I am considering an example where there are two goods and three budget sets $(\mathbf{p}^{(n)},w^{(n)}),n=1,2,3$. If we assume $\mathbf{p}^{(n)} \cdot \mathbf{x}(\mathbf{p}^{(n+1)},w^{(n+1)}) \leq w^{(n)}$ for $n=1,2$, and that WARP holds for any pair of bundles, how can I show that SARP holds?
Graphically, it is easy to convince myself that the statement holds, but I would like to explore a more rigorous proof.
That WARP is equivalent to SARP for two goods is a result from Rose (1958), "Consistency of Preference: The Two-Commodity Case". Although the proof is not so difficult is is rather too lengthy to put down here.
A few years ago, I wrote a paper (Transitivity of preferences: When does it matter?) that gives the exact conditions under which WARP and SARP are equivalent. You can find the published version here. A necessary and sufficient condition turns out to be what we call a triangle condition. It boils down to the requirement that for any three prices vectors, one of the three price vectors should be below or above a linear combination of the other two price vectors. This is always satisfied in the two commodity case. Again the proof is not very complicated but still rather too lengthy to replicate here.
