# Revealed preference if we know that the decisionmaker is rational?

In standard revealed preference, we don't assume that the agent has rational preferences over a choice set $X$, and we can then ask: under what conditions can $X$ be rationalized by a rational preference relation?

One answer turns out to be: if the choice set $\mathcal C$ contains all subsets $C\subset X$ of 3 elements or less, and the weak axiom of revealed preference is satisfied.

But now, what if we already know from the outset that the preference relation of the decisionmaker is rational, and we just want to pinpoint uniquely which preference relation this is? Then at the very least, we only need every 2-element subset of $X$, but it seems that this can be imporved. Is there work on this?

EDIT: Let me state the question more clearly: What are minimal conditions on a choice collection $\mathcal C\subseteq \mathcal P(X)$ such that for any preference relation that generates choices $c(C)$, that preference relation is the unique rationalization of those choices?

• I don't know any work on this (not really my field). But, conceptually, suppose we construct a network graph in which each alternative is a node and an edge connects two nodes whenever they are both contained within a single element of $\mathcal{C}$. It seems like a necessary and sufficient condition to be able to reconstruct the preference relation is that the resulting graph is connected. – Ubiquitous Sep 26 '18 at 11:34
• "The answer turns out to be: if and only if the choice set $\mathcal{C}$ contains all subsets $\mathcal{C}\subset X$ of 3 elements or less, and the weak axiom of revealed preference is satisfied." I don't think so. Let $X=\{1,2,3,4\}$, $\mathcal{C}=\big\{\{1,2\}.\{1,2,3\},\{1,2,3,4\}\big\}$ and the choice function always uniquely selects the largest number. There is exactly one compatible rational preference relation. – Michael Greinecker Sep 26 '18 at 11:36
• @MichaelGreinecker, edited. – user56834 Sep 26 '18 at 12:45
• You might want to look into the work of Michael Mandler; he did at least a lot of related stuff on the complexity and efficiency of rational preferences. – Michael Greinecker Sep 26 '18 at 22:05