The examples of social choice theory put forward in the first chapter have one common central feature. Each example involves the combination of information about individuals' preferences into some sort of social statement. We begin this chapter with a discussion of individual preferences which establishes both the basic ‘input’ of social choice theories and a convenient notation in which to express it. Virtually every book on social choice theory introduces notation first – one of the hazards of the literature is that each author's notation is different. We stay fairly close to that of Arrow (1951).

Individuals and alternatives

In every problem that we examine, there is a set of nindividuals, prosaically named 1 to n, and known collectively as the society. This society may be a whole community, some collection of electors, consumers or any other group of interest to us. In our illustrative examples, n is usually a fairly small number, though in reality an electorate may consist of millions of people.

The other ‘raw material’ of the theory is a set of alternatives. These are the things over which individuals have preferences, and could be, for example:

In general, the alternatives are any situations about which some judgement or choice is to be made, and, from a formal point of view, it does not matter what these alternatives are.