very intersting question.
For one, I'd never heard of The Identiacal....and I'm a fan of both Ray Liotta and Asley Judd. I think this raises a secondary question about promotion which for most "hollywood films" is larger than the budgets of your average movie.
In New York, you have theaters that won't/don't show blockbusters. In Austin, and other cities you have the same thing. They show "indies" or forigen films.
I have read about renting out theaters. I believe Stephen Spielberg did it when he was young for a film he made before he was hired on as a contract director. This was either late 60's or early 70s. I believe he was still a student, so it was most likely his father/mother who rented it out. I woudl think that in small towns outside of major cities, as long as the cinema is non-franchise, you have a good shot at doing something like this. Especially for a one night screening.
On the one hand it is disheartening to see a select few monster budget and overall Milquetoast films filling theaters, but then consider that people now have access to things like Netflix, Amazon, and of course DVD's.....
In the 90s, you had to travel to one of 2 or 3 theaters in NYC to see Kieslowski. It was a hard ticket to get...... Now, that same film would be avilable to millions on Netflix, Amazon and other services.
If the viewer is adept at knowing whats out there, then they actually have better chance of seeing it now.
For example....here are 3 of the best films I've seen in the last year. I don't think they had wide exposure in theaters. But on Amazon, Netflix, DVD's, etc...they have an audience.
----Upstream Color (never saw a commercial for this. Played in NYC and a few other places)
----Bullhead (great forigen film. never saw a commercial, it had a small run in the US.)
---Bellflower (older film/2011, but one I heard about and found on Netflix. It had a small run as well.
http://www.imdb.com/title/tt1242599/business?ref_=tt_ql_dt_4)
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I'll add films like Eraserhead to the Rockey Horror List. I saw it as recently as last year or so as being shown. Of course, that's a type of movie (cult) and not your regular movie.
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As for you're particular questions about theaters....the type of theater and area is important.
I would avoid big chains. I can't imagine a 10 screen multiplex would do this. However. the independent cinemas, which still exist, are more likely to do this. I think they would want a real movie, but these places will often do retrospectives put on by local cities/municipalities. these are essentially hiring out. The place that plays X Men and Transformers and then does a double billing of something like The Third Man and M for one night, is the type of place to approach.