Some games have, in fact, been solved. Tic-Tac-Toe is a very easy one for which to build an AI that will always win or tie. Recently, Connect 4 has been solved as well (and shown to be unfair to the second player, since a perfect play will cause him to lose).
Chess, however, has not been solved, and I don't think there's any proof that it is a fair game (i.e., whether the perfect play results in a draw). Speaking strictly from a theoretical perspective though, Chess has a finite number of possible piece configurations. Therefore, the search space is finite (albeit, incredibly large). Therefore, a deterministic Turing machine that could play perfectly does exist. Whether one could ever be built, however, is a different matter.
