The voting paradox introduced in section 1.3 shows that majority voting ‘does not work’ in all circumstances. It is not possible to use majority voting and the Condorcet criterion when there are three or more alternatives because, for some combinations of preferences, the social preference relation involves a voting cycle such as aPb, bPc, cPa, so that C(a,b,c) is empty. Arrow's theorem generalises this paradox to show that there are difficulties with any method that might be used to generate social choices.

One way of attempting to escape from the difficulties posed by Arrow's theorem is to examine the circumstances in which majority voting and the Condorcet criterion does not give a voting cycle, so that no social choice set is empty. If we remove the condition of unrestricted preferences so that we exclude consideration of combinations of preferences that give rise to the voting paradox, majority voting avoids dictators and is democratic to the full extent that each individual has equal weight in determining the social choice, and each alternative is treated equally with the other alternatives. To what extent must preferences be restricted to ensure that majority voting ‘works’? Is it possible to describe the restrictions in ways that allow us to judge whether they are plausible.