# A Mean field games approach to consensus problems

**Mojtaba Nouraian**, McGill University, Canada

**Abstract:**

In standard consensus (SC) algorithms: (i) agents need no a priori information on the initial state distribution of the overall population but require communication with other agents, (ii) consensus can be achieved if the union of the interaction graphs for the system is connected frequently enough as the system evolves. However, the connectivity of the network structure needed for the SC models may not hold, and even if it holds, the communications and computational complexity required for these SC algorithms may be high, particularly for systems with large populations. Therefore, we take a mean field games (MFG) approach to synthesize the consensus behavior in a set of agents. MFG control strategies display the possibly counterintuitive nature that the feedback control applied by each agent depends only on that agent's stochastic state and the initial mean state distribution of the population of agents, no observations of other agents' states are necessary. In the MFG consensus formulation: (i) each agent in the system has simple stochastic dynamics with inputs directly controlling the rate of change of that agent’s state, and (ii) each agent seeks to minimize its individual quadratic discounted cost function involving the mean of the states of all other agents. For this dynamic game model with mean field couplings, the limiting infinite population MFG system is derived and its unique solution is explicitly computed. The resulting MFG best response strategies steer each agent’s state toward the initial mean state distribution of the overall population, and by applying these decentralized strategies, the system reaches mean-consensus asymptotically as time and population size go to infinity. Furthermore, the MFG strategies possess an *ε*_{N}-Nash equilibrium property where *ε*_{N} goes to zero as the population size *N* goes to infinity. The analysis is extended to the case of agents with non-uniform mean field cost-couplings which corresponds to a heterogeneous system with homogeneous sub-populations.

* This is joint work with Peter E. Caines, Roland P. Malhamé and Minyi Huang.

#### Presentation Slides

**Biography:**

**Mojtaba Nourian **received dual B.Sc. degrees in applied mathematics and in electrical engineering, and the M.Sc. degree in applied mathematics from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2005 and 2007, respectively. He defended his Ph.D. thesis entitled “Mean Field Game Theory: Consensus, Leader-Follower and Major-Minor Agent Systems”, under the supervision of Professor Peter Caines, at the Department of Electrical and Computer Engineering, McGill University, Montreal, Canada, in September 2012. Mr. Nourian will start his new position as a Research Fellow at the Department of Electrical and Electronic Engineering, the University of Melbourne, Melbourne, Australia, in November 2012.