Control theory approach to continuous-time finite state mean field games

Yurii Averboukh

doi:10.3934/mcrf.2022029

Article Contents

2023, Volume 13, Issue 3: 1109-1130. Doi: 10.3934/mcrf.2022029

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Control theory approach to continuous-time finite state mean field games

Yurii Averboukh

Higher School of Economics, Moscow, Russia

Received: March 2021

Revised: January 2022

Early access: July 2022

Published: September 2023

The article was prepared in the framework of a research grant funded by the Ministry of Science and Higher Education of the Russian Federation (grant ID: 075-15-2020-928)

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Abstract

In the paper, we study the dependence of solutions of the continuous-time finite state mean field game on initial distribution of players. Our approach relies on the concept of value multifunction of the mean field game that is a mapping assigning to an initial time and an initial distribution a set of expected outcomes of the representative player corresponding to solutions of the mean field game. Using the reformulation of the finite state mean field game as a control problem with mixed constraints, we give the sufficient condition on a given multifunction to be a value multifunction in the terms of the viability theory. The maximal multifunction (i.e., the mapping assigning to an initial time and an initial distribution the whole set of values corresponding to solutions of the mean field game) is characterized via the backward attainability set for the certain control system.

Keywords:

Continuous-time finite state mean field game,
value multifunction,
viability,
backward attainability domain,
master equation.

Mathematics Subject Classification: Primary: 49N80, 91A16; Secondary: 49J45, 60J27, 34H05.

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References

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