Can anyone explain the Rules of the MiB/Jacob "game"?

In Ab Aeterno Jacob tells Richard that he brings people to the island to expose them to the "malevolence". MiB thinks if they have free will they will always be corrupted, Jacob says not. If we see this as a game, under what circumstances will a winner be declared?

MiB seemingly can kill of as many of the people brought to the island as he chooses. Why is his trivial solution to the game not just to kill every single one of the people that Jacob brings? In that way, Jacob can never be proved correct, and loses by default. It would certainly not be in MiB's interest to keep any of them alive. If I were MiB, I would just kill everyone until Jacob got fed up with the whole stupid business.

So, without this, how do we determine whether one side has won? As I see it,

For Jacob to win, At least one person would need to be shown to be incorruptible. However, just because they have not been corrupted (for example) by money, does not mean they would not be corrupted by power, lust, etc. The person involved would have be be shown to be incorruptible in every single situation that the person could conceivably find themselves in. By simply not being corrupted in one situation, does not mean that they cannot be corrupted in all situations. In other words, even if Jacob found the one person, the game would need to go on forever, inventing new scenarios testing that person to the end of time, or until he or she eventually becomes corrupted. Therefore Jacob cannot win the game, as there can never be conclusive proof that Jacob is correct.

For MiB to win, he would have to demonstrate that all possible people can be corrupted at least once in their lives. Again, as this involves a potentially infinite number of iterations (in this case people). Therefore MiB can never win, as Jacob can just keep on bringing more people to the island to test, and so there can never be conclusive proof that MiB is correct.

Surely this means that no side can possibly win this game. In one case, you have a single person being tested an infinite number of times, and in the second case you have an infinite number of people requiring to be tested. This is a classically undecidable proposition.

Am I overlooking something obvious here?

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