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# Communications in Information and Systems

## Volume 6 (2006)

### Number 3

### Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle

Pages: 221 – 252

DOI: http://dx.doi.org/10.4310/CIS.2006.v6.n3.a5

#### Authors

#### Abstract

We consider stochastic dynamic games in large population conditions where multiclass agents are weakly coupled via their individual dynamics and costs. We approach this large population game problem by the so-called Nash Certainty Equivalence (NCE) Principle which leads to a decentralized control synthesis. The McKean-Vlasov NCE method presented in this paper has a close connection with the statistical physics of large particle systems: both identify a consistency relationship between the individual agent (or particle) at the microscopic level and the mass of individuals (or particles) at the macroscopic level. The overall game is decomposed into (i) an optimal control problem whose Hamilton-Jacobi-Bellman (HJB) equation determines the optimal control for each individual and which involves a measure corresponding to the mass effect, and (ii) a family of McKean-Vlasov (M-V) equations which also depend upon this measure. We designate the NCE Principle as the property that the resulting scheme is consistent (or soluble), i.e. the prescribed control laws produce sample paths which produce the mass effect measure. By construction, the overall closed-loop behaviour is such that each agent's behaviour is optimal with respect to all other agents in the game theoretic Nash sense.

#### Keywords

Stochastic dynamic games; large populations; multi-class agents; interacting particle systems; statistical physics; decentralized control; Hamilton-Jacobi-Bellman equation; McKean-Vlasov equation; Nash equilibria