Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth

Abstract

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas et al [13] to model knowledge growth in an economy. Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have. The existence of balanced growth path solutions implies exponential growth of the overall production in time. We proof existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfies a Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange the knowledge level evolves by geometric Brownian motion. arXiv:1602.01423v2 [math.AP] 30 Mar 2016z 0 α(s)ρ x (y, t) dy with ρ x (z, t ) = δ x . Here S denotes the set of admissible controls given by S = {s : [0, ∞) × [0, T ] → [0, 1]}.

Weak and renormalized solutions to a hypoelliptic Mean Field Games system

“…As far as applications are concerned, we refer the reader to the flocking model in Carmona and Delarue [10]. Finally, we mention that the generatl form of (1) is reminiscent of Boltzmann type equations, which have been investigated in the MFG context by Burger, Lorz, Wolfram [12] in a setting quite different to the one used in this paper.…”

An Elo-type rating model for players and teams of variable strength

“…Toscani [9] was the first to introduce kinetic models in the context of opinion formation. His ideas were later generalised for more complex opinion dynamics [10][11][12][13][14][15][16][17], or in the context of wealth distribution [18,19] or knowledge growth in societies [20,21]. For a general overview on interacting multi-agent systems and kinetic equations we refer to the book of Pareschi and Toscani [22].…”

Mean-field selective optimal control via transient leadership

“…It can be generally intended as a measure of the influence of an agent, accounting for a number of different interpretations according to the context. Similar background variables have been used in recent literature to describe wealth distribution [26,28,43], degree of knowledge [16,17], degree of connectivity of an agent in a network [6,15], and also applications to opinion formation [29], just to name a few. Comparing to these other approaches, our mean-field approximation (1.3), (1.4) features a more profound interplay between the variable λ and the spatial distribution x of the agents, resulting in a higher flexibility of the model: not only is λ changing in time, but its variation is driven by an optimality principle steered by the controls.…”