Density Decomposition in Dual-Modular Optimization: Markets, Fairness, and Contracts

2025-05-26

AI Summary

This paper investigates the density decomposition problem in bi-modular optimization: given a finite ground set (V) and a monotone supermodular reward function (f) alongside a strictly monotone submodular cost function (g), it unifies classical problems including submodular utility allocation (SUA) markets and combinatorial contract design. We establish the first density decomposition theory for the bi-modular setting, proving the equivalence of fairness notions—lexicographic optimality, local max-min fairness—and showing they are jointly characterized by a unique density vector. This vector encodes market equilibria, fair allocations, and optimal contract responses, exhibiting invariance under convex optimization reformulations and an intrinsic information-theoretic interpretation. Leveraging a family of divergence-parameterized convex programs—including quadratic, logarithmic, and hinge divergences—we design Frank–Wolfe-type iterative algorithms with gradient oracles and provide convergence guarantees. Finally, we fully characterize the structure and cardinality of critical thresholds in combinatorial contracts.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We study a unified framework for optimization problems defined on dual-modular instances, where the input comprises a finite ground set $V$ and two set functions: a monotone supermodular reward function $f$ and a strictly monotone submodular cost function $g$. This abstraction captures and generalizes classical models in economics and combinatorial optimization, including submodular utility allocation (SUA) markets and combinatorial contracts. At the core of our framework is the notion of density decomposition, which extends classical results to the dual-modular setting and uncovers structural insights into fairness and optimality. We show that the density decomposition yields a canonical vector of reward-to-cost ratios (densities) that simultaneously characterizes market equilibria, fair allocations -- via both lexicographic optimality and local maximin conditions -- and best-response strategies in contract design. Our main result proves the equivalence of these fairness notions and guarantees the existence of allocations that realize the decomposition densities. Our technical contributions include the analysis of a broad family of convex programs -- parameterized by divergences such as quadratic, logarithmic, and hockey-stick functions -- whose minimizers recover the density decomposition. We prove that any strictly convex divergence yields the same canonical density vector, and that locally maximin allocations act as universal minimizers for all divergences satisfying the data processing inequality. As an application of our framework, we determine the structure and number of critical values in the combinatorial contracts problem. Additionally, we generalize a Frank-Wolfe-type iterative method for approximating the dual-modular density decomposition, establishing both convergence guarantees and practical potential through efficient gradient oracle design.

Problem

• Study dual-modular optimization with supermodular reward and submodular cost functions
• Characterize market equilibria and fair allocations via density decomposition
• Analyze convex programs to recover density decomposition and ensure fairness

Innovation

• Density decomposition in dual-modular optimization
• Convex programs parameterized by divergences
• Frank-Wolfe-type iterative method