Cases where the Penrose limit theorem does not hold
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Penrose's limit theorem (PLT, really a conjecture) states that the relative power measure of two voters tends asymptotically to their relative voting weight (number of votes). This property approximately holds in most of real life and in randomly generated WVGs for various measures of voting power. Lindner and Machover prove it for some special cases; amongst others they give a condition for this theorem to hold for the Banzhaf–Coleman index for a quota of 50%. We show here, by counterexamples, that the conclusion need not hold for other values of the quota. In doing this, we present an analytic proof of a counterexample recently given by Chang et al. using simulation techniques.
The article of record as published may be located at http://doi.org/10.1016/j.mathsocsci.2007.01.005Research performed while at the Naval Postgraduate School, Monterey, California, United States.