Wei Dai: Consider a program which when given the choices (A,B) outputs A. If you reset it and give it choices (B,C) it outputs B. If you reset it again and give it choices (C,A) it outputs C. The behavior of this program cannot be reproduced by a utility function.
I don’t know the proper rational-choice-theory terminology, but wouldn’t modeling this program just be a matter of describing the “space” of choices correctly? That is, rather than making the space of choices {A, B, C}, make it the set containing
(1) = taking A when offered A and B,
(2) = taking B when offered A and B,
(3) = taking B when offered B and C,
(4) = taking C when offered B and C,
(5) = taking C when offered C and A,
(6) = taking A when offered C and A.
Then the revealed preferences (if that’s the way to put it) from your experiment would be (1) > (2), (3) > (4), and (5) > (6). Viewed this way, there is no violation of transitivity by the relation >, or at least none revealed so far. I would expect that you could always “smooth over” any transitivity-violation by making an appropriate description of the space of options. In fact, I would guess that there’s a standard theory about how to do this while still keeping the description-method as useful as possible for purposes such as prediction.
Wei Dai: Consider a program which when given the choices (A,B) outputs A. If you reset it and give it choices (B,C) it outputs B. If you reset it again and give it choices (C,A) it outputs C. The behavior of this program cannot be reproduced by a utility function.
I don’t know the proper rational-choice-theory terminology, but wouldn’t modeling this program just be a matter of describing the “space” of choices correctly? That is, rather than making the space of choices {A, B, C}, make it the set containing
(1) = taking A when offered A and B, (2) = taking B when offered A and B,
(3) = taking B when offered B and C, (4) = taking C when offered B and C,
(5) = taking C when offered C and A, (6) = taking A when offered C and A.
Then the revealed preferences (if that’s the way to put it) from your experiment would be (1) > (2), (3) > (4), and (5) > (6). Viewed this way, there is no violation of transitivity by the relation >, or at least none revealed so far. I would expect that you could always “smooth over” any transitivity-violation by making an appropriate description of the space of options. In fact, I would guess that there’s a standard theory about how to do this while still keeping the description-method as useful as possible for purposes such as prediction.