scholarly journals Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

2021 ◽  
Author(s):  
Matteo Basei ◽  
Haoyang Cao ◽  
Xin Guo
Keyword(s):  
Nash Equilibria ◽  
Stochastic Games ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Mean Field Games ◽  
Model Parameters ◽  
Management Problem ◽  
Player Game ◽  
The Impact ◽  
Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to infinity, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with [Formula: see text]. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

scholarly journals Markov--Nash Equilibria in Mean-Field Games with Discounted Cost

2018 ◽  
Vol 56 (6) ◽  
pp. 4256-4287 ◽  
Author(s):  
Naci Saldi ◽  
Tamer Başar ◽  
Maxim Raginsky
Keyword(s):  
Nash Equilibria ◽  
Mean Field ◽  
Mean Field Games ◽  
Discounted Cost

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

2013 ◽  
Vol 24 (1) ◽  
pp. 93-115 ◽  
Author(s):  
Pierre Degond ◽  
Jian-Guo Liu ◽  
Christian Ringhofer
Keyword(s):  
Large Scale ◽  
Nash Equilibria ◽  
Mean Field ◽  
Mean Field Games

scholarly journals On Sparse Discretization for Graphical Games

2020 ◽  
Vol 69 ◽  
pp. 67-84
Author(s):  
Luis Ortiz
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Sufficient Conditions ◽  
Research Note ◽  
Graphical Games ◽  
Approximate Nash Equilibrium ◽  
Approximate Nash Equilibria ◽  
Polymatrix Games ◽  
The Impact ◽  
Game Representation

Graphical games are one of the earliest examples of the impact that the general field of graphical models have had in other areas, and in this particular case, in classical mathematical models in game theory. Graphical multi-hypermatrix games, a concept formally introduced in this research note, generalize graphical games while allowing the possibility of further space savings in model representation to that of standard graphical games. The main focus of this research note is discretization schemes for computing approximate Nash equilibria, with emphasis on graphical games, but also briefly touching on normal-form and polymatrix games. The main technical contribution is a theorem that establishes sufficient conditions for a discretization of the players’ space of mixed strategies to contain an approximate Nash equilibrium. The result is actually stronger because every exact Nash equilibrium has a nearby approximate Nash equilibrium on the grid induced by the discretization. The sufficient conditions are weaker than those of previous results. In particular, a uniform discretization of size linear in the inverse of the approximation error and in the natural game-representation parameters suffices. The theorem holds for a generalization of graphical games, introduced here. The result has already been useful in the design and analysis of tractable algorithms for graphical games with parametric payoff functions and certain game-graph structures. For standard graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. Combining the improved discretization result with old results on constraint networks in AI simplifies the derivation and analysis of algorithms for computing approximate Nash equilibria in graphical games.

Numerical study of the stock market crises based on mean field games approach

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolai V. Trusov
Keyword(s):  
Stock Market ◽  
Numerical Study ◽  
Mean Field ◽  
Planck Equation ◽  
Coupled System ◽  
Share Price ◽  
Mean Field Games ◽  
Chinese Stock Market ◽  
Bear Market ◽  
The Impact

Abstract We present an approach to describe the stock market crises based on Mean Field Games (MFGs) and Optimal Control theory with a turnpike effect. The impact of the large amount of high-frequency traders (HFTs) can be modeled via a mean field term. We introduce the turnpike as a function that relies on the changes of the asset share price. An MFG is a coupled system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time, and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The ill-posedness of this system comes from a turnpike effect. The numerical solution of an extremal problem that is dual to a PDE system is presented. We apply this approach to describe the Chinese stock market crash in 2015 considering the representative stock of CITIC Securities (ticker 600030). We consider HFTs that form a dominating bull and bear market. As a result, the bull strategy imitators do not make any profit.

scholarly journals Are Mean-field Games the Limits of Finite Stochastic Games?

2016 ◽  
Vol 44 (2) ◽  
pp. 18-20 ◽  
Author(s):  
Josu Doncel ◽  
Nicolas Gast ◽  
Bruno Gaujal
Keyword(s):  
Stochastic Games ◽  
Mean Field ◽  
Mean Field Games
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