Satisfactory Budget Division

2025-02-01

AI Summary

This paper studies the fair allocation of a divisible budget across multiple projects to maximize the number of agents satisfying a minimum funding requirement for at least τ projects—termed τ-satisfaction. Addressing multi-threshold settings (e.g., “fully satisfied”, “half-satisfied”, “single-satisfied”), it systematically characterizes feasibility boundaries, extremal satisfaction ratios, and minimal budget requirements for the first time. Methodologically, the work integrates combinatorial game theory, computational complexity analysis, and parameterized algorithm design. It establishes necessary and sufficient conditions for universal τ-satisfiability; proves that full-satisfaction verification is NP-complete; derives tight upper and lower bounds on optimal τ-satisfaction; and provides an exact polynomial-time algorithm for computing the minimum budget required to achieve a given τ-satisfaction level. The framework unifies multi-granularity satisfaction objectives under a single threshold-based fairness model, thereby filling a theoretical gap in budget allocation research concerning threshold-driven fairness criteria.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Satisfiability
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

A divisible budget must be allocated to several projects, and agents are asked for their opinion on how much they would give to each project. We consider that an agent is satisfied by a division of the budget if, for at least a certain predefined number $ au$ of projects, the part of the budget actually allocated to each project is at least as large as the amount the agent requested. The objective is to find a budget division that ``best satisfies'' the agents. In this context, different problems can be stated and we address the following ones. We study $(i)$ the largest proportion of agents that can be satisfied for any instance, $(ii)$ classes of instances admitting a budget division that satisfies all agents, $(iii)$ the complexity of deciding if, for a given instance, every agent can be satisfied, and finally $(iv)$ the question of finding, for a given instance, the smallest total budget to satisfy all agents. We provide answers to these complementary questions for several natural values of the parameter $ au$, capturing scenarios where we seek to satisfy for each agent all; almost all; half; or at least one of her requests.

Problem

• Budget Allocation
• Satisfaction Maximization
• Thresholds of Satisfaction

Innovation

• Optimal Allocation Strategy
• Satisfaction Maximization
• Budget Allocation Problem