230 research outputs found
A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market
We present the first combinatorial polynomial time algorithm for computing
the equilibrium of the Arrow-Debreu market model with linear utilities.Comment: Preliminary version in ICALP 201
A Combinatorial Polynomial Algorithm for the Linear {Arrow-Debreu} Market
We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities
Concave Generalized Flows with Applications to Market Equilibria
We consider a nonlinear extension of the generalized network flow model, with
the flow leaving an arc being an increasing concave function of the flow
entering it, as proposed by Truemper and Shigeno. We give a polynomial time
combinatorial algorithm for solving corresponding flow maximization problems,
finding an epsilon-approximate solution in O(m(m+log n)log(MUm/epsilon))
arithmetic operations and value oracle queries, where M and U are upper bounds
on simple parameters. This also gives a new algorithm for linear generalized
flows, an efficient, purely scaling variant of the Fat-Path algorithm by
Goldberg, Plotkin and Tardos, not using any cycle cancellations.
We show that this general convex programming model serves as a common
framework for several market equilibrium problems, including the linear Fisher
market model and its various extensions. Our result immediately extends these
market models to more general settings. We also obtain a combinatorial
algorithm for nonsymmetric Arrow-Debreu Nash bargaining, settling an open
question by Vazirani.Comment: Major revision. Instead of highest gain augmenting paths, we employ
the Fat-Path framework. Many parts simplified, running time for the linear
case improve
Non-Separable, Quasiconcave Utilities are Easy -- in a Perfect Price Discrimination Market Model
Recent results, establishing evidence of intractability for such restrictive
utility functions as additively separable, piecewise-linear and concave, under
both Fisher and Arrow-Debreu market models, have prompted the question of
whether we have failed to capture some essential elements of real markets,
which seem to do a good job of finding prices that maintain parity between
supply and demand.
The main point of this paper is to show that even non-separable, quasiconcave
utility functions can be handled efficiently in a suitably chosen, though
natural, realistic and useful, market model; our model allows for perfect price
discrimination. Our model supports unique equilibrium prices and, for the
restriction to concave utilities, satisfies both welfare theorems
An Improved Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market
We present an improved combinatorial algorithm for the computation of
equilibrium prices in the linear Arrow-Debreu model. For a market with
agents and integral utilities bounded by , the algorithm runs in time. This improves upon the previously best algorithm of Ye by a
factor of \tOmega(n). The algorithm refines the algorithm described by Duan
and Mehlhorn and improves it by a factor of \tOmega(n^3). The improvement
comes from a better understanding of the iterative price adjustment process,
the improved balanced flow computation for nondegenerate instances, and a novel
perturbation technique for achieving nondegeneracy.Comment: to appear in SODA 201
Ascending-Price Algorithms for Unknown Markets
We design a simple ascending-price algorithm to compute a
-approximate equilibrium in Arrow-Debreu exchange markets with
weak gross substitute (WGS) property, which runs in time polynomial in market
parameters and . This is the first polynomial-time
algorithm for most of the known tractable classes of Arrow-Debreu markets,
which is easy to implement and avoids heavy machinery such as the ellipsoid
method. In addition, our algorithm can be applied in unknown market setting
without exact knowledge about the number of agents, their individual utilities
and endowments. Instead, our algorithm only relies on queries to a global
demand oracle by posting prices and receiving aggregate demand for goods as
feedback. When demands are real-valued functions of prices, the oracles can
only return values of bounded precision based on real utility functions. Due to
this more realistic assumption, precision and representation of prices and
demands become a major technical challenge, and we develop new tools and
insights that may be of independent interest. Furthermore, our approach also
gives the first polynomial-time algorithm to compute an exact equilibrium for
markets with spending constraint utilities, a piecewise linear concave
generalization of linear utilities. This resolves an open problem posed by Duan
and Mehlhorn (2015).Comment: 33 page
On Computability of Equilibria in Markets with Production
Although production is an integral part of the Arrow-Debreu market model,
most of the work in theoretical computer science has so far concentrated on
markets without production, i.e., the exchange economy. This paper takes a
significant step towards understanding computational aspects of markets with
production.
We first define the notion of separable, piecewise-linear concave (SPLC)
production by analogy with SPLC utility functions. We then obtain a linear
complementarity problem (LCP) formulation that captures exactly the set of
equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production,
and we give a complementary pivot algorithm for finding an equilibrium. This
settles a question asked by Eaves in 1975 of extending his complementary pivot
algorithm to markets with production.
Since this is a path-following algorithm, we obtain a proof of membership of
this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of
existence of equilibrium (i.e., without using a fixed point theorem),
rationality, and oddness of the number of equilibria. We further give a proof
of PPAD-hardness for this problem and also for its restriction to markets with
linear utilities and SPLC production. Experiments show that our algorithm runs
fast on randomly chosen examples, and unlike previous approaches, it does not
suffer from issues of numerical instability. Additionally, it is strongly
polynomial when the number of goods or the number of agents and firms is
constant. This extends the result of Devanur and Kannan (2008) to markets with
production.
Finally, we show that an LCP-based approach cannot be extended to PLC
(non-separable) production, by constructing an example which has only
irrational equilibria.Comment: An extended abstract will appear in SODA 201
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