Avoiding Anscombe's paradox (Q794541)

As first noted by \textit{G. E. M. Anscombe} [Analysis 36, 161-168 (1976)], the disposition of a set of proposals by majority rule does not preclude the set of voters who disagree with a majority of the outcomes from comprising a majority. It is intuitively clear, however, that when proposals are adopted or rejected by a sufficiently strong consensus, the aforementioned possibility is precluded. In this paper, it is shown that, for N voters, K proposals, and \(0<\alpha\), \(\beta<1\), if the average number of voters in the prevailing coalitions is at least \((1- \alpha\beta)N,\) then no more than \(\beta N\) voters can disagree with the outcomes on more than \(\alpha K\) proposals. In particular, setting \(\alpha =\beta =1/2\) yields the previously known result of the author [Theory Decis. 15, 303-308 (1983; Zbl 0513.90006)] that when prevailing coalitions comprise, on average, at least three-fourths of those voting, the set of voters disagreeing with a majority of outcomes cannot comprise a majority. Examples are presented to illustrate the sharpness of these and related results.