The Efficiency of Resource Allocation Mechanisms for Budget-Constrained Users

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 [8], 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.

Simple combinatorial auctions with budget constraints

“…By considering instances in which the number of items is large (tends to infinity), we recover the bound of 2 − 1/n proved by Caragiannis and Voudouris [2018] for single divisible resource allocation mechanisms. Essentially, we can interpret eorem 6 as a discrete version of the corresponding theorem of Caragiannis and Voudouris [2018], which shows how the bound depends not only on the number of players, but also on the number of items. Consequently, it leaves room for improvement for special cases where the number of items is bounded.…”

“…To start with, the papers by Caragiannis and Voudouris [2016] and Christodoulou et al [2016b] focused on the proportional allocation mechanism, used for the allocation of divisible resources, and were the first to formally bound the liquid price of anarchy. Following their work, Caragiannis and Voudouris [2018] characterized the structure of worst-case pure equilibria and proved tight bounds on the liquid price of anarchy for almost all divisible resource allocation mechanisms. In a similar spirit, Voudouris [2019] showed tight bounds on the pure liquid price of anarchy and stability for ad auctions, including the generalized second price auction and VCG.…”

A note on the efficiency of position mechanisms with budget constraints

“…Consequently, in contrast to the no-budget setting, when we consider players with budget constraints and the liquid welfare benchmark, these mechanisms do not have socially efficient equilibria. Such a phenomenon was first observed by Caragiannis and Voudouris [5] for all single divisible resource allocation mechanisms, and it might be the case that this holds for any position mechanism as well.…”

“…Our upper bounds follow this technique as well, but it seems non-trivial to extend them to more general equilibrium concepts due to the particular form of the deviating bids used. For pure equilibria in particular, by exploiting the structure of worst-case equilibria, Caragiannis and Voudouris [5] were able to characterize the liquid price of anarchy of all mechanisms for the allocation of a single divisible resource, leading to tight bounds. Finally, Azar et al [2] refined the definition of the liquid welfare for randomized allocations and proved constant liquid price of anarchy bounds over general equilibrium concepts for simultaneous first price auctions.…”

Task Allocation and Utility Distribution Algorithms Based on Nash Bargaining Solution

“…For cooperative agents, utility distribution is not a problem that must be considered. At the same time, for a self-interested agent, utility distribution must also meet budget validity [6][7] . (4) When communicating, to obtain better individual revenue, self-interested agents may withhold personal information, such as concealing their geographical location, exaggerating their cost, and even hiding the contents of the task.…”