We now have a fairly clear sense of exactly what simple majority voting is and what a set of characteristic properties for that rule looks like. It is extremely important for you to see that simple majority voting has only been defined for the case of exactly two alternatives. Most of the problems with simple majority voting arise in a context of more than two alternatives and all interesting contexts have more than two alternatives. Sometimes this is disguised as when we are asked to make a final choice between two party-selected candidates. But the social choice process for filling office starts with many potential candidates and includes many preliminary narrowing procedures: decisions to run or not run, nominating petitions, primaries, withdrawals, conventions,.... So let’s explore what happens when we increase the number of alternatives to three or more, labeled x, y, z ....
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- On the discovery of the voting paradox and other early history of social choice, there is a historical part in Duncan Black’s The Theory of Committees and Elections (Cambridge University Press, 1958). For more on the calculation of the likelihood of the voting paradox, see William Gehrlein’s, “Condorcet’s Paradox,” Theory and Decision, Vol. 15 (1983), pp. 161–197, from which Table 1 was obtained.Google Scholar