It's not so bad if one person is playing by themselves, e.g. in order to prove they didn't modify the software. And you could expand that to people competing for a high score on an unmodified copy of Doom. But a big problem arises when you give two people the controls and have them do a death match. Presumably, both will commit to reaching a state where the controls they input result in their character killing their opponent some number of times. Whoever does it first wins.

So then, what happens if a player, beginning to lose and not wanting to, abandons the game and challenges the remaining player to show what inputs they provided to reach the end state? The remaining player cannot prove that any inputs he provides "make it" to his committed end state, because as soon as either player quits, the next frame of the game cannot load. The game's state, in each frame, depends on both players revealing hashes for whatever buttons they are pressing or not pressing, and it cannot compute how the frame should look without that data -- from both players. Since the victim of this attack cannot prove he reached the end state, which he committed to reach, he cannot respond to the challenge created by the "sore loser," and will therefore lose his money when a timelock expires. So it basically turns into "whoever challenges first wins." Which is no fun at all.

This doesn't apply to a game like chess or tic tac toe, where there are a relatively small number of turns, and you can construct a set of bitcoin transactions that "passes the turn" back and forth. In Doom, both players provide inputs simultaneously, so you cannot pass it back and forth without modifying the game engine. You could probably modify it so that player 1 provides inputs on "even" frames and player 2 provides inputs on "odd" frames, and that would probably mostly work out decently if neither player abandons the game. But if either player does abandon the game, you have to take every remaining frame of the game to the blockchain, giving each player multiple blocks worth of time to figure out what buttons they should press given what they see on the screen. That makes the game effectively impossible to win. So I just don't think Doom deathmatches work.