I have been thinking about this same mathematical object (although with a different orientation/motivation) as where I want to go with a weaker replacement for utility functions.
I get the impression that for Diffractor/Vanessa, the heart of a concave-value-function-on-lotteries is that it represents the worst case utility over some set of possible utility functions. For me, on the other hand, a concave value function represents the capacity for compromise—if I get at least half the good if I get what I want with 50% probability, then I have the capacity to merge/compromise with others using tools like Nash bargaining.
This brings us to the same mathematical object, but it feels like I am using the definition of convex set related to the line segment connecting any two points in the set is also in the set, where Diffractor/Vanessa is using the definition of convex set related to being an intersection of half planes.
I think this pattern where I am more interested in merging, and Diffractor and Vanessa are more interested in guarantees, but we end up looking at the same math is a pattern, and I think the dual definitions of convex set in part explains (or at least rhymes with) this pattern.
I have been thinking about this same mathematical object (although with a different orientation/motivation) as where I want to go with a weaker replacement for utility functions.
I get the impression that for Diffractor/Vanessa, the heart of a concave-value-function-on-lotteries is that it represents the worst case utility over some set of possible utility functions. For me, on the other hand, a concave value function represents the capacity for compromise—if I get at least half the good if I get what I want with 50% probability, then I have the capacity to merge/compromise with others using tools like Nash bargaining.
This brings us to the same mathematical object, but it feels like I am using the definition of convex set related to the line segment connecting any two points in the set is also in the set, where Diffractor/Vanessa is using the definition of convex set related to being an intersection of half planes.
I think this pattern where I am more interested in merging, and Diffractor and Vanessa are more interested in guarantees, but we end up looking at the same math is a pattern, and I think the dual definitions of convex set in part explains (or at least rhymes with) this pattern.