Choosing first or second in a triplet race

Two people each name a sequence of three coin outcomes (such as HTH). The second person picks after hearing the first person's choice, and must pick a different triplet. A fair coin is then flipped until one named triplet appears, and its owner wins. Would you prefer to choose first or second, and how likely are you to win?

Choose second; you can always win with probability at least $2/3$. This is Penney's game: for any first choice $x_1x_2x_3$, the reply $\bar{x}_2x_1x_2$ beats it with probability between $\tfrac23$ and $\tfrac78$ (for instance THH beats HHT with probability $\tfrac34$), so the game is non-transitive and moving second wins.