 As before, consider a set \(N = \{1, 2, \ldots ,n\}\) of individuals \((n \ge 2)\) and a set \(X = \{x, y, z, \ldots \}\) of social alternatives. Now each individual \(i \in N\) has a welfare function \(W_i\) over these alternatives, which assigns a real number \(W_i (x)\) to each alternative \(x \in X\), interpreted as a measure of \(i\)’s welfare under alternative \(x\). Any welfare function on \(X\) induces an ordering on \(X\), but the converse is not true: welfare functions encode more information. A combination of welfare functions across the individuals, \(\langle W_1, W_2 , \ldots ,W_n\rangle\), is called a profile.