A more general claim is: if something can predict your action with better than random accuracy no matter how hard you try to prevent them, you don’t have free will over that action. (I’m not addressing the question of whether free will exists in general, only whether a particular action is chosen freely.)
The whole framing here strikes me as confused (although maybe I am confused). The way you are phrasing it already assumes that you are in conflict with someone (“no matter how hard you try to prevent them”). Your setup already assumes away you have free will. Both Cooperate Bot and Defect Bot do in fact not have free will. The whole point of game-theory is that you have agents that can simulate each other to some extent. A more useful game theory bot is a function that takes the function of another agent as input. When you assume that you just exist in one place and time, then you are already assuming a setup where game theory is not useful. If you are predicted by someone else, then you (your computation) is being simulated, you don’t exist in just one place and time, you exist in different places and times (although not all of these versions are the full You). You don’t get to choose where you exist, but in this framing you do get to choose your output.
The whole framing here strikes me as confused (although maybe I am confused). The way you are phrasing it already assumes that you are in conflict with someone (“no matter how hard you try to prevent them”). Your setup already assumes away you have free will. Both Cooperate Bot and Defect Bot do in fact not have free will. The whole point of game-theory is that you have agents that can simulate each other to some extent. A more useful game theory bot is a function that takes the function of another agent as input. When you assume that you just exist in one place and time, then you are already assuming a setup where game theory is not useful. If you are predicted by someone else, then you (your computation) is being simulated, you don’t exist in just one place and time, you exist in different places and times (although not all of these versions are the full You). You don’t get to choose where you exist, but in this framing you do get to choose your output.