A geometric approach to revealed preference via Hamiltonian cycles

It is shown that a fundamental question of revealed preference theory, namely whether the weak axiom of revealed preference (WARP) implies the strong axiom of revealed preference (SARP), can be reduced to a Hamiltonian cycle problem: A set of bundles allows a preference cycle of irreducible length if and only if the convex monotonic hull of these bundles admits a Hamiltonian cycle. This leads to a new proof to show that preference cycles can be of arbitrary length for more than two but not for two commodities. For this, it is shown that a set of bundles satisfying the given condition exists if and only if the dimension of the commodity space is at least three. Preference cycles can be constructed by embedding a cyclic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(L-1)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>-polytope into a facet of a convex monotonic hull in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>L</mi> </mrow> </math> </EquationSource> </InlineEquation>-space, because cyclic polytopes always admit Hamiltonian cycles. An immediate corollary is that WARP only implies SARP for two commodities. The proof is intuitively appealing as this gives a geometric interpretation of preference cycles. Copyright Springer Science+Business Media New York 2014