Price of Fairness for Allocating a Bounded Resource (1508.05253v2)

Abstract: In this paper we study the problem of allocating a scarce resource among several players (or agents). A central decision maker wants to maximize the total utility of all agents. However, such a solution may be unfair for one or more agents in the sense that it can be achieved through a very unbalanced allocation of the resource. On the other hand fair/balanced allocations may be far from being optimal from a central point of view. So, in this paper we are interested in assessing the quality of fair solutions, i.e. in measuring the system efficiency loss under a fair allocation compared to the one that maximizes the sum of agents utilities. This indicator is usually called the Price of Fairness and we study it under three different definitions of fairness, namely maximin, Kalai-Smorodinski and proportional fairness. Our results are of two different types. We first formalize a number of properties holding for any general multi-agent problem without any special assumption on the agents utility sets. Then we introduce an allocation problem, where each agent can consume the resource in given discrete quantities (items), such that the maximization of the total utility is given by a Subset Sum Problem. For the resulting Fair Subset Sum Problem, in the case of two agents, we provide upper and lower bounds on the Price of Fairness as functions of an upper bound on the items size.

• The paper's main contribution is establishing theoretical bounds that quantify the efficiency loss when fair resource allocations are implemented.
• It introduces three fairness definitions—Maximin, Kalai-Smorodinski, and Proportional—to analyze trade-offs in multi-agent scenarios.
• The study applies the Fair Subset Sum Problem to derive concrete upper and lower bounds on the efficiency loss for two-agent resource allocation.

The paper "Price of Fairness for Allocating a Bounded Resource" addresses the problem of distributing a limited resource among multiple agents. The central issue this research tackles is the trade-off between fairness and overall utility maximization. Specifically, the central decision maker aims to maximize the total utility derived by all agents, but this can result in an inequitable distribution of resources. Conversely, fair or balanced allocations may lead to inefficient use of resources.

The paper introduces three definitions of fairness:

1. Maximin Fairness: Ensures that the least well-off agent is as well-off as possible. This approach prioritizes equalizing the minimum utility received by any agent.
2. Kalai-Smorodinski Fairness: Focuses on achieving outcomes where the agents' utilities are in proportion to their ideal but unattainable utilities.
3. Proportional Fairness: Each agent's utility is proportional to their entitlements.

The paper’s primary goal is to evaluate the "Price of Fairness" (PoF), which quantifies the efficiency loss incurred by adopting fair allocations compared to the utility-maximizing allocation. The primary contributions and findings of the paper are twofold:

1. General Properties for Multi-Agent Problems:
   • The authors establish several universal properties applicable to any multi-agent allocation problem, independent of specific utility structures.
   • These properties provide a theoretical foundation for understanding how fairness criteria impact overall system efficiency.
2. Fair Subset Sum Problem (FSSP):
   • To provide more concrete insights, the authors introduce a specific allocation problem where agents can consume resources in discrete quantities (items), which translates into a Subset Sum Problem.
   • The analysis is then focused on the scenario involving two agents, where they derive upper and lower bounds on the Price of Fairness. These bounds are defined as functions of an upper limit on item sizes.

This work highlights the inherent tension between fairness and efficiency in resource allocation. By formalizing the properties and bounds associated with the Price of Fairness, the authors provide a clearer understanding of how different fairness criteria can affect overall utility. Their theoretical insights and modelled scenarios offer a foundation for future research in multi-agent resource allocation schemes.