Duty, self-interest, and the chance of casting a pivotal vote

One votes from self-interest or from a sense of duty. Voting from self-interest requires there to be some chance, however small, that one's vote swings the outcome of an election from one political party to another. This paper is a discussion of three models of what that chance might be: the common sense model inferring the probability of a tied vote today from the distribution of outcomes in past elections, person-to-person randomization where each voter looks upon the rest of the electorate as analogous to drawings from an urn with given proportions of red and blue balls, and nation-wide randomization where voters are lined up according to their preferences for one party or the other but where chance shifts the entire schedule of preferences up or down. Emphasis is on the third model about which the paper may have something new to say. Nation-wide randomization may be helpful in connecting private benefits from a win for one's preferred party with a duty to vote, and in comparing the pros and cons of compulsory voting.