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jmmcd comments on Gains from trade: Slug versus Galaxy - how much would I give up to control you? - Less Wrong
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You are correct. But if both utilities are linear in resources, then the outcome space forms a line in the joint utility space. And when maximising utility versus a line, the generic outcome is to pick one end of the line or the other, never a midpoint.
This is a feature of any approach that maximises a summed utility: you (almost always) will pick the end point of any straight segment.
It is the fundamental theorem of linear programming.