 Consider a set \(N = \{1, 2, \ldots ,n\}\) of individuals \((n \ge 2)\). Let \(X = \{x, y, z, \ldots \}\) be a set of social alternatives, for example possible worlds, policy platforms, election candidates, or allocations of goods. Each individual \(i \in N\) has a preference ordering \(R_i\) over these alternatives: a complete and transitive binary relation on  \(X\).[4]  For any \(x, y \in X\), \(xR_i y\) means that individual \(i\) weakly prefers \(x\) to \(y\). We write \(xP_i y\) if \(xR_i y\) and not \(yR_i x\) (‘individual \(i\) strictly prefers \(x\) to \(y\)’), and \(xI_i y\) if \(xR_i y\) and \(yR_i x\) (‘individual \(i\) is indifferent between \(x\) and \(y\)’).