Optimal strategy for a simple game Let's say I'm playing a game that works likes this: I have n bowls that are each filled with a different number of marbles, and I know how many marbles are in each bowl. At each round, the dealer goes to each bowl one at a time, and decides to add or remove some number of marbles from that bowl. He decides how much to add or remove from each bowl based on a set of rules that only take into account the number of marbles currently in each bowl, though I have no idea what those rules are. I am able to see the changes he makes. After he has made his changes, I have to make a decision: I can either have him give me a dollar for each marble in each of the bowls, or I can have him undo all the changes he just made to the bowls. After I have made my decision and the dealer has either paid me or undid his changes to the number of marbles in each bowl, I begin the next round with the bowls as they currently are. My goal is to maximize the average amount of dollars I get per round.
My question is, what's the optimal strategy for playing this game?
 A: The dealer has enough power to make any strategy wortheless. I would argue that this falls under the no free lunch theorem.
As a small reasoning by the absurd, let's suppose that you have a strategy X, that is supposed to be optimal, and I'm the dealer. Because I'm the dealer I've thought long and hard about the problem and found that same strategy X. I start by removing every marble from every bowl.
Now, you follow your optimal strategy. But as I know that's the optimal one, I will do my utmost to fight it. Thus, if you do the moves that someone playing the optimal strategy does, I will keep you at 0 marbles all the time (or oscillate between 0 and 1 if I'm forced to add/remove). If you don't, then i'll choose to add a lot of marbles all the time, because I like people that disregard strategy and play less than perfect.
Thus non-optimal strategies are better than the optimal strategy, which is a contradiction. Hence there is no optimal strategy.
Edit : 
With the update to the question : the dealer now has a pre-determined f(number of marbles) strategy. Yes, there is an optimal strategy for the player : always take.
If you don't take, the number of marbles stays the same, the dealer will do the exact same thing, and you've just done a 0-round, which is always worst case (you can't have negative rounds), so you lost on the average $/round.
