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Publisher: Society for Industrial and Applied Mathematics
Languages: English
Types: Article
Subjects:
Emulation, mimicry, and herding behaviors are phenomena that are observed when multiple social groups interact. To study such phenomena, we consider in this paper a large population of homogeneous social networks. Each such network is characterized by a vector state, a vector-valued controlled input and a vector-valued exogenous disturbance. The controlled input of each network is to align its state to the mean distribution of other networks’ states in spite of the actions of the disturbance. One of the contributions of this paper is a detailed analysis of the resulting mean field game for the cases of both polytopic and L2 bounds on controls and disturbances. A second contribution is the establishment of a robust mean-field equilibrium, that is, a solution including the worst-case value function, the state feedback best-responses for the controlled inputs and worst-case disturbances, and a density evolution. This solution is characterized by the property that no player can benefit from a unilateral deviation even in the presence of the disturbance. As a third contribution, microscopic and macroscopic analyses are carried out to show convergence properties of the population distribution using stochastic stability theory.
Huang MY, Caines PE, Malham´e RP (2007) Large population cost-coupled LQG problems with non-uniform agents: individual-mass behaviour and decentralized ǫ-Nash equilibria. IEEE Trans. on Automatic Control 52(9):1560-1571.
Krause U (2000) A discrete nonlinear and non-autonomous model of consensus formation. Communications in Difference Equations: 227-236, S. Elaydi, G. Ladas, J. Popenda, and J. Rakowski editors, Gordon and Breach, Amsterdam, 2000.
Note that from (4.33) we have that kW − Mk < 1 which in turn implies that A is negative definite, i.e., ξT Aξ < 0 for all ξ ∈ Rn, ξ 6= 0. We use this fact to study the infinitesimal generator of the Lyapunov function V (e) = 12 eT e. Our aim now is to prove that there exists a finite scalar κ and a neighborhood of zero of size κ, denoted by Nκ = {e ∈ Rn| V (e) ≤ κ}, such that LV (e(t)) < 0 for all e(t) 6∈ Nκ, where L is the infinitesimal generator of the process e(t). Actually, Lemma 7.1, borrowed from (Gard 1988), p. 129, and reported also in (Thygesen 1997), Theorem 2, establishes that if the former condition holds, which is LV (e(t)) < 0, then V (e(t)) is a supermartingale whenever e(t) is not in Nκ and therefore by the martingale convergence theorem the system is stochastically sample path bounded.
With the above lemma at hand, let us first consider the SDE for the error vector, de(t) = Ae(t)dt + (I − M)σdB(t), and rewrite (I − M)σdB(t) = P bidBi(t), where 1 1 1 )2], 2 nσ2[(n − 1) n2 + (1 − n