A partial taxonomy of judgment aggregation rules and their properties

Abstract

The literature on judgment aggregation is moving from studying impossibility results regarding aggregation rules towards studying specific judgment aggregation rules. Here we give a structured list of most rules that have been proposed and studied recently in the literature, together with various properties of such rules. We first focus on the majority-preservation property, which generalizes Condorcet-consistency, and identify which of the rules satisfy it. We study the inclusion relationships that hold between the rules. Finally, we consider two forms of unanimity, monotonicity, homogeneity, and reinforcement, and we identify which of the rules satisfy these properties.

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Notes

  1. 1.

    The proofs can be found in Lang and Slavkovik (2013).

  2. 2.

    This result has been independently proven (and stated in a stronger way) in Nehring et al. (2014).

  3. 3.

    The proof—almost straightforward—can be found in Lang (2015).

  4. 4.

    Our name; no name was given of this distance in Duddy and Piggins (2012).

  5. 5.

    A weaker unanimity property has been defined by List and Puppe (2009), for resolute rules as well: \(F(P) = J\) whenever all the voters in \(P\) have the judgment set \(J\).

  6. 6.

    \({{\textsc {mc}}}\) failing to satisfy strong unanimity is also a consequence of Theorem 2.2 in Nehring et al. (2015), which can be reformulated as: \({{\textsc {mc}}}\) satisfies strong unanimity if and only if \(\mathcal {A}\) does not contain a minimal inconsistent subset of size 3 or more.

  7. 7.

    Recall that there is a constraint \(\gamma =q\rightarrow r\) for the agenda in this example.

  8. 8.

    This surprising result triggers further questions: are there interesting agendas, other than the preference agenda, for which \(F_{\text{ rev }}\) remains monotonic? Can we find a natural monotonic extension of the Borda rule? Such intriguing questions are left for further study.

  9. 9.

    There exists a recent axiomatization of the median rule (in the general judgement aggregation framework) (Nehring and Pivato 2016); we are not aware of axiomatic characterizations of other rules.

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Acknowledgements

Gabriella Pigozzi and Srdjan Vesic benefited from the support of the project AMANDE ANR-13-BS02-0004 of the French National Research Agency (ANR). Jérôme Lang benefited from the support of the ANR Project 14-CE24-0007-01 CoCoRICo-CoDec. The authors would like to thank Denis Bouyssou, as well as the anonymous reviewers and associate editor.

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Correspondence to Jérôme Lang.

Appendix: Proof of Proposition 3

Appendix: Proof of Proposition 3

Many of the non-inclusion relationships can be derived from the profile of our running example, introduced in Example 2, and used again in Example 3 for \({{\textsc {mcc}}}\) and \({{\textsc {mc}}}\), Example 4 for \({{\textsc {med}}}\), Example 5 for \({{\textsc {ra}}}\) and \({\textsc {leximax}}\), Example 6 for \({{\textsc {y}}}\), and Example 7 for \({{\textsc {mpc}}}\). This profile already shows that \({{\textsc {mc}}}\not \subseteq {{\textsc {mcc}}}\), \({{\textsc {mc}}}\not \subseteq {{\textsc {med}}}\), \({{\textsc {mc}}}\not \subseteq {{\textsc {ra}}}\), \({{\textsc {mc}}}\not \subseteq {\textsc {leximax}}\), \({{\textsc {y}}}\not \subseteq {{\textsc {ra}}}\), \({{\textsc {y}}}\not \subseteq {\textsc {leximax}}\), that \({{\textsc {med}}}\) and \({{\textsc {ra}}}\) are incomparable, as well as \({{\textsc {med}}}\) and \({\textsc {leximax}}\), that \({{\textsc {y}}}\) and each of \({{\textsc {mc}}}\), \({{\textsc {mcc}}}\), \({{\textsc {med}}}\) are incomparable, and that \({{\textsc {mpc}}}\) is incompatible with each of \({{\textsc {y}}}\), \({{\textsc {ra}}}\), and \({\textsc {leximax}}\).

The inclusion relationships \({{\textsc {mcc}}}\subseteq {{\textsc {mc}}}\), \({\textsc {leximax}}\subseteq {{\textsc {ra}}}\) are clear from their definitions, and a proof that \({{\textsc {med}}}\subseteq {{\textsc {mc}}}\) can be found in Nehring et al. (2014).

Table 19 A profile showing that \({{\textsc {ra}}}\not \subseteq {\textsc {leximax}}\)
Table 20 A profile showing that \({{\textsc {ra}}}\not \subseteq {{\textsc {y}}}\)
Table 21 A profile \(P\) showing that \({{\textsc {mpc}}}\ {\mathtt {inc}}\ {{\textsc {mc}}}\)
Table 22 A profile \(Q\) at a minimal \(D_H\) distance from \(P\) in Table 21
Table 23 A profile showing that \({{\textsc {mpc}}}\) does not satisfy homogeneity
Table 24 A profile showing that \({{\textsc {mpc}}}\ {\mathtt {inc}}\ {{\textsc {med}}}\)
Table 25 A profile showing that \({F^{d_H,\textsc {max}}}\), \(F_{\text{ rev }}\) and \({F^{d_G,\varSigma }}\) are mutually incomparable

We now prove what remains to be proven:

  1. 1.

    \({{\textsc {ra}}}\subseteq {{\textsc {mc}}}\): If \(J\in {{\textsc {ra}}}(P)\) then, by definition of \({{\textsc {ra}}}\), \(J\cap m(P)\) is a maximal consistent subset of \(m(P)\), thus \(J\in {{\textsc {mc}}}(P)\).

  2. 2.

    \({{\textsc {ra}}}\not \subseteq {\textsc {leximax}}\): Consider the profile \(P\) in Table 19.

    \({{\textsc {ra}}}(P) = \{ \{ p \wedge q, p, q, p \wedge r, q \wedge r, s\}, \{ \lnot (p \wedge q), p, \lnot q, p \wedge r, \lnot (q \wedge r), s\} \{ \lnot (p \wedge q), \lnot p, q, \lnot (p \wedge r), q \wedge r, s\}\}\) and leximax \((P) = \{ \{ p \wedge q, p, q, p \wedge r, q \wedge r, s\} \}\).

  3. 3.

    \({{\textsc {med}}}\not \subseteq {{\textsc {mcc}}}\): Consider the example from Table 6. We have \({{\textsc {mcc}}}(P) = \{ \lnot p, p \rightarrow (q \vee r), \lnot p, \lnot r, p \rightarrow (s \vee t), \lnot s, \lnot t ,p \rightarrow (u \vee v) ,\lnot u, \lnot v\}\) and \(\{ p, p \rightarrow (q \vee r), \lnot q, r, p\rightarrow (s \vee t), \lnot s, t, {p \rightarrow (u \vee v) ,\lnot u, v} \} \in {{\textsc {med}}}(P)\).

  4. 4.

    \({{\textsc {ra}}}\) and \({\textsc {leximax}}\) are incomparable with \({{\textsc {mcc}}}\).

    Consider again the example from Table 6. \({{\textsc {mcc}}}(P) = \{ \{ \lnot p, p \rightarrow (q \vee r), \lnot p, \lnot r, p \rightarrow (s \vee t), \lnot s, \lnot t \}, {p \rightarrow (u \vee v) ,\lnot u, \lnot v}\}\) and for every \(J\in {{\textsc {ra}}}(P)\), and a fortiori for every \(J\in {{\textsc {ra}}}(P)\), \(p \in J\). Thus \({{\textsc {mcc}}}\not \subseteq {{\textsc {ra}}}\) and \({\textsc {leximax}}\not \subseteq {{\textsc {mcc}}}\).

  5. 5.

    \({{\textsc {ra}}}\not \subseteq {{\textsc {y}}}\): Consider the example from Table 20. The minimal number of voters to remove to make the profile majority-consistent is two. These two voters are the two voters of the fourth row (light gray shaded). We have \({{\textsc {y}}}(P) = \{ \{ p, q, p \wedge q, r, s, r \wedge s, t\} \}\) and \({{\textsc {ra}}}(P) = \{ \{ p, q, p \wedge q, r, s, r \wedge s, t\}, \{ p, q, p \wedge q, r, s, r \wedge s, \lnot t\} \}\). Thus, \({{\textsc {ra}}}\not \subseteq {{\textsc {y}}}\).

  6. 6.

    \({{\textsc {mpc}}}\) is incomparable with \({{\textsc {mc}}}\): Consider the pre-agenda \([\mathcal {A}]=\{ p,q,p\wedge q,p\wedge \lnot q\wedge r,\alpha _1,\alpha _2, \lnot p\wedge q \wedge s,\alpha _3,\;\alpha _4, \alpha _5,\alpha _6, \alpha _7 \}\), where

    \(\;\alpha _1 = p\wedge \lnot q \wedge r \wedge \lnot q\), \(\alpha _2~=~p\wedge \lnot q \wedge r \wedge \lnot q\wedge \lnot q\), \(\alpha _3 = q\wedge \lnot p\wedge \lnot p \wedge s\), \(\alpha _4 = q\wedge \lnot p\wedge \lnot p\wedge \lnot p \wedge s\), \( \alpha _5 = (p \leftrightarrow q) \wedge t\), \( \alpha _6 = (p \leftrightarrow q) \wedge t \wedge t\) and \( \alpha _7 = (p \leftrightarrow q) \wedge t \wedge t \wedge t\). Let \(P\) be the profile from Table 21.

    We obtain

    $$\begin{aligned} {{\textsc {mc}}}(P) = \left\{ {\begin{array}{lrrrrrrrrrrr} \{p,&{} q,&{} p \wedge q, &{}\lnot (p\wedge \lnot q \wedge r),&{}\lnot \alpha _1,&{} \lnot \alpha _2, &{}\lnot (\lnot p\wedge q \wedge s),&{}\lnot \alpha _3, &{}\lnot \alpha _4, &{}\lnot \alpha _5, &{}\lnot \alpha _6,&{} \lnot \alpha _7\} \\ \{ p, &{}\lnot q, &{}\lnot (p \wedge q), &{}\lnot (p\wedge \lnot q \wedge r), &{}\lnot \alpha _1, &{}\lnot \alpha _2, &{}\lnot (\lnot p\wedge q \wedge s),&{}\lnot \alpha _3, &{}\lnot \alpha _4, &{}\lnot \alpha _5, &{}\lnot \alpha _6,&{} \lnot \alpha _7\} \\ \{ \lnot p, &{} q,&{} \lnot (p \wedge q),&{}\lnot (p\wedge \lnot q \wedge r),&{}\lnot \alpha _1, &{}\lnot \alpha _2, &{}\lnot (\lnot p\wedge q \wedge s), &{}\lnot \alpha _3,&{} \lnot \alpha _4,&{}\lnot \alpha _5, &{}\lnot \alpha _6, &{}\lnot \alpha _7\} \end{array}} \right\} \end{aligned}$$

    To obtain \({{\textsc {mpc}}}(P)\), we need to change the first three judgments of the first voter, obtaining the profile \(Q\) given in Table 22. This is the minimal change, since if either the second or the third agent change either their judgment on p or their judgment on q, they have to change additional other three judgments. We obtain \({{\textsc {mpc}}}(P)=\{ \{ \lnot p, \lnot q, \lnot (p~\wedge ~q),\lnot (p\wedge \lnot q \wedge r),\lnot \alpha _1, \lnot \alpha _2, \lnot (p\wedge q \wedge s), \lnot \alpha _3, \lnot \alpha _4,\lnot \alpha _5, \lnot \alpha _6, \lnot \alpha _7\} \}\). Thus, \({{\textsc {mc}}}\ {\mathtt {inc}}\ {{\textsc {mpc}}}\).

  7. 7.

    \({{\textsc {mpc}}}\) is incomparable with \({{\textsc {mcc}}}\): Consider the profile \(P\) from Table 23. We have \({{\textsc {mpc}}}(P) = \{ \{ \lnot (p \wedge r), \lnot (p \wedge s), q, \lnot (p \wedge q) \} \}\) and \({{\textsc {mcc}}}(P) = \{ \{ p \wedge r, p \wedge s, \lnot q, \lnot (p \wedge q) \}, \{ p \wedge r, p \wedge s, q, p \wedge q \} \}\); thus \({{\textsc {mpc}}}(P) \cap {{\textsc {mcc}}}(P) = \emptyset \).

  8. 8.

    \({{\textsc {mpc}}}\ {\mathtt {inc}}\ {{\textsc {med}}}\): Consider the pre-agenda \([\mathcal {A}]=\{ p,q,p~\wedge q,p~\wedge ~\lnot q,\alpha _1,\alpha _2,q\wedge \lnot p,\alpha _3,\;\alpha _4 \}\), where \(\;\alpha _1 = p\wedge \lnot q\wedge \lnot q\), \(\alpha _2~=~p\wedge \lnot q\wedge \lnot q\wedge \lnot q\), \(\alpha _3 = q\wedge \lnot p\wedge \lnot p\) and \(\alpha _4 = q\wedge \lnot p\wedge \lnot p\wedge \lnot p\).

    We obtain \({{\textsc {mpc}}}(P)\) by changing the first three judgments of the first voter. This is the minimal change, since if either the second or the third agent change either their judgment on p or their judgment on q, they have to change additional other three judgments. Observe that \({{\textsc {med}}}(P) = \{\{ p, q, p \wedge q, \lnot (p \wedge \lnot q), \lnot \alpha _1, \lnot \alpha _2, \lnot (q \wedge \lnot p), \lnot \alpha _3, \lnot \alpha _4 \}\}\) since for this judgment set the weight is 17, and for the remaining three other possible judgment sets the weights are: 14 for the set of the judgment sets of the second, and third voter and 16 for the judgment set \(\{\lnot p, \lnot q, \lnot (p~\wedge ~q),\lnot (p~\wedge ~\lnot q),\lnot \alpha _1, \lnot \alpha _2, \lnot (q\wedge \lnot p),\lnot \alpha _3, \lnot \alpha _4\}\).

    Thus \({{\textsc {mpc}}}\ {\mathtt {inc}}\ {{\textsc {med}}}\) (Table 24).

  9. 9.

    \({F^{d_H,\textsc {max}}}\), \(F_{\text{ rev }}\) and \({F^{d_G,\varSigma }}\) are pairwise incomparable: We give one counterexample for all three pairs. Let \(\mathcal {A}_{C}\) be the preference agenda for the set of alternatives \(C= \{c_1,c_2,c_3,c_4\}\), together with the transitivity constraint. Consider the profile given in Table 25. The collective judgments obtained by \(F_{\text{ rev }}\), \({F^{d_G,\varSigma }}\), and \({F^{d_H,\textsc {max}}}\) are represented in the last five rows of this table.

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Lang, J., Pigozzi, G., Slavkovik, M. et al. A partial taxonomy of judgment aggregation rules and their properties. Soc Choice Welf 48, 327–356 (2017). https://doi.org/10.1007/s00355-016-1006-8

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