Mean Field Linear Quadratic Games with Set Up Costs

This paper studies linear quadratic games with set up costs monotonic on the number of active players, namely, players whose action is non-null. Such games arise naturally in joint replenishment inventory systems. Building upon a preliminary analysis of the properties of the best response strategies and Nash equilibria for the given game, the main contribution is the study of the same game under large population. We also analyze the influence of an additional disturbance in the spirit of the literature on H∞ control. Numerical illustrations are provided.

[1]  E. Roxin.  Axiomatic approach in differential games , 1969 .

[2]  Hamidou Tembine,et al.  Robust Mean Field Games with Application to Production of an Exhaustible Resource , 2012, ROCOND.

[3]  Olivier Guéant,et al.  Mean Field Games and Applications , 2011 .

[4]  Martino Bardi,et al.  Explicit solutions of some linear-quadratic mean field games , 2012, Networks Heterog. Media.

[5]  A. Lachapelle,et al.  COMPUTATION OF MEAN FIELD EQUILIBRIA IN ECONOMICS , 2010 .

[6]  Minyi Huang,et al.  Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized $\varepsilon$-Nash Equilibria , 2007, IEEE Transactions on Automatic Control.

[7]  R. Elliott,et al.  The existence of value in differential games of pursuit and evasion , 1972 .

[8]  Quanyan Zhu,et al.  Mixed integer optimal compensation: Decompositions and mean-field approximations , 2012, 2012 American Control Conference (ACC).

[9]  P. Lions,et al.  Mean field games , 2007 .

[10]  I. Milchtaich,et al.  Congestion Games with Player-Specific Payoff Functions , 1996 .

[11]  Laura Giarré,et al.  Consensus in Noncooperative Dynamic Games: A Multiretailer Inventory Application , 2008, IEEE Transactions on Automatic Control.

[12]  Tamer Basar,et al.  H∞-Optimal Control and Related , 1991 .

[13]  T. Basar,et al.  H∞-0ptimal Control and Related Minimax Design Problems: A Dynamic Game Approach , 1996, IEEE Trans. Autom. Control..

[14]  Dario Bauso,et al.  Objective function design for robust optimality of linear control under state-constraints and uncertainty , 2011 .

[15]  Hamidou Tembine.  Risk-sensitive mean field stochastic games , 2011, IEEE Conference on Decision and Control and European Control Conference.

[16]  Monique Dauge,et al.  Koiter Estimate Revisited , 2010 .

[17]  T. Başar,et al.  Risk-sensitive mean field stochastic differential games , 2011 .

[18]  A. Watts.  Uniqueness of equilibrium in cost sharing games , 2002 .

[19]  P. Varaiya.  On the Existence of Solutions to a Differential Game , 1967 .

[20]  Sylvain Sorin,et al.  APPROACHABILITY THEORY, DISCRIMINATING DOMAIN AND DIFFERENTIAL GAMES , 2007 .

[21]  Sylvain Sorin,et al.  Repeated Games and Qualitative Differential Games: Approachability and Comparison of Strategies , 2009, SIAM J. Control. Optim..