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

    Duncan, Tyrone E. ; Hamidou, Tembine. / Linear–quadratic mean-field-type games : A direct method. In: Games. 2018 ; Vol. 9, No. 1.
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