Utility Functions

ah, utility. econ major lost me in those classes --- the teacher refused to make all axioms explicit, and when i asked why should it be anywhere near commonsense that utilities are always comparable & transitive, he told me to leave the classroom immediately. :-)(now i'm a liberal arts major, look what they did to me).great post as usual, Mark. you have a gift for exposing maths simply (intuitively even) but without ever letting go of the rigor. since this is applied maths, however, and can therefore, informally, be quite easily subverted into a mere rhetorical figure, i think it would be nice to point out a couple of drawbacks and quirks of mathematical utilitarism.i think the foremost problem here is that people forget to state their conclusions as "if preferences are universal, comparable and consistent and the agents are rational...". i remember reading a footnote somewhere (iirc, in Mas-Collel et al., Microeconomics) that referred to a couple of everyday examples on the contrary. another two fallacies that came up a lot in econ classes but which are also more specific to economics were, first, to expose a choice between good x1 and the bundle of "all possible goods you could afford with the same money" while tacitly deriving conclusions from the economical proposition of homogeneous goods. another --- this one made my teeth cringe --- was to assume long term (or even short term) consistency. consistency is only possible instantly. the moment you took whatever action or made whatever decision, your preferences are no longer supposed to remain as they were.these are the fallacies i can think of right now. there are a bunch more, of course. economics, game theory very much included, are used as a rhetorical device every day because people don't state their axioms clearly enough (as if they were commonsense) and then don't keep to them. (and as if the trivialization of classical systems wasn't bad enough, they manage to bend fuzzy and paraconsistent systems as well).again, excelent post, Mark. but for the sake of keeping the rigor, i'd like to see, before we move on, a post on cases in which mathematical utilitarism doesn't apply, or on usual economical & game-theoretical fallacies (they're probably as serious if not more dangerous than those bayesian proofs of god you hate so much). :-)

Count me among those who find human behavior to be too complicated. More specifically, there are so many terms (and second-order terms, and third-order, and...) to ever make any concrete calculation of utility values. I'd draw the analogy with nonrenormalizable quantum field theories and their completely unsummable series, but QFTs at least have observable effects we can measure. Humans are just ornery!

yup, Mas-Colell, Microeconomic Theory. (do notice i mis-spelt the author's name and got the title wrong, in my previous comment). footnote in page 7: "if we continue in this fashion, letting the paint colors get progressively lighter with each successive choice experiment, she may express indifference at each step" --- colours aren't always comparable.(sure, colours sound like a most useless counter-example for these axioms of preference, but then again, people tend to apply utilitarian theories to every possible thing without first checking at any level comparability and consistency).

This is nitpicky, but you have described Von Neumann-Morgenstern Utility or Expected Utility(better than lots of textbooks I might add) which makes use of lotteries to create cardinal utilities and there are other conceptions of utility, that maintain that utility is ordinal, that are more common outside of game theory in economics.

I agree with Bruno, however, there are some awful inconsistencies in the use of all theories of utility, and the theories should be treated as little more than fascinating mathematical curiosities. As a means of desciribing and explaining human behaviour they are just garbage.

Count me as someone who lands on the "Utility functions are ideal" side of things.

I don't think they can describe human behavior, but that's because humans don't always act rationally - even ignoring situations like altruism, there's still spite and simply doing things against our intentions.

A partway argument can be seen in the work of Ken Binmore (UCL).

Basically, when running game theory experiments on undergrads (a plentiful and renewable resource), he found that their behavior didn't follow game theoretical predictions.

When running the same games (on a new crop of undergrads) with the payoffs significantly higher (maximum about 50 pounds instead of 5), they suddenly became a *lot* more rational.

Other experiments also suggest that game theory has decent predictive power in situations where the payoffs matter, and they have time to learn the game.

Tht said, one core element of the examples given is that they assume the players are risk neutral. So the palyer is equally happy either getting $100 or getting a lottery ticket that is worth $100,000 but only pays off .1% of the time.

This makes the formulation of condition 4 (rationality) slightly incorrect (the final example, between a payoff of 4 or an expected payoff of 4, is wrong). A player will always select a strategy that maximizes her expected utility. However, the expected utility is not necessarily the expected payoff.

In my graduate micro theory course, the professor used the following to show the issues of comparability and transitivity:

Suppose you have a bottle of Sam Adams, and you replace one drop of beer with one drop of water. You probably could not detect the difference, and would thus be indifferent. This would be true as you continued this process. However, at some point, you would have a Bud Light, and clearly you are not indifferent between Sam Adams and Bud Light, even though you were indifferent between each pair.

I love reading this blog but like a few readers commented before, you left a few things out of the description.

So, like 2 guys have pointed out already, you've described vonNeumann-Morgenstern utilities, that is, preferences over lotteries. Originally, utility is used in basic rational choice theory to assign values to choices from an arbitrary set.

Where do the utility numbers come from?
Suppose we have some possible set of outcomes X, it can be finite or infinite. We're looking to be able to represent an individual's ordering of these outcomes as a real number.

This is ordinal utility. That is, it represents only an ordering of choice, not "strength of" ordering. In this case u(A) = 10 and u(B) = 5 it means A is preferred to B, not A is "twice as good as B", in fact, replacing u(A) with any number greater than 5 yields the same preferences.

There is a proof by Debreu of the Representation Theorem that says that if you have an infinite set of choices, say X with some kind of metric (to tell you how 'different' two choices are) and preferences obey the following conditions:

1) COMPLETENESS For each a, b in X either a is preferred to b or b to a.
2) TRANSITIVITY If a is preferred to b and b is preferred to c then a is preferred to c.
3) CONTINUITY If a is preferred to b then each other choice that is 'arbitrarily similar' to a is preferred to each choice 'arbitrarily similar' to b

If these 3 conditions are satisfied then there is a function which can assign to each choice a real number such that if a is preferred to b then u(a) >= u(b) and is in agreement with the preference relation on X for each member of X.

It's a pretty interesting result, and certainly much less controversial (though bordering on tautological) than vNM utility.

I took utility functions in economics. I agree with what you have said. I was wondering what your opinion was on applying calculus to utility functions. Measurements of utility are supposed use ordinal numbers. With ordinal numbers the difference between any two ordinal numbers is meaningless. A derivative is defined as a quotient with both parts having a subtraction sign. I think a derivative of a utility function contradicts the concept of a utility function but it is used nevertheless in economics. I asked my prof (years ago) about this and he never really answered my question. He has to teach the material as part of the course even if the math stinks. What is your opinion Mark?

Reply to Peter:

There's no reason for calculus not to work on a 'well behaved' utility function if indeed the properties I posed about hold on the preference relation on the space. A derivative (total, partial or as a linear operator) represents a local rate of change, which is why we define it as a limit, and thus does not impose any cardinality and leaves utility as a strictly ordinal relation. Indeed, you can see this by composing the original utility function with any monotonic transformation and comparing the locations of the maxima and minima.

Late to the party, but:

It's a pretty interesting result, and certainly much less controversial (though bordering on tautological) than vNM utility.

Ah, so there are different types of utility functions!

Good, because the posted one strikes me as removed from how we use money - as commenters said or indicated, we would like to avoid risks, and the actual amount of capital matters, as real life players wants to avoid bankruptcy.

That said, and being a layman here, I would deem the utility [sic] of utility functions comes from predicting market (player) behavior, so "experiments also suggest that game theory has decent predictive power in situations where the payoffs matter, and they have time to learn the game" sounds promising.