This paper investigates the effects of abstentions on election outcomes in two-candidate simply majority elections. It is primarily concerned with the probability that the winning candidate is the candidate who would have been elected if all eligible voters had voted, or with the complementary probability that the winner is not preferred to the loser by a majority of eligible voters. Although abstentions have been discussed by others (Aranson and Ordeshook 1972;Davis et al. 1970; Hinich and Ordeshook 1969, Hinich et al. 1972, Sen 1964;Shepsle 1972), the phenomenon examined here does not appear to have been considered in detail elsewhere.
Our interest in abstentions stems from a larger concern, represented by Fishburn and Gehrlein (1976a, 1976b, 1977) Gehdein and Fishburn (1976), that is motivated in part by a desire to identify voting procedures that tend to maximize the likelihood of electing the candidate who in some sense is socially most preferred by the eligible electorate. Although considerations of strategic voting (especially in multicandidate cases) and abstentions are relevant to this work, their effects have not yet been integrated into it. In the present paper we begin an examination of the abstention half of this pair by considering a siraple and common voting procedure that obviates concern about strategic voting. The main purpose of the investigation is to examine how different assumptions about the electorate influence the probability that the "minority candidate" wins the election or, equivalently, that the "majority candidate" loses the election. We shall not be concerned with predicting election outcomes from samples or polls although the models developed here might form a basis for statistical inference. (It may well be true that other models are better suited to the latter endeavor.) In