The goal of the paper is to develop the theory of finite state mean field games with major and minor players when the state space of the game is finite. We introduce the finite player games and derive a mean field game formulation in the limit when the number of minor players tends to infinity. In this limit, we prove that the value functions of the optimization problems are viscosity solutions of PIDEs of the HJB type, and we construct the best responses for both types of players. From there, we prove existence of Nash equilibria under reasonable assumptions. Finally we prove that a form of propagation of chaos holds in the present context and use this result to prove existence of approximate Nash equilibria for the finite player games from the solutions of the mean field games. this vindicate our formulation of the mean field game problem.
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On non-unique solutions in mean field games
The theory of mean field games is a tool to
understand noncooperative dynamic stochastic games
with a large number of players. Much of the theory has evolved
under conditions ensuring uniqueness of the mean field
game Nash equilibrium. However, in some situations, typically
involving symmetry breaking, non-uniqueness of solutions
is an essential feature. To investigate the nature of non-unique
solutions, this paper focuses on the technically simple setting where
players have one of two states, with continuous time dynamics, and
the game is symmetric in the players, and players are restricted
to using Markov strategies. All the mean field
game Nash equilibria are identified for a symmetric follow the crowd game.
Such equilibria correspond to symmetric $\epsilon$-Nash Markov
equilibria for $N$ players with $\epsilon$
converging to zero as $N$ goes to infinity.
In contrast to the mean field game, there is a unique Nash equilibrium
for finite $N.$ It is shown that fluid limits arising from the
Nash equilibria for finite $N$ as $N$ goes to infinity are mean field
game Nash equilibria, and evidence is given supporting
the conjecture that such limits, among all mean field game Nash equilibria,
are the ones that are stable fixed points of the mean field best response mapping.
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- Award ID(s):
- 1900636
- NSF-PAR ID:
- 10143110
- Date Published:
- Journal Name:
- 2019 58th Conference on Decision and Control
- Page Range / eLocation ID:
- 1219 to 1224
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/CDC40024.2019.9029906
