Many political scientists and commentators have been asking this question in recent days. For one answer, you can see Mike Munger's thoughts in Duke
Today yesterday. In this post I will consider one unlikely but interesting scenario: what could happen if elections in New Jersey or New York are delayed until after Tuesday?
According to Article 2, Clause 4 of the US constitution,
The Congress may determine the Time of chusing (sic) the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the United States.
Obviously the founders felt that it was important that polls be held simultaneously across the states. However, there is some leeway at the state level for special circumstances like a natural disaster.
How would voters in New York and New Jersey be affected if the election was already decided before they cast their ballots? These are fairly predictable states, unlikely to tip the election on their own. If one candidate already had enough electoral votes for victory and there is zero chance that your vote will matter, I can think of no other explanation than that voting is a form of self-expression.
There two examples that help shed light on this question. One that came to mind was a paper by Thomas Jensen and Asger Lau Andersen on how exit polls affect voter turnout. Their approach is a game theoretic model, but they cite a 2009 referendum vote in Denmark as a motivating example (ungated):
In order to pass, the proposal therefore had to overcome two obstacles: One, a majority of the votes cast in the referendum must be in favor of the proposal. And two, at least 40% of all eligible voters must vote in favor of the proposal. In the weeks preceding the June 7 election, there was no doubt that only the latter of these requirements had the potential to become binding. In a Gallup poll released a week before the election, 84% of respondents indicated that they approved of the proposal to change the law. However, only 40.2% responded that they would show up at polls and vote in favor of the proposal.
On the afternoon of the election day, TV2, a major Danish TV channel, published the results of an exit poll, which predicted that 37.9% of all eligible voters would cast a vote in favor of the proposal to change the law. However, during the evening the situation turned around with pollsters reporting a considerable increase in turnout. In the end, the official result was that 45.1% of all eligible voters had voted in favor of the proposal, which corresponded to 85.4% of all votes cast. Thus, the proposal passed with a comfortable margin.
So in this instance, receiving information about the likelihood of a particular outcome before polls closed affected the vote. But that still does not answer the question of how voters react when they have zero chance of changing the result.
For that, we turn to an example from France. The French government tries to avoid the fate of the Danish referendum by taking several strong measures. From Wikipedia:
Elections are always held on Sundays in France. The campaigns end at midnight the Friday before the election; then, on election Sunday, by law, no polls can be published, no electoral publication and broadcasts can be made. The voting stations open at 8 am and close at 6 pm in small towns or at 8 pm in cities, depending on prefectoral decisions. By law, publication of results or estimates is prohibited prior to that time; such results are however often available from the media of e.g. Belgium and Switzerland, or from foreign Internet sites, prior to that time. The first estimate of the results are thus known at Sunday, 8pm, Paris time; one consequence is that voters in e.g. French Guiana, Martinique and Guadeloupe knew the probable results of elections whereas they had not finished voting, which allegedly discouraged them from voting. For this reason, since the 2000s, elections in French possessions in the Americas, as well as embassies and consulates there, are held on Saturdays as a special exemption.
Gaining information about the expected outcome of an election does appear to affect turnout. It seems that if voters can still make a difference they are more likely to show up and vote for their desired outcome, while if the election is already decided they will stay home. As I said at the outset this is an unlikely scenario, but you can bet that political scientists will be keeping an eye on this chance for a true natural experiment.