Let's assume that the being that is supposed to find a strategy for this scenario operates in a universe whose laws of physics can be specified mathematically. Given this scenario, it will try to maximize the number it outputs. Its output cannot possibly surpass the maximum finite number that can be specified using a string no longer than its universes specification, so it need not try to surpass it, but it might come pretty close. Therefore, for each such universe, there is a best rational actor.

Edit: No, wait. Umm, you might want to find the error in the above reasoning yourself before reading on. Consider the universe with an actor for every natural number that always outputs that number. The above argument says that no actor from that universe could output a bigger number than can be specified using a string no longer than the laws of physics of the universe, but that only goes if the laws of physics include a pointer to that actor - to extract the number 100 from that universe, we need to know that we want to look at the hundredth actor. But your game didn't require that: Inside the universe, each actor knows that it is itself without any global pointers, and so there can be an infinite hierarchy of better-than-the-previous rational actors in a finitely specified universe.