
Contents
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1.1 From The Nash System to the Master Equation 1.1 From The Nash System to the Master Equation
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1.1.1 Statement of the Problem 1.1.1 Statement of the Problem
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1.1.2 Link with the Mean Field Theory 1.1.2 Link with the Mean Field Theory
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1.1.3 The Mean Field Game System 1.1.3 The Mean Field Game System
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1.1.4 The Master Equation 1.1.4 The Master Equation
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1.1.5 Well-posedness of the Master Equation 1.1.5 Well-posedness of the Master Equation
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1.1.6 The Convergence Result 1.1.6 The Convergence Result
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1.1.7 Conclusion and Further Prospects 1.1.7 Conclusion and Further Prospects
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1.1.8 Organization of the Text and Reading Guide 1.1.8 Organization of the Text and Reading Guide
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1.2 Informal Derivation of the Master Equation 1.2 Informal Derivation of the Master Equation
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1.2.1 The Differential Game 1.2.1 The Differential Game
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1.2.2 Derivatives in the Space of Measures 1.2.2 Derivatives in the Space of Measures
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1.2.3 Formal Asymptotic of the (υN, i) 1.2.3 Formal Asymptotic of the (υN, i)
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1.2.4 The Master Equation and the MFG System 1.2.4 The Master Equation and the MFG System
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Cite
Abstract
This chapter provides a background on recent advances in the theory of mean field games (MFGs). MFGs has met an amazing success since pioneering works of more than ten years ago. It gives a self-contained study of the so-called master equation and an answer to the convergence problem. MFGs should be understood as games with a continuum of players, each of them interacting with the whole statistical distribution of the population. In this regard, they are expected to provide an asymptotic formulation for games with finitely many players with mean field interaction. This chapter focuses on the converse problem, which may be formulated by confirming whether the equilibria of the finite games converge to a solution of the corresponding MFG as the number of players becomes very large.
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