Measuring Majority Tyranny: Axiomatic Approach

We study voting rules with respect to how they allow or limit a majority to dominate minorities. For this purpose we propose a novel quantitative criterion for voting rules: the quali ed mutual majority criterion (q; k)-MM. For a xed total number of m candidates, a voting rule satis es (q; k)-MM if whenever some k candidates receive top k ranks in an arbitrary order from a majority that consists of more than q 2 (0; 1) of voters, the voting rule selects one of these k candidates. The standard majority criterion is equivalent to (1=2; 1)-MM. The standard mutual majority criterion (MM) is equivalent to (1=2; k)-MM, where k is arbitrary. We nd the bounds on the size of the majority q for several important voting rules, including the plurality rule, the plurality with runo rule, Black's rule, Condorcet least reversal rule, Dodgson's rule, Simpson's rule, Young's rule and monotonic scoring rules; for most of these rules we show that the bound is tight.