Coin removing problem,

Here is a classic problem from combinatorial game theory/high school math contest:

There are 100 coins on the table, and there are two players that take turn, each taking 1, 2, or 3 coins from the pile. The person who can not move, loses. Who wins and what is the winning strategy?

If you haven't seen this before, I'm sure you can solve it after a little bit of playing around. Of course the 100 coins is just a red herring, and the winning strategy is all that matters.Furthermore, if the possible moves were 1,2,3,...,n coins, then the problem is about the same level of difficulty. However, I think this problem is a lot more tricky if the possible moves are chosen randomly. Say, each player can remove 3,5 or 8 coins. So, here is a problem that I like to know how to solve:

Given a set S, the set of possible moves, describe the set of positive integers so that player 1 has a winning strategy.

I'm not sure if there is a nice answer to this, however, I'm still curious if there is a way of computing this.