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Perplexed comments on In the Pareto-optimised crowd, be sure to know your place - Less Wrong
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Not true after my edit. The three-way split is the unique maximum when all players are valued equally; but the two way splits are also Pareto optimal, are each favored by a majority of players, and are bargaining solutions (for some bargaining protocols).
But, now that I look at it, these two-way splits are also scaled utility maxima - for example, using a (49, 49, 2) scaling.
You have me convinced that every point on the Pareto frontier is a maximum of some weighted sum of utilities. But you don't yet have me convinced that, in every case, all of the weights can be non-zero.
ETA: I am a bit more convinced after re-reading the last paragraph of your posting.
No, you are right. For certain setups (say when the set of outcomes is a perfect circle/sphere), then if one player gets everything they want, this only happens if the weight of the other player's utility is zero.
I didn't write up that case, because there I would have to let the thetas be zero and infinity, a subtlety that would distract from the main point.