The paradigmatic economic application I recall is consumer choice theory: You have a certain amount of money, m, and two goods you can buy. These goods have fixed prices p and q. Your choices are pairs (x,y) saying how much of each good you buy; the “feasible set” of choices is {(x,y) : x,y >= 0 and xp + yq <= m}. What’s your best choice in this set? We want to use calculus to solve this, so we’ll express your preferences as a differentiable utility function. The reasons VNM or Savage doesn’t enter into it is that actions lead to outcomes deterministically.
In UDT, we don’t even start with a natural definition of “outcome”; in principle, we need to specify (1) a set of outcomes, (2) an ordering on these outcomes, and (3) a deterministic, no-input program which does some complicated computation and returns one of these outcomes. (The intuition is that the complicated computation computes everything that happens in our universe, then picks out the morally relevant parts and prints them out.) It’s just simpler to skip parts (1) and (2) in the formal specification and say that the program (3) returns a number. Since the proof-based models have no notion of probability (even implicitly like in Savage’s theorem), this makes the program an order-only “utility function.”
(Thanks for adding the point about Savage’s theorem!)
Yes, but “we want to use calculus to solve this” isn’t a very natural constraint on the set of orderings. :) It’s a “we want to make the math easier” constraint, not a “we have reason to believe that any rational agent should act this way” constraint.
Not that it’s necessarily inappropriate in the example you give—it probably makes sense there. Just a bit surprising that UDT would restrict itself in such a way.
In the UDT case, the set of outcomes is finite (well, or at least the set of equivalence classes of outcomes under the preference relation is finite) and the utility functions don’t have any particular properties, so every possible preference relation the model can treat at all can be represented by a utility function!
(I should note that this is not UDT as such we’re talking about here, but one particular formal way of implementing some of the ideas of UDT.)
The paradigmatic economic application I recall is consumer choice theory: You have a certain amount of money,
m, and two goods you can buy. These goods have fixed pricespandq. Your choices are pairs (x,y) saying how much of each good you buy; the “feasible set” of choices is{(x,y) : x,y >= 0 and xp + yq <= m}. What’s your best choice in this set? We want to use calculus to solve this, so we’ll express your preferences as a differentiable utility function. The reasons VNM or Savage doesn’t enter into it is that actions lead to outcomes deterministically.In UDT, we don’t even start with a natural definition of “outcome”; in principle, we need to specify (1) a set of outcomes, (2) an ordering on these outcomes, and (3) a deterministic, no-input program which does some complicated computation and returns one of these outcomes. (The intuition is that the complicated computation computes everything that happens in our universe, then picks out the morally relevant parts and prints them out.) It’s just simpler to skip parts (1) and (2) in the formal specification and say that the program (3) returns a number. Since the proof-based models have no notion of probability (even implicitly like in Savage’s theorem), this makes the program an order-only “utility function.”
(Thanks for adding the point about Savage’s theorem!)
Yes, but “we want to use calculus to solve this” isn’t a very natural constraint on the set of orderings. :) It’s a “we want to make the math easier” constraint, not a “we have reason to believe that any rational agent should act this way” constraint.
Not that it’s necessarily inappropriate in the example you give—it probably makes sense there. Just a bit surprising that UDT would restrict itself in such a way.
In the UDT case, the set of outcomes is finite (well, or at least the set of equivalence classes of outcomes under the preference relation is finite) and the utility functions don’t have any particular properties, so every possible preference relation the model can treat at all can be represented by a utility function!
(I should note that this is not UDT as such we’re talking about here, but one particular formal way of implementing some of the ideas of UDT.)
Oh, OK then!