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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While originating in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity.Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players, as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit.This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

Convergence. --- Mean field theory. --- Many-body problem --- Statistical mechanics --- Functions --- A priori estimate. --- Approximation. --- Bellman equation. --- Boltzmann equation. --- Boundary value problem. --- C0. --- Chain rule. --- Compact space. --- Computation. --- Conditional probability distribution. --- Continuous function. --- Convergence problem. --- Convex set. --- Cooperative game. --- Corollary. --- Decision-making. --- Derivative. --- Deterministic system. --- Differentiable function. --- Directional derivative. --- Discrete time and continuous time. --- Discretization. --- Dynamic programming. --- Emergence. --- Empirical distribution function. --- Equation. --- Estimation. --- Euclidean space. --- Folk theorem (game theory). --- Folk theorem. --- Heat equation. --- Hermitian adjoint. --- Implementation. --- Initial condition. --- Integer. --- Large numbers. --- Linearization. --- Lipschitz continuity. --- Lp space. --- Macroeconomic model. --- Markov process. --- Martingale (probability theory). --- Master equation. --- Mathematical optimization. --- Maximum principle. --- Method of characteristics. --- Metric space. --- Monograph. --- Monotonic function. --- Nash equilibrium. --- Neumann boundary condition. --- Nonlinear system. --- Notation. --- Numerical analysis. --- Optimal control. --- Parameter. --- Partial differential equation. --- Periodic boundary conditions. --- Porous medium. --- Probability measure. --- Probability theory. --- Probability. --- Random function. --- Random variable. --- Randomization. --- Rate of convergence. --- Regime. --- Scientific notation. --- Semigroup. --- Simultaneous equations. --- Small number. --- Smoothness. --- Space form. --- State space. --- State variable. --- Stochastic calculus. --- Stochastic control. --- Stochastic process. --- Stochastic. --- Subset. --- Suggestion. --- Symmetric function. --- Technology. --- Theorem. --- Theory. --- Time consistency. --- Time derivative. --- Uniqueness. --- Variable (mathematics). --- Vector space. --- Viscosity solution. --- Wasserstein metric. --- Weak solution. --- Wiener process. --- Without loss of generality.

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The utility idea has had a long history in economics, especially in the explanation of demand and in welfare economics. In a comprehensive survey and critique of the Slutsky theory and the pattern to which it belongs in the economic context, S. N. Afriat offers a resolution of questions central to its main idea, including sufficient conditions as well.Originally published in 1980.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Demand functions (Economic theory) --- Utility theory --- 330.105 --- 338.5 --- Demand (Economic theory) --- Value --- Revealed preference theory --- Demand curves (Economic theory) --- Functions, Demand (Economic theory) --- Economics --- 330.105 Wiskundige economie. Wiskundige methoden in de economie --- Wiskundige economie. Wiskundige methoden in de economie --- 338.5 Prijsvorming. Prijskostenverhouding. Prijsbeweging. Prijsfluctuatie--macroeconomisch; prijsindex zie {336.748.12} --- Prijsvorming. Prijskostenverhouding. Prijsbeweging. Prijsfluctuatie--macroeconomisch; prijsindex zie {336.748.12} --- Mathematical models --- Quantitative methods (economics) --- E-books --- Utility theory. --- DEMAND FUNCTIONS (Economic theory) --- Adjoint. --- Aggregate supply. --- Arrow's impossibility theorem. --- Axiom. --- Big O notation. --- Bruno de Finetti. --- Chain rule. --- Coefficient. --- Commodity. --- Concave function. --- Continuous function. --- Convex cone. --- Convex function. --- Convex set. --- Corollary. --- Cost curve. --- Cost-effectiveness analysis. --- Cost–benefit analysis. --- Counterexample. --- Demand curve. --- Derivative. --- Determinant. --- Differentiable function. --- Differential calculus. --- Differential equation. --- Differential form. --- Divisia index. --- Economic equilibrium. --- Economics. --- Einstein notation. --- Equivalence relation. --- Explicit formulae (L-function). --- Factorization. --- Frobenius theorem (differential topology). --- Function (mathematics). --- Functional equation. --- General equilibrium theory. --- Heine–Borel theorem. --- Hessian matrix. --- Homogeneous function. --- Idempotence. --- Identity (mathematics). --- Identity matrix. --- Inequality (mathematics). --- Inference. --- Infimum and supremum. --- Integrating factor. --- Interdependence. --- Interval (mathematics). --- Inverse demand function. --- Inverse function theorem. --- Inverse function. --- Invertible matrix. --- Lagrange multiplier. --- Lagrangian (field theory). --- Lagrangian. --- Law of demand. --- Limit point. --- Line segment. --- Linear function. --- Linear inequality. --- Linear map. --- Linearity. --- Logical disjunction. --- Marginal cost. --- Mathematical induction. --- Mathematical optimization. --- Maxima and minima. --- Monotonic function. --- Ordinary differential equation. --- Orthogonal complement. --- Oskar Morgenstern. --- Pareto efficiency. --- Partial derivative. --- Permutation. --- Preference (economics). --- Price index. --- Principal part. --- Production function. --- Production theory. --- Quasiconvex function. --- Recursive definition. --- Reductio ad absurdum. --- Regular matrix. --- Requirement. --- Row and column vectors. --- Samuelson condition. --- Second derivative. --- Sign (mathematics). --- Special case. --- Statistic. --- Support function. --- Symmetric relation. --- Theorem. --- Theory. --- Transpose. --- Upper and lower bounds. --- Utility. --- Variable (mathematics). --- Welfare economics.