1,961 research outputs found
Computing Cournot-Nash equilibria / 1441
Includes bibliographical references (p. 22-23)
Computing equilibria of Cournot oligopoly models with mixed-integer quantities
We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called “2-groups partitionable” Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method
Equilibrium Computation in Resource Allocation Games
We study the equilibrium computation problem for two classical resource
allocation games: atomic splittable congestion games and multimarket Cournot
oligopolies. For atomic splittable congestion games with singleton strategies
and player-specific affine cost functions, we devise the first polynomial time
algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and
computes the exact equilibrium assuming rational input. The idea is to compute
an equilibrium for an associated integrally-splittable singleton congestion
game in which the players can only split their demands in integral multiples of
a common packet size. While integral games have been considered in the
literature before, no polynomial time algorithm computing an equilibrium was
known. Also for this class, we devise the first polynomial time algorithm and
use it as a building block for our main algorithm.
We then develop a polynomial time computable transformation mapping a
multimarket Cournot competition game with firm-specific affine price functions
and quadratic costs to an associated atomic splittable congestion game as
described above. The transformation preserves equilibria in either games and,
thus, leads -- via our first algorithm -- to a polynomial time algorithm
computing Cournot equilibria. Finally, our analysis for integrally-splittable
games implies new bounds on the difference between real and integral Cournot
equilibria. The bounds can be seen as a generalization of the recent bounds for
single market oligopolies obtained by Todd [2016].Comment: This version contains some typo corrections onl
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
Remarks on existence and uniqueness of Cournot-Nash equilibria in the non-potential case
This article is devoted to various methods (optimal transport, fixed-point,
ordinary differential equations) to obtain existence and/or uniqueness of
Cournot-Nash equilibria for games with a continuum of players with both
attractive and repulsive effects. We mainly address separable situations but
for which the game does not have a potential. We also present several numerical
simulations which illustrate the applicability of our approach to compute
Cournot-Nash equilibria
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