Paper: arXiv 2403.13138
Authors: Christopher Chambers, Alan Miller, Ruodu Wang, Qinyu Wu
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.
Complexity vs Empirical Score
- Math Complexity: 8.5/10
- Empirical Rigor: 1.5/10
- Quadrant: Lab Rats — theoretically deep, empirically untested
Why this score: The paper provides strong theoretical foundations with advanced mathematical proofs establishing representation theorems for max- and min-stability under stochastic dominance. However, it is entirely theoretical with no empirical data or backtesting, limiting its practical empirical rigor score.
Research Flowchart
flowchart TD
A[Research Goal: Characterize functionals satisfying max- and min-stability under FSD] --> B[Axiomatic Framework & Inputs]
B --> C[Inputs: FSD, Nondegeneracy, Lower Semicontinuity]
C --> D[Axiomatic Analysis & Mathematical Proofs]
D --> E[Max-Stability Characterization: Supremum of Bivariate Function]
D --> F[Min-Stability Characterization: Dual Infimum Representation]
E & F --> G[Combined Max & Min Stability: Characterization of Lambda-Quantiles]