[Paper Review] Testing Consumer Rationality using Perfect Graphs and Oriented Discs

Shant Boodaghians, Adrian Vetta|arXiv (Cornell University)|Jul 27, 2015

Advanced Graph Theory Research22 references4 citations

TL;DR

This paper establishes the computational complexity of testing consumer rationality by measuring the minimum number of data points whose removal makes a consumer's choices rational. It proves that the problem is polynomial-time solvable for two-commodity markets using perfect graphs and vertex cover on perfect graphs, but becomes NP-complete for three-commodity markets via a reduction from planar 3-SAT using oriented-disc drawings.

ABSTRACT

Given a consumer data-set, the axioms of revealed preference proffer a binary test for rational behaviour. A natural (non-binary) measure of the degree of rationality exhibited by the consumer is the minimum number of data points whose removal induces a rationalisable data-set.We study the computational complexity of the resultant consumer rationality problem in this paper. This problem is, in the worst case, equivalent (in terms of approximation) to the directed feedback vertex set problem. Our main result is to obtain an exact threshold on the number of commodities that separates easy cases and hard cases. Specifically, for two-commodity markets the consumer rationality problem is polynomial time solvable; we prove this via a reduction to the vertex cover problem on perfect graphs. For three-commodity markets, however, the problem is NP-complete; we prove thisusing a reduction from planar 3-SAT that is based upon oriented-disc drawings.

Motivation & Objective

• To develop a non-binary measure of consumer rationality based on the minimum number of data points whose removal rationalizes a consumer's behavior.
• To determine the computational complexity of this rationality testing problem across different numbers of commodities.
• To identify the exact threshold at which the problem transitions from tractable to intractable.
• To establish a connection between revealed preference theory and graph-theoretic problems such as feedback vertex set and vertex cover.
• To provide a novel reduction framework using oriented-disc graphs to prove NP-completeness for three-commodity markets.

Proposed method

• Represents consumer data as a directed revealed preference graph where arcs indicate revealed preference based on budget constraints.
• Reduces the consumer rationality problem to finding a minimum directed feedback vertex set (DFVS) in the revealed preference graph.
• For two-commodity markets, maps the problem to vertex cover on perfect graphs, leveraging the fact that perfect graphs allow polynomial-time solution of vertex cover.
• For three-commodity markets, constructs an oriented-disc drawing of a planar 3-SAT instance, embedding the problem into a directed graph with digons and acyclic auxiliary arcs.
• Uses the property that all cycles in the oriented-disc graph must correspond to digons, so hitting all digons via a feedback vertex set corresponds to a vertex cover in the underlying planar graph.
• Employs a gadget-based construction for variables and clauses, embedding the 3-SAT instance into a directed graph whose cycles are tied to edge coverage in a planar bipartite graph.

Experimental results

Research questions

• RQ1 What is the computational complexity of the consumer rationality problem when measured by the minimum number of data points whose removal induces rationalizability?
• RQ2 At what number of commodities does the consumer rationality problem transition from polynomial-time solvable to NP-complete?
• RQ3 Can the revealed preference graph's cycle structure be used to model and reduce to known NP-hard problems such as vertex cover or feedback vertex set?
• RQ4 Is it possible to represent a planar 3-SAT instance as a directed graph via oriented-disc drawings such that cycle detection corresponds to satisfiability?
• RQ5 Can perfect graph theory be leveraged to solve the rationality problem efficiently in low-dimensional markets (e.g., two commodities)?

Key findings

• The consumer rationality problem is solvable in polynomial time for two-commodity markets, as it reduces to vertex cover on perfect graphs.
• For three-commodity markets, the problem is NP-complete, proven via a reduction from planar 3-SAT using oriented-disc graph constructions.
• The reduction constructs an oriented-disc graph where all cycles are induced by digons, and hitting all cycles corresponds to finding a vertex cover in the underlying planar graph.
• The minimum feedback vertex set in the oriented-disc graph is equivalent to a vertex cover in the planar 3-SAT instance's associated graph.
• The construction ensures that the auxiliary arcs (not in digons) form an acyclic subgraph, so only the digons need to be hit to eliminate all cycles.
• The threshold between tractable and intractable cases is exactly two commodities: solvable in polynomial time for two, NP-complete for three.