 For any \(x, y \in X\), xRy means that \(x\) is socially weakly preferred to \(y\). We also write \(xPy\) if \(xRy\) and not \(yRx\) (‘\(x\) is strictly socially preferred to \(y\)’), and \(xIy\) if \(xRy\) and \(yRx\) (‘\(x\) and \(y\) are socially tied’). For generality, the requirement that \(R\) be complete and transitive is not built into the definition of a preference aggregation rule.