3,207 research outputs found
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
Monotonicity and Competitive Equilibrium in Cake-cutting
We study the monotonicity properties of solutions in the classic problem of
fair cake-cutting --- dividing a heterogeneous resource among agents with
different preferences. Resource- and population-monotonicity relate to
scenarios where the cake, or the number of participants who divide the cake,
changes. It is required that the utility of all participants change in the same
direction: either all of them are better-off (if there is more to share or
fewer to share among) or all are worse-off (if there is less to share or more
to share among).
We formally introduce these concepts to the cake-cutting problem and examine
whether they are satisfied by various common division rules. We prove that the
Nash-optimal rule, which maximizes the product of utilities, is
resource-monotonic and population-monotonic, in addition to being
Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium
condition. Moreover, we prove that it is the only rule among a natural family
of welfare-maximizing rules that is both proportional and resource-monotonic.Comment: Revised versio
Fair Knapsack
We study the following multiagent variant of the knapsack problem. We are
given a set of items, a set of voters, and a value of the budget; each item is
endowed with a cost and each voter assigns to each item a certain value. The
goal is to select a subset of items with the total cost not exceeding the
budget, in a way that is consistent with the voters' preferences. Since the
preferences of the voters over the items can vary significantly, we need a way
of aggregating these preferences, in order to select the socially best valid
knapsack. We study three approaches to aggregating voters' preferences, which
are motivated by the literature on multiwinner elections and fair allocation.
This way we introduce the concepts of individually best, diverse, and fair
knapsack. We study the computational complexity (including parameterized
complexity, and complexity under restricted domains) of the aforementioned
multiagent variants of knapsack.Comment: Extended abstract will appear in Proc. of 33rd AAAI 201
Nash Social Welfare Approximation for Strategic Agents
The fair division of resources is an important age-old problem that has led
to a rich body of literature. At the center of this literature lies the
question of whether there exist fair mechanisms despite strategic behavior of
the agents. A fundamental objective function used for measuring fair outcomes
is the Nash social welfare, defined as the geometric mean of the agent
utilities. This objective function is maximized by widely known solution
concepts such as Nash bargaining and the competitive equilibrium with equal
incomes. In this work we focus on the question of (approximately) implementing
the Nash social welfare. The starting point of our analysis is the Fisher
market, a fundamental model of an economy, whose benchmark is precisely the
(weighted) Nash social welfare. We begin by studying two extreme classes of
valuations functions, namely perfect substitutes and perfect complements, and
find that for perfect substitutes, the Fisher market mechanism has a constant
approximation: at most 2 and at least e1e. However, for perfect complements,
the Fisher market does not work well, its bound degrading linearly with the
number of players.
Strikingly, the Trading Post mechanism---an indirect market mechanism also
known as the Shapley-Shubik game---has significantly better performance than
the Fisher market on its own benchmark. Not only does Trading Post achieve an
approximation of 2 for perfect substitutes, but this bound holds for all
concave utilities and becomes arbitrarily close to optimal for Leontief
utilities (perfect complements), where it reaches for every
. Moreover, all the Nash equilibria of the Trading Post mechanism
are pure for all concave utilities and satisfy an important notion of fairness
known as proportionality
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