Abstract
A social choice rule (SCR) is monotonic if raising a single alternative in voters’ preferences while leaving the rankings otherwise unchanged is never detrimental to the prospects for winning of the raised alternative. Monotonicity is rather weak but well-known to discriminate against scoring elimination rules, such as plurality with a run off and single transferable vote. We define the minimal monotonic extension of an SCR as its unique monotonic supercorrespondence that is minimal with respect to set inclusion. After showing the existence of the concept, we characterize, for every non-monotonic SCR, the alternatives that its minimal monotonic extension must contain. As minimal monotonic extensions can entail coarse SCRs, we address the possibility of refining them without violating monotonicity provided that this refinement does not diverge from the original SCR more than the divergence prescribed by the minimal monotonic extension itself. We call these refinements monotonic adjustments and identify conditions over SCRs that ensure unique monotonic adjustments that are minimal with respect to set inclusion. As an application of our general findings, we consider plurality with a runoff, characterize its minimal monotonic extension as well as its (unique) minimal monotonic adjustment. Interestingly, this adjustment is not coarser than plurality with a runoff itself, hence we suggest it as a monotonic substitute to plurality with a runoff.
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Notes
Simple monotonicity is perhaps the oldest known monotonicity condition in the literature. It has been expressed under different names during its relatively long history that predates modern social choice theory. For a comprehensive account, see Black et al. (1958), Brams and Fishburn (2002) and comments on page 120 of Fishburn (1982).
The condition has a slightly stronger version which additionally requires that no new alternative is added to the chosen set. We chose to analyze the weaker version for reasons we discuss in Footnote 11.
See Doron and Kronick (1977) for arguments against using non-monotonic SCRs.
These instances, expressed by their Propositions 1 and 3, are special cases of the general characterization we give in our Theorem 5.1.
Our approach is similar to the approach in Sen (1995) for Maskin monotonicity.
Caragiannis et al. (2014) characterize what they call “the approximation with the least approximation ratio” for the non-monotonic Dodgson’s voting rule and this corresponds to the minimal monotonic extension as we define here.
We further address this point in Sect. 4. We thank an anonymous referee for raising the issue.
So precisely one of x \(P_{i}\) y and y \(P_{i}\) x holds for any distinct \(x,y\in A\) while x \(P_{i}\) x fails for all \(x\in A\). Moreover, x \(P_{i}\) y and y \(P_{i}\) z implies x \(P_{i}\) z for all \(x,y,z\in A\).
The stronger version which we mention in Footnote 2 would additionally impose \(F(P^{\prime })\subseteq F(P)\).
Our conclusion on the existence of a unique minimal monotonic extension would not be valid under the stronger version of monotonicity expressed in Footnote 10, as the non-emptiness of \(\mu (F)\) could not be ensured. We thank David Pennock for raising this issue.
This argument would fail in case we were refining a monotonic extension of F that is not minimal. We thank an anonymous referee for raising this issue.
One can see this through the profiles in Example 4.2 where \(\delta _{F_{2}}(Q)=\{x\}\) and \(R\in\) IMP\(_{x}(Q)\) but \(x\notin \delta _{F_{2}}(R)=\{y\}\).
We could define a more sophisticated measure that also takes into account the amount of disagreement at a given profile but we don’t wish to deal with details that are unnecessary for the argument we are about to make.
To see this, let \(A=\{x,y\}\) and \(N=\{1,2\}\). Define \(F(P)=\{x,\) \(y\}\) when x \(P_{i}\) y \(\forall i\in N\); \(F(P)=\{y\}\) when y \(P_{i}\) x \(\forall i\in N\); and \(F(P)=\{x\}\) when x \(P_{i}\) y and y \(P_{j}\) x for \(i\ne j\). Here, monotonicity is violated at the two profiles where the outcome is \(\{x\}\), hence \({\widetilde{F}}\) disagrees with F at those two profiles. On the other hand, the SCR G that agrees with F at every profile except the one where x is ranked first by both voters (G picks \(\{x\}\) rather than \(\{x,\) \(y\}\) at this profile) is monotonic. So, under the measure we described, G is closer to F than \({\widetilde{F}}\).
For example, let \(n=93\), \(A=\{a,b,c\}\) and P be a profile where 42 voters have the preference \(aP_{i}bP_{i}c\); 27 voters have the preference \(bP_{i}cP_{i}a\); and 24 voters have the preference \(cP_{i}aP_{i}b.\) So \(RO(P)=\left\{ \left\{ a,b\right\} \right\}\) and \(F_{PR}(P)=\{a\}\). Let \(P^{\prime }\) be a profile where 46 voters have the preference \(aP_{i}^{\prime }b P_{i}^{\prime } c\); 23 voters have the preference \(bP_{i}^{\prime }c P_{i}^{\prime } a\); and 24 voters have the preference \(c P_{i}^{\prime }aP_{i}^{\prime }b.\) Now \(RO(P^{\prime })=\left\{ \left\{ a,c\right\} \right\}\) and \(F_{PR}(P^{\prime })=\{c\}\) while \(P^{\prime }\in\) IMP\(_{a}(P).\)
Sertel and Kalaycıoğlu (1995) comprehensively discussed this issue while campaigning against the usage of the plurality rule in Turkish political elections.
A more formal and complete description of these concepts is beyond the scope of this paper but can be found in Saari (1990).
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Our work is partly supported by the projects ANR-14-CE24-0007-01 CoCoRICo-CoDec and IDEX ANR-10-IDEX-0001-02 PSL* MIFID, as well as the LAMSADE internal project programme. The main findings of this paper were discovered when H. Berkay Tosunlu was a masters student in economics at İ stanbul Bilgi University. We thank Jerome Lang, Vincent Merlin, Hervé Moulin and participants of the Dagstuhl Seminar 19381 on Application-Oriented Computational Social Choice for useful comments and discussions. The paper extensively benefited from the thoughtful comments of two anonymous reviewers and the anonymous associate editor to whom we are grateful.
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Keskin, U., Sanver, M.R. & Tosunlu, H.B. Recovering non-monotonicity problems of voting rules. Soc Choice Welf 56, 125–141 (2021). https://doi.org/10.1007/s00355-020-01272-0
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DOI: https://doi.org/10.1007/s00355-020-01272-0