Akrasia is a Greek term referring to choices made that are not in the best interest of the person making them. Standard utility theory does not accommodate such choices, of course – indeed, we often go back and “read” underlying preferences from WARP-satisfying choices. There has been a ton of work on preferences that allow choice cycling (A>B,B>C and C>A) or other violations of WARP published in (relatively) recently: Gul and Pesendorfer on temptation, of course, but also the “menu choice” papers leading up to Rubinstein and Salant’s “Choice from Lists” in TE 2006. Unlike most of those papers, Sandroni considers akratic choice only in a static, not a dynamic, context, and shows that with some assumptions it is possible to identify non-akratic choice even in the static context. Further, one can identify an agent whose preferences deviate from this akratic model using only choice data.
Now this sounds like some sort of trick: if non-akratic choice is choice made in accordance with an agent’s preferences, and akratic choice are choices violating those inherent preferences, how am I able to look at choice data and refute this model? The trick is the following. Let non-akratic choice follow some strict preference relation R that represents “deep thought” preferences. Let akratic choice follow from some other strict preference relation S that represents “instinct”. Assume that instinctual preferences satisfy WARP. Now give a list from the power set of all choice bundles which specifies which bundles trigger instinctive choice; this can be arbitrary (Correction: the list is a subset of the set of alternatives, not of the power set). If that list is the null set, then we are in the standard classical utility model since all choice is non-akratic. The researcher can see only final choice made; that is, I see the choice made using R if the choice is made from a set of alternatives not on the instinctive list, and otherwise, I see the choice made using S. In either case, I don’t know what that instinctive list contains. Choice is said to be revealed akratic if there is no preference relation R which can explain the observed choices.
Let a super-issue B* be a set of alternatives such that another set of alternatives (a sub-issue) B is a subset of B*. Sandroni shows that, within the instinct/reflection paradigm, choice C(B) is revealed akratic iff there is a super-issue B* of B such that C(B*) and C(B) together violate WARP. Further, choice function C is within the instinct/reflection paradigm iff given two pairs of nested issues (B1,B1*) and (B2,B2*), each of which evidence violations of WARP, there is some preference relation that resolves B1, B2 and the union of B1 and B2 in a consistent way. The proofs are fairly tricky, though by decision theory standards you might consider them basic!
This is an interesting result: essentially, Sandroni constructs a model of choice that allows for multiple selves where the “self” making the decision is exogenously determined by the choice being considered, rather than endogenously determined as in a temptation-avoidance model. Indeed, such a model has the nice property of being falsifiable in the same manner as standard choice theory.
I don’t see this mentioned in the paper, but my intuition is that if the agent has 3 (or more) separate preference relations depending on the choice set, the nice results in this paper will fall apart. Essentially, set inclusion is driving the representation theorem, and we need “larger” and “smaller” sets with WARP-violating anomalies. I would have liked to see a more direct connection to the choice under frames literature as well – I can’t really make the theoretical connection at first glance, though it must be fairly straightforward.
http://www.springerlink.com/content/y65477j3ln858584 (Final version in forthcoming issue of Synthese – GATED COPY, I’m afraid. I don’t see a working paper version online.)