291 research outputs found
Combinatorial Algorithms for General Linear Arrow-Debreu Markets
We present a combinatorial algorithm for determining the market clearing prices of a general linear Arrow-Debreu market, where every agent can own multiple goods. The existing combinatorial algorithms for linear Arrow-Debreu markets consider the case where each agent can own all of one good only. We present an O~((n+m)^7 log^3(UW)) algorithm where n, m, U and W refer to the number of agents, the number of goods, the maximal integral utility and the maximum quantity of any good in the market respectively. The algorithm refines the iterative algorithm of Duan, Garg and Mehlhorn using several new ideas. We also identify the hard instances for existing combinatorial algorithms for linear Arrow-Debreu markets. In particular we find instances where the ratio of the maximum to the minimum equilibrium price of a good is U^{Omega(n)} and the number of iterations required by the existing iterative combinatorial algorithms of Duan, and Mehlhorn and Duan, Garg, and Mehlhorn are high. Our instances also separate the two algorithms
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
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
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
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
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
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
Network Cournot Competition
Cournot competition is a fundamental economic model that represents firms
competing in a single market of a homogeneous good. Each firm tries to maximize
its utility---a function of the production cost as well as market price of the
product---by deciding on the amount of production. In today's dynamic and
diverse economy, many firms often compete in more than one market
simultaneously, i.e., each market might be shared among a subset of these
firms. In this situation, a bipartite graph models the access restriction where
firms are on one side, markets are on the other side, and edges demonstrate
whether a firm has access to a market or not. We call this game \emph{Network
Cournot Competition} (NCC). In this paper, we propose algorithms for finding
pure Nash equilibria of NCC games in different situations. First, we carefully
design a potential function for NCC, when the price functions for markets are
linear functions of the production in that market. However, for nonlinear price
functions, this approach is not feasible. We model the problem as a nonlinear
complementarity problem in this case, and design a polynomial-time algorithm
that finds an equilibrium of the game for strongly convex cost functions and
strongly monotone revenue functions. We also explore the class of price
functions that ensures strong monotonicity of the revenue function, and show it
consists of a broad class of functions. Moreover, we discuss the uniqueness of
equilibria in both of these cases which means our algorithms find the unique
equilibria of the games. Last but not least, when the cost of production in one
market is independent from the cost of production in other markets for all
firms, the problem can be separated into several independent classical
\emph{Cournot Oligopoly} problems. We give the first combinatorial algorithm
for this widely studied problem
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