I think there are some subtleties with the (non-infra) bayesian VNM version, which come down to the difference between “extreme point” and “exposed point” of D. If a point is an extreme point that is not an exposed point, then it cannot be the unique expected utility maximizer under a utility function (but it can be a non-unique maximizer).
For extreme points it might still work with uniqueness, if, instead of a VNM-decision-maker, we require a slightly weaker decision maker whose preferences satisfy the VNM axioms except continuity.
Another excellent catch, kudos. I’ve really been sloppy with this shortform. I corrected it to say that we can approximate the system arbitrarily well by VNM decision-makers. Although, I think it’s also possible to argue that a system that selects a non-exposed point is not quite maximally influential, because it’s selecting somethings that’s very close to delegating some decision power to chance.
Also, maybe this cannot happen when D is the inverse limit of finite sets? (As is the case in sequential decision making with finite action/observation spaces). I’m not sure.
I think there are some subtleties with the (non-infra) bayesian VNM version, which come down to the difference between “extreme point” and “exposed point” of D. If a point is an extreme point that is not an exposed point, then it cannot be the unique expected utility maximizer under a utility function (but it can be a non-unique maximizer).
For extreme points it might still work with uniqueness, if, instead of a VNM-decision-maker, we require a slightly weaker decision maker whose preferences satisfy the VNM axioms except continuity.
Another excellent catch, kudos. I’ve really been sloppy with this shortform. I corrected it to say that we can approximate the system arbitrarily well by VNM decision-makers. Although, I think it’s also possible to argue that a system that selects a non-exposed point is not quite maximally influential, because it’s selecting somethings that’s very close to delegating some decision power to chance.
Also, maybe this cannot happen when D is the inverse limit of finite sets? (As is the case in sequential decision making with finite action/observation spaces). I’m not sure.