If I am the first player and choose 9 as my first pick, then the other player will be forced to pick 6 because if not, I win. Then I pick 5 (sum is now 14) and the other player will be forced to pick 1, since if he does not, I win. His sum is now 7 (if he picks 8 I'm toast). Now I pick 10 and get a sum evaluating to 15 and I win.

So yes, the first player can win every time with the exact same strategy! If the first player implements this strategy, the second player cannot win, so in that case the second player cannot, in general, win. So yea, this is my answer.

Now, are there any additional restrictions?