Quantifying Inefficiency
Yannai A. Gonczarowski, Ella Segev
TL;DR
This paper axiomatizes a cardinal social inefficiency function $I(C,x)$ that depends solely on individuals' vNM preferences and is uniquely determined by seven axioms up to a global unit. It provides an explicit construction, showing $I(C,x)=c\cdot\hat{I}(C,x)$ with $\hat{I}(C,x)=\max_{x'\in X}V(C,x')-V(C,x)$ and $V(C,x)=\frac{1}{n}\sum_{i=1}^n\frac{u_i(x)-u_i^{\min}}{u_i^{\max}-u_i^{\min}}$. The framework is applied to object allocation without money, yielding a robust bound: no truthful ordinal mechanism can guarantee an inefficiency improvement over RSD by more than $28\%$, and RSD itself has an upper bound on its inefficiency of $\ln 2$. The paper also develops a computational approach to evaluate $I$ and discusses the broader potential of using cardinal social choice to derive approximation guarantees in econCS contexts. Overall, the work offers a microfoundational, cross-context measure of social inefficiency with practical implications for mechanism design and beyond.
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.