You might have already talked about this in the meeting (I couldn’t attend), but here goes
wSi=∑i∈R⊆S(−1)|S|−|R|uRi. This is around where I have problems. I just can’t quite manage to get myself to see how this quantity is the “slice of marginal utility that coalition S promises to player i”, so let me know in the comments if anyone manages to pull it off.
Let’s reason this out for a coalition of 3 members, the simplest case that is not readily understandable (as in your “Alice and Bob” example). We have w{1,2,3}1=u{1,2,3}1−u{1,2}1−u{1,3}1+u{1}1. We can interpret w{1,2,3}1 as the strategic gain obtained (for 1) thanks to this 3 member coalition, that is a direct product of this exact coalition’s capability for coordination and leverage, that is, that doesn’t stem from the player’s own potentials (u{1}1) neither was already present from subcoalitions (like u{1,2}1). The only way to calculate this exact strategic gain in terms of the uSi is to subtract from u{1,2,3}1 all these other gains that were already present. In our case, when we rewrite u{1,2,3}1=u{1,2}1+(u{1,3}1−u{1}1)+w{1,2,3}1, we’re only saying that w{1,2,3}1 is the supplementary gain missing from the sum if we only took into account the gain from the 1-2 coalition plus the further marginal gain added by being in a coalition with 3 as well, and didn’t consider the further strategic benefits that the 3 member coalition could offer. Or expressed otherwise, if we took into account the base potential u{1}1 and added the two marginal gains (u{1,2}1−u{1}1) and (u{1,3}1−u{1}1).
Of course, this is really just saying that u{1,2,3}1=w{1}1+w{1,2}1+w{1,3}1+w{1,2,3}1, which is justified by your (and Harsanyi’s) previous reasoning, so this might seem like a trivial rearrangement which hasn’t provided new explanatory power. One might hope, as you seem to imply, that we can get a different kind of justification for the formula, by for instance appealing to bargaining equilibria inside the coalition. But I feel like this is nowhere to be found. After all, you have just introduced/justified/defined uSi=∑i∈R⊆SwRi, and this is completely equivalent to wSi=∑i∈R⊆S(−1)|S|−|R|uRi. It’s an uneventful numerical-set-theoretic rearrangement. Not only that, but this last equality is only true in virtue of the “nice coherence properties” justification/definition you have provided for the previous one, and would not necessarily be true in general. So it is evident that any justification for it will be a completely equivalent reformulation of your previous argument. We will be treading water and ultimately need to resource back to your previous justification. We wouldn’t expect a qualitatively different justification for a+b=c than for a=c−b, so we shouldn’t either expect one in this situation (although here the trivial rearrangement is slightly less obvious than subtracting b, because to prove the equivalence we need to know those equalities hold for every i and S).
Of course, the same can be said of the expression for tSi, which is an equivalent rearrangement of that for the wSi: any of its justifications will ultimately stem from the same initial ideas, and applying definitions. It will be the disagreement point for a certain subgame because we have defined it/justified its expression just like that (and then trivially rearranged).
Please do let me know if I have misinterpreted your intentions in some way. After all, you probably weren’t expecting the controversial LessWrong tradition of dissolving the question :-)
You might have already talked about this in the meeting (I couldn’t attend), but here goes
Let’s reason this out for a coalition of 3 members, the simplest case that is not readily understandable (as in your “Alice and Bob” example). We have w{1,2,3}1=u{1,2,3}1−u{1,2}1−u{1,3}1+u{1}1. We can interpret w{1,2,3}1 as the strategic gain obtained (for 1) thanks to this 3 member coalition, that is a direct product of this exact coalition’s capability for coordination and leverage, that is, that doesn’t stem from the player’s own potentials (u{1}1) neither was already present from subcoalitions (like u{1,2}1). The only way to calculate this exact strategic gain in terms of the uSi is to subtract from u{1,2,3}1 all these other gains that were already present. In our case, when we rewrite u{1,2,3}1=u{1,2}1+(u{1,3}1−u{1}1)+w{1,2,3}1, we’re only saying that w{1,2,3}1 is the supplementary gain missing from the sum if we only took into account the gain from the 1-2 coalition plus the further marginal gain added by being in a coalition with 3 as well, and didn’t consider the further strategic benefits that the 3 member coalition could offer. Or expressed otherwise, if we took into account the base potential u{1}1 and added the two marginal gains (u{1,2}1−u{1}1) and (u{1,3}1−u{1}1).
Of course, this is really just saying that u{1,2,3}1=w{1}1+w{1,2}1+w{1,3}1+w{1,2,3}1, which is justified by your (and Harsanyi’s) previous reasoning, so this might seem like a trivial rearrangement which hasn’t provided new explanatory power. One might hope, as you seem to imply, that we can get a different kind of justification for the formula, by for instance appealing to bargaining equilibria inside the coalition. But I feel like this is nowhere to be found. After all, you have just introduced/justified/defined uSi=∑i∈R⊆SwRi, and this is completely equivalent to wSi=∑i∈R⊆S(−1)|S|−|R|uRi. It’s an uneventful numerical-set-theoretic rearrangement. Not only that, but this last equality is only true in virtue of the “nice coherence properties” justification/definition you have provided for the previous one, and would not necessarily be true in general. So it is evident that any justification for it will be a completely equivalent reformulation of your previous argument. We will be treading water and ultimately need to resource back to your previous justification. We wouldn’t expect a qualitatively different justification for a+b=c than for a=c−b, so we shouldn’t either expect one in this situation (although here the trivial rearrangement is slightly less obvious than subtracting b, because to prove the equivalence we need to know those equalities hold for every i and S).
Of course, the same can be said of the expression for tSi, which is an equivalent rearrangement of that for the wSi: any of its justifications will ultimately stem from the same initial ideas, and applying definitions. It will be the disagreement point for a certain subgame because we have defined it/justified its expression just like that (and then trivially rearranged).
Please do let me know if I have misinterpreted your intentions in some way. After all, you probably weren’t expecting the controversial LessWrong tradition of dissolving the question :-)