"Yeah, the thing I was trying to force you to do with the four students in twenty-four possible orders was sum over the 6 possible ways you could be standing second in line, to make the point about how the sum is defined as being over every permutation.  In retrospect, clearly, I should've started with the case of tokens labeled 2, 3, and 5, but I'm sort of making this up as I go along because it's been a few years and I don't remember some of the exercises let alone their ordering.  Sorry about that."

"When you're trying to see if there's a way to do what ideal agents would do - or gods, if you think gods are powerful enough to be ideal about that particular case - you want to distinguish the Law that defines what the solution is, and any clever ways you come up with to compute the Lawful solution faster."

"When you've got 12 identical tokens, such that any group of 11 or 12 of them will produce 12 jellychips, there's a symmetry argument which says that each token must get one jellychip.  If you thought there ought to be a coherence constraint on the Law of fairness saying that holders of identical tokens should end up with identical payouts, you could use that to compute the answer even if you had no idea what the actual Law was.  Often when you do see how the Law works, you can go back over a lot of your intuitions, and say, 'Oh, yes, that intuition I had previously was shadowing this coherence of the Law, even though I didn't know how the whole Law worked' and that's a kind of sanity check on whether you're reasoning correctly at all."

"But the Law of fairness that defines the target answer for the '11 tokens of 12' problem is in principle a sum over 479,001,600 marginal productions, of which all but 39,916,800 are zero, and 39,916,800 of which are 12, divided at the end by 479,001,600.  Which means that we can say there's a single ideal fairness formula that governs both the '11 of 12' game, and the '2, 3, 5' game, even if shortcuts or approximations for the particular cases of the formula can be different, in cases where a shortcut exists."

"Yeah, the thing I was trying to force you to do with the four students in twenty-four possible orders was sum over the 6 possible ways you could be standing second in line, to make the point about how the sum is defined as being over every permutation.  In retrospect, clearly, I should've started with the case of tokens labeled 2, 3, and 5, but I'm sort of making this up as I go along because it's been a few years and I don't remember some of the exercises let alone their ordering.  Sorry about that.  Anyways -"

"When you're trying to see if there's a way to do what ideal agents would do - or gods, if you think gods are powerful enough to be ideal about that particular case - you want to distinguish the Law that defines what the solution is, and any clever ways you come up with to compute the Lawful solution faster."

"When you've got 12 identical tokens, such that any group of 11 or 12 of them will produce 12 jellychips, there's a symmetry argument which says that each token must get one jellychip.  If you thought there ought to be a coherence constraint on the Law of fairness saying that holders of identical tokens should end up with identical payouts, you could use that to compute the answer even if you had no idea what the actual Law was.  Often when you do see how the Law works, you can go back over a lot of your intuitions, and say, 'Oh, yes, that intuition I had previously was shadowing this coherence of the Law, even though I didn't know how the whole Law worked' and that's a kind of sanity check on whether you're reasoning correctly at all."

"But the Law of fairness that defines the target answer for the '11 tokens of 12' problem is in principle a sum over 479,001,600 marginal productions, of which all but 39,916,800 are zero, and 39,916,800 of which are 12, divided at the end by 479,001,600.  Which means that we can say there's a single ideal fairness formula that governs both the '11 of 12' game, and the '2, 3, 5' game, even if shortcuts or approximations for the particular cases of the formula can be different, in cases where a shortcut exists."