Abstract
Noting the existence of social choice problems over which no scoring rule is Maskin monotonic, we characterize minimal monotonic extensions of scoring rules. We show that the minimal monotonic extension of any scoring rule has a lower and upper bound, which can be expressed in terms of alternatives with scores exceeding a certain critical score. In fact, the minimal monotonic extension of a scoring rule coincides with its lower bound if and only if the scoring rule satisfies a certain weak monotonicity condition (such as the Borda and antiplurality rule). On the other hand, the minimal monotonic extension of a scoring rule approaches its upper bound as its degree of violating weak monotonicity increases, an extreme case of which is the plurality rule with a minimal monotonic extension reaching its upper bound.
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Notes
For any x, y∈A, p i (x)>p i (y) means that voter i∈N prefers x to y.
Note that the definitions of an improvement and a worsening do not require strictness. Hence p can be both an improvement and worsening for x with respect to p′—which we call a reordering and formally define in Section 3.
Recall that F* uniquely exists for every F. Note also that F*=F if and only if F is Maskin monotonic.
Note that q is a reordering of p with respect to x if and only if q is both an improvement and worsening for x with respect to p.
Which does not mean that their lower and upper bounds coincide as c*(x, p) and c**(x, p) need not be equal for weakly monotonic scoring rules.
The plurality rule is a scoring rule induced by the score vector s=(s 1, 0,..., 0). We will take s 1=1, without loss of generality.
The Borda rule is a scoring rule B:P N→A induced by a score vector s=(s 1,..., s m ) with s i −s i+1=s i+1−s i+2 for all i∈{1,..., m−2}. The antiplurality rule is a scoring rule AP : P N→A induced by a score vector s = (1, 1,...1, 0).
This does not mean that the minimal monotonic extensions of the Borda rule and the antiplurality rule coincide, as the value of c**(x; p) generally differs among the rules.
except for k=m−1, which is the antiplurality rule.
Recall that ĉ* is the lowest integer no less than (n/m)·ŝ and ŝ=2 for the antiplurality rule in this three-by-three problem.
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Acknowledgements
This paper was written while Orhan Erdem was a graduate student in Economics at Boğaziçi University. It has been presented at the Conference of the Society for Economic Design, July 2002, New York and the Conference of the Society for Social Choice and Welfare, July 2002, Pasadena. We thank Vincent Merlin, İpek Özkal-Sanver, Murat R. Sertel and all the participants. Remzi Sanver acknowledges partial financial support from İstanbul Bilgi University and thanks Haluk Sanver and Serem Ltd. for their continuous moral and financial support. Last but not the least, we thank Maurice Salles and two anonymous referees. Of course, we are the sole responsible for all possible errors.
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Erdem, O., Sanver, M.R. Minimal monotonic extensions of scoring rules. Soc Choice Welfare 25, 31–42 (2005). https://doi.org/10.1007/s00355-005-0058-y
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DOI: https://doi.org/10.1007/s00355-005-0058-y