# Analyses of the paradox of voting — given that close elections are usually disputed?

In the standard analysis of the paradox of voting, votes are pivotal only in two scenarios (exact tie or win by 1).

But in reality, vote counting is difficult and messy. And in reality, close elections are often disputed and the final outcome of the election may have little to do with who won the actual vote count. (The most famous example was the 2000 US presidential election.)

In light of the fact that close elections are usually disputed and rarely settled simply by an accurate count of the vote, what do we mean by a vote being pivotal? How do we calculate this probability?

The only analysis I've come across is the one-page appendix in Gelman, Katz, and Bafumi (2004). They argue that the analysis of the paradox of voting is pretty much unchanged, even with disputed elections:

In fact, our decisive-vote calculations are reasonable, even for real elections with disputed votes, recounts and so forth. We show this by setting up a more elaborate model that allows for a grey area in vote counting, and then demonstrating that the simpler model of decisive votes is a reasonable approximation.

Are there any other analyses of the matter?

(Given the vast literature on the paradox of voting, I'm surprised the only thing I can find on this obviously-important matter is the above one-page appendix.)

• A few points of clarification (just so I can try and better help with the references): 1) Are you only interested in voting behavior for things like national elections, or are you interested in smaller groups (like juries) too? 2) I'm a bit unclear what your fundamental question is- the idea of a pivotal vote is often used more to describe optimal behavior when faced with a decision rather than the probability a vote actually matters. In other words, most literature examines how you should act as if your vote is pivotal, not whether it will be. Are you only interested in the latter? – AndrewC Jun 19 '17 at 10:50
• @AndrewC: (1) Smaller votes would be of interest too, except I've never heard of a small vote count being disputed. (2) I don't know where you come up with the claim that most of the literature is not concerned with the probability a vote matters. Try starting from Kremp and Gelman (2016), Gelman, Silver, and Edlin (2012), Mulligan and Hunter (2003), and check out also what they cite, plus who they're cited by, for many, many more. – Kenny LJ Jun 20 '17 at 1:18