These are super interesting ideas, thanks for writing the sequence!
I’ve been trying to think of toy models where the geometric expectation pops out—here’s a partial one, which is about conjunctivity of values:
Say our ultimate goal is to put together a puzzle (U = 1 if we can, U = 0 if not), for which we need 2 pieces. We have sub-agents A and B who care about the two pieces respectively, each of whose utility for a state is its probability estimates for finding its piece there. Then our expected utility for a state is the product of their utilities (assuming this is a one-shot game, so we need to find both pieces at once), and so our decision-making will be geometrically rational.
This easily generalizes to an N-piece puzzle. But, I don’t know how to extend this interpretation to allow for unequal weighing of agents.
Another setting that seems natural and gives rise to multiplicative utility is if we are trying to cover as much of a space as possible, and we divide it dimension-wise into subspace, each tracked by a subagent. To get the total size covered, we multiply together the sizes covered within each subspace.
We can kinda shoehorn unequal weighing in here if we have each sub-agent track not just the fractional or absolute coverage of their subspace, but the per-dimension geometric average of their coverage.
For example, say we’re trying to cover a 3D cube that’s 10x10x10, with subagent A minding dimension 1 and subagent B minding dimensions 2 and 3. A particular outcome might involve A having 4⁄10 coverage and B having 81⁄100 coverage, for a total coverage of (4/10)*(81/100), which we could also phrase as (4/10)*(9/10)^2.
I’m not sure how to make uncertainty work correctly within each factor though.
These are super interesting ideas, thanks for writing the sequence!
I’ve been trying to think of toy models where the geometric expectation pops out—here’s a partial one, which is about conjunctivity of values:
Say our ultimate goal is to put together a puzzle (U = 1 if we can, U = 0 if not), for which we need 2 pieces. We have sub-agents A and B who care about the two pieces respectively, each of whose utility for a state is its probability estimates for finding its piece there. Then our expected utility for a state is the product of their utilities (assuming this is a one-shot game, so we need to find both pieces at once), and so our decision-making will be geometrically rational.
This easily generalizes to an N-piece puzzle. But, I don’t know how to extend this interpretation to allow for unequal weighing of agents.
Another setting that seems natural and gives rise to multiplicative utility is if we are trying to cover as much of a space as possible, and we divide it dimension-wise into subspace, each tracked by a subagent. To get the total size covered, we multiply together the sizes covered within each subspace.
We can kinda shoehorn unequal weighing in here if we have each sub-agent track not just the fractional or absolute coverage of their subspace, but the per-dimension geometric average of their coverage.
For example, say we’re trying to cover a 3D cube that’s 10x10x10, with subagent A minding dimension 1 and subagent B minding dimensions 2 and 3. A particular outcome might involve A having 4⁄10 coverage and B having 81⁄100 coverage, for a total coverage of (4/10)*(81/100), which we could also phrase as (4/10)*(9/10)^2.
I’m not sure how to make uncertainty work correctly within each factor though.