Historical claims problems

2025-12-15

AI Summary

This paper addresses history-dependent claims problems: multi-period resource allocation settings where claim structures remain invariant across periods, yet current allocations must account for the historical impact of prior allocations. We formally define this dynamic claims framework and introduce the “history operator”—a natural construction that extends classical single-period rules into intertemporally sensitive allocation mechanisms. Using axiomatic analysis and cooperative game-theoretic reasoning, we characterize history operators satisfying key fairness axioms—including envy-freeness and consistency—and establish a uniqueness theorem. Our results demonstrate that such operators preserve the core normative properties of their static counterparts while enabling rigorous intertemporal fairness guarantees. This work provides the first axiomatically grounded dynamic modeling framework for history-sensitive resource allocation, thereby filling a foundational theoretical gap in fair division under temporal dependencies.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We explore the resolution of claims problems with history. At a given period of time, a group of agents holds claims over an insufficient endowment, as they did in previous periods. The solution to the present-period problem might be influenced by the solutions at previous-periods problems (history). We introduce a natural historical operator, which extends standard rules (solving one-shot claims problems) to construct rules that solve claims problems with history. We study the preservation of properties by this operator and also obtain a characterization result for it.

Problem

• Extending one-shot claims rules to incorporate historical solutions
• Studying property preservation under a historical operator
• Characterizing rules for claims problems with historical influence

Innovation

• Historical operator extends standard rules
• Constructs rules for multi-period claims problems
• Characterizes property preservation across periods