The idea that candidates are interested not solely in the attainment of office, but also care about the policy to which they pledge themselves, is natural and appealing. Indeed, it was not until such works as Davis, Hinich and Ordeshook's "An expository development of a mathematical model of the electoral process" (Davis et al., 1970) and Mayhew's Congress: The Electoral Connection (Mayhew, 1974) employed the assumption that Congressmen were "single-minded seekers of re-election" that the usefulness of an approach which ignored the personal policy preferences of Congressmen became apparent. Now, of course, the assumption of policy-indifferent candidates is used widely in formal models of the electoral process. While the specific behavior of candidates may vary from maximizing votes (e.g., Kramer, 1977), to maximizing plurality (e.g., Hinich and Ordeshook, 1970), to choosing according to a uniform distribution from amongst the winning strategies (Ferejohn et al., 1981), and so on, the basic assumption that candidates care chiefly about winning elections is generally retained. There have, however, been a few attempts to reinstate policy motivations for candidates (Wittman, 1973, 1977; Petry, 1982). Indeed, these attempts do more than reinstate an element of policy concern; they posit that candidates care not at all about which candidate wins but only about which policy wins (i.e., is advocated by the winning candidate). In the models both of Wittman and of Petry, a candidate chooses a position x (when his opponent is at y) which will maximize his expected utility--the probability of his winning times the utility of x plus the probability of his opponent winning times the utility of y. Since both use the traditional assumption that voters have ideal points and vote with certainty for the candidate closer to this ideal point, maximizing expected utility given y is equivalent to maximizing utility by choice of x subject to the constraint that x be in the set of points which defeat y by majority vote. Unfortunately, this maximization problem does not necessarily have a solution, since the set of points defeating a given point