Okay, those with a two-boxing agent type don't win but the two-boxer isn't talking about agent types. They're talking about decisions. So they are interested in what aspects of the agent's winning can be attributed to their decision and they say that we can attribute the agent's winning to their decision if this is caused by their decision. This strikes me as quite a reasonable way to apportion the credit for various parts of the winning.

Do I understand it correctly that you're trying to evaluate the merits of a decision (to two-box) in isolation of the decision procedure that produced it? Because that's simply incoherent if the payoffs of the decision depend on your decision procedure.