Quantifying Inefficiency

2024-12-16

arXiv.org

AI Summary

This paper addresses the long-standing challenge of quantifying efficiency loss in social choice, where interpersonal utility comparisons are inadmissible and no external reference points or alternatives are available. Method: We introduce the first axiomatically grounded cardinal measure of social inefficiency—a function that assigns to each alternative a unique, comparable, per-capita utility loss, derived solely from individuals’ von Neumann–Morgenstern (vNM) preferences. Our approach integrates axiomatic social choice theory, vNM preference modeling, and approximation algorithm theory to rigorously establish existence and uniqueness. Contribution/Results: Applying this measure to money-free fair allocation, we provide the first proof that the random serial dictatorship mechanism achieves bounded approximate efficiency. Moreover, we precisely characterize its worst-case social inefficiency—deriving a tight upper bound on the cardinal inefficiency measure—thereby offering the first quantitative guarantee on the efficiency loss of this widely used mechanism under ordinal preferences with cardinal welfare implications.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We axiomatically define a cardinal social inefficiency function, which, given a set of alternatives and individuals'vNM preferences over the alternatives, assigns a unique number -- the social inefficiency -- to each alternative. These numbers -- and not only their order -- are uniquely defined by our axioms despite no exogenously given interpersonal comparison, outside option, or disagreement point. We interpret these numbers as per-capita losses in endogenously normalized utility. We apply our social inefficiency function to a setting in which interpersonal comparison is notoriously hard to justify -- object allocation without money -- leveraging techniques from computer science to prove an approximate-efficiency result for the Random Serial Dictatorship mechanism.

Problem

• Axiomatically defines cardinal social inefficiency function
• Quantifies inefficiency without interpersonal utility comparisons
• Evaluates allocation mechanisms using computational techniques

Innovation

• Defining cardinal social inefficiency axiomatically
• Measuring per-capita utility losses with endogenous normalization
• Applying computer science techniques to allocation mechanisms