleanalgebraic negation/complementation (see footnote 10).
 12.
Property aggregation raises the question of what it means for the collective to ‘have’ a property. Presumably, collective properties are something quite different from individual properties, just as collective judgments (or desires, hopes, \(\ldots \)) do not have the same status as individual ones. More generally, one may of course doubt the meaningfulness and relevance of group attributes. From a behaviourist perspective, the two ‘Humean’ collective attitudes—group beliefs (judgments) and group desires (preferences)—can be given meaning based on group behaviour, at least in principle. From my own (nonbehaviourist) perspective, all sorts of nonstandard group attributes can be meaningful and relevant, despite possibly being underdetermined by group action. But even if such group attributes are deemed meaningless or irrelevant, one may reinterpret them as being no more than summaries of the group members’ attributes (rather than attributes of any separate group agent).
 13.
More generally, certain increasing transformations have no effect. As one may show, scorings \(s\) and \(s^{\prime }\) are equivalent (i.e., \(F_{s}=F_{s^{\prime }}\)) whenever there are coefficients \(a>0\) and \(b_{p}\in \mathbb R \, (p\in X)\) with \(b_{p}=b_{\lnot p}\) for all \(p\in X\) such that \(s^{\prime }\) is given by \(s_{J}^{\prime }(p)=as_{J} (p)+b_{p}\).
 14.
A distance function or metric over \(\mathcal J \) is a function \(d:\mathcal J \times \mathcal J \rightarrow [0,\infty )\) satisfying three conditions: for all \(J,K,L\in \mathcal J \), (i) \(d(J,K)=0\Leftrightarrow J=K\), (ii) \(d(J,K)=d(K,J)\) (‘symmetry’), and (iii) \(d(J,L)\le d(J,K)+d(K,L)\) (‘triangle inequality’).
 15.
A technical difference between the standard notion of a scoring rule in preference aggregation theory and our judgmenttheoretic rendition of it arises when there happen to exist distinct alternatives with identical sumtotal score. In such cases, the standard scoring rule returns collective indifferences, whereas our \(F_{{ SCO}}\) returns a tie between strict preferences. From a formal perspective, however, the two definitions are equivalent, since to any weak order corresponds the set (tie) of all strict linear orders which linearize the weak order by breaking its indifferences (in any cyclefree way). The structural asymmetry between input and output preferences of scoring rules as defined standardly (i.e., the possibility of indifferences at the collective level) may have been one of the obstacles—albeit only a small, mainly psychological one—for importing scoring rules and Borda aggregation into judgment aggregation theory.
 16.
One might at first be tempted to generally define \(\mathbf{V}_{cycle}\) as the linear span of those vectors which correspond to the agenda’s minimal inconsistent subsets. Unfortunately, this span is often the entire space \(\mathbb R ^{m}\), an example for this being our doctrinal paradox agenda.
 17.
By contrast, Condorcet’s rule of pairwise majority voting only cares about whether or not someone prefers an alternative over another, without attempting to extract strengthofpreference information from the person’s full preference relation. It is of course notoriously controversial whether strength or intensity of preference is a permissible concept—and even if it is, whether the score of \(xPy\) is an adequate proxy of the strength to which \(x\) is preferred to \(y\). The ordinalist approach takes a sceptical stance here. Note however that Borda aggregation can be defended for reasons unrelated to strength of preference. For instance, its axiomatic characterizations offer possible normative defences.
 18.
Judgment sets \(J,J^{\prime }\in \mathcal J \) are minimal consistent modifications of each other if the set \(S=J\backslash J^{\prime }\) of propositions in \(J\) which need to be negated to transform \(J\) into \(J^{\prime }\) is nonempty and minimal (i.e., \(J\) couldn’t have been transformed into a consistent set by negating only a strict nonempty subset of \(S\)). For our doctrinal paradox agenda, the judgment sets \(pqr\) and \(p\lnot q\lnot r\) are minimal consistent modifications of each other, and hence have Duddy–Pigginsdistance of 1.
 19.
The reason is that, given \(m\) mutually disjoint judgment subsets which each entail \(p\), the reversal score of \(p\) is at least \(m\) since one must negate at least one proposition from each of these \(m\) sets in order to consistently reject \(p\).
 20.
Assuming that \(p\) is not a tautology, i.e., that \(\{\lnot p\}\) is consistent. (Otherwise, \(\varnothing \) minimally entails \(p\).)
 21.
Assuming \(X\) contains no tautology, i.e., no \(p\) such that \(\{\lnot p\}\) is inconsistent.
 22.
They are the only independent rules which are anonymous, monotonic and unanimitypreserving.
 23.
This follows from Nehring and Puppe’s (2010) intersection property, generalized to possibly incomplete collective judgment sets (Dietrich and List 2007b).
 24.
 25.
Our analysis generalizes easily to any \(X\) and \(P\) such that (i) the premise propositions in \(P\) are logically independent, and (ii) complete judgments across the premise propositions in \(P\) uniquely determine the judgments on the conclusion propositions in \(X\backslash P\).
 26.
In the literature, the conclusionbased procedure is usually taken to be silent on the premises, i.e., to return an incomplete judgment set not in \(\mathcal J \). I have replaced this silence by a tie between all compatible judgments on the premise propositions.
 27.
It ignores the internal structure of judgment sets, hence ‘throws away’ much information.
 28.
In trying to remodel an arbitrary set scoring rule \(F_{\sigma }\) as a distancebased rule, one might be tempted to define the ‘distance’ between \(J \) and \(J^{\prime }\) as \(d_{\sigma }(J,J^{\prime } ):=\sigma _{J}(J)\sigma _{J}(J^{\prime })\). If \(d_{\sigma }\) turns out to define a proper distance function (see fn. 14), then we obtain a distancebased rule \(F_{d_{\sigma }}\), which coincides with the set scoring rule \(F_{\sigma }\). But for many plausible set scorings \(\sigma , d_{\sigma }\) has little in common with a distance function, violating up to all three axioms, notably symmetry and the triangle inequality.
 29.
This metric derives its name from the fact that it induces the discrete topology on whatever set it is defined on (such as \(\mathbb R \) instead of \(\mathcal J \)).
 30.
This identification is onetoone as long as the scoring has the (very plausible) property that \(s_{J}(p)>s_{J}(\lnot p)\) whenever \(p\in J\).
 31.
Formally, \(F(J_{1},\ldots ,J_{n})^{s}=\hbox {PROJ}_\mathcal{J ^{s}}(\frac{1}{n}\sum _{i}J_{i}^{s})\), where the projection of \(x\in \mathbb R ^{X}\) on \(Y\subseteq \mathbb R ^{X}\) is defined as \(\hbox {PROJ}_{Y}(x):={\arg \!\min }_{y\in Y}\left\ yx\right\ \).
 32.
Since \(\sigma _{J}(C)=\left( \sqrt{\left C\bigtriangleup J\right }\right) ^{2}=\left C\bigtriangleup J\right =2\left C\backslash J\right =2\left( \left C\right \left C\cap J\right \right) =\left X\right +2\left C\cap J\right \).
 33.
Zwicker’s Theorem 4.2.1 (more precisely, its proof) essentially implies our Proposition 6 after some translation work.
 34.
The epistemic perspective is usually contrasted with the procedural perspective, which takes the goal of aggregation to be to generate collective judgments which reflect the individuals’ judgments in a procedurally fair way. To illustrate the contrast between the two perspectives, suppose that all individuals hold the same judgment set \(J\). Then \(J\) is clearly the right collective judgment set from the perspective of procedural fairness. But from an epistemic perspective, all depends on whether people’s unanimous endorsement of \(J\) is sufficient evidence for \(J\) being true.
 35.
 36.
The classical Condorcet jury theorem is essentially concerned with a simple judgment aggregation problem with a binary agenda \(X=\{p,\lnot p\}\).
 37.
For technical correctness, I also note two details about how to read Fig. 1. First, for trivial agendas, such as a singleissue agenda \(X=\{p,\lnot p\}\), several rules of course become equivalent, and distinctions drawn in
Fig. 1 disappear. More precisely, by positioning a rule outside a class of rules (e.g., by positioning plurality rule outside the class of scoring rules), I am of course not implying that for all agendas the rule does not belong to the class, but that for some (in fact, most) agendas this is so. Second, in placing propositionwise scoring rules among the distancebased rules, I made a very plausible restriction: \(s_{+}(p)>s_{}(p)\) for each \(p\in X\).
 38.
Notes
Acknowledgments
I am grateful for very useful suggestions received from two anonymous referees and the editor.
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