An official website of the United States government
We study the allocation of divisible goods to competing agents via a market mechanism, focusing on agents with Leontief utilities. The majority of the economics and mechanism design literature has focused on \emph{linear} prices, meaning that the cost of a good is proportional to the quantity purchased. Equilibria for linear prices are known to be exactly the maximum Nash welfare allocations. \emph{Price curves} allow the cost of a good to be any (increasing) function of the quantity purchased. We show that price curve equilibria are not limited to maximum Nash welfare allocations with two main results. First, we show that an allocation can be supported by strictly increasing price curves if and only if it is \emph{group-domination-free}. A similarly characterization holds for weakly increasing price curves. We use this to show that given any allocation, we can compute strictly (or weakly) increasing price curves that support it (or show that none exist) in polynomial time. These results involve a connection to the \emph{agent-order matrix} of an allocation, which may have other applications. Second, we use duality to show that in the bandwidth allocation setting, any allocation maximizing a CES welfare function can be supported by price curves.
more »
« less
Robust Allocations with Diversity Constraints
We consider the problem of allocating divisible items among multiple agents, and consider the setting where any agent is allowed to introduce {\emph diversity constraints} on the items they are allocated. We motivate this via settings where the items themselves correspond to user ad slots or task workers with attributes such as race and gender on which the principal seeks to achieve demographic parity. We consider the following question: When an agent expresses diversity constraints into an allocation rule, is the allocation of other agents hurt significantly? If this happens, the cost of introducing such constraints is disproportionately borne by agents who do not benefit from diversity. We codify this via two desiderata capturing {\em robustness}. These are {\emph no negative externality} -- other agents are not hurt -- and {\emph monotonicity} -- the agent enforcing the constraint does not see a large increase in value. We show in a formal sense that the Nash Welfare rule that maximizes product of agent values is {\emph uniquely} positioned to be robust when diversity constraints are introduced, while almost all other natural allocation rules fail this criterion. We also show that the guarantees achieved by Nash Welfare are nearly optimal within a widely studied class of allocation rules. We finally perform an empirical simulation on real-world data that models ad allocations to show that this gap between Nash Welfare and other rules persists in the wild.
more »
« less
- PAR ID:
- 10317081
- Date Published:
- Journal Name:
- Advances in Neural Information Processing Systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon.more » « less
-
We study the problem of allocating indivisible items to budget-constrained agents, aiming to provide fairness and efficiency guarantees. Specifically, our goal is to ensure that the resulting allocation is envy-free up to any item (EFx) while minimizing the amount of inefficiency that this needs to introduce. We first show that there exist two-agent problem instances for which no EFx allocation is Pareto-efficient. We, therefore, turn to approximation and use the (Pareto-efficient) maximum Nash welfare allocation as a benchmark. For two-agent instances, we provide a procedure that always returns an EFx allocation while achieving the best possible approximation of the optimal Nash social welfare that EFx allocations can achieve. For the more complicated case of three-agent instances, we provide a procedure that guarantees EFx, while achieving a constant approximation of the optimal Nash social welfare for any number of items.more » « less
-
We investigate the fair allocation of indivisible goods to agents with possibly different entitlements represented by weights. Previous work has shown that guarantees for additive valuations with existing envy-based notions cannot be extended to the case where agents have matroid-rank (i.e., binary submodular) valuations. We propose two families of envy-based notions for matroid-rank and general submodular valuations, one based on the idea of transferability and the other on marginal values. We show that our notions can be satisfied via generalizations of rules such as picking sequences and maximum weighted Nash welfare. In addition, we introduce welfare measures based on harmonic numbers, and show that variants of maximum weighted harmonic welfare offer stronger fairness guarantees than maximum weighted Nash welfare under matroid-rank valuations.more » « less
-
Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)Online allocation is a broad class of problems where items arriving online have to be allocated to agents who have a fixed utility/cost for each assigned item so to maximize/minimize some objective. This framework captures a broad range of fundamental problems such as the Santa Claus problem (maximizing minimum utility), Nash welfare maximization (maximizing geometric mean of utilities), makespan minimization (minimizing maximum cost), minimization of π_p-norms, and so on. We focus on divisible items (i.e., fractional allocations) in this paper. Even for divisible items, these problems are characterized by strong super-constant lower bounds in the classical worst-case online model. In this paper, we study online allocations in the learning-augmented setting, i.e., where the algorithm has access to some additional (machine-learned) information about the problem instance. We introduce a general algorithmic framework for learning-augmented online allocation that produces nearly optimal solutions for this broad range of maximization and minimization objectives using only a single learned parameter for every agent. As corollaries of our general framework, we improve prior results of Lattanzi et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan minimization, and obtain the first learning-augmented nearly-optimal algorithms for the other objectives such as Santa Claus, Nash welfare, π_p-minimization, etc. We also give tight bounds on the resilience of our algorithms to errors in the learned parameters, and study the learnability of these parameters.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
