Quality & Quantity 38: 637–647, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Local Monotonicity of Power: Axiom or just a
MANFRED J. HOLLER
and STEFAN NAPEL
Institute of SocioEconomics (IAW), University of Hamburg, Von-Melle-Park 5, D-20146 Hamburg,
Abstract. This paper discusses whether Local Monotonicity (LM) should be regarded as a prop-
erty of the power distribution of a speciﬁc voting game under consideration, indicated by a power
measure, or as a characteristic of power per se. The latter would require reasonable power measures
to satisfy a corresponding LM axiom. The former suggests that measures which do not allow for a
violation of LM fail to account for dimensions of power which can cause nonmonotonicity in voting
weight. Only if a measure is able to indicate nonmonotonicity, it can help design voting games for
which power turns out to be monotonic. The argument is discussed in the light of recent extensions
of traditional power indices.
Key words: Power measures, monotonicity, voting
Assume that there are three parties, A, B and C, which have a share of parliament
seats of 45%, 35%, and 20%, respectively. Given that decisions are made by simple
majority it seems not very likely that the distribution of power, however deﬁned,
coincides with the distribution of votes. Power indices have been developed to
discuss issues of assigning power values to the resources (e.g., votes) of decision
makers and to explain how these values change if the resource distribution changes
or a new decision rule is applied. They seem to be valuable instruments to analyze
institutional changes and potential effects of alternative institutional design. The
two volumes, “Power, Voting, and Voting Power” (Holler 1982a) and “Power In-
dices and Coalition Formation” (Holler & Owen 2001) not only contain original
contributions to this discussion but also illustrate the development in this ﬁeld over
the last twenty years. A recent monograph by Felsenthal and Machover (1998),
“The Measurement of Voting Power”, gives a comprehensive formal treatment.
There is a growing interest in power measures such as the Shapley–Shubik
index and the Banzhaf index, to name the two most popular measures. Their
application to political institutions, in particular to the analysis of the European
has thrived. There are also new theoretical instruments and perspectives
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