Common Information based Markov Perfect Equilibria for Linear-Gaussian Games with Asymmetric Information

We consider a class of two-player dynamic stochastic nonzero-sum games where the state transition and observation equations are linear, and the primitive random variables are Gaussian. Each controller acquires possibly different dynamic information about the state process and the other controller's past actions and observations. This leads to a dynamic game of asymmetric information among the controllers. Building on our earlier work on finite games with asymmetric information, we devise an algorithm to compute a Nash equilibrium by using the common information among the controllers. We call such equilibria common information based Markov perfect equilibria of the game, which can be viewed as a refinement of Nash equilibrium in games with asymmetric information. If the players' cost functions are quadratic, then we show that under certain conditions a unique common information based Markov perfect equilibrium exists. Furthermore, this equilibrium can be computed by solving a sequence of linear equations. We also show through an example that there could be other Nash equilibria in a game of asymmetric information, not corresponding to common information based Markov perfect equilibria.

[1]  R. Bellman,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[2]  Tamer Basar,et al.  A dynamic transmitter-jammer game with asymmetric information , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[3]  Y. Ho,et al.  On a class of linear stochastic differential games , 1968 .

[4]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[5]  W. Willman.  Formal solutions for a class of stochastic pursuit-evasion games , 1968 .

[6]  J. Filar,et al.  Competitive Markov Decision Processes , 1996 .

[7]  T. Başar.  An equilibrium theory for multiperson decision making with multiple probabilistic models , 1985, IEEE Transactions on Automatic Control.

[8]  Basar Tamer,et al.  A multistage pursuit-evasion game that admits a gaussian random process as a maximin control policy , 1975 .

[9]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[10]  Richard Bellman,et al.  Introduction to matrix analysis (2nd ed.) , 1997 .

[11]  Eitan Altman,et al.  Stochastic games with one step delay sharing information pattern with application to power control , 2009, 2009 International Conference on Game Theory for Networks.

[12]  Y. Ho.  On the minimax principle and zero sum stochastic differential games , 1972, CDC 1972.

[13]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[14]  Tamer Basar,et al.  Equilibrium solutions in two-person quadratic decision problems with static information structures , 1975 .

[15]  T. Başar.  On the saddle-point solution of a class of stochastic differential games , 1981, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[16]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[17]  J. Hespanha,et al.  Nash equilibria in partial-information games on Markov chains , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[18]  D. Luenberger,et al.  Differential games with imperfect state information , 1969 .

[19]  Tamer Basar,et al.  Two-Criteria LQG Decision Problems with One-Step Delay Observation Sharing Pattern , 1978, Inf. Control..

[20]  T. Başar.  Decentralized multicriteria optimization of linear stochastic systems , 1978 .

[21]  Tamer Basar,et al.  Common Information Based Markov Perfect Equilibria for Stochastic Games With Asymmetric Information: Finite Games , 2014, IEEE Transactions on Automatic Control.

[22]  Eric Maskin,et al.  Markov Perfect Equilibrium: I. Observable Actions , 2001, J. Econ. Theory.

[23]  John C. Harsanyi,et al.  Games with Incomplete Information Played by "Bayesian" Players, I-III: Part I. The Basic Model& , 2004, Manag. Sci..

[24]  L. Shapley,et al.  Stochastic Games* , 1953, Proceedings of the National Academy of Sciences.

[25]  D. Catlin.  Estimation, Control, and the Discrete Kalman Filter , 1988 .

[26]  Harold L. Cole,et al.  Dynamic Games with Hidden Actions and Hidden States , 2001, J. Econ. Theory.

[27]  M. J. Sobel.  Noncooperative Stochastic Games , 1971 .