But utility functions are only defined up to positive affine transformation.
As far as I understand, it’s worse than that, linearity is not required, any (continuous) monotonic rescaling would do, since the only thing that needs to be preserved is the ranking of outcomes.
Linearity is required… what’s preserved is the ranking of lotteries over outcomes. Preserving the order of “a cookie” and “two cookies but no dollar” and “three cookies but a dollar in debt” isn’t enough, you also have to preserve “40% chance of a cookie and 60% chance of two cookies but no dollar”.
There may be some confusion over terms, because economists do in fact also have use for utility functions that only express an ordering of outcomes. (Incidentally, this is also true of some of the decision theory work that has appeared on LessWrong: the utility functions in our proof-based versions of UDT only express an ordering; these models don’t have a notion of probabilities at all.) The OP and the parent comment are about the utility functions given by the von Neumann-Morgenstern theorem; these are left invariant by any affine rescaling and (by the uniqueness part of the theorem) are changed by any non-affine rescaling.
The OP and the parent comment are about the utility functions given by the von Neumann-Morgenstern theorem; these are left invariant by any affine rescaling and (by the uniqueness part of the theorem) are changed by any non-affine rescaling.
It’s worth mentioning that all three kinds of utility functions can be constructed: ordinal scale, interval scale, and ratio scale. For an overview of ratio scale utility functions, see Peterson (2009), pp. 106-110.
Yes, to be absolutely clear, I’m talking about the sort of utility functions you get from the VNM theorem or Savage’s Theorem.
It’s not really clear to me what the use is for a utility function if all you have is ordering; why not just use an ordering? Seems that using a utility function then would just be needlessly restricting what sort of orderings you can have. Well, depending on what requirements you want that ordering to satisfy… after all if you have all of Savage’s axioms then you do get a utility function! But that requires ordering actions, not just outcomes...
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.)
As far as I understand, it’s worse than that, linearity is not required, any (continuous) monotonic rescaling would do, since the only thing that needs to be preserved is the ranking of outcomes.
Linearity is required… what’s preserved is the ranking of lotteries over outcomes. Preserving the order of “a cookie” and “two cookies but no dollar” and “three cookies but a dollar in debt” isn’t enough, you also have to preserve “40% chance of a cookie and 60% chance of two cookies but no dollar”.
There may be some confusion over terms, because economists do in fact also have use for utility functions that only express an ordering of outcomes. (Incidentally, this is also true of some of the decision theory work that has appeared on LessWrong: the utility functions in our proof-based versions of UDT only express an ordering; these models don’t have a notion of probabilities at all.) The OP and the parent comment are about the utility functions given by the von Neumann-Morgenstern theorem; these are left invariant by any affine rescaling and (by the uniqueness part of the theorem) are changed by any non-affine rescaling.
It’s worth mentioning that all three kinds of utility functions can be constructed: ordinal scale, interval scale, and ratio scale. For an overview of ratio scale utility functions, see Peterson (2009), pp. 106-110.
Yes, to be absolutely clear, I’m talking about the sort of utility functions you get from the VNM theorem or Savage’s Theorem.
It’s not really clear to me what the use is for a utility function if all you have is ordering; why not just use an ordering? Seems that using a utility function then would just be needlessly restricting what sort of orderings you can have. Well, depending on what requirements you want that ordering to satisfy… after all if you have all of Savage’s axioms then you do get a utility function! But that requires ordering actions, not just outcomes...
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!
No, you don’t. Risk-aversion is legal.