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    Recovering non-monotonicity problems of voting rules
    (İstanbul Bilgi Üniversitesi, 2019) Tosunlu, Halil Berkay; Keskin, Umut
    ABSTRACT: Social choice rule (SCR) is monotonic if an elected alternative still gets elected when it is improved for at least one voter and the ranking among other alternatives remain the same. 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 a SCR as its unique monotonic supercorrespondence that is minimal with respect to set inclusion. After showing the existence of the concept, we charac- terize, for every non-monotonic SCR, the alternatives that its minimal mono- tonic extension must contain. We also discuss measures of non-monotonicity based on the divergence between the minimal monotonic extension of a SCR and the SCR itself. As minimal monotonic extensions can entail coarse SCRs, we address the possibility of maximally refining them without violating mono- tonicity provided that this refinement does not diverge from the original SCR more than the divergence prescribed by the minimal monotonic extension it- self. We call these refinements minimal monotonic adjustments and identify conditions over SCRs that ensure unique minimal monotonic adjustments. As an application of our general findings, we consider plurality with a runoff, charaterize its minimal monotonic extension as well as its (unique) minimal monotonic adjustment. Interestingly, this adjustment is not coarser than plu- rality with a run off itself, hence we suggest it as a monotonic rectification of plurality with a runoff.