Paradigm Shift: A Mean Field Game Approach

Besancenot, Damien; Dogguy, Habib

doi:10.1111/boer.12024

Cited by 10 publications

(22 citation statements)

References 25 publications

(29 reference statements)

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“…Our claim is that Eqs. (7)(8)(9)(10)(11)(12)(13), together with many results known in the context of the non-linear Schrödinger equation, can form the basis of the analysis of a very large class of mean field games for various associated potentials, including some long range interactions. In the following, we will illustrate our point of view, restricting ourselves to the one dimensional case and to potentials of the form Eq.…”

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confidence: 99%

Schrödinger Approach to Mean Field Games

2016

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Mean Field Games (MFG) provide a theoretical frame to model socio-economic systems. In this letter, we study a particular class of MFG which shows strong analogies with the non-linear Schrödinger and Gross-Pitaevskii equations introduced in physics to describe a variety of physical phenomena. Using this bridge many results and techniques developed along the years in the latter context can be transferred to the former, which provides both a new domain of application for the non-linear Schrödinger equation and a new and fruitful approach in the study of mean field games.As an illustration, we analyze in some details an example in which the "players" in the mean field game are under a strong incentive to coordinate themselves. [8][9][10]. Phrased in the language of macroeconomy, it makes it possible to go beyond the "representative agent" description [10] and introduce, through its game-theory component, some of the complexity associated with the variability of economic agents' situations. It does so while keeping some reasonable degree of simplicity thanks to the "mean-field" point of view taken. In engineering science, it also proposes a manageable framework to approach complex optimization problems involving a large number of coupled subsystems [3].This relatively new field has witnessed a very rapid development in the last few years, and has followed two major avenues. The first one is a mathematical approach in which one aims at proving the internal consistency of the theory [11][12][13] as well as deriving other rigorous results such as existence and uniqueness of solutions for some classes of models [14,15]. The other direction taken was to develop efficient numerical schemes [5,16,17]. One thing which has, however, prevented the diffusion of this tool at a significantly larger scale is the lack of effective approximation schemes. In fact, in spite of the "mean-field-type" assumptions, the constitutive equations of these models remain rather difficult to analyze, in particular because of their atypical forward-backward structure, and only a few simple models admit an analytical solution [6,[18][19][20]. On the other hand, full fledged numerical analyses of the mean field games equations leave much to be understood.We show here that there is a strong and deep relationship between mean field games (or at least a large class of them), and the non-linear Schrödinger (or GrossPitaevskii) equation, which has been studied for almost a century by physicists to describe various physical systems ranging from interacting bosons in the mean field approximation to gravity waves in inviscid fluids. The goal of this paper is to show that this identification allows to transfer to mean field games (or at least to a class of them) a vast array of knowledge and techniques that have been developed through the years in this field (see e.g. [21][22][23][24][25]). In particular, this opens the way to very effective approximation schemes leading both to a qualitative understanding and a good quantitative description of the solutio...

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confidence: 99%

Schrödinger Approach to Mean Field Games

2016

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“…Indeed, for non-trivial models with finite action spaces there are always population distributions for which more than one optimizer of the Hamiltonian exists, in which case most of the techniques presented in the literature so far are not applicable. For this reason for the previously mentioned examples the authors develop their own tools to solve their particular model: Kolokoltsov and Bensoussan(2016) and Kolokoltsov and Malafeyev(2017) only analyse stationary equilibria in deterministic strategies, Gueant(2009a) and Besancenot and Dogguy(2015) set up a dynamics equation only after analysing optimal decisions given a certain population distribution. They then also focus on stationary equilibria, as well as the dynamic behaviour close to these stationary equilibria and the effect of shocks.…”

Section: Introductionmentioning

confidence: 99%

Stationary Equilibria of Mean Field Games with Finite State and Action Space

Neumann

2020

Dyn Games Appl

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Mean field games formalize dynamic games with a continuum of players and explicit interaction where the players can have heterogeneous states. As they additionally yield approximate equilibria of corresponding N -player games, they are of great interest for socio-economic applications. However, most techniques used for mean field games rely on assumptions that imply that for each population distribution there is a unique optimizer of the Hamiltonian. For finite action spaces, this will only hold for trivial models. Thus, the techniques used so far are not applicable. We propose a model with finite state and action space, where the dynamics are given by a time-inhomogeneous Markov chain that might depend on the current population distribution. We show existence of stationary mean field equilibria in mixed strategies under mild assumptions and propose techniques to compute all these equilibria. More precisely, our results allow -given that the generators are irreducible -to characterize the set of stationary mean field equilibria as the set of all fixed points of a map completely characterized by the transition rates and rewards for deterministic strategies. Additionally, we propose several partial results for the case of non-irreducible generators and we demonstrate the presented techniques on two examples.

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“…The Brock and Durlauf mechanism is quite different from the mechanism below, focusing on the role of social conformity in the adoption of competing theories. Besancenot and Dogguy () offer a model closely related to the first part of the model presented below, in which there is a positive production externality in which more researchers in a field make newly entering researchers more productive. In their model, paradigm shifts are likely to be socially efficient.…”

Section: Introductionmentioning

confidence: 99%

How Research Goes Astray: Paths and Equilibria

Startz

2020

Economic Inquiry

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Some research disciplines persist for long periods despite contributing very little to our understanding of the world. In the presence of imperfect information about the quality of research, multiple equilibria can exist in which path dependence leads to disciplines going astray because researchers are incentivized to invest in low value lines of research. Research fields which have high social value are more likely to thrive, as are fields in which results are subject to evidence. Survivorship bias can lead to a distribution across disciplines in which weak disciplines have more talented researchers than do stronger disciplines. (JEL A11, A12, B40)

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Paradigm Shift: A Mean Field Game Approach

Cited by 10 publications

References 25 publications

Schrödinger Approach to Mean Field Games

Schrödinger Approach to Mean Field Games

Stationary Equilibria of Mean Field Games with Finite State and Action Space

How Research Goes Astray: Paths and Equilibria

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