ABSTRACT
L S Penrose’s Limit Theorem – which is implicit in Penrose [7, p. 72] and
for which he gave no rigorous proof – says that, in simple weighted voting
games, if the number of voters increases indefinitely and the relative quota is
pegged, then – under certain conditions – the ratio between the voting powers
of any two voters converges to the ratio between their weights. Lindner and
Machover [4] prove some special cases of Penrose’s Limit Theorem. They
give a simple counter-example showing that the theorem does not hold in
general even under the conditions assumed by Penrose; but they conjecture,
in effect, that under rather general conditions it holds ‘almost always’ – that
is with probability 1 – for large classes of weighted voting games, for various
values of the quota, and with respect to several measures of voting power.
We use simulation to test this conjecture. It is corroborated with respect to
the Penrose–Banzhaf index for a quota of 50% but not for other values; with
respect to the Shapley–Shubik index the conjecture is corroborated for all
values of the quota (short of 100%).
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