To vote or not to vote: The paradox of nonvoting

One paradox of voting states that, in a general election, in which many citizens vote, the probability that a single voter can affect the outcome is so small that in general citizens have no rational reason for voting. However, if all citizens accept this reasoning, then none will vote, and so each vote has a large probability of affecting the outcome. Hence all should vote after all. The adoption of mixed strategies resolves this paradox: if each citizen adopts a certain (small) probability of voting, then the actual number of citizens voting will be just enough to make it worth those citizens' while to vote. A Nash equilibrium point thus occurs. Here we compute Nash equilibria for the simple case of majority voting; for the more complicated case of composite voting (for example, as in a presidential election), we draw certain qualitative inferences.