19,140 research outputs found
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
The Max-Distance Network Creation Game on General Host Graphs
In this paper we study a generalization of the classic \emph{network creation
game} in the scenario in which the players sit on a given arbitrary
\emph{host graph}, which constrains the set of edges a player can activate at a
cost of each. This finds its motivations in the physical
limitations one can have in constructing links in practice, and it has been
studied in the past only when the routing cost component of a player is given
by the sum of distances to all the other nodes. Here, we focus on another
popular routing cost, namely that which takes into account for each player its
\emph{maximum} distance to any other player. For this version of the game, we
first analyze some of its computational and dynamic aspects, and then we
address the problem of understanding the structure of associated pure Nash
equilibria. In this respect, we show that the corresponding price of anarchy
(PoA) is fairly bad, even for several basic classes of host graphs. More
precisely, we first exhibit a lower bound of
for any . Notice that this implies a counter-intuitive lower
bound of for very small values of (i.e., edges can
be activated almost for free). Then, we show that when the host graph is
restricted to be either -regular (for any constant ), or a
2-dimensional grid, the PoA is still , which is proven to be tight for
. On the positive side, if , we show
the PoA is . Finally, in the case in which the host graph is very sparse
(i.e., , with ), we prove that the PoA is , for any
.Comment: 17 pages, 4 figure
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 a Bounded Budget Network Creation Game
We consider a network creation game in which each player (vertex) has a fixed
budget to establish links to other players. In our model, each link has unit
price and each agent tries to minimize its cost, which is either its local
diameter or its total distance to other players in the (undirected) underlying
graph of the created network. Two versions of the game are studied: in the MAX
version, the cost incurred to a vertex is the maximum distance between the
vertex and other vertices, and in the SUM version, the cost incurred to a
vertex is the sum of distances between the vertex and other vertices. We prove
that in both versions pure Nash equilibria exist, but the problem of finding
the best response of a vertex is NP-hard. We take the social cost of the
created network to be its diameter, and next we study the maximum possible
diameter of an equilibrium graph with n vertices in various cases. When the sum
of players' budgets is n-1, the equilibrium graphs are always trees, and we
prove that their maximum diameter is Theta(n) and Theta(log n) in MAX and SUM
versions, respectively. When each vertex has unit budget (i.e. can establish
link to just one vertex), the diameter of any equilibrium graph in either
version is Theta(1). We give examples of equilibrium graphs in the MAX version,
such that all vertices have positive budgets and yet the diameter is
Omega(sqrt(log n)). This interesting (and perhaps counter-intuitive) result
shows that increasing the budgets may increase the diameter of equilibrium
graphs and hence deteriorate the network structure. Then we prove that every
equilibrium graph in the SUM version has diameter 2^O(sqrt(log n)). Finally, we
show that if the budget of each player is at least k, then every equilibrium
graph in the SUM version is k-connected or has diameter smaller than 4.Comment: 28 pages, 3 figures, preliminary version appeared in SPAA'1
Mass conserved elementary kinetics is sufficient for the existence of a non-equilibrium steady state concentration
Living systems are forced away from thermodynamic equilibrium by exchange of
mass and energy with their environment. In order to model a biochemical
reaction network in a non-equilibrium state one requires a mathematical
formulation to mimic this forcing. We provide a general formulation to force an
arbitrary large kinetic model in a manner that is still consistent with the
existence of a non-equilibrium steady state. We can guarantee the existence of
a non-equilibrium steady state assuming only two conditions; that every
reaction is mass balanced and that continuous kinetic reaction rate laws never
lead to a negative molecule concentration. These conditions can be verified in
polynomial time and are flexible enough to permit one to force a system away
from equilibrium. In an expository biochemical example we show how a
reversible, mass balanced perpetual reaction, with thermodynamically infeasible
kinetic parameters, can be used to perpetually force a kinetic model of
anaerobic glycolysis in a manner consistent with the existence of a steady
state. Easily testable existence conditions are foundational for efforts to
reliably compute non-equilibrium steady states in genome-scale biochemical
kinetic models.Comment: 11 pages, 2 figures (v2 is now placed in proper context of the
excellent 1962 paper by James Wei entitled "Axiomatic treatment of chemical
reaction systems". In addition, section 4, on "Utility of steady state
existence theorem" has been expanded.
Kinetic Inductance of Josephson Junction Arrays: Dynamic and Equilibrium Calculations
We show analytically that the inverse kinetic inductance of an
overdamped junction array at low frequencies is proportional to the admittance
of an inhomogeneous equivalent impedance network. The bond in this
equivalent network has an inverse inductance
, where is the Josephson
coupling energy of the bond, is the ground-state phase
of the grain , and is the usual magnetic phase factor. We use this
theorem to calculate for square arrays as large as .
The calculated is in very good agreement with the low-temperature
limit of the helicity modulus calculated by conventional equilibrium
Monte Carlo techniques. However, the finite temperature structure of ,
as a function of magnetic field, is \underline{sharper} than the
zero-temperature , which shows surprisingly weak structure. In
triangular arrays, the equilibrium calculation of yields a series of
peaks at frustrations , where is an integer , consistent with experiment.Comment: 14 pages + 6 postscript figures, 3.0 REVTe
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
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