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Abstract
In the theory of economic design and voting, incompatibilities between axioms have given rise to a myriad of notions of the degree to which a given axiom is satisfied. However, these are, generally, model-and-axiom-specific. In contrast, this paper explores the potential of defining such a notion as the probability with which an axiom, as well as a set of axioms, is satisfied, without restricting the analysis to particular types of properties or problems.
The formal framework we provide is consistent with the use of simulation models, which aim at assessing the empirical performance of rules but focus, in general, on a single axiom, of a particular type.
We propose and axiomatize a criterion to evaluate and compare the performance of rules given a set of axioms, based on two key components: the probabilities of satisfaction and the normative desirability of axioms, and, crucially, that of their combinations. Finally, we propose and axiomatize a criterion to measure axioms’ compatibility between each other for a given rule, or a given family of rules, building on an analogy with cooperative game theory.