Analyzing the Effects of Voting
I haven't seen a persuasive analysis of whether and how one should vote in a US presidential election; every analysis I've seen either concludes that voting is pointless (at least outside of a swing state), or ignores the Electoral College completely. I intend to fill this gap.
Tipping the Election
Obviously one vote is statistically extremely unlikely to affect the outcome of the election. The naive analysis suggests that the probability your vote tips the election is the probability of a tie assuming you don't vote.
Some people would object to this analysis here, contending that a single voter never decides an election, because if an election is within a single vote then it ultimately gets decided by court battles over recounts. There is some truth to this, but I don't think it significantly effects the probability of your vote deciding the election. Almost certainly there is some negative margin against which your candidate could not possibly win a legal challenge, and some positive margin that would make their victory immune to one. If we let denote the probability that your candidate will win given votes, then the sum of (which is the impact of your vote) over this margin is 1. If we further assume that the probability of any given within the margin is constant, then the expected value of the impact of your vote, should it fall in the margin, is where is the size of the margin. Since the probability of any given outcome within the margin is assumed to be constant, it is the same as the probability of a tie, so the probability of your vote falling in the margin is . Thus the probability of your vote affecting the outcome is , as in the naive analysis.
The low influence of a single vote on the outcome of the election is counter-balanced by the large impact affecting its outcome would have. I'm inclined to believe that the two roughly cancel, so that a perhaps one-in-several-million chance of affecting the outcome is worth the effort of voting.
Effects in a Safe State
But I live in California, where my vote is almost 1,000 times less likely than the average vote to affect the outcome. So then what are the effects of my vote? It nudges various statistics a little.
The Popular Vote
My vote increases my candidate's popular vote total. If they win, it slightly improves their electoral mandate. If they lose, it slightly erodes their opponent's mandate. It also increases the chance of your candidate winning the popular vote but losing the electoral college, and decreases the chance of the reverse.
Given George W. Bush's presidency, I think the impact of electoral mandates is relatively minor, and can mostly be ignored for this analysis. Similarly, we seem to have handled the electoral/popular vote mismatch without much consequence, although this upcoming election may be different given its ugly tone.
Increased Turnout Among My Demographic Groups
My vote also increases turnout among members of demographic groups I belong to (white, college-educated, 18-29 years old), probably increasing their political clout and furthering causes they support. Do I agree with them on the whole, or in particular on issues where their political clout is likely to tip the scales? Perhaps. At least the 18-29 year-old demographic is badly under-represented, and has serious long-term concerns which most (older) voters do not.
Increased Total Turnout
Any vote increases total voter turnout, which probably increases confidence in our Democracy. In my view, this is the strongest argument for voting.
Democracy is famously "the worst form of Government except all those other forms that have been tried". Voters are often short-sighted, poorly informed and easily misled, while politicians lie and pander to them. But while rule by strength is oft contested, and heirs have long raised armies to contest each others' claim to the throne, Democracy has given the United States peaceful transitions of power for 150 years. I believe this is because Democracy grants legitimacy to a leader in a way that is uniquely hard to overcome by martial force, based on the understanding that the results of an election reflect the will of the people. When more than half of adult citizens vote, this is a strong claim. But if this number slides to 40%, 30%? At some point the loser, or anyone else with a popular following, can credibly reject the President's authority.
This effect is inherently hard to measure. But the collapse of our Democracy would, I think, be at least 100x worse than the average difference between presidential candidates, likely resulting in Third and Final World War given the US's dominant role in global politics. Thus even a weak relationship between voting and peaceful transitions of power would be on par with a one-in-a-million chance to determine the next President. So even if you live in a safe state, if you are eligible, please vote.
I'm only considering the presidential election because it's the most universal. State and local elections are often more important, but those dynamics vary heavily between state and locality.
This should be fairly accurate for small margins, say 0.5%.
This number comes from FiveThirtyEight's election forecast, which computes the "voter power index" of a voter in each state, defined as the probability of both their state being decided by a single vote and their state tipping the electoral college. I use the "now-cast" version of the forecast, because it assumes the difference between the current polls and the results is entirely polling error, rather than allowing for some movement in the polls before the election. This more accurately reflects the knowledge I will have on or near election day, when I actually decide whether to vote. Note also that the voting power indices don't actually display values as low as 0.001; values below 0.1 are displayed as "<0.1". However the actual value can be inferred from the width of the bar used to illustrate it. At time of writing, Alaska has a voting power index of 4.3 and a bar with width 45%, while California has a bar with width 0.01156%, giving a voting power index of 0.0011.
Of course the popular vote count is not precise, and in the event of a result within the margin of error it is unlikely an "official" winner would be designated since it makes no legal difference. So to be more precise, your vote increases the strength of your candidate's claim to having won the popular vote.
Admittedly, it is difficult to tell how large the effect of 9/11 was on Bush's power.
Winston Churchill, speech to the House of Commons, November 11, 1947. Appears in Winston S. Churchill: His Complete Speeches, 1897–1963, ed. Robert Rhodes James, vol. 7, p. 7566 (1974).
Admittedly, given some of Trump's statements about disputing the election, jailing his opponent, admiring dictators, and bombing families of militants, this election may be an exception.