Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

Degond, Pierre; Liu, Jian Guo; Ringhofer, Christian

doi:10.1007/s00332-013-9185-2

Cited by 39 publications

(48 citation statements)

References 27 publications

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“…The first equation describes the evolution of the economic configuration, which is driven by the local Nash equilibrium and it is related to mean-field games [70,12]. The second equation describes the evolution of the wealth, which contains two mechanisms: the trading model proposed by Bouchaud and Mezart [71], and the geometric Brownian motion in finance proposed by Bachelier in 1900 [72].…”

Section: Stochastic Dynamics Of Wealthmentioning

confidence: 99%

Random Batch Methods (RBM) for interacting particle systems

Jin

Liu

2020

Journal of Computational Physics

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115

189

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We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N 2 ) per time step to O(N ), for a system with N particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing N -independent time steps and also capture, in the N → ∞ limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.

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Section: Stochastic Dynamics Of Wealthmentioning

confidence: 99%

Random Batch Methods (RBM) for interacting particle systems

Jin

Liu

2020

Journal of Computational Physics

Self Cite

115

189

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“…However, relation (14) is not enough to uniquely define the weak solution for (12) with Riemann initial data, since s * is unknown. Sheng and Zhang in [37] derive this speed by constructing a delta distribution solution as a vanishing viscosity solution of (12) with a Dafermos regularization.…”

Section: And Choose An Open Regionmentioning

confidence: 99%

“…The weights could correspond to a straight average (w ij constant) or depend on some environmental variable such as distance between agents. We note that the basic contagion model is a variant of a classical consensus model in control theory for which there is an extensive literature [22,19,14,34].…”

Section: Discrete Contagion Modelsmentioning

confidence: 99%

Contagion Shocks in One Dimension

Bertozzi¹,

Rosado²,

Short

et al. 2014

J Stat Phys

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We consider an agent-based model of emotional contagion coupled with motion in one dimension that has recently been studied in the computer science community. The model involves movement with a speed proportional to a "fear" variable that undergoes a temporal consensus averaging based on distance to other agents. We study the effect of Riemann initial data for this problem, leading to shock dynamics that are studied both within the agent-based model as well as in a continuum limit. We examine the behavior of the model under distinguished limits as the characteristic contagion interaction distance and the interaction timescale both approach zero. The limiting behavior is related to a classical model for pressureless gas dynamics with "sticky" particles. In comparison, we observe a threshold for the interaction distance vs. interaction timescale that produce qualitatively different behavior for the system -in one case particle paths do not cross and there is a natural Eulerian limit involving nonlocal interactions and in the other case particle paths can cross and one may consider only a kinetic model in the continuum limit.

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“…See figure 3 for a two-dimensional example. A dynamical extension of this best-reply scheme was recently developed in [18].…”

Section: (A) Dimension Onementioning

confidence: 99%

From Nash to Cournot–Nash equilibria via the Monge–Kantorovich problem

Blanchet

Carlier

2014

Phil. Trans. R. Soc. A.

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The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of N player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article, we emphasize the role of optimal transport theory in (i) the passage from Nash to Cournot–Nash equilibria as the number of players tends to infinity and (ii) the analysis of Cournot–Nash equilibria.

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Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

Cited by 39 publications

References 27 publications

Random Batch Methods (RBM) for interacting particle systems

Random Batch Methods (RBM) for interacting particle systems

Contagion Shocks in One Dimension

From Nash to Cournot–Nash equilibria via the Monge–Kantorovich problem

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