Can we crank this in reverse: given a utility function, design a market that whose representative agent has this utility function?
It seems like trivially, we could just have the market where a singleton agent has the desired utility function. But I wonder if there’s some procedure where we can “peel off” sub-agents within a market and end up at a market composed of the simplest possible sub-agents for some metric of complexity.
Either there is some irreducible complexity there, or maybe there is a Universality theorem proving that we can express any utility function using a market of agents who all have some extremely simple finite state, similar to how we can show any form of computation can be expressed using Turing Machines.
We can get part of the way with Harsanyi’s utilitarian theorem: any market/veto committee of subagents with VNM utility functions satisfies the pareto indifference principle (if all members are indifferent, the aggregate is indifferent), and so by Harsanyi’s theorem if the aggregate preference is given by a VNM utility it must be a linear combination of the subagents’ utilties. (emphasis on the parts where we need the utilities to satisfy the VNM axioms—though if each subagent satisfies them then after completing the veto committee’s preferences (viewed as preferences on lotteries over states) you automatically get every other axiom besides continuity. idk if continuity is preserved by the completion mechanism of the post—i suspect not).
I’d guess that for pareto efficiency the coefficients have to be positive, but I have no proof.
This isn’t enough, because this still includes e.g. doing whatever John says and ignoring David. But it does constrain the search space.
Since we’re given aggregate utilty w, we then need to find U and c such that c • U = w.
Once we fix U, finding the hyperplane of possible c is just a linear algebra problem. (likewise if we restrict to c≥0). So we can make a U by taking any linear subspace that contains w. Each one can just be made by taking w and some additional vectors and looking at their span; by rotating this subbasis we can get whatever collections of utility functions in that subspace we desire. (though we must be careful that the coefficients are positive, that is, that w lies in the cone of our candidate vectors)
The only question is how to identify which ways of having a w arise from some U never show up as a (randomized) completion of a veto committee.
I weakly suspect that we can attain most of the points on the Pareto frontier with some nonzero probability, where the exceptions are only the boundary of the surface (as there at least one agent is getting their worst possible deal; e.g. David and John never agree to just follow John’s preferences all the time)
Can we crank this in reverse: given a utility function, design a market that whose representative agent has this utility function?
It seems like trivially, we could just have the market where a singleton agent has the desired utility function. But I wonder if there’s some procedure where we can “peel off” sub-agents within a market and end up at a market composed of the simplest possible sub-agents for some metric of complexity.
Either there is some irreducible complexity there, or maybe there is a Universality theorem proving that we can express any utility function using a market of agents who all have some extremely simple finite state, similar to how we can show any form of computation can be expressed using Turing Machines.
We can get part of the way with Harsanyi’s utilitarian theorem: any market/veto committee of subagents with VNM utility functions satisfies the pareto indifference principle (if all members are indifferent, the aggregate is indifferent), and so by Harsanyi’s theorem if the aggregate preference is given by a VNM utility it must be a linear combination of the subagents’ utilties. (emphasis on the parts where we need the utilities to satisfy the VNM axioms—though if each subagent satisfies them then after completing the veto committee’s preferences (viewed as preferences on lotteries over states) you automatically get every other axiom besides continuity. idk if continuity is preserved by the completion mechanism of the post—i suspect not).
I’d guess that for pareto efficiency the coefficients have to be positive, but I have no proof.
This isn’t enough, because this still includes e.g. doing whatever John says and ignoring David. But it does constrain the search space.
Since we’re given aggregate utilty w, we then need to find U and c such that c • U = w.
Once we fix U, finding the hyperplane of possible c is just a linear algebra problem. (likewise if we restrict to c≥0). So we can make a U by taking any linear subspace that contains w. Each one can just be made by taking w and some additional vectors and looking at their span; by rotating this subbasis we can get whatever collections of utility functions in that subspace we desire. (though we must be careful that the coefficients are positive, that is, that w lies in the cone of our candidate vectors)
The only question is how to identify which ways of having a w arise from some U never show up as a (randomized) completion of a veto committee.
I weakly suspect that we can attain most of the points on the Pareto frontier with some nonzero probability, where the exceptions are only the boundary of the surface (as there at least one agent is getting their worst possible deal; e.g. David and John never agree to just follow John’s preferences all the time)