[Paper Review] Rationalizing Path-Independent Choice Rules

Koji Yokote, Isa E. Hafalir+2 more|arXiv (Cornell University)|Mar 2, 2023

Economic theories and modelsEconomics, Econometrics and Finance3 citations

TL;DR

This paper establishes that a choice rule is path independent if and only if it is rationalizable by a utility function satisfying ordinal concavity—a concept closely tied to discrete convex analysis. It further shows that path-independent choice rules satisfying the law of aggregate demand are rationalizable by utility functions satisfying both ordinal concavity and size-restricted concavity, providing the first representation theorems for path-independent choice rules in economic theory.

ABSTRACT

Path independence is arguably one of the most important choice rule properties in economic theory. We show that a choice rule is path independent if and only if it is rationalizable by a utility function satisfying ordinal concavity, a concept closely related to concavity notions in discrete mathematics. We also provide a rationalization result for choice rules that satisfy path independence and the law of aggregate demand.

Motivation & Objective

• To address the long-standing gap in representation theorems for path-independent choice rules, despite their central role in economic theory.
• To establish a formal link between path independence and a novel utility-based rationalizability condition—ordinal concavity.
• To extend the characterization to choice rules satisfying both path independence and the law of aggregate demand, introducing size-restricted concavity as a key condition.
• To clarify the relationship between path independence, the substitutes condition, and established concepts like M♮-concavity and submodularity.
• To provide a foundation for modeling rational behavior in market design and decision theory where utility functions are unobservable.

Proposed method

• The paper introduces the concept of ordinal concavity, defined as a condition on utility functions where utility does not decrease when sets are made closer via symmetric differences.
• It proves that path independence is equivalent to rationalizability by a utility function satisfying ordinal concavity, using a constructive proof based on the choice rule's behavior on unions of sets.
• For the law of aggregate demand extension, the paper defines a utility function ˜u that rationalizes the choice rule and shows it satisfies size-restricted concavity under the given conditions.
• The proof leverages properties of the choice rule under path independence, including the substitutes condition and the uniqueness of maximizers.
• It uses contradiction arguments to show that violations of size-restricted concavity lead to violations of the maximization property of the choice rule.
• The analysis draws on tools from discrete convex analysis, particularly M♮-concavity and its relationship to gross substitutes, to contextualize the new conditions.

Experimental results

Research questions

• RQ1 Is there a utility-based rationalization for path-independent choice rules, and what properties must the utility function satisfy?
• RQ2 How does ordinal concavity relate to established notions like M♮-concavity and submodularity in discrete convex analysis?
• RQ3 Can path-independent choice rules that also satisfy the law of aggregate demand be rationalized by a utility function with additional structural properties?
• RQ4 What is the precise relationship between path independence, the substitutes condition, and the irrelevance of rejected contracts?
• RQ5 How do the new conditions—ordinal concavity and size-restricted concavity—characterize the structure of rationalizable choice rules?

Key findings

• A choice rule is path independent if and only if it is rationalizable by a utility function satisfying ordinal concavity.
• Ordinal concavity is a weaker condition than M♮-concavity, and the difference between path independence and the gross substitutes property can be attributed to this distinction.
• Path-independent choice rules satisfying the law of aggregate demand are rationalizable by a utility function that satisfies both ordinal concavity and size-restricted concavity.
• The utility function ˜u defined in the proof uniquely maximizes utility over all subsets of any given set, ensuring rationalizability.
• The proof shows that violations of size-restricted concavity lead to contradictions with the maximization property of the choice rule, thereby establishing the necessity of the condition.
• The results clarify that submodularity and ordinal concavity are logically unrelated, as the former does not imply the latter and vice versa.