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Exactly that has already been going on since 2010 :)

There's a livestream of bots facing off here: https://www.twitch.tv/certicky

There's a pretty interesting Starcraft AI tournament going on right now actually, though I'm not certain whether any of the bots are using deep learning techniques: http://sscaitournament.com/

Corewars, where the program is written by a neural net.

Sounds like RoboCode.