250 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
On the Proximity of Markets with Integral Equilibria
We study Fisher markets that admit equilibria wherein each good is integrally
assigned to some agent. While strong existence and computational guarantees are
known for equilibria of Fisher markets with additive valuations, such
equilibria, in general, assign goods fractionally to agents. Hence, Fisher
markets are not directly applicable in the context of indivisible goods. In
this work we show that one can always bypass this hurdle and, up to a bounded
change in agents' budgets, obtain markets that admit an integral equilibrium.
We refer to such markets as pure markets and show that, for any given Fisher
market (with additive valuations), one can efficiently compute a "near-by,"
pure market with an accompanying integral equilibrium.
Our work on pure markets leads to novel algorithmic results for fair division
of indivisible goods. Prior work in discrete fair division has shown that,
under additive valuations, there always exist allocations that simultaneously
achieve the seemingly incompatible properties of fairness and efficiency; here
fairness refers to envy-freeness up to one good (EF1) and efficiency
corresponds to Pareto efficiency. However, polynomial-time algorithms are not
known for finding such allocations. Considering relaxations of proportionality
and EF1, respectively, as our notions of fairness, we show that fair and Pareto
efficient allocations can be computed in strongly polynomial time.Comment: 17 page
Algorithms for Competitive Division of Chores
We study the problem of allocating divisible bads (chores) among multiple
agents with additive utilities, when money transfers are not allowed. The
competitive rule is known to be the best mechanism for goods with additive
utilities and was recently extended to chores by Bogomolnaia et al (2017). For
both goods and chores, the rule produces Pareto optimal and envy-free
allocations. In the case of goods, the outcome of the competitive rule can be
easily computed. Competitive allocations solve the Eisenberg-Gale convex
program; hence the outcome is unique and can be approximately found by standard
gradient methods. An exact algorithm that runs in polynomial time in the number
of agents and goods was given by Orlin.
In the case of chores, the competitive rule does not solve any convex
optimization problem; instead, competitive allocations correspond to local
minima, local maxima, and saddle points of the Nash Social Welfare on the
Pareto frontier of the set of feasible utilities. The rule becomes multivalued
and none of the standard methods can be applied to compute its outcome.
In this paper, we show that all the outcomes of the competitive rule for
chores can be computed in strongly polynomial time if either the number of
agents or the number of chores is fixed. The approach is based on a combination
of three ideas: all consumption graphs of Pareto optimal allocations can be
listed in polynomial time; for a given consumption graph, a candidate for a
competitive allocation can be constructed via explicit formula; and a given
allocation can be checked for being competitive using a maximum flow
computation as in Devanur et al (2002).
Our algorithm immediately gives an approximately-fair allocation of
indivisible chores by the rounding technique of Barman and Krishnamurthy
(2018).Comment: 38 pages, 4 figure
The Equilibrium Existence Duality: Equilibrium with Indivisibilities & Income Effects
We show that, with indivisible goods, the existence of competitive
equilibrium fundamentally depends on agents' substitution effects, not their
income effects. Our Equilibrium Existence Duality allows us to transport
results on the existence of competitive equilibrium from settings with
transferable utility to settings with income effects. One consequence is that
net substitutability---which is a strictly weaker condition than gross
substitutability---is sufficient for the existence of competitive equilibrium.
We also extend the ``demand types'' classification of valuations to settings
with income effects and give necessary and sufficient conditions for a pattern
of substitution effects to guarantee the existence of competitive equilibrium.Comment: 46 pages, 1 figur
On the Efficiency of the Walrasian Mechanism
Central results in economics guarantee the existence of efficient equilibria
for various classes of markets. An underlying assumption in early work is that
agents are price-takers, i.e., agents honestly report their true demand in
response to prices. A line of research in economics, initiated by Hurwicz
(1972), is devoted to understanding how such markets perform when agents are
strategic about their demands. This is captured by the \emph{Walrasian
Mechanism} that proceeds by collecting reported demands, finding clearing
prices in the \emph{reported} market via an ascending price t\^{a}tonnement
procedure, and returns the resulting allocation. Similar mechanisms are used,
for example, in the daily opening of the New York Stock Exchange and the call
market for copper and gold in London.
In practice, it is commonly observed that agents in such markets reduce their
demand leading to behaviors resembling bargaining and to inefficient outcomes.
We ask how inefficient the equilibria can be. Our main result is that the
welfare of every pure Nash equilibrium of the Walrasian mechanism is at least
one quarter of the optimal welfare, when players have gross substitute
valuations and do not overbid. Previous analysis of the Walrasian mechanism
have resorted to large market assumptions to show convergence to efficiency in
the limit. Our result shows that approximate efficiency is guaranteed
regardless of the size of the market
Dividing bads under additive utilities
We compare the Egalitarian rule (aka Egalitarian Equivalent) and the Competitive rule (aka Comeptitive Equilibrium with Equal Incomes) to divide bads (chores). They are both welfarist: the competitive disutility profile(s) are the critical points of their Nash product on the set of efficient feasible profiles. The C rule is Envy Free, Maskin Monotonic, and has better incentives properties than the E rule. But, unlike the E rule, it can be wildly multivalued, admits no selection continuous in the utility and endowment parameters, and is harder to compute. Thus in the division of bads, unlike that of goods, no rule normatively dominates the other
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