In revealed preference (RP), is any two points $x,y$ related by the indirect revealed preference relation?

Let $$X$$ be the closed compact convex set of alternative and $$B$$ be a closed compact convex subset of $$X$$. $$C$$ is defined on all closed compact convex set $$B\subseteq X$$.

$$X$$ is ordered by a strictly convex preference (not necessarily continuous).

$$C$$ is a well-defined choice function choosing the maximal element from $$B$$. Function $$C:2^B\to B$$. If follows that $$C(B)$$ is unique for each $$B$$

Define $$xPy$$ if $$x=C(B)$$ and $$y\in B$$. If $$xPy$$ and $$yPz$$ , we say $$xP'z$$.

$$P$$ is the direct revealed preference relation. $$P'$$ is the indirect one?

Can we say $$P'$$ is complete, in a sense that if $$x\succ y$$ then $$xP'y$$? This means any pair of $$x,y$$ can be ranked by the indirect revealed preference.

To prove that $$xP'y$$, we need to find a finite sequence that $$z_1,z_2...z_n$$ such that $$xPz_1P...Pz_nPy$$.

If you draw two indifference curve, and choose one point from each, you can see that it is usually not the case the $$xPy$$, but usually you can draw a new IC in between, find a $$z$$, to make $$xPzPy$$.

Side-Note that in classic RP theory, the data set is usually finite, and therefore $$P'$$ cannot be complete. Here $$C$$ is a function so completeness might be possible.

Strong Axiom of revealed preference: $$xP' z$$ and $$x\neq z$$ implies $$z\not P' x$$. SARP implies the existence of $$u$$ such that $$x=\arg\max_{x\in B}u(x)$$. It seems to me that SARP is not enough as it is possible that $$x\sim y$$ and then $$x\not P'y$$ and $$y\not P'x$$.

• You might want to consult the papers of Mas-Collel(1977) and Mas-Collel(1978).
– tdm
Dec 6, 2023 at 15:25
• You probably need some substitute for continuity to guarantee $C(B)\neq\emptyset$. Dec 6, 2023 at 15:42
• @MichaelGreinecker Yes you are right. I think we define $C$ as well-defined function. Dec 6, 2023 at 18:39

No. Let $$X=[-1,1]$$ and $$u:X\to\mathbb{R}$$ be given by $$u(x)=-|x|$$. Then, $$u$$ represents strictly convex preferences. Moreover, $$0.5\succ -1$$. Now, the only points that $$0.5$$ is revealed preferred to are larger points, since every convex set containing $$0.5$$ and smaller points must contain strictly better points. Now, every larger point in turn can only be revealed preferred to even larger points, and so on. It follows that $$0.5$$ is only indirectly revealed preferred to larger points but not $$-1$$.
• If preferences are continuous and strictly monotone, something like the following might work: Pick a utility representation $u$. If $u(x)>u(y)$, find a sufficiently well-behaved path from $y$ to $x$ along which $u$ increases. Replace the path by a piecewise-linear path with the same property. The linear pieces are the budget sets used to verify that $x P' y$. Dec 7, 2023 at 14:41
• I think strict monotonicity might be insufficient. Consider a utility function $u(x,y)=x+y$. Let the revealed demand at wealth=1, price=1 be (x,y)=(1/2,1/2). It follows that (1/2,1/2) is revealed preferred to (1/3,1/3). However, the point (1,0) is not revealed preferred to (1/3,1/3), as (1,0) was never chosen in any budget set. Am I correct? Dec 8, 2023 at 2:38