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Kolokoltsov, Vassili N. (2022) Quantum mean-field games. The Annals of Applied Probability, 32 (3). pp. 2254-2288. doi:10.1214/21-aap1733 ISSN 1050-5164.
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Official URL: https://doi.org/10.1214/21-aap1733
Abstract
In this paper we are merging the two new branches of game theory: quantum games and mean-field games (MFG). Building a quantum analog of MFGs requires the full reconstruction of its foundations and methodology, because in N-particle quantum evolution particles are not separated in individual dynamics and the key concept of the classical MFG theory, the empirical measure defined as the sum of Dirac masses of the positions of the players, is not applicable in quantum setting.
As a preliminary result we derive the new nonlinear stochastic Schrödinger equation, as the limit of the quantum filtering equation describing continuously observed and controlled system of a large number of interacting particles, the result that may have an independent value. We then show that to a control quantum system of interacting particles there corresponds a special system of classical interacting particles with the identical limiting MFG system, defined on an appropriate Riemanian manifold. Solutions of this system are shown to specify approximate Nash equilibria for N-agent quantum games.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||||
SWORD Depositor: | Library Publications Router | ||||||||
Library of Congress Subject Headings (LCSH): | Game theory , Schrödinger equation, Quantum theory, Riemannian manifolds , Hamilton-Jacobi equations, Stochastic processes -- Mathematical models, System theory | ||||||||
Journal or Publication Title: | The Annals of Applied Probability | ||||||||
Publisher: | Institute of Mathematical Statistics | ||||||||
ISSN: | 1050-5164 | ||||||||
Official Date: | 1 June 2022 | ||||||||
Dates: |
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Volume: | 32 | ||||||||
Number: | 3 | ||||||||
Page Range: | pp. 2254-2288 | ||||||||
DOI: | 10.1214/21-aap1733 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Copyright Holders: | Rights: Copyright © 2022 Institute of Mathematical Statistics | ||||||||
Date of first compliant deposit: | 20 July 2022 | ||||||||
Date of first compliant Open Access: | 20 July 2022 |
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