Scoring rules for judgment aggregation

This paper studies a class of judgment aggregation rules, to be called (cid:145)scoring rules(cid:146) after their famous counterpart in preference aggregation theory. A scoring rule delivers the collective judgments which reach the highest total (cid:145)score(cid:146) across the individuals, subject to the judgments having to be rational. Depending on how we de(cid:133)ne (cid:145)scores(cid:146), we obtain several (old and new) solutions to the judgment aggregation problem, such as distance-based aggregation, premise- and conclusion-based aggregation, truth-tracking rules, and a generalization of Borda rule to judgment aggregation. Scoring rules are shown to generalize the classical scoring rules of preference aggregation theory.


Introduction
The judgment aggregation problem consists in merging many individuals' yes/no judgments on some interconnected propositions into collective yes/no judgments on these propositions. The classical example, born in legal theory, is that three jurors in a court trial disagree on which of the following three propositions are true: the defendant has broken the contract (p); the contract is legally valid (q); the defendant is liable (r). According to a universally accepted legal doctrine, r (the 'conclusion') is true if and only if p and q (the two 'premises') are both true. So, r is logically equivalent to p^q. The simplest rule to aggregate the jurors' judgments -namely propositionwise majority voting -may generate logically inconsistent collective judgments, as Table 1 illustrates. There are of course nupremise p premise q conclusion r (,  ', i.e., kinds of interconnected propositions a group might face. Preference aggregation is a special case with propositions of the form 'x is better than y' (for many alternatives x and y), where these propositions are interconnected through standard conditions such as transitivity. In this context, Condorcet's classical voting paradox about cyclical majority preferences is nothing but another example of inconsistent majority judgments. Starting with List and Pettit's (2002) seminal paper, a whole series of contributions have explored which judgment aggregation rules can be used, depending on, …rstly, the agenda in question, and, secondly, the requirements placed on aggregation, such as anonymity, and of course the consistency of collective judgments. Some theorems generalize Arrow's Theorem from preference to judgment aggregation List 2007, Dokow andHolzman 2010; both build on Nehring and Puppe 2010a and strengthen Wilson 1975). Other theorems have no immediate counterparts in classical social choice theory (e.g., List 2004, Dietrich 2006a, 2010, Nehring and Puppe 2010b, Dietrich and Mongin 2010. It is fair to say that judgment aggregation theory has until recently been dominated by 'impossibility' …ndings, as is evident from the Symposium on Judgment Aggregation in Journal of Economic Theory (C. List and B. Polak eds., 2010, vol. 145 (2)). The recent conference 'Judgment aggregation and voting' (Freudenstadt, 2011) however marks a visible shift of attention towards constructing concrete aggregation rules and …nding 'second best' solutions in the face of impossibility results. The new proposals range from a …rst Borda-type aggregation rule (Zwicker 2011) to, among others, new distance-based rules (Duddy and Piggins 2011) and rules which approximate the majority judgments when these are inconsistent (Nehring, Pivato and Puppe 2011). The more traditional proposals include premise-and conclusion-based rules (e.g., Kornhauser and Sager 1986, Pettit 2001, List & Pettit 2002, Dietrich 2006, Dietrich and Mongin 2010, sequential rules (e.g., List 2004, Dietrich andList 2007b), distance-based rules (e.g., Konieszni and Pino-Perez 2002, Pigozzi 2006, Miller and Osherson 2008, Eckert and Klamler 2009, Hartmann et al. 2010, Lang et al. 2011, and quota rules with well-calibrated 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' e¤ort 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 sum-total 'score' it receives from the group members, where the 'score' could be de…ned in di¤erent ways, leading to di¤erent classical scoring rules such as Borda rule (see Smith 1973, Young 1975, Zwicker 1991, and for abstract generalizations Myerson 1995, Zwicker 2008and 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 -di¤ers from standard paradigms in judgment aggregation, such as the premise-, conclusion-or distance-based paradigms. Nonetheless, it will turn out that several existing rules can be re-modelled 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' -di¤ers strongly across rules; for instance, the Kemeny rule and the premise-based rule can each be viewed as a scoring rule, but with respect to two very di¤erent scorings. This paper explores various plausible scorings. It uncovers the scorings which implicitly underlie several well-known 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 Borda rule from preference aggregation to judgment aggregation. The problem of how to generalize Borda rule has been a long-lasting open question in judgment aggregation theory. Recently, an interesting (so far incomplete) proposal was made by William Zwicker (2011). Surprisingly, his and the present Borda generalizations are distinctively di¤erent, as detailed below. 2 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, by Nehring, Pivato and Puppe 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). Set scoring rules are for instance interesting in the context of epistemic ('truth-tracking') aggregation models, where they have recently been studied by Pivato (2011a).
After this introduction, Section 2 de…nes the general framework, Section 3 analyses various scoring rules, Section 4 goes on to analyse several set scoring rules, and Section 5 draws some conclusions about where we stand in terms of concrete aggregation procedures.

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 ( 2) individuals, denoted N = f1; :::; ng. 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 exactly one proposition from each issue ('completeness'), while respecting any logical interconnections between propositions ('consistency'). We write ':p' for the negation of a proposition 'p', so that the agenda takes the form X = fp; :p; q; :q; :::g, with issues fp; :pg, fq; :qg, etc. It is worth de…ning the present notion of an agenda formally: De…nition 1 An agenda is a set X (containing the propositions) which is: (a) partitioned into pairs fp; p 0 g (the issues, where the members p and p 0 of an issue are the negations of each other, written p :p 0 and p 0 :p); (b) endowed with logical interconnections, i.e., a notion of which subsets of X are consistent, or formally, a system C C X of subsets (the consistent sets). 4 A simple example is the agenda given by X = fp; :p; q; :q; p^q; :(p^q)g; where p and q are two atomic sentences, for instance 'it rains' and 'it is cold', and p^q is their conjunction. Here, propositions are formulated as logical sentences. This is an example of the logical, or more precisely syntactic, approach of de…ning an agenda. This approach is particularly natural, partly because the structure of the agenda -i.e., the partition into issues and the interconnections -need not be speci…ed explicitly as it is directly inherited from logic. Logic for instance tells us that the set fp; p^qg is consistent while the set f:p; p^qg is not.
Given an agenda X, an individual's judgment set is the set J X of propositions he accepts. It is complete if it contains a member of each pair p; :p 2 X, and (fully) rational if it is complete and consistent. The set of all rational judgment sets is denote by J . Notationally, a judgment set J X is often abbreviated by concatenating its members in any order (so, p:q:r is short for fp; :q; :rg); and the negation-closure of a set Y X is denoted Y fp; :p : p 2 Y g.
We now introduce the two lead examples of this paper, the …rst one being isomorphic to the previous example (1).
The systems C of consistent and J of fully rational judgment sets are thus interde…nable, so that we could use J instead of C to concisely characterize the logical interconnections of a well-behaved agenda. 5 Aggregation rules. A (multi-valued) aggregation rule is a correspondence F which to every pro…le of 'individual' judgment sets (J 1 ; :::; J n ) (from some domain, usually J n ) assigns a set F (J 1 ; :::; J n ) of 'collective' judgment sets. Typically, the output F (J 1 ; :::; J n ) is a singleton set fCg, in which case we identify this set with C and write F (J 1 ; :::; J n ) = C. If F (J 1 ; :::; J n ) contains more than one judgment set, there is a 'tie' between these judgment sets. An aggregation rule is called single-valued or tie-free if it always generates a single judgment set. A standard (single-valued) aggregation rule is majority rule; it is given by F (J 1 ; :::; J n ) = fp 2 X : p 2 J i for more than half of the individuals ig and generates inconsistent collective judgment sets for many agendas and pro…les. If both individual and collective judgment sets are rational (i.e., in J ), the aggregation rule de…nes a correspondence J n J , and in the case of single-valuedness a function J n ! J . 6 Approaches and interpretations. For interested readers, let me add some considerations about the present model and its ‡exibility. Firstly, I mention three salient ways of specifying an agenda in practice. All three approaches could qualify broadly as 'logical': Under the syntactic approach mentioned above, the propositions are logical sentences, i.e., X is a subset of the set L of sentences of some logic, where X is negation-closed. 7 Such an agenda inherits its partition into issues (i.e., its negation operator) and its interconnections (i.e., its consistency notion) from the logic. The logic is general: it could for instance be standard propositional logic, standard predicate logic, or various modal or conditional logics (see Dietrich 2007). Many real-life agendas draw on non-standard logics by involving for instance modal operators or non-material conditionals. Fortunately, most relevant logics are well-behaved, i.e., satisfy the conditions C1-C3 (now read as conditions on sentences in L), so that the agenda is automatically well-behaved. Under the semantic approach, the propositions are subsets of some set of possibilities or worlds, i.e., X 2 , where X is closed under taking complements in ('negations'). The issues are simply the pairs fp; npg X, and the consistent sets are the sets S X which are satis…able, i.e., \ p2S p 6 = ?. Notice that, just as in the syntactic approach, the agenda's structure (i.e., the issues and interconnections) is inherited and thus need not be introduced explicitly. 8 5 So, algebraically speaking, a well-behaved agenda X could be de…ned as the structure (X; I; J ) instead of (X; I; C) (where I is the partition of X into issues, replaceable by the negation operator : on X). To see why C and J are indeed interde…nable, note that if we start from a system J (any non-empty system of sets containing exactly one member from each pair p; :p 2 X) then we can derive the system C as [ J2J fC : C Jg (using that C must be well-behaved). A future challenge is to relax well-behavedness by studying, e.g., judgment aggregation in non-monotonic logics. 6 More generally, dropping the requirement of collective rationality, we have a correspondence J n 2 X , where 2 X is the set of all judgment sets, rational or not. As usual, I write ' ' instead of '!' to indicate a multi -function. 7 Negation-closure means that if X contains a sentence p then it also contains the sentence 'not p' (or, if p is already a negated sentence 'not q', the sentence q). Technically, we also exclude doube-negated sentences 'not not p' from the agenda. In summary, the agenda thus consists of pairs of an unnegated sentence p and its negation 'not p'. (Negation-closure of course implicitly assumes the negation symbol 'not' to belong to the logic, a minimal requirement of expressiveness.) 8 Nehring and Puppe's (2010a) property spaces are essentially semantically de…ned agendas.
Under an algebraic (or abstract semantic) approach, the agenda is a subset of an arbitrary Boolean algebra 9 , where this subset is closed under taking (Boolean-algebraic) complements. Here again the agenda directly inherits a structure of issues and interconnections.
Secondly, I mention an interpretational point (orthogonal to the formal question of whether one works with a syntactic, semantic or abstract agenda). The standard interpretation of judgment aggregation is of course an aggregation of 'judgments', i.e., belief-type attitudes towards propositions. But one may re-interpret 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 attributes other than propositional attitudes. Here the agenda contains not propositions which one may or may not believe (or desire, or hope etc.), but arbitrary attributes which one may or may not have. For instance, the agenda might contain the attributes of liking piece, being successful, and so on, each of which someone may or may not have. This leads to general attribute aggregation rather than propositional attitude aggregation. 10

Scoring rules
Scoring rules are particular judgment aggregation rules, de…ned on the basis of a so-called scoring function. A scoring function -or simply a scoring -is a function s : X J ! R which to each proposition p and rational judgment set J assigns a number s J (p), called the score of p given J and measuring how p performs ('scores') from the perspective of holding judgment set J. As an elementary example, so-called simple scoring is given by: so that all accepted propositions score 1, whereas all rejected propositions score 0. This and many other scorings will be analysed. Let us think of the score of a set of propositions as the sum of the scores of its members. So, the scoring s is extended to a function which (given the agent's judgment set J 2 J ) assigns to each set C X the score A scoring s gives rise to an aggregation rule, called the scoring rule w.r.t. s and denoted F s . Given a pro…le (J 1 ; :::; J n ) 2 J n , this rule determines the collective judgments by 9 A Boolean algebra is a lattice L (with its operations of join and meet) in which there exists a top element | (tautology) and a bottom element ? (contradiction) and in which every element p has a complement (i.e., an element whose join with p is | and whose meet with p is ?). An important example is a concrete Boolean algebra L 2 (for some underlying set of 'worlds' ), in which the join is given by the union, the meet by the intersection, the top by , the bottom by ?, and the complement by the standard settheoretic complement. In this case the algebraic approach reduces to the standard semantic approach. Another example is the Boolean algebra generated from a logic by considering 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).
1 0 Attribute aggregation raises the question of what it means for the collective to 'have' an attribute. Presumably, collective attributes are something quite di¤erent from individual attributes (just as collective judgments di¤er in status from individual judgments). selecting the rational judgment set(s) with the highest sum-total score across all judgments and all individuals: F s (J 1 ; :::; J n ) = judgment set(s) in J with highest total score By a scoring rule simpliciter we of course mean an aggregation rule which is a scoring rule w.r.t. some scoring. Di¤erent scorings s and s 0 can generate the same scoring rule F s = F s 0 , in which case they are called equivalent. For instance, s is equivalent to s 0 = 2s. 11

Simple scoring and Kemeny rule
We …rst consider the most elementary de…nition of scoring, namely simple scoring (2). Table 2 illustrates the corresponding scoring rule F s for the case of the agenda and pro…le of our doctrinal paradox example. The entries in Table 2 are derived as follows. First, enter Score of... p :p q :q r :r pqr p:q:r :pq:r :p:q:r Indiv. 1 (pqr) 1 0 1 0 1 0 3 1 1 0 Indiv. 2 (p:q:r) 1 0 0 1 0 1 1 3 1 2 Indiv. 3 (:pq:r) 0 1 1 0 0 1 1 1 3 2 Group 2 1 2 1 1 2 5* 5* 5* 4 Table 2: Simple scoring (2) for the doctrinal paradox agenda and pro…le the score of each proposition (p; :p; q; :::) from each individual (1, 2 and 3). Second, enter each individual's score of each judgment set by taking the row-wise sum. For instance, individual 1's score of pqr is 1 + 1 + 1 = 3, and his score of p:q:r is 1 + 0 + 0 = 1. Third, enter the group's score of each proposition by taking the column-wise sum. For instance, the group's score of p is 1 + 1 + 0 = 2. Finally, enter the group's score of each judgment set, by taking either a vertical or a horizontal sum (the two give the same result), and add a star '*' in the …eld(s) with maximal score to indicate the winning judgment set(s). For instance, the group's score of pqr using a vertical sum is 3 + 1 + 1 = 5, and using a horizontal sum it is 2 + 2 + 1 = 5. Since the judgment sets pqr, p:q:r and :pq:r all have maximal group score, the scoring rule delivers a tie: F (J 1 ; J 2 ; J 3 ) = fpqr; p:q:r; :pq:rg: This is a tie between the premise-based outcome pqr and the conclusion-based outcomes p:q:r and :pq:r. Were we to add more individuals, the tie would presumably be broken in one way or the other. In large groups, ties are a rare coincidence.
To link simple scoring to distance-based aggregation, suppose we measure the distance between two rational judgment sets by using some distance function ('metric') d over J . 12 The most common example is Kemeny distance d = d Kem eny , de…ned as follows (where by a 'judgment reversal' I mean the replacement of an accepted proposition by its negation): For instance, the Kemeny-distance between pqr and p:q:r (for our doctrinal paradox agenda) is 2.
Now the distance-based rule w.r.t. distance d is the aggregation rule F d which for any pro…le (J 1 ; :::; J n ) 2 J n determines the collective judgment set(s) by minimizing the sum-total distance to the individual judgment sets: The most popular example, Kemeny rule F d K e m e n y , can be characterized as a scoring rule: Proposition 1 The simple scoring rule is the Kemeny rule.

Classical scoring rules for preference aggregation
I now show that our scoring rules generalize the classical scoring rules of preference aggregation theory. Consider the preference agenda X for a given set of alternatives A of …nite size k. Classical scoring rules (such as Borda rule) are de…ned by assigning scores to alternatives in A, not to propositions xP y in X. Given a strict linear order over A, each alternative x 2 A is assigned a score SCO (x) 2 R. The most popular example is of course Borda scoring, for which the highest ranked alternative in A scores k, the secondhighest k 1, the third-highest k 2, ..., and the lowest 1. Given a pro…le ( 1 ; :::; n ) of individual preferences (strict linear orders), the collective ranks the alternatives x 2 X according to their sum-total score P i2N SCO i (x). To translate this into the judgment aggregation formalism, recall that each strict linear order over A uniquely corresponds to a rational judgment set J 2 J (given by xP y 2 J , x y); we may therefore write SCO J (x) instead of SCO (x), and view the classical scoring SCO as a function of (x; J) in A J . Formally, I de…ne a classical scoring as an arbitrary function SCO : A J ! R, and the classical scoring rule w.r.t. it as the judgment aggregation rule F F SCO for the preference agenda which for every pro…le (J 1 ; :::; J n ) 2 J n returns the rational judgment set(s) that rank an alternative x over another y whenever x has a higher sum-total score than y: 13 F (J 1 ; ::: 1 3 A technical di¤erence between the standard notion of a scoring rule in preference aggregation theory and our judgment-theoretic rendition of it arises when there happen to exist distinct alternatives with identical sum-total score. In such cases, the standard scoring rule returns collective indi¤erences, whereas our F SCO returns a tie between strict preferences. From a formal perspective, however, the two de…nitions are equivalent, since to any weak order corresponds the set (tie) of all strict linear orders which linearize the weak order by breaking its indi¤erences (in any cycle-free way). The structural asymmetry between input and output preferences of scoring rules as de…ned standardly (i.e., the possibility of indi¤erences at the collective level) may have been one of the obstacles -albeit only a small, mainly psychological onefor importing scoring rules and Borda aggregation into judgment aggregation theory. Now, any given classical scoring SCO induces a scoring s in our (proposition-based) sense. In fact, there are two canonical (and, as we will see, equivalent) ways to de…ne s: one might de…ne s either by or, if one would like the lowest achievable score to be zero, by (where the last equality assumes that SCO J (x) > SCO J (y) , xP y 2 J for all x, y and J, a property that is so natural that we might have built it into the de…nition of a 'classical scoring' SCO). This allows us to characterize classical scoring rules in terms of proposition-based rather than alternative-based scoring: Proposition 2 In the case of the preference agenda (for any …nite set of alternatives), every classical scoring rule is a scoring rule, namely one with respect to a scoring s derived from the classical scoring SCO via (3) or via (4).

Reversal scoring and a Borda rule for judgment aggregation
Given the agent's judgment set J, let us think of the score of a proposition p 2 X as a measure of how 'distant' the negation :p is from J; so, p scores high if :p is far from J, and low if :p is contained in J. More precisely, let the score of a proposition p given J 2 J be the number of judgment reversals needed to reject p, i.e., the number of propositions in J that must (minimally) be negated in order to obtain a consistent judgment set containing :p. So, denoting the judgment set arising from J by negating the propositions in a subset R J by J :R = (JnR) [ f:r : r 2 Rg, so-called reversal scoring is de…ned by s J (p) = number of judgment reversals needed to reject p (5) = min For instance, a rejected proposition p 6 2 J scores zero, since J itself contains :p so that it su¢ces to negate zero propositions (R = ?). An accepted proposition p 2 J scores 1 if J remains consistent by negating p (R = fpg), and scores more than 1 otherwise (R ) fpg). Table 3 illustrates reversal scoring for our doctrinal paradox example. For instance, individual 1's judgment set pqr leads to a score of 2 for proposition p, since in order for him to reject p he needs to negate not just p (as :pqr is inconsistent), but also r (where :pq:r is consistent). The scoring rule delivers a tie between the judgment sets Score of... p :p q :q r :r pqr p:q:r :pq:r :p:q:r Indiv. 1 (pqr) 2 0 2 0 2 0 6 2 2 0 Indiv. 2 (p:q:r) 1 0 0 2 0 2 1 5 2 4 Indiv. 3 (:pq:r) 0 2 1 0 0 2 1 2 5 4 Group 3 2 3 2 2 4 8 9* 9* 8 Table 3: Reversal scoring (5) for the doctrinal paradox agenda and pro…le p:q:r and :pq:r. This is a tie between two conclusion-based outcomes; the premise-based outcome pqr is rejected (unlike for simple scoring in Section 3.1).
The remarkable feature of reversal scoring rule is that it generalizes Borda rule from preference to judgment aggregation. Borda rule is initially only de…ned for the preference agenda X (for a given …nite 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 …nite set of alternatives), reversal scoring s is given by (4) with SCO de…ned as classical Borda scoring.
Let me sketch the simple argument -it should sound familiar to social choice theorists. Let s be reversal scoring, X the preference agenda for a set of alternatives A of size k < 1, and SCO classical Borda scoring. Consider any xP y 2 X and J 2 J . If xP y 2 XnJ, then :xP y = yP x 2 J, which implies s J (xP y) = 0, as required by (4). Now suppose xP y 2 J. Clearly, SCO J (x) > SCO J (y). Consider the alternatives in the order established by J: Step by step, we now move y up in the ranking, where each step consists in raising the position (score) of y by one. Each step corresponds to negating one proposition in J, namely the proposition zP y where z is the alternative that is currently being 'overtaken' by y. After exactly SCO J (x) SCO J (y) steps, y has 'overtaken' x, i.e., xP y has been negated. So, , since, as the reader may check, no smaller number of judgment reversals allows y to 'overtake' x in the ranking.
Remark 1 and Proposition 2 imply that reversal scoring allows us to extend Borda rule to arbitrary judgment aggregation problems: Proposition 3 The reversal scoring rule generalizes Borda rule, i.e., matches it in the case of the preference agenda (for any …nite 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 Reversal scoring s is equivalent (in terms of the resulting scoring rule) to the scoring s 0 given by and in the case of the preference agenda (for any …nite set of alternatives) this scoring is given by For comparison, I now sketch Zwicker's (2011) interesting approach to extending Borda rule to judgment aggregation -let me call such an extension a 'Borda-Zwicker' rule. The motivation derives from a geometric characterization of Borda preference aggregation obtained by Zwicker (1991). Let me write the agenda as X = fp 1 ; :p 1 ; p 2 ; :p 2 ; :::; p m ; :p m g, where m is the number of 'issues'. Each pro…le gives rise to a vector v (v 1 ; :::; v m ) in R m whose j th entry v j is the net support for p j , i.e., the number of individuals accepting p j minus the same number for :p j . Now if X is the preference agenda for any …nite set of alternatives A, then each p j takes the form xP y for certain alternatives x; y 2 A. Each preference cycle can be mapped to a vector in R m ; for instance, if p 1 = xP y, p 2 = yP z and p 3 = xP z, then the cycle x y z x becomes the vector (1; 1; 1; 0; :::; 0) 2 R m . The linear span of all vectors corresponding to preference cycles de…nes the so-called 'cycle space' V cycle R m , and its orthogonal complement de…nes the 'cocycle space' V cocycle R m . Let v cocycle be the orthogonal projection of v on the cocycle space V cocycle . Intuitively, v cocycle contains the 'consistent' or 'acyclic' part of v. The upshot is that the Borda outcome can be read o¤ from v cocyle : for each p j = xP y, the Borda group preference ranks x above (below) y if the j th entry of v cocyle is positive (negative). Zwicker's strategy for extending Borda rule to judgment aggregation is to de…ne a subspace V cycle analogously for agendas other than the preference agenda; one can then again project v on the orthogonal complement of 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 de…ne V cycle . Whether the approach is viable for general agendas (i.e., whether V cycle has a useful general de…nition) seems to be open so far. 14 A Borda-Zwicker rule is not just constructed di¤erently 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 pro…le's 'net support vector' v 2 R m . (Even more does the required information go beyond the projection of v on the orthogonal complement of V cycle .) In summary, there seem to exist two quite di¤erent approaches to generalizing Borda aggregation. One approach, taken by Zwicker, seeks to …lter out the pro…le's 'inconsistent component' along the lines of the just-described geometric technique. The other approach, taken here, seeks to retain the principle of score-maximization inherent in Borda aggregation (with scoring now de…ned at the level of propositions, not alternatives, as these do not exist outside the world of preferences). The normative core of the scoring approach is to use information about someone's strength of accepting a proposition (as measured by the score), just as Borda preference aggregation uses information about someone's strength of preferring one alternative x over another y (as measured by the score of xP y, i.e., the di¤erence between x's and y's score). Whether strength or intensity of preference is a permissible or even meaningful concept is a notoriously controversial question; the purely ordinalist approach takes a sceptical stance here. This is where Borda preference aggregation di¤ers from Condorcet's rule of pairwise majority voting, which uses only the (ordinal) information of whether someone prefers an alternative over another, without attempting to extract strength-of-preference information from that person's full preference relation.

A generalization of reversal scoring
Recall that the reversal score of a proposition p can be characterized as the distance by which one must deviate from the current judgment set in order to reject p -where 'distance' is understood as Kemeny-distance. It is natural to also consider other kinds of a distance.
Relative to any given distance function d over J , one may de…ne a corresponding scoring by s J (p) = distance by which one must depart from J to reject p (6) This provides us with a whole class of scoring rules, all of which are variants of our judgment-theoretic Borda rule. In the special case of the preference agenda, we thus obtain new variants of classical Borda rule.
Interestingly, if we adopt Duddy and Piggins' (2011) distance function, i.e., if d(J; J 0 ) is the number of minimal consistent modi…cations needed to transform J into J 0 , 15 then scoring (6) reduces to simple scoring (2), and so the scoring rule reduces to the Kemeny rule by Proposition 1. So, ironically, while Duddy and Piggins had introduced their distance in the di¤erent context of distance-based aggregation to develop an alternative to Kemeny rule, when we use their distance (instead of Kemeny's) in our context of scoring rules we are led back to Kemeny rule.

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 (2 X) given the judgment set J (2 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 J which entails p, i.e., for which S [ f:pg 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 fpg, and perhaps also by very di¤erent judgment subsets; so the strength of entailment (score) of p is positive and more or less high.
There are di¤erent 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 Borda rule from preference to judgment aggregation. These generalizations di¤er from that based on reversal scoring in Section 3.3.
Our …rst, naive approach is to count each judgment subset which entails p as a separate, full- ‡edged 'way' in which p is entailed. This leads to so-called entailment scoring, de…ned by: s J (p) = number of judgment subsets which entail p = jfS J : S entails pgj .
If p 6 2 J then s J (p) = 0, while if p 2 J then s J (p) 2 jXj=2 1 since p is entailed by at least all sets S J which contain p, i.e., by at least 2 jJj 1 = 2 jXj=2 1 sets. One might object 1 5 Judgment sets J; J 0 2 J are minimal consistent modi…cations of each other if the set S = JnJ 0 of propositions in J which need to be negated to transform J into J 0 is non-empty and minimal (i.e., J couldn't have been transformed into a consistent set by negating only a strict non-empty subset of S). For our doctrinal paradox agenda, the judgment sets pqr and p:q:r are minimal consistent modi…cations of each other, and hence have Duddy-Piggins-distance of 1. that this de…nition of scoring involves redundancies, i.e., 'multiple counting'. Suppose for instance p belongs to J and is logically independent of all other propositions in J. Then p is entailed by several subsets S of J -all S J which contain p -and yet these entailments are essentially identical since all premises in S other than p are irrelevant.
I now present three re…nements of scoring (7), each of which responds di¤erently to the mentioned redundancy objection. In the …rst re…nement, we count two entailments of p as di¤erent only if they have no premise in common. This leads to what I call disjointentailment scoring, formally de…ned by: s J (p) = number of mutually disjoint judgment subsets entailing p = maxfm : J has m mutually disjoint subsets each entailing pg.
Disjoint-entailment scoring turns out to match reversal scoring for our doctrinal paradox agenda (check that Tables 3 and 4 coincide), as well as for the preference agenda (as shown later). Is this pure coincidence? The general relationship is that the disjoint-entailment score of a proposition p is always at most the reversal score, as one may show. 16 While this re…nement of naive entailment scoring (7) avoids 'multiple counting' by only counting entailments with mutually disjoint sets of premises, the next two re…nements use a di¤erent strategy to avoid 'multiple counting'. The new strategy is to count only those entailments whose sets of premises are minimal -with minimality understood either in the sense that no premises can be removed, or in the sense that no premises can be logically weakened. To begin with the …rst sense of minimality, I say that a set minimally entails p (2 X) if it entails p but no strict subset of it entails p, and I de…ne minimal-entailment scoring by s J (p) = number of judgment subsets which minimally entail p = jfS J : S minimally entails pgj .
If for instance p is contained in J, then fpg minimally entails p, 17 but strict supersets of fpg do not and are therefore not counted. For our doctrinal paradox agenda, this scoring happens to coincide with reversal scoring and disjoint-entailment scoring. Indeed, Table 3 resp. 4 still applies; e.g., for individual 2 with judgment set p:q:r, p still scores 1 (it is minimally entailed only by fpg), :q still scores 2 (it is minimally entailed by f:qg and by fp; :rg), :r still scores 2 (it is minimally entailed by f:rg and by f:qg), and all rejected propositions still score zero (they are not minimally entailed by any judgment subsets).
Scoring (9) is certainly appealing. Nonetheless, one might complain that it still allows for certain redundancies, albeit of a di¤erent kind. Consider the preference agenda with set of alternatives A = fx; y; z; wg, and the judgment set J = fxP y; yP z; zP w; xP z; yP w; xP wg (2 J ). The proposition xP w is minimally entailed by the subset S = fxP y; yP z; zP wg. While this entailment is minimal in the (set-theoretic) sense that we cannot remove premises, it is non-minimal in the (logical) sense that we can weaken some of its premises: if we replace xP y and yP z in S by their logical implication xP z, then we obtain a weaker set of premises S 0 = fxP z; zP wg which still entails xP w. We shall say that S fails to 'irreducibly' entail xP w, in spite of minimally entailing it. In general, a set of propositions is called weaker than another one (which is called stronger ) if the second set entails each member of the …rst set, but not vice versa. A set S ( X) is de…ned to irreducibly (or logically minimally) entail p if S entails p, and moreover there is no subset Y ( S which can be weakened (i.e., for which there is a weaker set Y 0 X such that (SnY ) [ Y 0 still entails p). Each irreducible entailment is a minimal entailment, as is seen by taking Y 0 = ?. 18 In the previous example, the set fxP y; yP z; zP wg minimally, but not irreducibly entails xP w, and the set fxP z; zP wg irreducibly entails xP w. Irreducible-entailment scoring is naturally de…ned by s J (p) = number of judgment subsets which irreducibly entail p (10) = jfS J : S irreducibly entails pgj .
This scoring matches reversal scoring and both previous scorings in the case of our doctrinal paradox example: Table 3 resp. 4 still applies. But for many other agendas these scorings all deviate from one another, resulting in di¤erent collective judgments. As for the preference agenda, we have already announced the following result: Proposition 4 Disjoint-entailment scoring (8) and irreducible-entailment scoring (10) match reversal scoring (5) in the case of the preference agenda (for any …nite set of alternatives).

Propositions 3 and 4 jointly have an immediate corollary.
Corollary 1 The scoring rules w.r.t. scorings (8) and (10) both generalize Borda rule, i.e., match it in the case of the preference agenda (for any …nite set of alternatives).

Propositionwise scoring and a way to repair quota rules with non-rational outputs
We now consider a special class of scorings: propositionwise scorings. This will allow us to relate scoring rules to the well-known judgment aggregation rules called quota rules -in fact, to 'repair' these rules by rendering their outcomes rational across all pro…les.
I call scoring s propositionwise if the score of a proposition p 2 X only depends on whether p is accepted, i.e., if s J (p) = s K (p) whenever J and K (in J ) both contain p or both do not contain p. Equivalently, scoring is propositionwise just in case for each p 2 X there is a pair of real numbers s + (p); s (p) such that s J (p) = s + (p) for all J 2 J containing p s (p) for all J 2 J not containing p.
Intuitively, s + (p) is the score of an accepted proposition p, and s (p) is the score of a rejected proposition p. Typically, of course, s + (p) > s (p). An example is simple scoring: there, s + (p) = 1 and s (p) = 0.
How do propositionwise scoring rules behave? They derive a proposition p's sum-total score 'locally', i.e., based only on people's judgments about p. This property stands in obvious analogy to a well-studied axiom on aggregation rules, namely the axiom of propositionwise or independent aggregation, which prescribes that the collective judgment about any given proposition p is derived 'locally', i.e., again based only on people's judgments about p. Can we therefore relate propositionwise scoring to independent aggregation? The paradigmatic independent aggregation rules are the quota rules. 19 A quota rule is a (single-valued) aggregation rule which is given by an acceptance threshold m p 2 f1; :::; ng for each proposition p 2 X. The quota rule corresponding to the so-called threshold family (m p ) p2X is denoted F (mp) p2X and accepts those propositions p which are supported by at least m p individuals: for each pro…le (J 1 ; :::; J n ) 2 J n , F (mp) p2X (J 1 ; :::; J n ) = fp 2 X : jfi : p 2 J i gj m p g.
Special cases are unanimity rule (given by m p = n for all p), majority rule (given by the majority threshold m p = d(n + 1)=2e for all p), and more generally, uniform quota rules (given by a uniform threshold m p m for all p). A uniform quota rules is also referred to as a supermajority rule if m exceeds the majority threshold, and a submajority rule if m is below the majority threshold. Note that supermajority rules may generate incomplete collective judgment sets, while submajority rule may accept both members of a pair p; :p 2 X, a drastic form of inconsistency. If one wishes that exactly one member of each pair p; :p 2 X is accepted, the thresholds of p and :p should be 'complements' of each other: m p = n + 1 m :p .
A non-trivial question is how the acceptance thresholds would have to be set to ensure that the collective judgment set satis…es 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 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 = fp; q; rg only the extreme thresholds m p = m q = m r = n and m :p = m :q = m :r = 1 achieve (iv), and for the preference agenda (with more than two alternatives) no thresholds achieve (iv).
Given that quota rules with 'reasonable' thresholds typically violate many of the conditions (i)-(iv), one may want to depart from ordinary quota rules by modifying ('repairing') them so that they always generate rational outputs. This can be done by using propositionwise scoring rules. Given an arbitrary quota rule with threshold family (m p ) p2X , one can specify a propositionwise scoring such that the scoring rule replicates the quota rule whenever the quota rule generates a rational output, while 'repairing' the output otherwise. How must we calibrate s + (p) and s (p) in order to achieve this? The idea is that individuals who accept p should contribute a positive score s + (p) > 0, while those who reject p should contribute a negative score s (p) < 0. The absolute sizes of s + (p) and s (p) should be calibrated such that the sum-total score of p becomes positive (helping the scoring rule to accept p) exactly when the quota rule accepts p, i.e., when at least m p individuals accept p. Speci…cally, we set: Intuitively, the higher the acceptance threshold m p is, the smaller the positive contribution s + (p) is and the larger the negative contribution s (p) is (in absolute value); hence, the more individuals accepting p are needed for p's sum-total score to get positive, and the harder it becomes for the scoring rule to accept p. This scoring does the intended job: Proposition 5 For every threshold family (m p ) p2X , the scoring rule w.r.t. scoring (12) matches the quota rule F (mp) p2X at all pro…les where the quota rule generates rational outputs (and still generates rational outputs at all other pro…les).
How does our scoring rule 'repair' those special quota rules which use a uniform threshold m m p (p 2 X), such as majority rule?
Remark 3 For a uniform threshold m 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 m p , scoring (12) is equivalent to simple scoring by footnote 11.
Finally, I note that the scoring rules w.r.t. (12) is not the only scoring rule which can 'repair' the quota rule F (mp) p2X -though it might be the most plausible one, as long as we do not wish to introduce additional parameters. If, however, we are prepared to introduce additional parameters, scoring (12) can be generalized: for each p 2 X let p > 0 be a coe¢cient measuring how important it is that the scoring rule is faithful to the quota rule's collective judgment on p; and let scoring be de…ned by The earlier scoring (12) is obviously a special case in which all p are 1. Proposition 5 still holds for this generalized kind of propositionwise scoring. The scoring rule will tend to match the quota rule on propositions p with high importance coe¢cient p , while modifying ('repairing') the quota rule at propositions p with low p .

Premise-and conclusion-based aggregation
I have just mentioned the possibility of a di¤erential treatment of propositions when 'repairing' a quota rule. This possibility is particularly salient in the popular context of premise-or conclusion-based aggregation. 21 One may indeed view the classical premiseand conclusion-based rules as two (rival) ways of repairing the simplest of all quota rulesmajority rule -by privileging certain propositions over others, namely premise propositions or conclusion propositions, respectively.
Let me put this precisely. Consider majority voting, i.e., the quota rule with a uniform majority threshold m m p (the smallest integer above n=2). To restore collective rationality, we again endow each proposition p 2 X with a 'coe¢cient of importance', but now let this coe¢cient be determined by whether p has a 'premise' or 'conclusion' status. Formally, suppose the agenda is partitioned into two negation-closed sets, the set P of 'premise propositions' and the set XnP of 'conclusion propositions'. In the case of our doctrinal paradox agenda X = fp; q; rg , we have P = fp; qg . Each premise proposition p 2 P has the importance coe¢cient p premise , and each conclusion proposition p 2 XnP has the importance coe¢cient p conclusion , for …xed parameters premise ; conclusion 0. In this scenario, the scoring (13) becomes equivalent (by footnote 11) to the scoring given by By calibrating the two importance coe¢cients, we can in ‡uence the relative weights of premises and conclusions. If we give far more importance to premise propositions ( premise conclusion ) or to conclusion propositions ( conclusion premise ), the scoring rule reduces to the premise-or conclusion-based rule, respectively. To substantiate this claim, one needs to de…ne both rules. For simplicity, I restrict attention to our doctrinal paradox agenda X = fp; q; rg with P = fp; qg (though more general X and P could be considered 22 ). In this case, assuming for simplicity that the group size n is odd, the premise-based rule is the aggregation rule which for each pro…le in J n delivers the (unique) judgment set in J containing each premise proposition accepted by a majority; the conclusion-based rule is the aggregation rule which for each pro…le in J n delivers the judgment set (or sets) in J containing the conclusion proposition accepted by a majority. 23 These two rules have the following characterizations as scoring rules: Remark 4 For our doctrinal paradox agenda X = fp; q; rg with set of premise propositions P = fp; qg , and for an odd group size, the scoring rule w.r.t. scoring (14) is the premise-based rule if and only if premise > (n 2) conclusion , the conclusion-based rule if and only if conclusion > premise = 0.
This result lets premise-and conclusion-based aggregation appear in a rather extreme light: each rule is based on somewhat unequal importance coe¢cients premise and 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 coe¢cients, 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 An interesting generalization of scoring rules is obtained by assigning scores directly to entire judgment sets rather than single propositions. A set scoring function -or simply set scoring -is a function which to every pair of rational judgment sets C and J assigns a real number J (C), the score of C given J, which measures how well C performs ('scores') from the perspective of holding the judgment set J. Formally, : J J ! R. The most elementary example, to be called naive set scoring, is given by Any set scoring gives rise to an aggregation rule F , the set scoring rule (or generalized scoring rule) w.r.t. , which for each pro…le (J 1 ; :::; J n ) 2 J n selects the collective judgment set(s) C in J having maximal sum-total score across individuals: F (J 1 ; :::; J n ) = argmax C2J X i2N Ji (C).
An aggregation rule is a set scoring rule simpliciter if it is the set scoring rule w.r.t. to some set scoring . Set scoring rules generalize ordinary scoring rules, since to any ordinary scoring s corresponds a set scoring , given by and the ordinary scoring rule w.r.t. s coincides with the set scoring rule w.r.t. .

Naive set scoring and plurality voting
Plurality rule is the aggregation rule F which for every pro…le (J 1 ; :::; J n ) 2 J n declares the most often submitted judgment set(s) as the collective judgment set(s): F (J 1 ; :::; J n ) = most frequently submitted judgment set(s) This rule is of course normatively questionable; 24 but it deserves our attention, if only because of its simplicity and the recognized importance of plurality voting in social choice theory more broadly. Plurality rule can be construed as a set scoring rule: Remark 5 The naive set scoring rule is plurality rule.

Distance-based set scoring
Set scoring rules generalize distance-based aggregation. Given an arbitrary distance function d over J (not necessarily the Kemeny-distance), all that is needed is to consider what I call distance-based set scoring, de…ned by So, C scores high if it is close to the judgment set held, J. This renders sum-scoremaximization equivalent to sum-distance-minimization: Remark 6 For every given distance function over J , the distance-based set scoring rule is the distance-based rule.
So, all distance-based rules can be modelled as set scoring rules (but not vice versa 25 ). As an example, consider the so-called discrete distance, 26 de…ned by Here, distance-based set scoring (16) is equivalent to naive set scoring (15), since the two di¤er only by a constant (of one). So, joining Remarks 5 and 6, we may view plurality rule either as the naive set scoring rule or as the discrete-distance-based rule.

Approximating the 'average voter'
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 di¤erent 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 an 'average individual's 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 di¤erent intuitive appeal. But is it really totally di¤erent? As will turn out, aggregation rules which follow this approach -I call them 'average-score rules' as opposed to 'scoring rules' -can be viewed as a particular kind of set scoring rules. This result is in fact a special case of a powerful precursor result by Zwicker (2008), as Marcus Pivato kindly pointed out to me. 27 Given an ordinary scoring s, we can represent judgment sets in J as vectors in R X , by identifying each judgment set J in J with its score vector, i.e., the vector in R X whose p th component is the score of p, s J (p). 28 The score vector corresponding to J 2 J is denoted J s (s J (p)) p2X 2 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 J may be 'infeasible', i.e., not correspond to any judgment set in J .
The average-score rule w.r.t. scoring s is de…ned as the aggregation rule F which for every pro…le (J 1 ; :::; J n ) 2 J n chooses the collective judgment set(s) whose score vector comes closest to the group's average score vector 1 n P i2N J s i in the sense of Euclidean distance in R X : F (J 1 ; :::; J n ) = j.s. closest to the average individual j.s. in score vector terms Viewed geometrically as an operation in R X , the collective score vector is the orthogonal projection of the average score vector 1 n P i J s i on the set J s fJ s : J 2 J g R X of feasible score vectors. 29 As an illustration, consider once again reversal scoring for our doctrinal paradox agenda. Table 6 reports the score vector of each judgment set (including the one not submitted by any individual), and its distance to the group's average score vector. By minimizing this distance, the rule delivers a tie between the two conclusion-based outcomes p:q:r and 2 7 Average-score rules are special cases of Zwicker's 'mean proximity rules' in his abstract, more general aggregation framework. Zwicker's Theorem 4.2.1 (more precisely, its proof) reveals that any 'mean proximity rule' can be given a representation which essentially corresponds to our representation of an average-score rule in Proposition 6.
2 8 This identi…cation is one-to-one as long as the scoring has the (very plausible) property that s J (p) > s J (:p) whenever p 2 J.   :pq:r. The premise-based outcome pqr looks worse than ever: it is even farther from the average than the never-submitted outcome :p:q:r.
Now that we have two rival ways of aggregating based on a scoring s -namely, the scoring rule and the average-score rule -the question is whether any connection can be established. The average-score rule can be construed as a set scoring rule, namely in virtue of the set scoring given by Here, C is taken to score high if it is close to J in terms of the squared Euclidean distance of score vectors.
Proposition 6 For any scoring s, the average-score rule w.r.t. s is the set scoring rule w.r.t. set scoring (17).
As an application, let s be simple scoring (2). Here, the set scoring (17) is expressible as an increasing a¢ne transformation of the set scoring corresponding to simple scoring, i.e., of the set scoring 0 given by 30 So, the set scoring rule F coincides with the simple scoring rule F s , and hence with the Kemeny rule F d K e m e n y by Proposition 1. Thus, as a corollary of Propositions 1 and 6, the Kemeny rule can be characterized not just as a scoring rule but also as an average-score rule, both times using the same scoring: Corollary 2 The Kemeny rule is the scoring rule and the average-score rule, both times w.r.t. simple scoring.

Probability-based set scoring
I close the analysis by taking a brief (skippable) excursion into an important, but di¤erent approach to judgment aggregation: the epistemic or truth-tracking approach. In this approach, each proposition p 2 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. 31 The truth-tracking perspective has a long history elsewhere in social choice theory (e.g., Condorcet 1785, Grofman et al. 1983, Austen-Smith and Banks 1996, Dietrich 2006b, Pivato 2011a; but within judgment aggregation theory speci…cally, rather little work has been done on the epistemic side (e.g., Bovens and Rabinowicz 2006b, 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 maximum-likelihood 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). 32 For each combination (J 1 ; :::; J n ; T ) 2 J n J of n+1 judgment sets, let Pr(J 1 ; :::; J n ; T ) > 0 measure the probability that people submit the pro…le (J 1 ; :::; J n ) and the set of true propositions is T , where of course P (J1;:::;Jn;T )2J n J Pr(J 1 ; :::; 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 2 J , Pr(T ) = P (J1;:::;Jn)2J n Pr(J 1 ; :::; J n ; T ), the probability that the pro…le is (J 1 ; :::; J n ), Pr(J 1 ; :::; J n ) = (called the likelihood of the 'data' J 1 ; :::; J n given T ).
The maximum-likelihood rule is the aggregation rule F : J n J which for each pro…le (J 1 ; :::; J n ) 2 J n de…nes the collective judgments such that their truth would make the observed pro…le ('data') maximally likely: F (J 1 ; :::; J n ) = argmax T 2J Pr(J 1 ; :::; J n jT ).
The maximum-posterior rule is the aggregation rule F : J n J which for each pro…le (J 1 ; :::; J n ) 2 J n de…nes the collective judgments such that they have maximal posterior probability of truth conditional on the observed pro…le ('data'): F (J 1 ; :::; J n ) = argmax T 2J Pr(T jJ 1 ; :::; J n ).
Both of these rules correspond to well-established statistical estimation procedures.
Let us now make two standard, but restrictive assumptions on probabilities. We assume that voters are 'independent' and 'equally competent' (in analogy to the assumptions of Condorcet's classical jury theorem 33 ). Formally, for every T 2 J , (IND) the individual judgment sets are independent conditional on T being the true judgment set, i.e., Pr(J 1 ; :::; J n jT ) = Pr(J 1 jT ) Pr(J n jT ) for all J 1 ; :::; J n 2 J ('independence') (COM) for each J 2 J , each individual has the same probability, denoted Pr(JjT ), of submitting the judgment set J conditional on T being the true judgment set ('equal competence').
Condition (COM) in particular implies that individuals have the same (conditional) probability of holding the true judgment set; but nothing is assumed about the size of this probability of 'getting it right'. The just-de…ned aggregation rules turn out to be set scoring rules in virtue of de…ning the score of T 2 J given J 2 J by, respectively, 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 so-called Condorcet-admissibility rule, which generates rational judgment set(s) that 'approximate' the majority judgment set). Other counterexamples are non-anonymous 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 'non-anonymous scoring rules'. Secondly, the search for a collective judgment set with maximal total score might be done within a larger set than the set 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. Much work is ahead of us. Proof of Proposition 1. The Kemeny-distance between J; C 2 J can be written as d Kem eny (J; C) = 1 2 jJ 4 Cj = 1 2 (jXj (jJ \ Cj + J \ C )).

References
Now, since J and C each contains exactly one member of each pair fp; :pg X, we have p 2 J \C , :p 2 J \C, and so, jJ \ Cj = J \ C . Hence, d Kem eny (J; C) = 1 2 jXj jJ \ Cj. So, for each pro…le (J 1 ; :::; J n ) 2 J n , minimizing P i2N d Kem eny (J i ; C) is equivalent to maximizing P i2N jJ i \ Cj. Hence, rewriting each jJ i \ Cj as P p2C s Ji (p) where s is simple scoring (2), it follows that F d K e m e n y (J 1 ; :::; J n ) = F s (J 1 ; :::; J n ).
Before proving Proposition 2, I start with a lemma.
Lemma 1 Consider the preference agenda (for any …nite set of alternatives A), any classical scoring SCO, and the scoring s given by (4). For all distinct x; y 2 A and all J 2 J , Proof. This follows easily from (4).
Two elements of a set of alternatives A are called neighbours w.r.t. a strict linear order over A if they di¤er and no alternative in A is ranked strictly between them. In the case of the preference agenda (for a set of alternatives A), the strict linear order over A corresponding to any J 2 J is denoted J .
Proof of Proposition 2. Consider the preference agenda X for a set of alternatives A of …nite size k, and let SCO be any classical scoring. I show that F SCO = F s for each scoring s satisfying (20), and hence for the scoring (4) (since it satis…es (20) by Lemma 1) and the scoring (3) (since a half times it satis…es (20)).
Claim 1. For all a; b 2 A and C; C 0 2 J , if CnC 0 = faP bg, then Consider a; b 2 A and C; C 0 2 J such that CnC 0 = faP bg. For each individual i 2 N , we by (20)  Summing over all individuals, the claim follows, q.e.d.
Consider any C 2 F s (J 1 ; :::; J n ). We have to show that C 2 F SCO (J 1 ; :::; J n ), i.e., that for all distinct x; y 2 A, Said in yet another way, we have to show that where I have labelled the alternatives x 1 ; x 2 ; :::; x k such that x k C x k 1 C C x 1 . Consider any t 2 f1; :::; k 1g, and write a for x t+1 and b for x t . Let C 0 be the judgment set arising from C by replacing aP b with its negation bP a. Now C 0 2 J ; this is because a and b are neighbours w.r.t. C , which guarantees that C 0 corresponds to a strict linear order (namely to the same one as for C except that b now ranks above a). Since C 2 F s (J 1 ; :::; J n ), C has maximal sum-total score within J ; in particular, which by Claim 1 implies the desired inequality, Claim 3. F SCO (J 1 ; :::; J n ) F s (J 1 ; :::; J n ).
Consider any C 2 F SCO (J 1 ; :::; J n ). To show that C 2 F s (J 1 ; :::; J n ); we consider an arbitrary C 0 2 J nfCg and have to show that C has an at least as high sum-total score as To prove this, we …rst transform C gradually into C 0 in m jC 0 nCj steps, where each step consists in a single judgment reversal, i.e., in the replacement of a single proposition xP y (2 CnC 0 ) by its negation yP x (2 C 0 nC). This de…nes a sequence of judgment sets C 0 ; :::; C m , where C 0 = C and C m = C 0 , and where for each step t 2 f1; :::; mg there is a proposition x t P y t such that C t = (C t 1 nfx t P y t g) [ fy t P x t g. Note that fx t P y t : t = 1; :::; mg = CnC 0 . By a standard relation-theoretic argument, we may assume that in each step t the judgment reversal consists in switching the relative order of two neighbouring alternatives; i.e., x t ; y t are neighbours w.r.t. the old and new relations Ct 1 and Ct . This guarantees that each step t generates a set C t such that Ct is still a strict linear order, i.e., such that C t 2 J . Now for each step t, by Claim 1 we have s Ji (p), and also, since y t P x t 6 2 C and C 2 F SCO (J 1 ; :::; J n ), we have it follows that X i2N;p2Ct 1 s Ji (p) X i2N;p2Ct s Ji (p) 0.
Summing this inequality over all steps t 2 f1; :::; mg, we obtain X i2N;p2C0 s Ji (p) 0, which is equivalent to the desired inequality (21) since C 0 = C and C m = C 0 .
Proof of Remark 2. Let s 0 be de…ned from reversal scoring s in the speci…ed way.
Claim 1. s 0 and s are equivalent.
Hence, using the de…nition of s 0 , So, that y 2 = x i j . Hence, S is the set fx i P x i j ; x i j P x i 0 g = S j , and we are done again. Finally, m cannot exceed 3, since otherwise the set S (= fx i P y 2 ; y 2 P y 3 ; :::; y m 1 P x i 0 g) would entail p (= x i P x i 0 ) non-irreducibly, since the set arising from S by replacing x i P y 2 and y 2 P y 3 with their implication x i P y 3 still entails p.
Proof of Proposition 5. Consider any threshold family (m p ) p2X (2 f1; :::; ng X ), and de-…ne scoring s by (12). Consider a pro…le (J 1 ; :::; J n ) 2 J n for which C F (mp) p2X (J 1 ; :::; J n ) belongs to J . We have to show that F s (J 1 ; :::; J n ) = C . For each proposition p 2 X, writing the number of individuals accepting p as n p jfi : p 2 J i gj, the sum-total score of p is given by = n(n p m p ) + n p ; and so, X i2N s Ji (p) > 0 if n p m p , i.e., if p 2 C < 0 if n p < m p , i.e., if p 6 2 C .
Now we have fC g = argmax C2J P p2C;i2N s Ji (p), because for each C 2 J nfC g, X So, F s (J 1 ; :::; J n ) = fC g C .
Proof of Remark 4. Consider this X and P , let n be odd, and let s be scoring (14). I write pr for premise and co for conclusion . Whenever I consider a pro…le (J 1 ; :::; J n ) 2 J n , I write N t := fi : t 2 J i g for all t 2 X, and I write MAJ , PRE, CON and SCO for the outcome of majority rule, premise-based rule, conclusion-based rule, and the scoring rule w.r.t. (14), respectively. Note that for all (J 1 ; :::; J n ) 2 J n the sum-total score of a C = fp 0 ; q 0 ; r 0 g 2 J (where p 0 2 fp; :pg, q 0 2 fq; :qg and r 0 2 fr; :rg) is given by X i2N;t2C s Ji (t) = (jN p 0 j + jN q 0 j) pr + jN r j co : Claim 1. [PRE = SCO for all pro…les in J n ] if and only if pr > (n 2) co .
Conversely, assume pr > (n 2) co . Consider any pro…le. We have to show that PRE = SCO.
Case 1 : MAJ 2 J . Check that it follows that PRE = MAJ , and also that SCO = MAJ . So, PRE = SCO.
Case 2 : MAJ 6 2 J . Check that it follows that MAJ = fp; q; :rg. Hence PRE = fp; q; rg. We thus have to show that SCO = fp; q; rg, i.e., that By (23), 1 = (jN p j jN :p j) pr + (jN r j jN :r j) co = (2 jN p j n) pr + (2 jN r j n) co : (24) In this, as p 2 MAJ we have jN p j (n + 1)=2; and further, as p; q 2 MAJ the sets N p and N q each contain a majority, so that N p \ N q 6 = ?, which (since N p \ N q N r ) implies jN r j 1. Using these lower bounds for jN p j and jN r j, we obtain 1 ((n + 1) n) pr + (2 n) co = pr + (2 n) co > 0.
Unlike in the proof of the Claim, there may be ties, and so we treat CON and SCO as subsets of J, not elements. First, if co > pr = 0, then it is easy to show that CON = SCO for each pro…le. Conversely, suppose it is not the case that co > pr = 0. Then either co = pr = 0 or pr > 0. In the …rst case, clearly CON 6 = SCO for some pro…les, since SCO is always J. In the second case, again CON 6 = SCO for some pro…les: for instance, if each individual submits :pq:r then SCO = f:pq:rg while CON = f:pq:r; p:q:r; :p:q:rg.
Proof of Proposition 6. It will sometimes be convenient to write a vector D = (D 1 ; :::; D n ) 2 R n as hD i i. The mean and variance of this vector D are denoted and de…ned by, respectively,