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Don’t say “utility function,” say “value function”

Like Dave Krantz, I’m down on the decision-theoretic concept of “utility” because it doesn’t really exist.

The utility function doesn’t exist

You cannot, in general, measure utility directly, and attempts to derive it based on preferences (based on the Neumann-Morgenstern theory) won’t always work either because:

1. Actual preferences aren’t necessarily coherent, meaning that there is no utility function that can produce all these preferences.

2. Preferences themselves don’t in general exist until you ask people (or, to be even more rigorous, place them in a decision setting).

So, yeah, utility theory is cool, but I don’t see utility as something that’s Platonically “out there” in the sense that I can talk about Joe’s utility function for money, or whatever.

Call it value, not utility

The above is commonplace (although perhaps not as well known as it should be). But my point here is something different, a point about terminology. I would prefer to follow the lead of some decision analysis books and switch from talking about “utility” to talking about “value.” To the extent the utility function has any meaning, it’s about preferences, or how you value things. I don’t think it’s about utility, or how useful things are. (Yes, I understand the idea of utility in social choice theory, where you’re talking about what’s useful to society in general, but even there I’d say you’re really talking about what society values, or what you value for society.)

Just play around with the words for a minute. Instead of “my utility function for money” or “my utility for a washer and a dryer, compared to my utility for two washers or two dryers” (to take a standard example of a nonadditive utility function) or “my utility for a Picasso or for an SUV,” try out “my value function for money” or “the value I assign to a washer and a dryer, compared to the value I assign to two washers or two dryers” or “the value I assign to a Picasso or to an SUV.” This terminology sounds much better to me.

P.S. See Dave’s comments here.

23 Comments

  1. Tom S. says:

    Economists will promise to use "value functions" when statisticians promise to find a better phrase than "degrees of freedom."

  2. pat toche says:

    In common terminology, the value function is the maximized utility subject to constraints,
    V = max U(c) s.t. c in C (say).

  3. Except that we already have another use for the term 'value function'.

  4. pat toche says:

    In common terminology, the value function is the maximized utility subject to constraints,
    V = max U(c) s.t. c in C (say).

  5. Andrew says:

    Stephen: I think that double-use of the term would be ok here.

    Pat: I guess a bit more renaming is in order.

    To both: I conjecture that the term "utility" creates problems because it is separated from its common English meaning in a way that allows people to consider it as a Platonic quantity without stepping back and thinking about what it's really supposed to mean.

  6. Anonymous says:

    But that's not how economists use it. Or at least, that's not how I use it. If preferences have a certain structure, then they can be characterised by a function with the property that U(a,b) > U(x,y) iff the combination (a,b) is preferred to that of (x,y). That's all. The actual value of U(.,.) is of no importance; we can always apply any monotonic transformation to U and obtain an equivalent utility function that.

    And yes, not all preferences allow for the existence of utility functions (lexicographic preferences are the usual counter-example). So we don't use utility functions for those cases.

  7. Anonymous says:

    But that's not how economists use it. Or at least, that's not how I use it. If preferences have a certain structure, then they can be characterised by a function with the property that U(a,b) > U(x,y) iff the combination (a,b) is preferred to that of (x,y). That's all. The actual value of U(.,.) is of no importance; we can always apply any monotonic transformation to U and obtain an equivalent utility function.

    Not all preferences allow for the existence of utility functions, of course: lexicographic preferences are the usual counter-example.

  8. Andrew says:

    Anon,

    1. Your definition is ok but I'd prefer to call it a value function.

    2. Once you start taking expectations, you can't just do any monotonic function. I think the numerical values have some meaning. For example, if outcomes A,B,C have values 1,2,3, I think that's different from having values 1,2,100.

    3. It's not just lexicographic preferences or even nontransitive preferences. It's also that many preferences and preference orderings don't even exist until you ask the question (or, to be even more rigorous, place them in a decision setting).

    4. The theory of deriving utilities from preferences doesn't make complete sense to me. On one hand, the utility-deriver is saying: (a) I don't trust survey responses (i.e., directly asking people what their values or utilities are for each outcome), but (b) I'll trust people's statements about hypothetical decision options. I don't get it. When you can do it based on real decisions (e.g., hedonic regression from amounts people actually pay for things), that makes sense, but I don't buy the general idea of figuring out someone's utility function from their implicit web of decision preferences.

    5. I do think that utility theory (or, I should say, value theory) has some normative advantages, for sure. This is discussed (with examples) in Chapter 22, I believe, of the 2nd edition of Bayesian Data Analysis. For example, lexicographic orderings typically involve the fallacy of the one-sided bet.

  9. Anonymous says:

    Economists are usually quite cautious when handling self-reported survey preferences, and many (most?) are downright dismissive. We're much more comfortable drawing inferences about preferences from actual decisions.

    And perhaps preferences don't exist. Neither does probability, come to that. But models based on those concepts are still pretty useful analytical tools.

  10. My apologies; I seem to have posted twice as 'anonymous' with realising it.

  11. OneEyedMan says:

    Andrew, my understanding is that Roy's linear utility model is used to justify probit and logit modeling. Does your opinion on utility theory cause you to reject these statistical approaches in situations involving human decisions?

    "So, yeah, utility theory is cool, but I don't see utility as something that's Platonically "out there" in the sense that I can talk about Joe's utility function for money, or whatever."

    If one buys into the existence of the axioms of choice, then it doesn't matter that the utility function can't be observed, they must exist.

    Now sure, those axioms are often violated, but I'm not sure they are violated over matters of serious import. It seems like a decent first level approximation.

    "It's not just lexicographic preferences or even nontransitive preferences."

    These are not compatible with the requirements to establish a utility function (if I recall correctly they have an if and only if relationship) so I don't see this as a criticism of utility functions.

    "It's also that many preferences and preference orderings don't even exist until you ask the question (or, to be even more rigorous, place them in a decision setting)."
    I don't understand this criticism. There is no requirement that the utility function be visible to decider. All that is required is that the utility function operate on different bundles and tell the agent what is the prefered bundle.They don't have to know what they'd decide in advance on every possible pairing of bundles. That strikes me as a more implausible requirement then the potential to evaluate any two bundles.

  12. Andrew says:

    Anonymous,

    I think probability exists, in the sense that I can define a clear reference set and then count successes and failures. In practice, though, I agree that in many settings, probabilities have to be computed partly through models. (See this paper as well as chapter 1 of Bayesian Data Analysis for more discussion.)

    OneEyedMan,

    I think utility (or, I would now say, "value") models are just fine. We indeed review the connection between choice models and logit/probit in chapter 6 of Data Analysis Using Regression… I like utility theory, I just think it's useful to recognize its limitations, otherwise you can get tied into knots trying to make general statements about something that's merely a sometimes-useful concept.

  13. My apologies; for some reason my last two comments were signed 'Anonymous'.

    I think the comparison between the existence/non-existence of probabilities still holds. We observe neither the decision-maker's preferences nor her subjective probabilities. We assume that one or the other or both exist for modeling convenience. Not because they (even in principle) 'exist'.

  14. Aleks says:

    How about: Utility is one theory of value (it assumes value can be modeled with a function). Value is a theory of good and bad (it assumes it is reducible to a real quantity).

    We sometimes create terms that make it harder to understand, but by this we are also separating the simplified models from more truthful ones.

  15. Phil says:

    One-Eye,
    How did the Axiom of Choice get into it? "Axiom of Choice: Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S."

    Was your use of the term "axiom of choice" a reference to this, or does it mean something else?

  16. scott Cunningham says:

    Pat, that's the indirect utility function, and it's just the optimal utility given a budget constraint. It's still based on modern utility theory, which seems to be what Gelman's objecting to (right?).

  17. scott Cunningham says:

    Andrew – my understanding is that you are right that you cannot take expectations with the kind of utility function described in consumer theory. You've got to go into von Neumann-Morganstern world with cardinal utility which is unique only up to linear transformations (vs. monotonic transformations).

    Is your complaint in the word utility – that you prefer value to utility (preferences! hah!)? Or is it more fundamental than that?

  18. scott Cunningham says:

    Andrew – for #3. Sorry I'm answering these out of order. But isn't this just revealed preference theory? That is, these preferences are revealed in the decision context. We can look at this person and determine if his revealed preferences are consistent over time, too. But just because he doesn't know those preferences explicitly doesn't mean they don't exist. Maybe his 'id' knew all along, but his 'ego' wasn't brought in on that knowledge until he had to act (only halfway joking).

  19. Corey Yanofsky says:

    Andrew,

    Since "value function" already has an assigned meaning, how about "preference function"? This might even be closer in colloquial meaning to the concept you want to describe.

  20. pat toche says:

    It is a fact that the concept of "utility" has a bad reputation (I do not understand why), in everyday conversation I usually say "preferences" or "reward" or "satisfaction" or "pleasure" or "happiness", depending on context and audience, and what I have in mind is an increasing and concave transformation of the inputs (payoff, income, consumption, whatever) — much like a production function to be frank. I do not understand why some are so bothered about measurement or existence, perhaps someone can enlighten me: while we may not be able to quantify our enjoyment of a good wine or to compare the intensity of the enjoyment across different people, but we still have a pretty good idea of how to model the demand for wine and compare the demand of different people, and the cause of the difference must be something which exists, where do I err?

  21. Anonymous says:

    Side note: The axiom of choice is not necessary to prove the existence of a utility function, at least for the original proofs (Debreu 1954 and Rader 1963).

    I believe you need the axiom of choice to prove the existence of a differentiable utility function (Mas-Colell 1985).

  22. Andrew Gelman says:

    Scott,

    Regarding revealed preference theory: the point is that if these "revealed preferences" aren't consistent, then maybe it doesn't make sense to think of them as revealed preferences at all!

  23. Anon says:

    Economists try to model human transactional behavior under terms of scarcity and uncertainty. Why would one expect preferences to be consistent over time and space? People grow up, their tastes change. People have religious conversions, people go green, people realize they were wrong, people find something better to do.

    In practice, one of the main values of utility is that it reminds economists that people have different preferences and they have to be careful not to assume that their own utility function applies to other people.