Three stories about the chance of casting a pivotal vote

People vote 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 the election from the political party one opposes to the political party one favours. This paper is a discussion of three models of how that chance might arise: the common sense model inferring the probability of a tied vote today from the distribution of outcomes in past elections, person-toperson randomization where each voter looks upon the political preferences of 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 valuations (positive or negative) of a win for one of the two competing parties, but where chance shifts the entire schedule of preferences up or down. Emphasis is on the third model about which this paper may have something new to say. -- Pivotal voting ; duty to vote ; compulsory voting