 Gibbard (1969) proved that even if we replace the requirement of transitivity with what he called quasi-transitivity, the resulting possibilities of aggregation are still very limited. Call a preference relation \(R\) quasi-transitive if the induced strict relation \(P\) is transitive (while the indifference relation \(I\) need not be transitive). Call an aggregation rule oligarchic if there is a subset \(M \subseteq N\) (the ‘oligarchs’) such that (i) if, for all \(i \in M\), \(xP_i y,\) then \(xPy,\) and (ii) if, for some \(i \in M\), \(xP_i y\), then \(xRy.\) The Pareto extension rule is an example of an oligarchic aggregation rule with \(M = N\). In an oligarchy, the oligarchs are jointly decisive and have individual veto power. Gibbard proved the following: