The election model in this chapter generalizes that of the preceding chapter in two important ways: (i) This model allows much greater flexibility in the specification of the number of groups in the electorate and the distribution of voters across groups, and (ii) the set of possible locations for the candidates is not assumed to be convex. Significantly, (as in Chapter 5) the remaining convexity assumptions used in the equilibrium existence results in Theorem 4.1 and Corollaries 5.1–5.3 are again treated as possible special cases for the model. These and the remaining features of this more general model are identified in precise terms in Section 6.1.

Because the convexity assumptions used to obtain the equilibrium existence results in Chapters 4 and 5 are not required in this chapter, it immediately follows that there need not be an electoral equilibrium (as defined in Section 2.3), as Example 5.5 (in Section 5.5) made clear. In this chapter, therefore, we study some alternative concepts of what constitutes an electoral equilibrium that are weaker (i.e., less demanding) than the concept used so far. The specific feature of the earlier concept that is relaxed is the assumption that the candidates are free to choose any location in the set S. Relaxing this feature may be desirable in a particular context for reasons that have been discussed in the literature (e.g., see Kramer and Klevorick, 1974, pp. 540–1; Kats and Nitzan, 1976; and/or Matthews, 1979, pp. 142–3).