Social Choice and Elections

The electoral models based on the early work of Hotelling (1929) and Downs (1957) essentially suppose that the motivation of parties is to win a majority of the votes or seats. A very considerable literature developed in the period up to 1973 (ably summarized by Riker and Ordeshook, 1973), focusing on two party competition and the existence of convergent equilibrium at the electoral median. McKelvey’s thesis (1972) and his first technical paper (McKelvey, 1975) made a significant contribution to this literature. A feature of this literature was that symmetry in the electoral distribution was the sufficient condition for existence. Plott’s (1967) previous analysis indicated that symmetry was also a necessary condition. Indeed, Gerry Kramer’s (1972, 1973) papers suggested that equilibria might generally not even exist. The papers by McKelvey (1976) and Schofield (1977) though independently arriving at somewhat similar conclusions on the existence of voting cycles, used entirely different formal methods. McKelvey supposed that the electoral distribution was not symmetric, so that the electoral equilibrium, or core, was empty, and then showed that disconnected preference cycles could wander throughout the preference space. Schofield (1977) first generalized Kramer’s result by showing that there was a local electoral condition sufficient to generate voting cycles near the point, and then demonstrated that this condition could be expected to hold somewhere, whenever two dimensions were involved. Extensions of these two papers (Schofield, 1978; McKelvey, 1979) showed that very stringent symmetry conditions on voter preferences were necessary to avoid the kind