[karlson]

fnord had it right - you can do better by letting N/e of the prizes go by and then picking the first one that's better than any of the others. In fact, N/e is optimal for the strategies of that type. It also seems obvious that these are the only type of strategies to consider.
I don't feel like typing the proof that N/e is optimal, but I'm sure you can find it if you feel like it.