Nitpick: if one’s behavior cannot be described by a utility function, then one will have preferences that are intransitive, incomplete, or violate continuity.
I’m with you on “incomplete” (thanks for the catch!) but I’m not so sure about “violate continuity”. Can you give an example of preferences that are transitive and complete but violate continuity and are therefore not encodable in a utility function?
Lexicographic preferences are the standard example: they are complete and transitive but violate continuity, and are therefore not encodable in a standard utility function (i.e. if the utility function is required to be real-valued; I confess I don’t know enough about surreals/hyperreals etc. to know whether they will allow a representation).
I’d heard that mentioned before around these parts, but I didn’t recall it because I don’t really understand it. I think I must be making a false assumption, because I’m thinking of lexicographic ordering as the ordering of words in a dictionary, and the function that maps words to their ordinal position in the list ought to qualify. Maybe the assumption I’m missing is a countably infinite alphabet? English lacks that.
Lexicographic preferences (lexicographical order based on the order of amount of each good) describe comparative preferences where an economic agent infinitely prefers one good (X) to another (Y). Thus if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie of Xs between bundles will the agent start comparing Ys.
Lexicographic preferences (lexicographical order based on the order of amount of each good) describe comparative preferences where an economic agent) infinitely prefers one good (X) to another (Y). Thus if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie of Xs between bundles will the agent start comparing Ys.
(Obviously, one could have lexicographic preferences over more than two goods.)
Nitpick: if one’s behavior cannot be described by a utility function, then one will have preferences that are intransitive, incomplete, or violate continuity.
I’m with you on “incomplete” (thanks for the catch!) but I’m not so sure about “violate continuity”. Can you give an example of preferences that are transitive and complete but violate continuity and are therefore not encodable in a utility function?
Lexicographic preferences are the standard example: they are complete and transitive but violate continuity, and are therefore not encodable in a standard utility function (i.e. if the utility function is required to be real-valued; I confess I don’t know enough about surreals/hyperreals etc. to know whether they will allow a representation).
I’d heard that mentioned before around these parts, but I didn’t recall it because I don’t really understand it. I think I must be making a false assumption, because I’m thinking of lexicographic ordering as the ordering of words in a dictionary, and the function that maps words to their ordinal position in the list ought to qualify. Maybe the assumption I’m missing is a countably infinite alphabet? English lacks that.
The wikipedia entry on lexicographic preferences isn’t great, but gives the basic flavour:
That entry says,
So my intuition above was not correct—an uncountably infinite alphabet is what’s required.
The wikipedia entry on lexicographic preferences isn’t great, but gives the basic flavour:
(Obviously, one could have lexicographic preferences over more than two goods.)