Existence and Uniqueness of Solutions for Bertrand and Cournot Mean Field Games

Graber, P. Jameson; Bensoussan, Alain

doi:10.1007/s00245-016-9366-0

Cited by 34 publications

(19 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(See also the work of Bauso, Tembine, and Basar [6].) A well-posedness result for a related system of partial differential equations appears in a paper by Bensoussan and the present author [31]. The basic structure of the model is as follows.…”

Section: Introductionmentioning

confidence: 99%

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

Graber

2016

Appl Math Optim

Self Cite

View full text Add to dashboard Cite

We study a general linear quadratic mean field type control problem and connect it to mean field games of a similar type. The solution is given both in terms of a forward/backward system of stochastic differential equations and by a pair of Riccati equations. In certain cases, the solution to the mean field type control is also the equilibrium strategy for a class of mean field games. We use this fact to study an economic model of production of exhaustible resources.

show abstract

Section: Introductionmentioning

confidence: 99%

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

Graber

2016

Appl Math Optim

Self Cite

View full text Add to dashboard Cite

show abstract

“…Ever since, MFGs have become an important driver of mathematical innovation, especially in the areas of PDEs and backward stochastic equations. Complementing the theoretical work on MFGs, there is a by now substantial literature where anonymous and mean field games have been successfully applied to an array economic and engineering problems, ranging from network security and traffic networks [14,26] and systemic risk management [5], to portfolio liquidation [10,11,13,21] and oil and energy production in competitive markets [6,7,15,16,17].…”

Section: Introductionmentioning

confidence: 99%

A Maximum Principle approach to deterministic Mean Field Games of Control with Absorption

Graewe¹,

Horst²,

Sircar³

2021

Preprint

View full text Add to dashboard Cite

We study a class of deterministic mean field games on finite and infinite time horizons arising in models of optimal exploitation of exhaustible resources. The main characteristic of our game is an absorption constraint on the players' state process. As a result of the state constraint the optimal time of absorption becomes part of the equilibrium. This requires a novel approach when applying Pontyagin's maximum principle. We prove the existence and uniqueness of equilibria and solve the infinite horizon models in closed form. As players may drop out of the game over time, equilibrium production rates need not be monotone nor smooth.

show abstract

“…Quantity competition and price competition of oligopoly was debated for many years in an imperfectly competitive market. Some theorists [1][2][3][4][5][6][7] pointed out that the competitive equilibrium points, respectively, exist in the Cournot (quantity) model or Bertrand (price) model and compared the equilibrium with each other. Such as Singh and Vives first compared Cournot competition and Bertrand competition.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium Further Studied for Combined System of Cournot and Bertrand: A Differential Approach

Gao

2020

Complexity

View full text Add to dashboard Cite

In general, quantity competition and price competition exist simultaneously in a dynamic economy system. Whether it is quantity competition or price competition, when there are more than three companies in one market, the equilibrium points will become chaotic and are very difficult to be derived. This paper considers generally dynamic equilibrium points of combination of the Bertrand model and Cournot model. We analyze general equilibrium points of the Bertrand model and Cournot model, respectively. A general equilibrium point of the combination of the Cournot model and Bertrand model is further investigated in two cases. The theory of spatial agglomeration and intermediate value theorem are introduced. In addition, the stability of equilibrium points is further illustrated on celestial bodies motion. The results show that at least a general equilibrium point exists in combination of Cournot and Bertrand. Numerical simulations are given to support the research results.

show abstract

Existence and Uniqueness of Solutions for Bertrand and Cournot Mean Field Games

Cited by 34 publications

References 18 publications

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

A Maximum Principle approach to deterministic Mean Field Games of Control with Absorption

Equilibrium Further Studied for Combined System of Cournot and Bertrand: A Differential Approach

Contact Info

Product

Resources

About