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This is an electronic version of an Article published inThe first will apply from 2005 to the present 15-member EU, if it will not have been enlarged by then. The second or third will apply to an enlarged 27-member EU. We conclude that the first of these is an improvement on the current decision rule; but the other two have extremely undesirable features.

L.S. Penrose was the first to propose a measure of voting power (which later came to be known as 'the [absolute] Banzhaf (Bz) index'). His limit theorem-which is implicit in his booklet (1952) and for which he gave no rigorous proof-says that in simple weighted voting games (WVGs), if the number of voters increases indefinitely while the quota is pegged at half the total weight, then-under certain conditions-the ratio between the voting powers (as measured by him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined WVGs, other values of the quota, and other measures of voting power. We provide proofs for some special cases.

We define ternary voting games (TV Gs), a generalization of simple voting games (SVGs). In a play of an SVG each voter has just two options: voting 'yes' or 'no'. In a TVG a third option is added: abstention. Every SVG can be regarded as a (somewhat degenerate) TVG; but the converse is false. We define appropriate generalizations of the Shapley-Shubik and Banzhaf indices for TVGs. Wc define also the responsiveness (or degree of democratic participation) ofa TVG and determine, for each n, the most responsive TVGs with n voters. We show that these maximally responsive TVGs are more responsive than the corresponding SVGs.