The efficiency of resource allocation mechanisms for budget-constrained users

TL;DR

This paper analyzes the efficiency of resource allocation mechanisms under realistic budget constraints, showing a tight bound on the liquid price of anarchy for the Kelly mechanism and designing nearly optimal two-player mechanisms.

Contribution

It introduces the liquid welfare benchmark for budget-constrained users, establishes a tight bound on the Kelly mechanism's efficiency, and designs nearly optimal two-player mechanisms.

Findings

Kelly mechanism has a tight bound of 2 on liquid price of anarchy.

Budget constraints significantly affect the efficiency of resource allocation.

Nearly optimal two-player mechanisms are derived through differential equations.

Abstract

We study the efficiency of mechanisms for allocating a divisible resource. Given scalar signals submitted by all users, such a mechanism decides the fraction of the resource that each user will receive and a payment that will be collected from her. Users are self-interested and aim to maximize their utility (defined as their value for the resource fraction they receive minus their payment). Starting with the seminal work of Johari and Tsitsiklis [Mathematics of Operations Research, 2004], a long list of papers studied the price of anarchy (in terms of the social welfare --- the total users' value) of resource allocation mechanisms for a variety of allocation and payment rules. Here, we further assume that each user has a budget constraint that invalidates strategies that yield a payment that is higher than the user's budget. This subtle assumption, which is arguably more realistic, constitutes the traditional price of anarchy analysis meaningless as the set of equilibria may change drastically and their social welfare can be arbitrarily far from optimal. Instead, we study the price of anarchy using the liquid welfare benchmark that measures efficiency taking budget constraints into account. We show a tight bound of 2 on the liquid price of anarchy of the well-known Kelly mechanism and prove that this result is essentially best possible among all multi-user resource allocation mechanisms. This comes in sharp contrast to the no-budget setting where there are mechanisms that considerably outperform Kelly in terms of social welfare and even achieve full efficiency. In our proofs, we exploit the particular structure of worst-case games and equilibria, which also allows us to design (nearly) optimal two-player mechanisms by solving simple differential equations.