The probability of satisfying axioms: a non-binary perspective on economic design

TL;DR

This paper introduces a probabilistic framework to evaluate the likelihood of axioms being satisfied in economic design rules, enabling nuanced comparison and compatibility analysis of axioms based on input distributions.

Contribution

It develops a formal probabilistic approach to measure axiom satisfaction and compatibility, extending traditional binary assessments in economic design.

Findings

Defines a probability-based measure of rule satisfaction for axioms

Provides criteria for comparing rules considering axiom importance

Introduces a compatibility measure for axioms inspired by cooperative game theory

Abstract

We provide a formal framework accounting for a widespread idea in the theory of economic design: analytically established incompatibilities between given axioms should be qualified by the likelihood of their violation. We define the degree to which rules satisfy an axiom, as well as several axioms, on the basis of a probability measure over the inputs of the rules. Armed with this notion of degree, we propose and characterize i) a criterion to evaluate and compare rules given a set of axioms, allowing the importance of each combination of axioms to differ, and ii) a criterion to measure the compatibility between given axioms, building on a analogy with cooperative game theory.