# revealed preference given an uncertain environment?

classical revealed preference is about a situation where we assume an agent has a preference over some set of possible choices $X$. We then construct a revealed preference relation on $X$ from choices made given choice sets $C\subseteq X$.

But is there a theory of revealed preference for the case where the agent doesn't inherently care about his choice $x\in X$, but cares about outcome $o\in O$, where there is a function $f:X\to O$? This function $f$ may possibly be uncertain, or may vary over time?

EDIT: I am asking for a literature on the question of how we infer $u$ in this context, not on how do we infer the induced preference on $X$.

Revealed preference theory is essentially about the observable empirical content of theories. Now if we look at the classical model of utility maximization, if there is a fixed function $f:X\to O$ and a utility function $u:O\to\mathbb{R}$, utility maximization amounts to maximizing with respect to the utility function $v:X\to\mathbb{R}$ given by $v(x)=u\big(f(x)\big)$ for all $x\in X$. There is nothing new here.
If $f$ can change or is uncertain, you are looking at a form of changing preferences or uncertain preferences. Since revealed preference theory is about the observable empirical content of theories, you need some theory about changing preferences or uncertain preferences. The task is then to identify the empirical content of the theory. For this to be an interesting exercise, there needs to be some relation between different decision problems. One example is the work of Gul and Pesendorfer on the empirical implications of the Strotz multiple self model of changing preferences with consistent planning in
• I don't see why you say that if $f$ changes, then this means that there are "changing preferences". The preferences over outcomes are the same. e.g. if I care about the amount of money I receive, and a change in $f$ captures a change in the effect of my actions on my salary, etc, then a change (uncertainty) in $f$ should not be interpreted as a change in preference but as a change (uncertainty) in the environment. – user56834 Aug 28 '18 at 15:25
• @Programmer2134 The function $f$ induces preferences over choices as explained above and there is no way to differentiate the two things, unless you can observe and modify the function $f$. – Michael Greinecker Aug 28 '18 at 15:27
• This observing and modifying $f$ (as the entity who is trying to discover the preferences of the agent) is an example of the type of revealed preference that I'm interested in. – user56834 Aug 28 '18 at 15:38
• If you can change $f$ arbitrarily, you are basically in a situation in which the decisionmaker can choose directly from $O$. – Michael Greinecker Aug 28 '18 at 15:41
• I’m not sure why that is relevant? I am interested in the question: how can we pinpoint a utility function/preference relation on $O$, given that the agent chooses from $X$ and is faced with function $f:X\to O$ (over which he may have uncertainty). I’m interested in things like: necessary and/or sufficient conditions on the choice structure (i.e. a collection of chosen actions: given a subset of $X$, and a particular $f$ or probability distribution on $f$, an action chosen $x\in X$), in order to uniquely pinpoint a utility function up to e.g. monotonic or linear transformations. – user56834 Aug 28 '18 at 15:53