This is not an easy problem for me either. But this is how I would approach it:

1) Observe that you must roll three fours in a row at some point, and I must roll four threes in a row at some point. The game ends after you roll your third four in a row or I roll my fourth three.

2) Pretend that the game continues forever. Your rolls will be a list of randomly generated numbers in order, and so will mine. Let m represent the number of the roll in which you roll your third four in a row for the first time, and let n represent the number of the roll in which I roll my fourth three in a row for the first time. Then you win if m is less than or equal to n. So you must calculate the probability that m is less than or equal to n.