To obtain a contradiction, suppose that there is a cycle; this implies that so that there are q ≥ 2 different governments H1, . . . ,Hq such that φ (Hj) = Hj+1 for all 1 ≤ j < q, and φ (Hq) = H1. Without loss of generality, let H1 be the least competent of these governments. Take H = φ (H2) (if q > 2 then H = H3 and if q = 2 then H = H1). As φ is a political equilibrium, Vi (H) > Vi (H2) holds ...