Satisfactory Budget Division

TL;DR

This paper investigates the problem of dividing a divisible budget among projects to satisfy agents' requests, analyzing the maximum satisfaction, existence of perfect divisions, computational complexity, and minimal total budget needed.

Contribution

It introduces a formal framework for budget satisfaction, studies various satisfaction thresholds, and addresses computational and existence questions for different scenarios.

Findings

Maximum proportion of satisfiable agents identified

Conditions for perfect satisfaction in certain instances established

Complexity results for satisfaction decision problems provided

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 $\tau$ 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 $\tau$, capturing scenarios where we seek to satisfy for each agent all; almost all; half; or at least one of her requests.