48 research outputs found
Nash Social Welfare in Selfish and Online Load Balancing (Short Paper)
In load balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones, in order to execute a certain task. Each resource has a latency function, which depends on its workload, and a client's cost is the completion time of her chosen resource. Two fundamental variants of load balancing problems are selfish load balancing (aka. load balancing games), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and online load balancing, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both problems under the objective of minimizing the Nash Social Welfare, i.e., the geometric mean of the clients' costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal, as it matches the performance of any possible online algorithm
Fair and Efficient Allocations under Subadditive Valuations
We study the problem of allocating a set of indivisible goods among agents
with subadditive valuations in a fair and efficient manner. Envy-Freeness up to
any good (EFX) is the most compelling notion of fairness in the context of
indivisible goods. Although the existence of EFX is not known beyond the simple
case of two agents with subadditive valuations, some good approximations of EFX
are known to exist, namely -EFX allocation and EFX allocations
with bounded charity.
Nash welfare (the geometric mean of agents' valuations) is one of the most
commonly used measures of efficiency. In case of additive valuations, an
allocation that maximizes Nash welfare also satisfies fairness properties like
Envy-Free up to one good (EF1). Although there is substantial work on
approximating Nash welfare when agents have additive valuations, very little is
known when agents have subadditive valuations. In this paper, we design a
polynomial-time algorithm that outputs an allocation that satisfies either of
the two approximations of EFX as well as achieves an
approximation to the Nash welfare. Our result also improves the current
best-known approximation of and to
Nash welfare when agents have submodular and subadditive valuations,
respectively.
Furthermore, our technique also gives an approximation to a
family of welfare measures, -mean of valuations for ,
thereby also matching asymptotically the current best known approximation ratio
for special cases like while also retaining the fairness
properties
Developments in Multi-Agent Fair Allocation
Fairness is becoming an increasingly important concern when designing
markets, allocation procedures, and computer systems. I survey some recent
developments in the field of multi-agent fair allocation
Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods
The fair allocation of mixed goods, consisting of both divisible and
indivisible goods, among agents with heterogeneous preferences, has been a
prominent topic of study in economics and computer science. In this paper, we
investigate the nature of fair allocations when agents have binary valuations.
We define an allocation as fair if its utility vector minimizes a symmetric
strictly convex function, which includes conventional fairness criteria such as
maximum egalitarian social welfare and maximum Nash social welfare. While a
good structure is known for the continuous case (where only divisible goods
exist) or the discrete case (where only indivisible goods exist), deriving such
a structure in the hybrid case remains challenging. Our contributions are
twofold. First, we demonstrate that the hybrid case does not inherit some of
the nice properties of continuous or discrete cases, while it does inherit the
proximity theorem. Second, we analyze the computational complexity of finding a
fair allocation of mixed goods based on the proximity theorem. In particular,
we provide a polynomial-time algorithm for the case when all divisible goods
are identical and homogeneous, and demonstrate that the problem is NP-hard in
general. Our results also contribute to a deeper understanding of the hybrid
convex analysis
Envy-freeness and maximum Nash welfare for mixed divisible and indivisible goods
We study fair allocation of resources consisting of both divisible and
indivisible goods to agents with additive valuations. When only divisible or
indivisible goods exist, it is known that an allocation that achieves the
maximum Nash welfare (MNW) satisfies the classic fairness notions based on
envy. In addition, properties of the MNW allocations for binary valuations are
known. In this paper, we show that when all agents' valuations are binary and
linear for each good, an MNW allocation for mixed goods satisfies the
envy-freeness up to any good for mixed goods. This notion is stronger than an
existing one called envy-freeness for mixed goods (EFM), and our result
generalizes the existing results for the case when only divisible or
indivisible goods exist. Moreover, our result holds for a general fairness
notion based on minimizing a symmetric strictly convex function. For the
general additive valuations, we also provide a formal proof that an MNW
allocation satisfies a weaker notion than EFM
Computing large market equilibria using abstractions
Computing market equilibria is an important practical problem for market
design (e.g. fair division, item allocation). However, computing equilibria
requires large amounts of information (e.g. all valuations for all buyers for
all items) and compute power. We consider ameliorating these issues by applying
a method used for solving complex games: constructing a coarsened abstraction
of a given market, solving for the equilibrium in the abstraction, and lifting
the prices and allocations back to the original market. We show how to bound
important quantities such as regret, envy, Nash social welfare, Pareto
optimality, and maximin share when the abstracted prices and allocations are
used in place of the real equilibrium. We then study two abstraction methods of
interest for practitioners: 1) filling in unknown valuations using techniques
from matrix completion, 2) reducing the problem size by aggregating groups of
buyers/items into smaller numbers of representative buyers/items and solving
for equilibrium in this coarsened market. We find that in real data
allocations/prices that are relatively close to equilibria can be computed from
even very coarse abstractions
Existence of EFX for Two Additive Valuations
Fair division of indivisible items is a well-studied topic in Economics and
Computer Science.The objective is to allocate items to agents in a fair manner,
where each agent has a valuation for each subset of items. Envy-freeness is one
of the most widely studied notions of fairness. Since complete envy-free
allocations do not always exist when items are indivisible, several relaxations
have been considered. Among them, possibly the most compelling one is
envy-freeness up to any item (EFX), where no agent envies another agent after
the removal of any single item from the other agent's bundle. However, despite
significant efforts by many researchers for several years, it is known that a
complete EFX allocation always exists only in limited cases. In this paper, we
show that a complete EFX allocation always exists when each agent is of one of
two given types, where agents of the same type have identical additive
valuations. This is the first such existence result for non-identical
valuations when there are any number of agents and items and no limit on the
number of distinct values an agent can have for individual items. We give a
constructive proof, in which we iteratively obtain a Pareto dominating
(partial) EFX allocation from an existing partial EFX allocation.Comment: 14 pages, 2 figure
Computing Equilibria in Markets with Budget-Additive Utilities
We present the first analysis of Fisher markets with buyers that have
budget-additive utility functions. Budget-additive utilities are elementary
concave functions with numerous applications in online adword markets and
revenue optimization problems. They extend the standard case of linear
utilities and have been studied in a variety of other market models. In
contrast to the frequently studied CES utilities, they have a global satiation
point which can imply multiple market equilibria with quite different
characteristics. Our main result is an efficient combinatorial algorithm to
compute a market equilibrium with a Pareto-optimal allocation of goods. It
relies on a new descending-price approach and, as a special case, also implies
a novel combinatorial algorithm for computing a market equilibrium in linear
Fisher markets. We complement these positive results with a number of hardness
results for related computational questions. We prove that it is NP-hard to
compute a market equilibrium that maximizes social welfare, and it is PPAD-hard
to find any market equilibrium with utility functions with separate satiation
points for each buyer and each good.Comment: 21 page