Revealed preference test and shortest path problem; graph theoretic structure of the rationalizability test

Abstract

This paper presents some substantial relationships between the revealed preference test for a data set and the shortest path problem of a weighted graph. We give a unified perspective of several forms of rationalizability tests based on the shortest path problem and an additional graph theoretic structure, which we call the shortest path problem with weight adjustment. Furthermore, the proposed structure is used to extend the result of Quah (2014), which sharpened the classical Afriat's Theorem-type result.

Afriat and arbitrage

“…Teo and Vohra (2003) give a simpler combinatorial proof and bring out clearly the connection with finding negative length cycles in graphs, a theme initially explored by Varian (1982). Shiozawa (2016) further extends the link made by Teo and Vohra (2003) to shortest path problems and the condition that there should exist weights so that that are no negative length cycles. His proofs are combinatorial in nature.…”

“…The existence of weights so that there are no negative length cycles can be thought of as implying the existence of marginal utilities so that arbitrage cycles cannot exist, so (2) holds. Shiozawa (2016) however concentrates on connections to shortest path problems rather than links to conditions for absence of arbitrage.…”

Revealed preference theory: An algorithmic outlook

“…An alternative, simple statement of the O(n 2 ) test is derived in Talla Nobibon et al (2016) from the observation that a dataset S satisfies garp if and only if p i q i = p i q j for each arc (i, j) contained in a strongly connected component of G R 0 (see Condition 4 of Theorem 1). Shiozawa (2016) describes yet another way to test garp in O(n 2 ) time, using shortest path algorithms. Talla Nobibon et al (2015) prove a lower bound on testing garp, showing that no algorithm can exist with time complexity smaller than O(n log n).…”