Linear–quadratic mean-field-type games: A direct method

Tyrone E. Duncan, Tembine Hamidou

Research output: Contribution to journalArticle

Abstract

In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control actions of all decision-makers. We propose a direct method to solve the game, team, and bargaining problems. This solution approach does not require solving the Bellman–Kolmogorov equations or backward–forward stochastic differential equations of Pontryagin’s type. The proposed method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory. The optimal strategies for decision-makers are shown to be in a state-and-mean-field feedback form. The optimal strategies are given explicitly as a sum of the well-known linear state-feedback strategy for the associated deterministic linear–quadratic game problem and a mean-field feedback term. The equilibrium cost of the decision-makers are explicitly derived using a simple direct method. Moreover, the equilibrium cost is a weighted sum of the initial variance and an integral of a weighted variance of the diffusion and the jump process. Finally, the method is used to compute global optimum strategies as well as saddle point strategies and Nash bargaining solution in state-and-mean-field feedback form.

Original languageEnglish (US)
Article number7
JournalGames
Volume9
Issue number1
DOIs
StatePublished - Mar 1 2018

Fingerprint

Direct Method
Mean Field
Game
Feedback
Game theory
Optimal Strategy
State feedback
Costs
Differential equations
Nash Bargaining Solution
Engineers
Linear Diffusion
Jump Diffusion
Jump Process
Bargaining
Type Theory
Global Optimum
Game Theory
Saddlepoint
Weighted Sums

Keywords

  • Direct method
  • Mean-field equilibrium
  • Nash bargaining solution
  • Variance

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

Cite this

Linear–quadratic mean-field-type games : A direct method. / Duncan, Tyrone E.; Hamidou, Tembine.

In: Games, Vol. 9, No. 1, 7, 01.03.2018.

Research output: Contribution to journalArticle

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