Voting and rationality

Andrew Gelman had a series of posts on voting and rationality: here, here, here, and here for instance. Wading into very unknown territory for me is the following question: Why isn't the Nash equilibrium in a voting game to vote?

Consider an N-person game with two pure strategies of vote or not vote over two candidates.
1. Suppose nobody votes. This cannot possibly be a Nash equilibrium since any one person can vote and the vote will be decisive.
2. Thus the NE must be to vote.

Consider the N1+N2=N person game where N1 supports candidate 1 and N2 supports candidate 2.
1. Suppose no one from either or N1 or N2 votes. Again any one player from the N1 or N2 coalition will deviate from this strategy of not voting and the vote will be decisive.
2. Again the NE must be to vote.

Unfotunately, I have not been able to find out whether my reasoning is correct in this simple game. The economics literature is full of papers on costly voting where each voter incurs a cost C of voting (either uniformly distributed or not), imperfect information on the size of N1 and N2, strategic voting, coalition formation, stability of the core, etc.

I find the above simple and straightforward enough for me to explain why people vote. Why do economists have to cloud the issue? I guess it must make for fancier math and more publications.