### G-2015-102

# A class of interference induced games: Asymptotic Nash equilibria and parameterized cooperative solutions

## Mehdi Abedinpour-Fallah, Roland P. Malhamé, and Francesco Martinelli

We consider a multi-agent system with linear stochastic individual dynamics,
and individual linear quadratic ergodic cost functions. The agents partially
observe their own states. Their cost functions and initial statistics are a priori
independent but they are coupled through an interference term (the mean of all agent states),
entering each of their individual measurement equations.
While in general for a finite number of agents, the resulting optimal control law
may be a non linear function of the available observations, we establish that for certain classes
of cost and dynamic parameters, optimal separated control laws obtained by ignoring the
interference coupling, are asymptotically optimal when the number of agents goes to infinity,
thus forming for finite `\(N\)`

, an `\(\epsilon\)`

-Nash equilibrium.
More generally though, optimal separated control laws may not be asymptotically optimal,
and can in fact result in unstable overall behavior. Thus we consider a class of parameterized
decentralized control laws whereby the separated Kalman gain is treated as the arbitrary gain
of a Luenberger like observer. System stability regions are characterized and the nature of
optimal cooperative control policies within the considered class is explored.
Numerical results and an application example for wireless communications are reported.

Published **October 2015**
,
29 pages