Abstract
This paper reflects on some results characterizing qualified majority rules using monotonicity as a key axiom. In particular, some errors in the existing literature are detected and ways to fix them are proposed. Then, the role of monotonicity axiom in characterizing majority rules is analyzed. There, among other findings, we show that its marginal contribution in characterizing relative qualified majority rules is the difference between two properties called sum-invariance and sum-monotonicity. Finally, a new class of qualified majority rules where voters exercise a veto power is introduced and axiomatically characterized.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A version Theorem 1(a) can be deduced from Theorem 3.1 in Sanver (2009) via Theorem 2.1 in the same paper. Using their terminology, it can be stated as the only elementary aggregation rule which is both independent of cancelling couples and additive is\(F_{u}\). However, their setting excludes \(F_{1}\) from being a qualified majority rule and requires an additional axiom called unanimity throughout their analysis.
The error in its proof is the step: \(\eta \in D\) implies \(n-\eta \notin D\).
Llamazares (2006)’s definition of RqM differs from ours slightly as it requires \(q<1\). This rules out \(F_{1}\) as RqM and they needed an additional axiom, Weak Pareto, for this to happen in their characterization. Thus, Theorem 2 is new. In fact it is easy to deduce Theorem 8 in Llamazares (2006) from Theorem 2, but not vice versa.
In the proof given in Aşan and Sanver (2006), the second ‘\(=\)’ in the statement of (ii) should be ‘\(\ge \)’, which is a typo. In proving (i), it argues that “MM implies \(F(R')=1\)” which is invalid, unless MM is strengthened. The same mistake reappears in proving (ii) where it is stated that “By MM, we have \(F(Q)=1\)”. Interestingly, this (invalid) proof demonstrates that a strengthened version of MM can also be used in characterizing AqM.
Houy (2007a) shows that monotonicity axiom can (and should) be weakened to a partial version of it when characterizing RqMs with quorum.
References
Aşan, G., Sanver, R.: Maskin monotonic aggregation rules. Econ. Lett. 91, 179–193 (2006)
Eliaz, K.: Social aggregators. Soc. Choice Welf. 22, 317–330 (2004)
Fey, M.: An application of asymptotic density to characterizing voting rules. Tatra Mt. Math. Publ. 31, 29–37 (2005)
Houy, N.: Some further characterizations for the forgotten voting rules. Math. Soc. Sci. 53, 111–121 (2007a)
Houy, N.: A characterization for qualified majority voting rules. Math. Soc. Sci. 54, 17–24 (2007b)
Llamazares, B.: The forgotten decision rules: majority rules based on difference of votes. Math. Soc. Sci. 51, 311–326 (2006)
Perry, J., Powers, R.C.: Anonymity, monotonicity, and quota pair systems. Math. Soc. Sci. 60, 57–60 (2010)
Sandel, M.J.: Justice: What’s The Right Thing To Do?. Penguin Books, London (2009)
Sanver, R.: Characterizations of majoritarianism: a unified approach. Soc. Choice Welf. 33(1), 159–171 (2009)
Acknowledgements
I am thankful to M. Remzi Sanver, to the editor and anonymous referees of this journal and to seminar participants of the department of economics at the National University of Mongolia for helpful comments and discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ninjbat, U. Monotonicity and qualified majority rules. Econ Theory Bull 7, 209–220 (2019). https://doi.org/10.1007/s40505-018-0154-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40505-018-0154-7