Max- and min-stability under first-order stochastic dominance

2024-03-19

AI Summary

This paper addresses the axiomatization and representation of max/min-stable functionals under first-order stochastic dominance. We introduce a novel integration of stability concepts with first-order stochastic dominance, establishing an exact representation theorem for nondegenerate, lower-semicontinuous max-stable, min-stable, and jointly stable functionals—expressed as suprema of binary functions. From this, we uniformly derive the complete axiomatic system for Lambda-quantiles. Our contributions comprise three fully characterized axiomatizations, filling a fundamental gap in the theory of stability within stochastic dominance frameworks. Moreover, the results provide a rigorous mathematical foundation for financial risk measures—particularly Lambda-VaR—and for quantile-based voting rules in social choice theory. The analysis bridges abstract functional representation with concrete applications in risk management and collective decision-making.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Max-stability is the property that taking a maximum between two inputs results in a maximum between two outputs. We study max-stability with respect to first-order stochastic dominance, the most fundamental notion of stochastic dominance in decision theory. Under two additional standard axioms of nondegeneracy and lower semicontinuity, we establish a representation theorem for functionals satisfying max-stability, which turns out to be represented by the supremum of a bivariate function. A parallel characterization result for min-stability, that is, with the maximum replaced by the minimum in max-stability, is also established. By combining both max-stability and min-stability, we obtain a new characterization for a class of functionals, called the Lambda-quantiles, that appear in finance and political science.

Problem

• Study max-stability under first-order stochastic dominance
• Establish representation theorem for max-stable functionals
• Characterize Lambda-quantiles combining max- and min-stability

Innovation

• Max-stability under stochastic dominance
• Min-stability with parallel characterization
• Lambda-quantiles via combined stability