This chapter is concerned exclusively with individual choice. Let X be the universal set of alternatives. We wish to represent the individual's preferences in such a way that his choice from a subset B of X can be obtained from that representation. We begin with choice over two-element sets and build from there.

Definition: A binary relation Q on X is a subset of X × X. If (x, y) belongs to Q, we write xQy, Relation Q is complete if, for all x, y ∈ X, either xQy or yQx holds; Q is transitive if, for all x, y, z ∈ X, xQy and yQz imply xQz. A complete and transitive binary relation is called a weak order and is represented by the letter R.

Since R is complete, we have xRx (set y = x). Since x cannot be strictly preferred to itself, the statement xRy must mean that x is at least as desirable as y: It may be that x is equally desirable as y as in the case y = x, or that x is strictly preferred to y. We can be sure that y is not strictly preferred to x if xRy holds.

If x belongs to B and xRy for all y ∈ B, then x belongs to the set of best (or most-preferred) elements in B. A weak order allows us to determine choice on any finite set from choice on two-element sets.