Of the marketing tools I've utilized, writing contests rank very near the top. Why?
1. They attract attention
Contests are fun, and nothing is quite as alluring as the prospect of shiny prizes. People will not only enter the contest, but will cheerfully tell their friends about it, link to it, and so forth.
2. They encourage new visitors
Many marketing efforts are aimed at encouraging previous visitors to come back to the web site. Contests reach a new audience because they're interesting to groups who have had no prior exposure to your product.
3. They are not pushy
I hate pushy sales-people, blinking web banners, and pesky spam emails saying "Buy! Buy! Buy!"
Contests are a pleasant marketing tool because they are totally laid-back. Entrants don't have to make a purchase or ever come back to the site. It serves the noble goal of marketing ('making people aware of your product') without falling into the sleazy pitfalls ('pressuring people to buy something they don't really want').
4. They give something back
Writing contests are a great way to support and enourage aspiring writers. (I'm not sure why, but new writers seem far more likely to enter contests than they are to submit to established magazines. Perhaps it's because contests feel less threatening.) It gives something back to the community, and that makes me feel all warm-fuzzy inside.
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Short version: I heart contests. If I ever manage to (a) finish A New Kind of Sunrise and (b) sell it, I am sure as heck going to find a way to tie a writing contest into the marketing strategy.