A transaction will quickly propagate throughout the network, so if two versions of the same transaction were reported at close to the same time, the one with the head start would have a big advantage in reaching many more nodes first. Nodes will only accept the first one they see, refusing the second one to arrive, so the earlier transaction would have many more nodes working on incorporating it into the next proof-of-work. In effect, each node votes for its viewpoint of which transaction it saw first by including it in its proof-of-work effort. If the transactions did come at exactly the same time and there was an even split, it's a toss up based on which gets into a proof-of-work first, and that decides which is valid.