Computation of Mean Field Equilibria in Economics

Zhu

Başar

2016

J Optim Theory Appl

Mixed integer optimal compensation deals with optimization problems with integer-and real-valued control variables to compensate disturbances in dynamic systems. The mixed integer nature of controls could lead to intractability in problems of large dimensions. To address this challenge, we introduce a decomposition method which turns the original n-dimensional optimization problem into n independent scalar problems of lot sizing form. Each of these problems can be viewed as a two-player zero-sum game, which introduces some element of conservatism. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon, a step that mirrors a standard procedure in mixed integer programming. We apply the decomposition method to a mean-field coupled multi-agent system problem, where each agent seeks to compensate a combination of an exogenous signal and the local state average. We discuss a large population mean-field type Communicated by

Section: Highlights Of the Main Results And Relationship With The Relmentioning

confidence: 99%

Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

Zhu

Başar

2016

J Optim Theory Appl

“…Several application domains, such as economic, physics, biology and network engineering accommodate mean-field game theoretical models (see [16,[18][19][20]). …”

Section: Related Literaturementioning

confidence: 99%

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

Bagagiolo

Maggistro

et al. 2017

J Optim Theory Appl

This paper studies a decentralized routing problem over a network, using the paradigm of mean-field games with large number of players. Building on a state space extension technique, we turn the problem into an optimal control one for each single player. The main contribution is an explicit expression of the optimal decentralized control which guarantees the convergence both to local and global equilibrium points. Furthermore, we study the stability of the system also in the presence of a delay which we model using an hysteresis operator. As a result of the hysteresis, we prove existence of multiple equilibrium points and analyze convergence conditions. The stability of the system is illustrated via numerical studies .

“…Mean field games arise in several applicative domains such as economics, physics, biology, and network engineering (see [1,17,20,22,37]). …”

Section: Introductionmentioning

confidence: 99%

Mean-Field Games and Dynamic Demand Management in Power Grids

Bagagiolo

2013

Dyn Games Appl

This paper applies mean field game theory to dynamic demand management. For a large population of electrical heating or cooling appliances (called agents), we provide a mean field game that guarantees desynchronization of the agents thus improving the power network resilience. Second, for the game at hand, we exhibit a mean field equilibrium, where each agent adopts a bang-bang switching control with threshold placed at a nominal temperature. At the equilibrium, through an opportune design of the terminal penalty, the switching control regulates the mean temperature (computed over the population) and the mains frequency around the nominal value. To overcome Zeno phenomena we also adjust the bang-bang control by introducing a thermostat. Third, we show that the equilibrium is stable in the sense that all agents' states, initially at different values, converge to the equilibrium value or remain confined within a given interval for an opportune initial distribution.