I read Yudkowsky’s post on cheerful price and came away confused as to why there was no reference to shadow price. It seems the two are effectively the same modulo a perspective change. Am I missing something, are Yudkowsky (and others) somehow unaware of shadow price, or is it something else entirely?
The shadow price for a constrained optimization problem is the change in the optimal value of the primal utility function with respect to a change in constraints. More technically it’s the dual variables associated with the dual optimization problem.
Yudkowsky’s Tl;dr is, well comparable to the length of this post so to compare I’ll just list the first bullet,
The cheerful price for doing something is the price that gives you a cheerful feeling about the transaction.
People do what they do to achieve their ends (utility) while obeying constraints, physical or otherwise. If they already were going to do the ‘something’ to be proposed the cheerful price would be zero. This is also true for the shadow price. If the optima lies within the set after perturbing the constraints there is no shadow price.
Say we ask someone to do something they would not naturally do. This means we are going to propose a change in constraints. The shadow price will compensate for the decrease in the person’s utility, compared to no intervention.
This is premised on the idea that we can substitute utility with some other quantity. If it’s money then the shadow price for doing something (to constraints) is the price that leaves the person’s optimality unchanged. It seems clear that if the person’s objective value is their utility then the change in value is their cheerfulness to the request. Hence, the shadow price is the cheerful price.
The most immediate reaction I expect is that somehow the cheerful price has to be ‘more’ than the fair price or ‘catered’ towards a specific part of the inner utility optimizer. However, these all seem to be straight-forward modifications of the concept of shadow price. The sorts of things that immediately come up when you apply the concept. It doesn’t make sense to treat this as a novel concept.