Yeah, but unless you actually end up at that point, that's hardly relevant. If people donated rationally, we would always be at that point, but people don't, and we aren't.

I don't understand how what you just said relates to my example. To recap, I meant my example, where the maximum is at the even split, to refute the claim that any smooth utility function will obtain its maximum along one "most efficient" axis. The whole argument is only about the rational behavior.

We're normally only dealing with one person. If you play videogames, you quickly get to the point where you don't want to play anymore nearly as much, so you do something else. If you save someone's life, there's still another guy that needs saving, and another guy after that, etc.

While this is true, and does point at an interesting difference between charity and many other behaviors, it can't isolate charity, not by a long shot. There are many, many things we do that we stop doing not because of satiety or exhaustion, but because of other priorities.

To give the first example that comes to mind, a personal one, I'm learning piano and I also play table tennis. I enjoy both activities immensely and would like to do either of them a lot more (but can't because of other commitments). There's no question of satiety of exhaustion at the level I currently invest in either. I could stop doing one of them and use that time for the other, but I explicitly don't want do to that and consider that an inferior outcome. I don't think this preference of mine is either irrational or very unusual.

consider: If you wanted a shirt... If you wanted to donate $100 to X charity...

Here's a closer "personal spending" analogy to charity: I commit to putting aside $500 every month for a future downpayment on a house (a goal far in the future). A family friend gives me an unexpected present of $500, putting it right into the fund. Am I likely to forego my own deduction this month and use it for other fun things? Depends on the kind of person I am, but probably not.