### G-2006-57

# Large Population Stochastic Dynamic Games: Closed-Loop McKean-Vlasov Systems and the Nash Certainty Equivalence Principle

## , , and BibTeX reference

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 key feature of this methodology lies in specifying individual controls in the population limit context so that not only are they each optimal with respect to the mass effect, but also they collectively produce the mass effect.

Our method has a close connection with the statistical physics of large particle systems, and they share the common feature by identifying a consistency relationship between the individual-mass (uncontrolled or controlled) interaction via the study of the behavior of an individual agent or particle at the microscopic level. By the NCE Methodology, we mean the overall game decomposition into an optimal control problem whose Hamilton-Jacobi-Bellman (HJB) equation involves a mass effect of measure and a closed-loop McKean-Vlasov (M-V) equation; these two parts are related to each other by the optimal control law derived from the former. In this setting, we designate the NCE Principle as the property that the resulting scheme is consistent in the sense that the prescribed control laws produce sample paths which produce the mass effect with respect to which the optimal control is derived via the HJB equation. It is a property of this overall closed-loop behaviour that each agent’s optimal behaviour with respect to all other agents holds in the game theoretic Nash sense.

Published **October 2006**
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36 pages