Yep, following the footsteps of the likes of Fable II and Hitman: Absolution, BioShock Infiniteis jumping on the "Here's a little game as a teaser while you wait for the real thing" bandwagon, with the announcement of BioShock Infinite: Industrial Revolution.
Being a puzzle game of 60 levels, BI:IR puts you in the role of a factory worker in the game's flying city of Columbia. By solving puzzles - which will no doubt take the form of some thinly contrived simulation of your character's in-game job - you'll apparently learn more about BioShock Infinite's warring factions and earn some 'in-game rewards' of an unspecified nature.
The cost of this downloadable wonder of the modern age? Absolutely nothing. As long as you don't pay attention to the cost of the full-price game you have to pre-order in order to get it. It's definitely free once you've bought a BioShock Infinite on pre-release. Totally gratis. And if you're interested in making that pre-order, you can do so right here.
