 A profile \(\langle R_1, R_2 , \ldots ,R_n\rangle\) is single-peaked if the alternatives can be aligned from ‘left’ to ‘right’ (e.g., on some cognitive or ideological dimension) such that each individual has a most preferred position on that alignment with decreasing preference as alternatives get more distant (in either direction) from the most preferred position. Formally, this requires the existence of a linear ordering \(\Omega\) on \(X\) such that, for every triple of alternatives \(x, y, z \in X\), if \(y\) lies between \(x\) and \(z\) with respect to \(\Omega\), it is not the case that \(xR_i y\) and \(zR_i y\) (this rules out a ‘cave’ between \(x\) and \(z\), at \(y)\). Single-peakedness is plausible in some democratic contexts. If the alternatives in \(X\) are different tax rates, for example, each individual may have a most preferred tax rate (which will be lower for a libertarian individual than for a socialist) and prefer other tax rates less as they get more distant from the ideal.